General relativity

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the form of the boundary conditions appropriate for asymptotically flat gravitational fields. What they found was that the asymptotic symmetry transformations actually do form a group and the structure of this group does not depend on the particular gravitational field that happens to be present. This means that, as expected, one can separate the kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich infinite-dimensional group (the so-called BMS group) as the asymptotic symmetry group, instead of the finite-dimensional Poincaré group, which is a subgroup of the BMS group. Not only are the Lorentz transformations asymptotic symmetry transformations, there are also additional transformations that are not Lorentz transformations but are asymptotic symmetry transformations. In fact, they found an additional infinity of transformation generators known as

4632:, which form the basis of modern elementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth. In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime. Using this formalism, it can be shown that black holes emit a blackbody spectrum of particles known as 1251:, Einstein said "The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated." 21575: 772: 4148:, an additional phase of strongly accelerated expansion at cosmic times of around 10 seconds, was hypothesized in 1980 to account for several puzzling observations that were unexplained by classical cosmological models, such as the nearly perfect homogeneity of the cosmic background radiation. Recent measurements of the cosmic background radiation have resulted in the first evidence for this scenario. However, there is a bewildering variety of possible inflationary scenarios, which cannot be restricted by current observations. An even larger question is the physics of the earliest universe, prior to the inflationary phase and close to where the classical models predict the big bang 3019:, both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity. An extension of this expansion is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories. 4815: 4395: 21563: 4446:). There is strong evidence that the laws of black hole mechanics are, in fact, a subset of the laws of thermodynamics, and that the black hole area is proportional to its entropy. This leads to a modification of the original laws of black hole mechanics: for instance, as the second law of black hole mechanics becomes part of the second law of thermodynamics, it is possible for the black hole area to decrease as long as other processes ensure that entropy increases overall. As thermodynamical objects with nonzero temperature, black holes should emit 102: 2937:(usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present. 4500:, take on infinite values. Well-known examples of spacetimes with future singularities—where worldlines end—are the Schwarzschild solution, which describes a singularity inside an eternal static black hole, or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole. The Friedmann–Lemaître–Robertson–Walker solutions and other spacetimes describing universes have past singularities on which worldlines begin, namely Big Bang singularities, and some have future singularities ( 3691: 4233:, which is a ten-dimensional group of three Lorentz boosts, three rotations, and four spacetime translations. It is logical to ask what symmetries if any might apply in General Relativity. A tractable case might be to consider the symmetries of spacetime as seen by observers located far away from all sources of the gravitational field. The naive expectation for asymptotically flat spacetime symmetries might be simply to extend and reproduce the symmetries of flat spacetime of special relativity, 3840: 4511:, proved using the methods of global geometry, say otherwise: singularities are a generic feature of general relativity, and unavoidable once the collapse of an object with realistic matter properties has proceeded beyond a certain stage and also at the beginning of a wide class of expanding universes. However, the theorems say little about the properties of singularities, and much of current research is devoted to characterizing these entities' generic structure (hypothesized e.g. by the 3043: 3255: 3555: 20491: 19148: 785: 1273: 2855:), correspond to inertial motion within a curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow. The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist 47: 3905: 4713: 1360:, illustrated in the figure on the right: for an observer in an enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration. 1410: 21527: 21551: 3104: 4681: 4292: 21539: 7838:...redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. To be published Princeton University Press, 158 pages. 3753: 4112: years from a hot, early Big Bang phase. Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation, further observational data can be used to put the models to the test. Predictions, all successful, include the initial abundance of chemical elements formed in a period of 4697:

interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics. Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.

20035: 4171:(CTCs), which allow for loops in time. The solutions require extreme physical conditions unlikely ever to occur in practice, and it remains an open question whether further laws of physics will eliminate them completely. Since then, other—similarly impractical—GR solutions containing CTCs have been found, such as the 1101:'s 1905 theory of the dynamics of the electron was a relativistic theory which he applied to all forces, including gravity. While others thought that gravity was instantaneous or of electromagnetic origin, he suggested that relativity was "something due to our methods of measurement". In his theory, he showed that 3871:, the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of active galactic nuclei on galactic scales and stellar-size objects such as microquasars. In particular, accretion can lead to 1497:). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity. The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the 3003:, powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes. In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as 1493:. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. 3586:: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period. Because neutron stars are immensely compact, significant amounts of energy are emitted in the form of gravitational radiation. 50: 4822:

All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope

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Each solution of Einstein's equation encompasses the whole history of a universe—it is not just some snapshot of how things are, but a whole, possibly matter-filled, spacetime. It describes the state of matter and geometry everywhere and at every moment in that particular universe. Due to its general

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assumptions about the nature of the asymptotic symmetry group—not even the assumption that such a group exists. Then after designing what they considered to be the most sensible boundary conditions, they investigated the nature of the resulting asymptotic symmetry transformations that leave invariant

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Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class

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Even taken as is, general relativity is rich with possibilities for further exploration. Mathematical relativists seek to understand the nature of singularities and the fundamental properties of Einstein's equations, while numerical relativists run increasingly powerful computer simulations (such as

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General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed many unambiguous observational and experimental tests. However, there are strong indications that the theory is incomplete. The problem of quantum gravity and the question of the reality

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Another general feature of general relativity is the appearance of spacetime boundaries known as singularities. Spacetime can be explored by following up on timelike and lightlike geodesics—all possible ways that light and particles in free fall can travel. But some solutions of Einstein's equations

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Astronomical observations of the cosmological expansion rate allow the total amount of matter in the universe to be estimated, although the nature of that matter remains mysterious in part. About 90% of all matter appears to be dark matter, which has mass (or, equivalently, gravitational influence),

3187:, making their description difficult. However, linear approximations of gravitational waves are sufficiently accurate to describe the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing by 1476:

Special relativity is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, an analogous reasoning as in the previous section applies: there are no global

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General relativity can be understood by examining its similarities with and departures from classical physics. The first step is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity

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or otherwise. Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation also show that the evolution of our universe is significantly influenced by a cosmological constant resulting in an acceleration of cosmic expansion or, equivalently,

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The deflection of light by gravity is responsible for a new class of astronomical phenomena. If a massive object is situated between the astronomer and a distant target object with appropriate mass and relative distances, the astronomer will see multiple distorted images of the target. Such effects

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there was an "element of revelation" in the manner in which Einstein arrived at his theory. Other elements of beauty associated with the general theory of relativity are its simplicity and symmetry, the manner in which it incorporates invariance and unification, and its perfect logical consistency.

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Grossmann for the mathematical part and Einstein for the physical part (1913). Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation (Outline of a Generalized Theory of Relativity and of a Theory of Gravitation), Zeitschrift für Mathematik und Physik, 62, 225–261.

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matter and motion) which had previously been considered as entirely independent. Chandrasekhar also noted that Einstein's only guides in his search for an exact theory were the principle of equivalence and his sense that a proper description of gravity should be geometrical at its basis, so that

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of lightspeed in special relativity. As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion), several effects of gravity on light propagation emerge. Although the bending of light can also be derived by

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The demand for consistency between a quantum description of matter and a geometric description of spacetime, as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the

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To understand Einstein's equations as partial differential equations, it is helpful to formulate them in a way that describes the evolution of the universe over time. This is done in "3+1" formulations, where spacetime is split into three space dimensions and one time dimension. The best-known

3875:, focused beams of highly energetic particles that are being flung into space at almost light speed. General relativity plays a central role in modelling all these phenomena, and observations provide strong evidence for the existence of black holes with the properties predicted by the theory. 4700:

Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems. Some have argued that at low energies, this approach proves successful, in that it results in an acceptable

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The notion of evolution equations is intimately tied in with another aspect of general relativistic physics. In Einstein's theory, it turns out to be impossible to find a general definition for a seemingly simple property such as a system's total mass (or energy). The main reason is that the

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as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory;

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According to general relativity, objects in a gravitational field behave similarly to objects within an accelerating enclosure. For example, an observer will see a ball fall the same way in a rocket (left) as it does on Earth (right), provided that the acceleration of the rocket is equal to

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states that all realistic future singularities (no perfect symmetries, matter with realistic properties) are safely hidden away behind a horizon, and thus invisible to all distant observers. While no formal proof yet exists, numerical simulations offer supporting evidence of its validity.

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Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced, there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being the lattice theory of gravity based on the Feynman

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black hole, and introducing interesting features such as the ergosphere). Using global geometry, later studies have revealed more general properties of black holes. With time they become rather simple objects characterized by eleven parameters specifying: electric charge, mass–energy,

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The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (animated image to the right). Since Einstein's equations are

3890:" to deduce the distance to the merger events–and hence serve as a probe of cosmic expansion at large distances. The gravitational waves produced as a stellar black hole plunges into a supermassive one should provide direct information about the supermassive black hole's geometry. 4836:

those describing merging black holes). In February 2016, it was announced that the existence of gravitational waves was directly detected by the Advanced LIGO team on 14 September 2015. A century after its introduction, general relativity remains a highly active area of research.

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studied in differential geometry. With this additional condition—the covariant divergence of the energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero—the simplest nontrivial set of equations are what are called Einstein's (field) equations:

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was discovered in 1979; since then, more than a hundred gravitational lenses have been observed. Even if the multiple images are too close to each other to be resolved, the effect can still be measured, e.g., as an overall brightening of the target object; a number of such

4063: 52: 3151: 2262: 1512:. As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The 1118:

in November 1915 of what are now known as the Einstein field equations, which form the core of Einstein's general theory of relativity. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present. A version of

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A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of

5647:. The tensors on both side are of second rank, that is, they can each be thought of as 4×4 matrices, each of which contains ten independent terms; hence, the above represents ten coupled equations. The fact that, as a consequence of geometric relations known as 3228:, varieties of an expanding cosmos filled with gravitational waves. But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models. 3131:

Closely related to light deflection is the Shapiro Time Delay, the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction. In the

4427:: "black holes have no hair", that is, no distinguishing marks like the hairstyles of humans. Irrespective of the complexity of a gravitating object collapsing to form a black hole, the object that results (having emitted gravitational waves) is very simple. 3062:; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect is known as gravitational time dilation. 3158:

Predicted in 1916 by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between weak-field gravity and electromagnetism in that, they are analogous to

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The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how the theory can be used for model-building.

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Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source is mass. In special relativity, mass turns out to be part of a more general quantity called the

5643:. The Einstein tensor is the only divergence-free tensor that is a function of the metric coefficients, their first and second derivatives at most, and allows the spacetime of special relativity as a solution in the absence of sources of gravity, cf. 4496:, where the paths of light and falling particles come to an abrupt end, and geometry becomes ill-defined. In the more interesting cases, these are "curvature singularities", where geometrical quantities characterizing spacetime curvature, such as the 1694: 2735: 15964:

Spergel, D. N.; Verde, L.; Peiris, H. V.; Komatsu, E.; Nolta, M. R.; Bennett, C. L.; Halpern, M.; Hinshaw, G.; et al. (2003), "First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters",

4387:), no light from inside can escape to the outside. Since no object can overtake a light pulse, all interior matter is imprisoned as well. Passage from the exterior to the interior is still possible, showing that the boundary, the black hole's 3847:

Whenever the ratio of an object's mass to its radius becomes sufficiently large, general relativity predicts the formation of a black hole, a region of space from which nothing, not even light, can escape. In the currently accepted models of

2570: 1469:-independent. In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the spacetime's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a 16061:

Springel, Volker; White, Simon D. M.; Jenkins, Adrian; Frenk, Carlos S.; Yoshida, Naoki; Gao, Liang; Navarro, Julio; Thacker, Robert; et al. (2005), "Simulations of the formation, evolution and clustering of galaxies and quasars",

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The restriction to future singularities naturally excludes initial singularities such as the big bang singularity, which in principle be visible to observers at later cosmic time. The cosmic censorship conjecture was first presented in

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General relativity predicts that the path of light will follow the curvature of spacetime as it passes near a star. This effect was initially confirmed by observing the light of stars or distant quasars being deflected as it passes the

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Hobbs, George; Archibald, A.; Arzoumanian, Z.; Backer, D.; Bailes, M.; Bhat, N. D. R.; Burgay, M.; Burke-Spolaor, S.; et al. (2010), "The international pulsar timing array project: using pulsars as a gravitational wave detector",

1146:. This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to 3027:

General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed Einstein's initial publication.

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Bennett, C. L.; Halpern, M.; Hinshaw, G.; Jarosik, N.; Kogut, A.; Limon, M.; Meyer, S. S.; Page, L.; et al. (2003), "First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results",

3077:. All results are in agreement with general relativity. However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid. 2394:, meaning that repeated indices are summed (i.e. from zero to three). The Christoffel symbols are functions of the four spacetime coordinates, and so are independent of the velocity or acceleration or other characteristics of a 3311:

of Einstein's equations). Relativistic precession has been observed for all planets that allow for accurate precession measurements (Mercury, Venus, and Earth), as well as in binary pulsar systems, where it is larger by five

923:, which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond 3635:

in free fall in curved spacetime will change when compared, for instance, with the direction of light received from distant stars—even though such a gyroscope represents the way of keeping a direction as stable as possible

4256:. Their first step was to decide on some physically sensible boundary conditions to place on the gravitational field at light-like infinity to characterize what it means to say a metric is asymptotically flat, making no 1113:

involving an observer in free fall (FFO), he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the

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quantities, such as the mass of an isolated system formulated using only quantities defined within a finite region of space containing that system. The hope is to obtain a quantity useful for general statements about

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Blanchet, L.; Faye, G.; Iyer, B. R.; Sinha, S. (2008), "The third post-Newtonian gravitational wave polarisations and associated spherical harmonic modes for inspiralling compact binaries in quasi-circular orbits",

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Valtonen, M. J.; Lehto, H. J.; Nilsson, K.; Heidt, J.; Takalo, L. O.; Sillanpää, A.; Villforth, C.; Kidger, M.; et al. (2008), "A massive binary black-hole system in OJ 287 and a test of general relativity",

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as one of these objects' astrophysical manifestations. Ever more precise solar system tests confirmed the theory's predictive power, and relativistic cosmology also became amenable to direct observational tests.

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Coc, A.; Vangioni-Flam, Elisabeth; Descouvemont, Pierre; Adahchour, Abderrahim; Angulo, Carmen (2004), "Updated Big Bang Nucleosynthesis confronted to WMAP observations and to the Abundance of Light Elements",

14052:; Cheng, E. S.; Cottingham, D. A.; Eplee, R. E.; Fixsen, D. J.; Hewagama, T.; Isaacman, R. B.; Jensen, K. A.; et al. (1994), "Measurement of the cosmic microwave spectrum by the COBE FIRAS instrument", 7560:

Namely, some physicists have questioned whether or not the evidence for dark matter is, in fact, evidence for deviations from the Einsteinian (and the Newtonian) description of gravity cf. the overview in

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Kramer, M.; Stairs, I. H.; Manchester, R. N.; McLaughlin, M. A.; Lyne, A. G.; Ferdman, R. D.; Burgay, M.; Lorimer, D. R.; et al. (2006), "Tests of general relativity from timing the double pulsar",

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Shapiro, S. S.; Davis, J. L.; Lebach, D. E.; Gregory, J. S. (2004), "Measurement of the solar gravitational deflection of radio waves using geodetic very-long-baseline interferometry data, 1979–1999",

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and shear. Using the equivalence principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the

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O'Meara, John M.; Tytler, David; Kirkman, David; Suzuki, Nao; Prochaska, Jason X.; Lubin, Dan; Wolfe, Arthur M. (2001), "The Deuterium to Hydrogen Abundance Ratio Towards a Fourth QSO: HS0105+1619",

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of a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling particle always moves along a geodesic.

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Gravitational redshift has been measured in the laboratory and using astronomical observations. Gravitational time dilation in the Earth's gravitational field has been measured numerous times using

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have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework for

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Brandenberger, Robert H. (2008), "Conceptual problems of inflationary cosmology and a new approach to cosmological structure formation", in Lemoine, Martin; Martin, Jerome; Peter, Patrick (eds.),

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lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.

16015:; Doré, O.; Nolta, M. R.; Bennett, C. L.; Dunkley, J.; Hinshaw, G.; Jarosik, N.; et al. (2007), "Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology", 2121: 51: 7585:, sec. 10; aforementioned modifications need not be modifications of general relativity, they could, for example, be modifications in the way we treat the inhomogeneities in the universe, cf. 15143:

Remillard, Ronald A.; Lin, Dacheng; Cooper, Randall L.; Narayan, Ramesh (2006), "The Rates of Type I X-Ray Bursts from Transients Observed with RXTE: Evidence for Black Hole Event Horizons",

2328: 3663:, rotation is inevitable. Such effects can again be tested through their influence on the orientation of gyroscopes in free fall. Somewhat controversial tests have been performed using the 4345:. In global geometry, the object of study is not one particular solution (or family of solutions) to Einstein's equations. Rather, relations that hold true for all geodesics, such as the 2016: 4379:, which demarcate one region from the rest of spacetime. The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space (as specified in the 1813: 1575:, which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, 1162:

as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the

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but does not interact electromagnetically and, hence, cannot be observed directly. There is no generally accepted description of this new kind of matter, within the framework of known

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Astronomically, the most important property of compact objects is that they provide a supremely efficient mechanism for converting gravitational energy into electromagnetic radiation.

1127:, enabled Einstein to develop general relativity by providing the key mathematical framework on which he fit his physical ideas of gravity. This idea was pointed out by mathematician 13125: 3908:

This blue horseshoe is a distant galaxy that has been magnified and warped into a nearly complete ring by the strong gravitational pull of the massive foreground luminous red galaxy.

3136:(PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on the geometry of space. 5286:, pp. 1–6 "General relativity is not just an extraordinarily beautiful physical theory providing the best description of the gravitational interaction we have so far. It is more." 4507:

Given that these examples are all highly symmetric—and thus simplified—it is tempting to conclude that the occurrence of singularities is an artifact of idealization. The famous

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reduce to special relativity in the case of weak fields at long distances. It turns out that the BMS symmetry, suitably modified, could be seen as a restatement of the universal

1601: 4593:, it can be shown that these masses are positive. Corresponding global definitions exist for momentum and angular momentum. There have also been a number of attempts to define 7492:(WMAP). Future measurements could also reveal evidence about gravitational waves in the early universe; this additional information is contained in the background radiation's 2099: 3570:

will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the

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models, in which our universe has evolved from an extremely hot and dense earlier state. Einstein later declared the cosmological constant the biggest blunder of his life.

18040:; Pettengill, Gordon; Ash, Michael; Stone, Melvin; Smith, William; Ingalls, Richard; Brockelman, Richard (1968), "Fourth test of general relativity: preliminary results", 4097: 1937: 1761: 1731: 1516:

that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or

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Anderson, J. D.; Campbell, J. K.; Jurgens, R. F.; Lau, E. L. (1992), "Recent developments in solar-system tests of general relativity", in Sato, H.; Nakamura, T. (eds.),

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If general relativity were considered to be one of the two pillars of modern physics, then quantum theory, the basis of understanding matter from elementary particles to

4559:, and are uniquely defined, once suitable initial conditions have been specified. Such formulations of Einstein's field equations are the basis of numerical relativity. 3886:

should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and the phase directly before the merger ("chirp") could be used as a "

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With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each

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are known, although only a few have direct physical applications. The best-known exact solutions, and also those most interesting from a physics point of view, are the

2430: 1401:—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass. 1189:, being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein showed in 1915 how his theory explained the 3934: 3215: 19347: 11511:

Everitt, C. W. F.; Buchman, S.; DeBra, D. B.; Keiser, G. M. (2001), "Gravity Probe B: Countdown to launch", in Lämmerzahl, C.; Everitt, C. W. F.; Hehl, F. W. (eds.),

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Barstow, M.; Bond, Howard E.; Holberg, J. B.; Burleigh, M. R.; Hubeny, I.; Koester, D. (2005), "Hubble Space Telescope Spectroscopy of the Balmer lines in Sirius B",

5581:, ch. 2; there are, however, some differences between the modern version and Einstein's original concept used in the historical derivation of general relativity, cf. 3337: 2364: 1968: 1907: 2384: 1328:

Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as

18851: 6216:

From the standpoint of Einstein's theory, these derivations take into account the effect of gravity on time, but not its consequences for the warping of space, cf.

4105: 2957: 543: 12199:

Gowdy, Robert H. (1974), "Vacuum spacetimes with two-parameter spacelike isometry groups and compact invariant hypersurfaces: Topologies and boundary conditions",

4902: 20102: 7512:

Evidence for this comes from the determination of cosmological parameters and additional observations involving the dynamics of galaxies and galaxy clusters cf.

6680:

Kramer, M.; Stairs, I. H.; Manchester, R. N.; Wex, N.; Deller, A. T.; Coles, W. A.; Ali, M.; Burgay, M.; Camilo, F.; Cognard, I.; Damour, T. (13 December 2021).

4438:. For instance, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, analogous to the 3856:, and stellar black holes with a few to a few dozen solar masses, are thought to be the final state for the evolution of massive stars. Usually a galaxy has one 6411:

B. P. Abbott; et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2016). "Observation of Gravitational Waves from a Binary Black Hole Merger".

4402:

Early studies of black holes relied on explicit solutions of Einstein's equations, notably the spherically symmetric Schwarzschild solution (used to describe a

3540: 3518: 3493: 3468: 3288:-like shape (see image). Einstein first derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body as a 2285: 2060: 2036: 4586: 3128:

extending the universality of free fall to light, the angle of deflection resulting from such calculations is only half the value given by general relativity.

1439:, which includes translations, rotations, boosts and reflections.) The differences between the two become significant when dealing with speeds approaching the 1340:. However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of 4832:

of spacetime singularities remain open. Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics.

20406: 11089: 4891: 3702:

are known as gravitational lensing. Depending on the configuration, scale, and mass distribution, there can be two or more images, a bright ring known as an

9026:

Arun, K.G.; Blanchet, L.; Iyer, B. R.; Qusailah, M. S. S. (2008), "Inspiralling compact binaries in quasi-elliptical orbits: The complete 3PN energy flux",

5814:

For the (conceptual and historical) difficulties in defining a general principle of relativity and separating it from the notion of general covariance, see

20431: 2062:

the speed of light in vacuum. When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations,

816: 6603:

In consequence, in the parameterized post-Newtonian formalism (PPN), measurements of this effect determine a linear combination of the terms β and γ, cf.

4574:

gravitational field—like any physical field—must be ascribed a certain energy, but that it proves to be fundamentally impossible to localize that energy.

7581:. Here, too, scientists have argued that the evidence indicates not a new form of energy, but the need for modifications in our cosmological models, cf. 4775:) which, regrettably, turns out to be ill-defined without a proper ultraviolet (lattice) cutoff. However, with the introduction of what are now known as 1138:

and considered difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But in 1916, the astrophysicist

11853: 6325: 6284: 20017: 12954:

Iorio, L. (2009), "An Assessment of the Systematic Uncertainty in Present and Future Tests of the Lense–Thirring Effect with Satellite Laser Ranging",

13255: 10010:; Ciufolini, Ignazio; Bender, Peter L. (1987), "New test of general relativity: Measurement of de Sitter geodetic precession rate for lunar perigee", 7294:, and indications that at least some of the compact objects in question appear to have no solid surface, which can be deduced from the examination of 4705:

of gravity. At very high energies, however, the perturbative results are badly divergent and lead to models devoid of predictive power ("perturbative

3864:

solar masses in its center, and its presence is thought to have played an important role in the formation of the galaxy and larger cosmic structures.

3299:

The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass) or the much more general

19182: 11539: 11042:

Dalal, Neal; Holz, Daniel E.; Hughes, Scott A.; Jain, Bhuvnesh (2006), "Short GRB and binary black hole standard sirens as a probe of dark energy",

1356:

in free fall depends only on its position and initial speed, but not on any of its material properties. A simplified version of this is embodied in

10712:(1979), "The general theory of the mechanical, electromagnetic and thermodynamic properties of black holes", in Hawking, S. W.; Israel, W. (eds.), 8463:

Put simply, matter is the source of spacetime curvature, and once matter has quantum properties, we can expect spacetime to have them as well. Cf.

7916:, sec. 12.2; there are also more intuitive definitions for isolated systems that do not require knowledge of spacetime properties at infinity, cf. 4913: 3609:, where both stars are pulsars and which was last reported to also be in agreement with general relativity in 2021 after 16 years of observations. 2814: 1481:. Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight 195: 18026: 4469:"), and some regions of the future cannot be influenced (event horizon). Even in flat Minkowski space, when described by an accelerated observer ( 3817:. They are expected to yield information about black holes and other dense objects such as neutron stars and white dwarfs, about certain kinds of 3246:) of planetary orbits, as well as orbital decay caused by the emission of gravitational waves and effects related to the relativity of direction. 1394: 1293:) motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's second 18160: 15701:

Gravitation and Relativity: At the turn of the millennium (Proceedings of the GR-15 Conference, held at IUCAA, Pune, India, December 16–21, 1997)

10510:

Buras, R.; Rampp, M.; Janka, H.-Th.; Kifonidis, K. (2003), "Improved Models of Stellar Core Collapse and Still no Explosions: What is Missing?",

7748:

Bondi, H.; Van der Burg, M.G.J.; Metzner, A. (1962). "Gravitational waves in general relativity: VII. Waves from axisymmetric isolated systems".

12468:

Havas, P. (1964), "Four-Dimensional Formulation of Newtonian Mechanics and Their Relation to the Special and the General Theory of Relativity",

1053:, where the bending of light results in multiple images of the same distant astronomical phenomenon. Other predictions include the existence of 20759: 20042: 19484: 18140: 17209: 17184: 17024: 11863: 6080:

GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistic effects, see

4529: 3345: 1508:, to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the 1247: 17014: 20529: 18156: 10931:

Ciufolini, Ignazio; Pavlis, Erricos C.; Peron, R. (2006), "Determination of frame-dragging using Earth gravity models from CHAMP and GRACE",

9478:

Barack, Leor; Cutler, Curt (2004), "LISA Capture Sources: Approximate Waveforms, Signal-to-Noise Ratios, and Parameter Estimation Accuracy",

108: 20359: 15530:

Schödel, R.; Ott, T.; Genzel, R.; Eckart, A.; Mouawad, N.; Alexander, T. (2003), "Stellar Dynamics in the Central Arcsecond of Our Galaxy",

13715:. General Relativity and Gravitation: Proceedings of the 16th International Conference, Durban, South Africa, 15–21 July 2001. p. 210. 3242:

General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation (

1364:

of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific

10801:

Chandrasekhar, Subrahmanyan (1984), "The general theory of relativity – Why 'It is probably the most beautiful of all existing theories'",

9849:

Belinsky, V. A.; Lifshits, I. M.; Khalatnikov, E. M. (1970), "Колебательный Режим Приближения К Особой Точке В Релятивистской Космологии",

4994: 400: 17: 8300:, sec. 11.2; although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes, cf. 4465:

There are many other types of horizons. In an expanding universe, an observer may find that some regions of the past cannot be observed ("

1524:, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable 20848: 18898: 15699:

Seidel, Edward (1998), "Numerical Relativity: Towards Simulations of 3D Black Hole Coalescence", in Narlikar, J. V.; Dadhich, N. (eds.),

8584:

At the energies reached in current experiments, these strings are indistinguishable from point-like particles, but, crucially, different

8258:; for a review of the basics of numerical relativity, including the problems arising from the peculiarities of Einstein's equations, see 3772: 3601:

they had discovered in 1974. This was the first detection of gravitational waves, albeit indirect, for which they were awarded the 1993

20436: 19962: 19290: 19133: 4862: 753: 11188:

Theory and Practice of Natural Computing: Fourth International Conference, TPNC 2015, Mieres, Spain, December 15–16, 2015. Proceedings

3292:. For him, the fact that his theory gave a straightforward explanation of Mercury's anomalous perihelion shift, discovered earlier by 2940:

Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly. Nevertheless, a number of

19992: 19852: 18922: 18668: 13179:

Jaranowski, Piotr; Królak, Andrzej (2005), "Gravitational-Wave Data Analysis. Formalism and Sample Applications: The Gaussian Case",

10887:

Ciufolini, Ignazio; Pavlis, Erricos C. (2004), "A confirmation of the general relativistic prediction of the Lense–Thirring effect",

3123:—a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the 2930: 563: 405: 4585:). If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the 2774:

built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are

21238: 20476: 19212: 14326: 11930:

Giulini, Domenico (2006), "Algebraic and Geometric Structures in Special Relativity", in Ehlers, Jürgen; Lämmerzahl, Claus (eds.),

9434:

Bania, T. M.; Rood, R. T.; Balser, D. S. (2002), "The cosmological density of baryons from observations of 3He+ in the Milky Way",

4623: 4442:

of a thermodynamic system. This limits the energy that can be extracted by classical means from a rotating black hole (e.g. by the

4152:. An authoritative answer would require a complete theory of quantum gravity, which has not yet been developed (cf. the section on 2933:. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi- 2749: 1828: 924: 920: 868: 809: 518: 11982:

Giulini, Domenico (2007), "Remarks on the Notions of General Covariance and Background Independence", in Stamatescu, I. O. (ed.),

4577:

Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (

3843:

Simulation based on the equations of general relativity: a star collapsing to form a black hole while emitting gravitational waves

20365: 19788: 19608: 17808: 13330:

Kennefick, Daniel (2007), "Not Only Because of Theory: Dyson, Eddington and the Competing Myths of the 1919 Eclipse Expedition",

12606: 3224:

Some exact solutions describe gravitational waves without any approximation, e.g., a wave train traveling through empty space or

1058: 19461: 6654: 4896: 1498: 1313:

of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are

1170:

and others had shown that our universe is expanding. This is readily described by the expanding cosmological solutions found by

20727: 20220: 19767: 19716: 19275: 18704: 18658: 12256:

Greenstein, J. L.; Oke, J. B.; Shipman, H. L. (1971), "Effective Temperature, Radius, and Gravitational Redshift of Sirius B",

11659:

Fourès-Bruhat, Yvonne (1952), "Théoréme d'existence pour certains systémes d'équations aux derivées partielles non linéaires",

4058:{\displaystyle R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }+\Lambda \ g_{\mu \nu }={\frac {8\pi G}{c^{4}}}\,T_{\mu \nu }} 3237: 3133: 2844:

contained in that spacetime. Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as

2407: 431: 265: 21387: 16356:

Thiemann, Thomas (2003). "Lectures on Loop Quantum Gravity". In Domenico J.W. Giulini; Claus Kiefer; Claus Lämmerzahl (eds.).

13312:

Kennefick, Daniel (2005), "Astronomers Test General Relativity: Light-bending and the Solar Redshift", in Renn, Jürgen (ed.),

11006:

Cutler, Curt; Thorne, Kip S. (2002), "An overview of gravitational wave sources", in Bishop, Nigel; Maharaj, Sunil D. (eds.),

2386:. The quantity on the left-hand-side of this equation is the acceleration of a particle, and so this equation is analogous to 19793: 19260: 19175: 18842: 17840: 17818: 17699: 17678: 17660: 17635: 17614: 17586: 17563: 17526: 17508: 17486: 17465: 17444: 17426: 17405: 17380: 17362: 17337: 17316: 17271: 17250: 17229: 17194: 17158: 17076: 17055: 16987: 16965: 16943: 16923: 16740: 16714: 16509: 16489: 16453: 16391: 16339: 16204: 16183: 15909: 15870: 15726: 15603: 15585: 15389: 15325: 15307: 15285: 15216: 15198: 15066: 14955: 14937: 14916: 14768: 14706: 14591: 14540: 14522: 14499: 14353: 14247: 14165: 14116: 14098: 13994: 13794: 13746: 13656: 13563: 13545: 13471: 13381: 13321: 13115: 13058: 12717: 12691: 12616: 12592: 12313: 12247: 12017: 11965: 11921: 11843: 11520: 11466: 11303: 11285: 11267: 11196: 11176: 11033: 10792: 10721: 10639: 10455: 10332: 10288: 10270: 10154: 10131: 10046:

Bertotti, Bruno; Iess, L.; Tortora, P. (2003), "A test of general relativity using radio links with the Cassini spacecraft",

9998: 9730: 9658: 9592: 9425: 9267: 9073: 8997: 8976: 5711: 5686: 5330: 4195: 2941: 2875: 2257:{\displaystyle {d^{2}x^{\mu } \over ds^{2}}+\Gamma ^{\mu }{}_{\alpha \beta }{dx^{\alpha } \over ds}{dx^{\beta } \over ds}=0,} 1420: 1185:

During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to

470: 4555:. These decompositions show that the spacetime evolution equations of general relativity are well-behaved: solutions always 19538: 19469: 19445: 19418: 18326: 15852: 7695:

Finazzi, Stefano; Liberati, Stefano; Barceló, Carlos (15 June 2009). "Semiclassical instability of dynamical warp drives".

3090: 2775: 13938:

MacCallum, M. (2006), "Finding and using exact solutions of the Einstein equations", in Mornas, L.; Alonso, J. D. (eds.),

11149:

Donoghue, John F. (1995), "Introduction to the Effective Field Theory Description of Gravity", in Cornet, Fernando (ed.),

3825:. In February 2016, the Advanced LIGO team announced that they had detected gravitational waves from a black hole merger. 19592: 15735:

Seljak, Uros̆; Zaldarriaga, Matias (1997), "Signature of Gravity Waves in the Polarization of the Microwave Background",

14600:

Peebles, P.J.E.; Schramm, D.N.; Turner, E.L.; Kron, R.G. (1991), "The case for the relativistic hot Big Bang cosmology",

13266:

Kamionkowski, Marc; Kosowsky, Arthur; Stebbins, Albert (1997), "Statistics of Cosmic Microwave Background Polarization",

4885: 4398:

The ergosphere of a rotating black hole, which plays a key role when it comes to extracting energy from such a black hole

3655:

effects. A distant observer will determine that objects close to the mass get "dragged around". This is most extreme for

3258:

Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star. The influence of other planets is ignored.

802: 16807:

Walsh, D.; Carswell, R. F.; Weymann, R. J. (1979), "0957 + 561 A, B: twin quasistellar objects or gravitational lens?",

7539:; in particular, observations indicate that all but a negligible portion of that matter is not in the form of the usual 7306:, sec. 5. Observations of the "shadow" of the Milky Way galaxy's central black hole horizon are eagerly sought for, cf. 5079: 1583:-free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved- 1214: 21422: 21101: 20426: 20249: 20091: 19932: 19773: 19321: 19304: 19245: 18579: 13248:

Was Einstein right? Scientists provide first public peek at Gravity Probe B results (Stanford University Press Release)

9703:

Beig, Robert; Chruściel, Piotr T. (2006), "Stationary black holes", in Françoise, J.-P.; Naber, G.; Tsou, T.S. (eds.),

6140:

for the classic early measurements by Arthur Eddington's expeditions. For an overview of more recent measurements, see

4857: 4851: 4450:. Semiclassical calculations indicate that indeed they do, with the surface gravity playing the role of temperature in 3803: 3757: 2771: 2765: 1950:

or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant

1513: 980: 182: 38: 13436:

Kramer, Michael (2004). "Millisecond Pulsarsas Tools of Fundamental Physics". In Karshenboim, S. G.; Peik, E. (eds.).

8864:

for an account up to that year; up-to-date news can be found on the websites of major detector collaborations such as

3296:

in 1859, was important evidence that he had at last identified the correct form of the gravitational field equations.

2949: 2294: 20881: 20522: 20110: 19987: 19426: 19255: 18950: 18584: 17775: 17731: 17294: 4879: 4823:

for this to change as future data from cosmological observations and particle physics experiments becomes available.

4249: 4220: 3821:

implosions, and about processes in the very early universe, including the signature of certain types of hypothetical

3578:, the effect is too small to be observable. This is not the case for a close binary pulsar, a system of two orbiting 2886:

relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.

2810: 2039: 1541: 1093: 208: 6257:

that is part of a binary system, the gravitational field causing the time delay being that of the other pulsar, cf.

21303: 20225: 19168: 14362:

Olive, K. A.; Skillman, E. A. (2004), "A Realistic Determination of the Error on the Primordial Helium Abundance",

5068:

Moshe Carmeli (2008).Relativity: Modern Large-Scale Structures of the Cosmos. pp.92, 93.World Scientific Publishing

4184: 1190: 21574: 18904: 13942:(A Century of Relativity Physics: ERE05, the XXVIII Spanish Relativity Meeting), vol. 841, pp. 129–143, 11138: 9575:

Barish, Barry (2005), "Towards detection of gravitational waves", in Florides, P.; Nolan, B.; Ottewil, A. (eds.),

6367: 4614:, would be the other. However, how to reconcile quantum theory with general relativity is still an open question. 3046:

Schematic representation of the gravitational redshift of a light wave escaping from the surface of a massive body

1151: 771: 488: 101: 74:. The star field behind the black holes is being heavily distorted and appears to rotate and move, due to extreme 20270: 19227: 18617: 15683:(1916b), "Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie", 13000:

Iorio, L. (2010), "On the Lense–Thirring test with the Mars Global Surveyor in the gravitational field of Mars",

4319: 2902: 2342:

coefficients) which is symmetric in the two lower indices. Greek indices may take the values: 0, 1, 2, 3 and the

2112: 1973: 1135: 395: 13711:

Lehner, Luis (2002). "Numerical Relativity: Status and Prospects". In Nigel T. Bishop; Sunil D. Maharaj (eds.).

11557:

Falcke, Heino; Melia, Fulvio; Agol, Eric (2000), "Viewing the Shadow of the Black Hole at the Galactic Center",

8118:

Here one should remind to the well-known fact that the important "quasi-optical" singularities of the so-called

4302:

In general relativity, no material body can catch up with or overtake a light pulse. No influence from an event

1769: 1520:

metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the

21602: 20675: 20080: 19952: 19533: 19455: 19222: 18720: 18229: 18217: 17752: 10210:

Blanchet, Luc (2006), "Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries",

4800: 4671: 4590: 4327: 4206:

but these solutions requires exotic matter distribution, and generally suffers from semiclassical instability.

3164: 2413: 18132: 13106:

Israel, Werner (1987), "Dark stars: the evolution of an idea", in Hawking, Stephen W.; Israel, Werner (eds.),

4966:...was established by Einstein, and represents probably the most beautiful of all existing physical theories." 3767:, above). Detection of these waves is a major goal of current relativity-related research. Several land-based 3763:

Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (see

3050:

Assuming that the equivalence principle holds, gravity influences the passage of time. Light sent down into a

2893:: its laws—and further laws formulated within the general relativistic framework—take on the same form in all 21154: 21086: 19845: 19726: 19250: 19151: 18893: 18798: 18728: 18207: 11867: 9232:

Ashtekar, Abhay (2007), "Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions",

8552:; for a recent comprehensive review of the failure of perturbative renormalizability for quantum gravity see 7234:; this, however, can be explained as an optical illusion that does not violate the tenets of relativity, see 6341: 6302: 6052: 4741: 3913: 3726:, provide a "natural telescope" for observing distant galaxies, and to obtain an independent estimate of the 3589:

The first observation of a decrease in orbital period due to the emission of gravitational waves was made by

3154:

Ring of test particles deformed by a passing (linearized, amplified for better visibility) gravitational wave

3120: 1397:

concepts, i.e. a description which is valid in any desired coordinate system. In this geometric description,

1089: 914: 482: 190: 18861: 528: 21179: 20515: 20058: 19947: 19942: 18715: 16594:

Trautman, Andrzej (2006), "Einstein–Cartan theory", in Françoise, J.-P.; Naber, G. L.; Tsou, S. T. (eds.),

13246: 9109: 8910: 7904:. A more exact mathematical description distinguishes several kinds of horizon, notably event horizons and 6289: 5651:, the Einstein tensor satisfies a further four identities reduces these to six independent equations, e.g. 4663: 4516: 4362: 3747: 1689:{\displaystyle G_{\mu \nu }\equiv R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }=\kappa T_{\mu \nu }\,} 1206: 548: 18187: 11530: 2995:

Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by

21597: 21417: 20812: 20215: 20174: 20118: 19967: 19892: 19511: 19405: 19395: 19270: 18756: 18490: 18475: 12625:

Heckmann, O. H. L.; Schücking, E. (1959), "Newtonsche und Einsteinsche Kosmologie", in Flügge, S. (ed.),

12602: 11428: 9815: 7371:, ch. 9–11; use of these models is justified by the fact that, at large scales of around hundred million 5974:, pp. 164–172. Einstein derived these effects using the equivalence principle as early as 1907, cf. 4988: 4873: 4768: 4121: 3768: 3037: 2829: 1115: 976: 964: 940: 789: 280: 200: 11513:

Gyros, Clocks, and Interferometers: Testing Relativistic Gravity in Space (Lecture Notes in Physics 562)

7262:, sec. 4.2. Also, relativistic lensing effects are thought to play a role for the signals received from 5850:

A review showing Einstein's equation in the broader context of other PDEs with physical significance is

5031:; an up-to-date collection of current research, including reprints of many of the original articles, is 4737: 2390:

which likewise provide formulae for the acceleration of a particle. This equation of motion employs the

1528:, the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish). 21517: 21228: 21048: 20643: 20616: 20142: 19957: 19711: 19353: 19217: 19043: 18485: 18438: 18193: 18139: – Articles on a variety of aspects of relativistic physics for a general audience; hosted by the 17983: 17284: 16651:

Veltman, Martinus (1975), "Quantum Theory of Gravitation", in Balian, Roger; Zinn-Justin, Jean (eds.),

12726:

Hough, Jim; Rowan, Sheila (2000), "Gravitational Wave Detection by Interferometry (Ground and Space)",

10780: 9342:

Ashtekar, Abhay; Lewandowski, Jerzy (2004), "Background Independent Quantum Gravity: A Status Report",

7485: 6007: 5605: 4736:

of all particles and interactions, including gravity; the price to pay is unusual features such as six

4689: 4568: 4322:) yields key information about the spacetime's causal structure. This structure can be displayed using 3802:

frequency range, which originate from binary supermassive blackholes. A European space-based detector,

2837: 2798: 2387: 1548: 1294: 1226: 1080:, general relativity has often been described as the most beautiful of all existing physical theories. 693: 241: 18211: 3605:

in physics. Since then, several other binary pulsars have been found, in particular the double pulsar

2730:{\displaystyle F_{f}(r)=-{\frac {GMm}{r^{2}}}+{\frac {L^{2}}{mr^{3}}}-{\frac {3GML^{2}}{mc^{2}r^{4}}}} 1205:

confirmed general relativity's prediction for the deflection of starlight by the Sun during the total

20900: 20050: 19997: 19977: 19505: 19500: 19073: 18699: 18380: 17765: 16997: 16704: 16261: 13161: 12829: 12227: 11783:

Futamase, T.; Itoh, Y. (2006), "The Post-Newtonian Approximation for Relativistic Compact Binaries",

8880:

For the most recent papers on gravitational wave polarizations of inspiralling compact binaries, see

7969:, ch. 2. A thorough, book-length introduction including an introduction to the necessary mathematics 6157: 5000: 4792: 4149: 3594: 3567: 3070: 2910: 2897:. Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is 2883: 2068: 1105:

propagate at the speed of light. Soon afterwards, Einstein started thinking about how to incorporate

1070: 956: 493: 305: 31: 21382: 8540:, an integral part of deriving predictions which take into account higher-energy contributions, cf. 3730:. Statistical evaluations of lensing data provide valuable insight into the structural evolution of 3119:

This and related predictions follow from the fact that light follows what is called a light-like or

1237:. elements that excite wonderment and surprise). It juxtaposes fundamental concepts (space and time 21612: 21607: 21484: 21402: 21356: 21063: 20749: 20495: 20446: 20441: 20421: 20411: 20371: 20066: 19922: 19838: 19721: 19602: 19431: 19098: 18648: 18408: 18319: 17028: 15448: 13574:"Galilei and Lorentz Structures on spacetime: comparison of the corresponding geometry and physics" 11206: 9284:

Ashtekar, Abhay; Krishnan, Badri (2004), "Isolated and Dynamical Horizons and Their Applications",

4113: 3868: 3814: 3300: 3016: 2918: 2879: 2841: 1819: 1537: 1390: 1365: 910: 728: 718: 568: 385: 14279:

Nordtvedt, Kenneth (2003). "Lunar Laser Ranging—a comprehensive probe of post-Newtonian gravity".

8588:

of oscillation of one and the same type of fundamental string appear as particles with different (

7214:, sec. 17.2; for more about the different types of astronomical objects associated with this, cf. 4072: 3107:

Deflection of light (sent out from the location shown in blue) near a compact body (shown in gray)

2565:{\displaystyle U_{f}(r)=-{\frac {GMm}{r}}+{\frac {L^{2}}{2mr^{2}}}-{\frac {GML^{2}}{mc^{2}r^{3}}}} 1912: 1736: 1706: 21454: 21141: 21058: 21028: 20451: 19636: 19569: 18827: 18749: 18369: 18360: 13049:(1994), "Prima facie questions in quantum gravity", in Ehlers, Jürgen; Friedrich, Helmut (eds.), 11532:

The Gravity Probe B experiment. Post Flight Analysis—Final Report (Preface and Executive Summary)

10140: 7278:

The evidence includes limits on compactness from the observation of accretion-driven phenomena ("

4685: 3857: 3719: 3217:

or less. Data analysis methods routinely make use of the fact that these linearized waves can be

3124: 3086: 2964:, each describing an expanding cosmos. Exact solutions of great theoretical interest include the 2898: 2833: 1940: 1419:

As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a

1373: 1046: 1038: 713: 447: 12583:

Hawking, Stephen W. (1987), "Quantum cosmology", in Hawking, Stephen W.; Israel, Werner (eds.),

12327:"An Example of a New Type of Cosmological Solution of Einstein's Field Equations of Gravitation" 9388:

Ashtekar, Abhay; Magnon-Ashtekar, Anne (1979), "On conserved quantities in general relativity",

5004: 4772: 1501:, a crucial guiding principle for generalizing special-relativistic physics to include gravity. 1309:. The preferred inertial motions are related to the geometry of space and time: in the standard 21412: 21268: 21223: 20833: 20695: 20383: 20291: 20263: 20166: 20025: 19553: 19437: 19053: 18916: 18822: 18780: 18454: 18418: 17717: 17325: 17043: 16498:

Thorne, Kip (2003). "Warping spacetime". In G.W. Gibbons; E.P.S. Shellard; S.J. Rankin (eds.).

15443: 15382:

General Relativity: The most beautiful of theories (de Gruyter Studies in Mathematical Physics)

9778:

Bekenstein, Jacob D. (1974), "Generalized Second Law of Thermodynamics in Black-Hole Physics",

4760: 4702: 4637: 4435: 2945: 2856: 2779: 2339: 1576: 1521: 1345: 1120: 948: 880: 683: 668: 513: 275: 6664: 6494:

Most advanced textbooks on general relativity contain a description of these properties, e.g.

6156:

This is not an independent axiom; it can be derived from Einstein's equations and the Maxwell

3919: 3190: 71: 63: 21494: 21449: 20929: 20874: 20838: 20807: 20670: 20591: 20456: 20416: 20256: 20205: 20200: 20126: 19937: 19917: 19907: 19685: 19664: 19657: 19597: 19058: 18663: 18602: 14492:

Lonely Hearts of the Cosmos: the story of the scientific quest for the secret of the Universe

11151:

Effective Theories: Proceedings of the Advanced School, Almunecar, Spain, 26 June–1 July 1995

10731: 8332:

states, then flat, empty Minkowski space, which has mass zero, could evolve into these states

8123: 8119: 7826:

Strominger, Andrew (2017). "Lectures on the Infrared Structure of Gravity and Gauge Theory".

5207: 4763:

procedures of quantum theory. Using the initial-value-formulation of general relativity (cf.

4493: 4486: 4349:, and additional non-specific assumptions about the nature of matter (usually in the form of 4346: 4270: 4168: 3685: 3322: 3218: 2996: 2977: 2956:, each corresponding to a certain type of black hole in an otherwise empty universe, and the 2349: 1953: 1892: 1525: 1352:, or the universal equality of inertial and passive-gravitational mass): the trajectory of a 1349: 1163: 1050: 1025:—regions of space in which space and time are distorted in such a way that nothing, not even 841: 658: 226: 17890: 16501:

The future of theoretical physics and cosmology: celebrating Stephen Hawking's 60th birthday

11422: 11402: 11380: 10864: 10103: 5337: 4330:") so as to fit onto a finite map, while light still travels along diagonals as in standard 4277:(QFT), which relates universal infrared (soft) QFT with GR asymptotic spacetime symmetries. 3303:. It is due to the influence of gravity on the geometry of space and to the contribution of 2369: 1461:

in the image), and a set of events for which such an influence is impossible (such as event

1341: 67: 21469: 21397: 21283: 21149: 21111: 21043: 20797: 20621: 20581: 20335: 20284: 20277: 20230: 19902: 19875: 19752: 19671: 19650: 19643: 19518: 19373: 19331: 18963: 18889: 18733: 18335: 18088: 18049: 18008: 17955: 17915: 17873: 17107: 16870: 16816: 16768: 16728: 16676: 16632: 16609: 16583: 16553: 16477: 16421: 16371: 16319: 16277: 16222: 16135: 16081: 16034: 15984: 15937: 15897: 15858: 15829: 15790: 15754: 15714: 15688: 15669: 15634: 15549: 15503: 15435: 15353: 15234: 15162: 15124: 15084: 15033: 15003: 14973: 14904: 14859: 14796: 14731: 14681: 14648: 14609: 14561: 14466: 14427: 14381: 14310: 14268: 14193: 14061: 14022: 13953: 13919: 13871: 13825: 13782: 13726: 13684: 13608: 13501: 13451: 13417: 13345: 13285: 13188: 13137: 13079: 13019: 12973: 12927: 12881: 12841: 12786: 12735: 12705: 12643: 12556: 12508: 12477: 12432: 12378: 12338: 12265: 12208: 12180: 12137: 12089: 12043: 11997: 11945: 11900: 11792: 11756: 11698: 11668: 11620: 11576: 11446: 11325: 11228: 11164: 11104: 11061: 11021: 10979: 10940: 10896: 10860: 10810: 10753: 10670: 10603: 10529: 10483: 10411: 10385: 10365: 10312: 10219: 10183: 10099: 10055: 10019: 9946: 9897: 9858: 9831: 9787: 9751: 9718: 9684: 9620: 9580: 9548: 9497: 9443: 9397: 9361: 9303: 9247: 9205: 9169: 9118: 9088: 9045: 8940: 8328:, p. 295 and refs therein; this is important for questions of stability—if there were 8136: 7800: 7757: 7714: 7279: 7072: 6703: 6430: 6333: 6294: 4780: 4733: 4667: 4629: 4540: 4508: 4431: 4411: 4384: 4370: 4274: 3787: 3668: 3543: 3184: 3160: 3058:, whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is 3000: 2989: 2985: 2343: 1588: 1143: 999: 860: 478: 17861: 16915:

Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity

13589:

Lahav, Ofer; Suto, Yasushi (2004), "Measuring our Universe from Galaxy Redshift Surveys",

12304: 11412: 11392: 11366: 11346: 11125: 8848:

A review of the various problems and the techniques being developed to overcome them, see

5679:

Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity

4814: 2867: 1504:

The same experimental data shows that time as measured by clocks in a gravitational field—

1213:

and astrophysics until developments between approximately 1960 and 1975, now known as the

8: 21346: 21169: 21159: 21008: 20993: 20949: 20772: 20631: 20626: 20611: 20586: 20563: 20539: 19038: 19033: 19023: 18855: 18804: 18622: 18607: 18433: 18364: 18312: 18201: 15664:(1916a), "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie", 15515: 11316:(1997), "Local and Global Light Bending in Einstein's and other Gravitational Theories", 11008:

Proceedings of 16th International Conference on General Relativity and Gravitation (GR16)

10195: 7540: 7231: 4744:, it was conjectured that both string theory and a unification of general relativity and 4611: 4176: 4109: 3861: 3791: 3641: 3627:

Several relativistic effects are directly related to the relativity of direction. One is

3618: 3008: 2999:

on a computer, or by considering small perturbations of exact solutions. In the field of

2934: 2331: 1470: 1432: 1282: 1210: 1171: 1124: 1102: 1077: 673: 643: 638: 390: 19028: 18092: 18053: 18012: 17959: 17919: 17896:

Flanagan, Éanna É.; Hughes, Scott A. (2005), "The basics of gravitational wave theory",

17877: 17537: 17111: 16874: 16820: 16772: 16732: 16680: 16636: 16613: 16587: 16568:(2000), Bytsenko, A.; Williams, F. (eds.), "An Introduction to Black Hole Evaporation", 16557: 16481: 16425: 16375: 16323: 16281: 16226: 16139: 16085: 16038: 15988: 15941: 15901: 15862: 15833: 15794: 15758: 15718: 15692: 15673: 15638: 15553: 15507: 15439: 15357: 15238: 15166: 15128: 15088: 15037: 15007: 14977: 14908: 14863: 14800: 14735: 14685: 14652: 14613: 14565: 14470: 14431: 14385: 14314: 14295: 14272: 14256: 14197: 14065: 14026: 13957: 13923: 13875: 13829: 13786: 13730: 13688: 13612: 13505: 13455: 13421: 13349: 13289: 13192: 13141: 13083: 13023: 12977: 12931: 12885: 12845: 12790: 12777:(1929), "A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae", 12739: 12709: 12647: 12560: 12512: 12501:

Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences

12481: 12436: 12382: 12342: 12269: 12212: 12184: 12141: 12093: 12047: 12001: 11949: 11904: 11796: 11760: 11702: 11672: 11624: 11580: 11450: 11329: 11232: 11168: 11108: 11065: 11025: 10983: 10944: 10900: 10814: 10757: 10674: 10607: 10533: 10487: 10415: 10369: 10316: 10223: 10187: 10059: 10023: 9950: 9901: 9862: 9835: 9791: 9755: 9722: 9688: 9624: 9584: 9552: 9501: 9447: 9401: 9365: 9307: 9251: 9209: 9173: 9122: 9092: 9049: 8944: 7804: 7761: 7750:

Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences

7718: 7403:

These tests involve the separate observations detailed further on, see, e.g., fig. 2 in

7084: 7076: 6707: 6477: 6434: 6337: 6298: 1457:

via signals or interactions that do not need to travel faster than light (such as event

1453:, there is a set of events that can, in principle, either influence or be influenced by 1175: 653: 623: 21567: 21555: 21479: 21336: 21189: 20939: 20828: 20700: 20653: 20298: 20182: 20158: 20002: 19880: 19814: 19280: 19103: 19003: 18927: 18785: 18766: 18760: 18711: 18653: 18480: 18398: 18375: 18343: 18254: 18112: 18078: 17998: 17971: 17945: 17905: 17804: 17721: 17576: 17497: 17130: 17097: 16913: 16893: 16860: 16840: 16791: 16758: 16692: 16599: 16573: 16543: 16520: 16467: 16409: 16397: 16361: 16345: 16309: 16293: 16245: 16192: 16158: 16125: 16105: 16071: 16050: 16024: 16000: 15974: 15953: 15927: 15887: 15770: 15744: 15704: 15680: 15661: 15650: 15624: 15565: 15539: 15519: 15493: 15468: 15425: 15402: 15369: 15343: 15296: 15257: 15178: 15152: 15100: 14882: 14849: 14837: 14812: 14786: 14757: 14696: 14625: 14454: 14443: 14417: 14397: 14371: 14280: 14221: 14209: 14183: 14155: 14087: 14038: 14012: 13969: 13943: 13894: 13861: 13841: 13815: 13772: 13752: 13716: 13700: 13674: 13631: 13598: 13573: 13525: 13491: 13441: 13361: 13335: 13301: 13275: 13211: 13153: 13095: 13035: 13009: 12989: 12963: 12943: 12917: 12897: 12871: 12825: 12758: 12666: 12572: 12532: 12524: 12496: 12456: 12418: 12402: 12364: 12157: 12112: 12069: 12023: 11987: 11971: 11935: 11890: 11815: 11772: 11746: 11721: 11643: 11600: 11566: 11500: 11472: 11436: 11351: 11244: 11218: 11154: 11130: 11077: 11051: 11011: 10995: 10969: 10920: 10876: 10850: 10826: 10769: 10765: 10743: 10693: 10660: 10619: 10593: 10553: 10519: 10499: 10473: 10435: 10401: 10338: 10302: 10242: 10199: 10173: 10079: 9987: 9969: 9936: 9913: 9887: 9803: 9767: 9708: 9638: 9610: 9564: 9513: 9487: 9467: 9377: 9351: 9326: 9293: 9273: 9237: 9141: 9061: 9035: 8956: 8928: 8605: 7827: 7773: 7730: 7704: 7088: 7062: 6727: 6693: 6454: 6420: 6383: 5648: 4776: 4706: 4342: 4253: 4226: 3899: 3795: 3743: 3637: 3606: 3590: 3525: 3503: 3478: 3453: 3313: 3285: 3277: 3263: 3145: 3094: 3012: 2973: 2961: 2890: 2270: 2045: 2021: 1466: 1447: 1428: 1386: 1377: 1310: 1155: 1139: 1110: 1066: 1054: 1042: 1003: 952: 944: 864: 723: 608: 533: 380: 285: 270: 231: 75: 17967: 16213:

Szabados, László B. (2004), "Quasi-Local Energy–Momentum and Angular Momentum in GR",

15882:(2005), "Causal Sets: Discrete Gravity", in Gomberoff, Andres; Marolf, Donald (eds.), 15365: 13696: 13332:

Proceedings of the 7th Conference on the History of General Relativity, Tenerife, 2005

12939: 12809: 11116: 9373: 6681: 4756:, which would constitute a uniquely defined and consistent theory of quantum gravity. 4394: 1209:, instantly making Einstein famous. Yet the theory remained outside the mainstream of 21562: 21474: 21243: 21218: 21033: 20944: 20924: 20843: 20792: 20737: 20717: 20603: 20389: 19798: 19706: 19450: 19378: 19113: 18945: 18937: 18538: 18495: 18104: 17975: 17836: 17814: 17781: 17771: 17748: 17727: 17709: 17695: 17674: 17656: 17631: 17610: 17582: 17572: 17559: 17543: 17522: 17504: 17482: 17461: 17440: 17422: 17401: 17376: 17358: 17333: 17312: 17290: 17267: 17259: 17246: 17225: 17203: 17190: 17154: 17135: 17072: 17051: 16983: 16976: 16961: 16954: 16939: 16933: 16919: 16898: 16832: 16796: 16736: 16710: 16696: 16505: 16485: 16449: 16401: 16387: 16349: 16335: 16297: 16250: 16200: 16179: 16163: 16097: 15905: 15866: 15806: 15722: 15599: 15581: 15473: 15385: 15373: 15321: 15303: 15281: 15262: 15212: 15194: 15062: 14951: 14933: 14925: 14912: 14887: 14764: 14702: 14587: 14581: 14536: 14518: 14512: 14495: 14349: 14322: 14243: 14161: 14143: 14126: 14112: 14094: 13990: 13984: 13899: 13845: 13837: 13790: 13742: 13652: 13636: 13559: 13541: 13517: 13467: 13377: 13365: 13317: 13216: 13111: 13099: 13054: 13046: 12947: 12893: 12814: 12763: 12713: 12687: 12671: 12612: 12588: 12536: 12448: 12423: 12394: 12369: 12367:(14 July 1972). "Around-the-World Atomic Clocks: Predicted Relativistic Time Gains". 12309: 12243: 12220: 12149: 12117: 12061: 12013: 11961: 11917: 11839: 11820: 11726: 11648: 11604: 11592: 11516: 11504: 11476: 11462: 11355: 11299: 11281: 11263: 11192: 11172: 11134: 11029: 10912: 10880: 10830: 10788: 10717: 10698: 10635: 10623: 10615: 10545: 10451: 10439: 10427: 10328: 10284: 10266: 10247: 10150: 10127: 10071: 10035: 9994: 9974: 9807: 9771: 9726: 9654: 9633: 9588: 9459: 9421: 9381: 9331: 9277: 9263: 9221: 9185: 9146: 9014: 8993: 8972: 7777: 6731: 6719: 6458: 6446: 6387: 6229:

For the Sun's gravitational field using radar signals reflected from planets such as

6161: 5707: 5682: 5474:

Good introductions are, in order of increasing presupposed knowledge of mathematics,

5326: 4867: 4808: 4633: 4556: 4455: 4447: 4350: 4331: 4134: 4117: 3883: 3878:

Black holes are also sought-after targets in the search for gravitational waves (cf.

3849: 3293: 3172: 3168: 3004: 2894: 2753: 2391: 2335: 1517: 1186: 1147: 991: 984: 928: 844: 553: 328: 18983: 17936:

Landgraf, M.; Hechler, M.; Kemble, S. (2005), "Mission design for LISA Pathfinder",

17928: 16435:(1972), "Nonspherical Gravitational Collapse—A Short Review", in Klauder, J. (ed.), 16004: 15774: 15654: 15569: 15523: 15182: 15104: 14816: 14447: 14401: 14213: 14205: 14042: 13973: 13910:

Lovelock, David (1972), "The Four-Dimensionality of Space and the Einstein Tensor",

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Kuchař, Karel (1973), "Canonical Quantization of Gravity", in Israel, Werner (ed.),

13305: 13157: 13039: 12993: 12901: 12576: 12460: 12421:(14 July 1972). "Around-the-World Atomic Clocks: Observed Relativistic Time Gains". 12406: 12027: 11975: 11776: 11248: 11081: 10999: 10773: 10577: 10557: 10503: 10342: 10261:(1987), "Astrophysical Black Holes", in Hawking, Stephen W.; Israel, Werner (eds.), 10203: 9577:

General Relativity and Gravitation. Proceedings of the 17th International Conference

9568: 9517: 9065: 7734: 7092: 4326:

in which infinitely large regions of space and infinite time intervals are shrunk ("

4230: 4116:, the large-scale structure of the universe, and the existence and properties of a " 1733:, which is symmetric and a specific divergence-free combination of the Ricci tensor 1436: 1198: 1098: 603: 21531: 21489: 21164: 21131: 21116: 20998: 20867: 20782: 20722: 20648: 20347: 20329: 20007: 19780: 19474: 19368: 19018: 19008: 18955: 18932: 18510: 18197: 18116: 18096: 18057: 18037: 18016: 17963: 17923: 17881: 17687: 17623: 17474: 17125: 17115: 17001: 16888: 16878: 16844: 16824: 16786: 16776: 16684: 16640: 16379: 16327: 16285: 16240: 16230: 16153: 16143: 16109: 16089: 16054: 16042: 15992: 15957: 15945: 15837: 15817: 15798: 15762: 15642: 15557: 15511: 15463: 15453: 15361: 15273: 15252: 15242: 15209:

Albert Einstein—Chief Engineer of the Universe: Einstein's Life and Work in Context

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Pound, R. V.; Rebka, G. A. (1959), "Gravitational Red-Shift in Nuclear Resonance",

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Narayan, Ramesh; Bartelmann, Matthias (1997). "Lectures on Gravitational Lensing".

14201: 14069: 14030: 13961: 13927: 13889: 13879: 13833: 13756: 13734: 13704: 13692: 13626: 13616: 13529: 13509: 13459: 13425: 13353: 13293: 13206: 13196: 13145: 13087: 13027: 12981: 12935: 12889: 12849: 12804: 12794: 12753: 12743: 12661: 12651: 12564: 12516: 12485: 12440: 12414: 12386: 12360: 12346: 12299: 12291: 12273: 12216: 12188: 12145: 12107: 12097: 12073: 12051: 12005: 11953: 11810: 11800: 11764: 11716: 11706: 11676: 11638: 11628: 11584: 11492: 11454: 11341: 11333: 11313: 11236: 11120: 11112: 11069: 10987: 10948: 10924: 10904: 10868: 10838: 10818: 10761: 10688: 10678: 10611: 10573: 10537: 10491: 10419: 10373: 10320: 10237: 10227: 10191: 10111: 10107: 10090:

Bertschinger, Edmund (1998), "Simulations of structure formation in the universe",

10083: 10063: 10027: 9964: 9954: 9905: 9870: 9866: 9839: 9795: 9759: 9739: 9692: 9642: 9628: 9556: 9524: 9505: 9471: 9451: 9405: 9369: 9321: 9311: 9255: 9213: 9177: 9136: 9126: 9096: 9053: 9006: 8960: 8948: 8593: 7905: 7808: 7765: 7722: 7622: 7614: 7298:

for which the central compact object is either a neutron star or a black hole; cf.

7080: 6711: 6442: 6438: 6375: 6111:

General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36;

4845: 4466: 4420: 4403: 4286: 4199: 4145: 4129: 3872: 2745: 1560: 1509: 1329: 1322: 1202: 1194: 1128: 1011: 1007: 678: 633: 613: 558: 19078: 16412:; Veltman, Martinus (1974), "One Loop Divergencies in the Theory of Gravitation", 15802: 15113:"Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie" 13852:

Loll, Renate (1998), "Discrete Approaches to Quantum Gravity in Four Dimensions",

11255: 10541: 9917: 9234:

Proceedings of the Eleventh Marcel Grossmann Meeting Meeting on General Relativity

6253:, table 4.4 on p. 200; for more recent measurements using signals received from a 4252:

problem in order to investigate the flow of energy at infinity due to propagating

1225:

General relativity has acquired a reputation as a theory of extraordinary beauty.

1065:

and other observatories. In addition, general relativity has provided the base of

698: 21459: 21407: 21351: 21331: 21233: 21121: 20988: 20959: 20802: 20712: 20658: 20558: 20377: 20309: 20074: 19982: 19927: 19897: 19861: 19479: 19400: 19360: 19326: 19093: 19068: 18993: 18988: 18871: 18832: 18794: 18738: 18612: 18548: 18299: 18136: 17857: 17800: 17740: 17648: 17602: 17180: 17146: 17019: 16909: 16664: 16565: 16499: 16331: 15612: 14235: 14151: 12908:

Iorio, L. (2006), "A note on the evidence of the gravitomagnetic field of Mars",

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Images of all the known lenses can be found on the pages of the CASTLES project,

6201: 6064: 5320: 4653: 4599: 4512: 4451: 4443: 4424: 4416: 4380: 4375:

Using global geometry, some spacetimes can be shown to contain boundaries called

4366: 4323: 4296: 4245: 4180: 4172: 3887: 3807: 3727: 3645: 3628: 3554: 3471: 3073:(GPS). Tests in stronger gravitational fields are provided by the observation of 2914: 2783: 1700: 1572: 1490: 995: 876: 852: 733: 708: 593: 588: 452: 333: 295: 20317: 18876: 18061: 17000:(2003), "The Relativistic Binary Pulsar B1913+16"", in Bailes, M.; Nice, D. J.; 16570:

Mathematical Methods of Physics (Proceedings of the 1999 Londrina Winter School)

16383: 15766: 13463: 13357: 12009: 11458: 10031: 9822:(1971), "Oscillatory approach to the singular point in relativistic cosmology", 9181: 4644:, this radiation plays an important role for the thermodynamics of black holes. 4187:, which is an assumption beyond those of standard general relativity to prevent 3272:

of any orbit (the point of the orbiting body's closest approach to the system's

3069:, while ongoing validation is provided as a side effect of the operation of the 2965: 1818:

is the curvature scalar. The Ricci tensor itself is related to the more general

1142:

found the first non-trivial exact solution to the Einstein field equations, the

503: 21579: 21499: 21464: 21361: 21194: 21184: 21174: 21096: 21068: 21053: 21038: 20954: 20732: 20687: 20665: 20553: 20134: 19912: 19679: 19118: 18790: 18774: 18770: 18673: 18643: 18500: 18403: 17238: 16660: 15841: 15075:

Rees, Martin (1966), "Appearance of Relativistically Expanding Radio Sources",

15046: 14752: 14487: 14049: 14034: 13738: 12192: 11073: 10709: 10007: 9924: 9528: 9509: 9259: 9157: 9057: 8677: 8673: 8362: 8354: 7726: 7375:

and more, our own universe indeed appears to be isotropic and homogeneous, cf.

6715: 5601: 4907: 4796: 4729: 4536: 3697:: four images of the same astronomical object, produced by a gravitational lens 3694: 3652: 3622: 3496: 3273: 3225: 2752:, which is described by the inverse-square law. The second term represents the 1568: 1552: 1478: 1440: 1435:

as in classical mechanics. (The defining symmetry of special relativity is the

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Sorkin, Rafael D. (1997), "Forks in the Road, on the Way to Quantum Gravity",

15024:

Pound, R. V.; Snider, J. L. (1964), "Effect of Gravity on Nuclear Resonance",

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Engler, Gideon (2002), "Einstein and the most beautiful theories in physics",

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The fact that black holes radiate, quantum mechanically, was first derived in

6379: 6330:

Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin

6290:

Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin

6144:, ch. 4.3. For the most precise direct modern observations using quasars, cf. 5902:

Brief descriptions of these and further interesting solutions can be found in

4732:, but of minute one-dimensional extended objects. The theory promises to be a 4108:, allow physicists to model a universe that has evolved over the past 14 3839: 3813:

Observations of gravitational waves promise to complement observations in the

3640:"). For the Moon–Earth system, this effect has been measured with the help of 2580: 1217:. Physicists began to understand the concept of a black hole, and to identify 994:

remains a problem, however, as there is a lack of a self-consistent theory of

21591: 21436: 21341: 21253: 21126: 20742: 20353: 20323: 20210: 19616: 19285: 19108: 19088: 19083: 18998: 18866: 18694: 18638: 18470: 18423: 17885: 17713: 17217: 17085: 17064: 15879: 15398: 15136: 15054: 14692: 14669: 14636: 13297: 13067: 12235: 12156:

Gourgoulhon, Eric (2007). "3+1 Formalism and Bases of Numerical Relativity".

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Friedrich, Helmut (2005), "Is general relativity 'essentially understood'?",

11431:; Van Elst, Henk (1999), "Cosmological Models", in Lachièze-Rey, Marc (ed.), 10349: 9217: 9196:

Ashtekar, Abhay (1987), "New Hamiltonian formulation of general relativity",

9010: 8358: 8329: 7812: 7254:, sec. 4.1; for supernovae, there are still major problems to be solved, cf. 6723: 5040: 4745: 4725: 4659: 4552: 4473:), there will be horizons associated with a semiclassical radiation known as 4470: 4407: 4376: 4338: 4241: 3822: 3703: 3656: 3289: 3098: 3074: 3066: 3042: 2395: 1286: 1230: 1018: 703: 618: 598: 523: 421: 300: 59: 20507: 17785: 17671:

General Relativity: An Introduction to the Theory of the Gravitational Field

17355:

The Geometry of Spacetime: an Introduction to Special and General Relativity

16644: 16174:

Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; Herlt, E. (2003),

13513: 13429: 12322: 11957: 10423: 10377: 9799: 9763: 8672:

These variables represent geometric gravity using mathematical analogues of

5639:, pp. 19–22; for similar derivations, see sections 1 and 2 of ch. 7 in 5363: 4882: – Albert Einstein's hypothetical situations to argue scientific points 4164: 3936:

since it has important influence on the large-scale dynamics of the cosmos,

2988:(which has recently come to prominence in the context of what is called the 663: 21543: 21504: 21308: 21293: 21258: 21106: 21091: 20767: 20461: 20341: 19574: 19383: 19128: 19048: 19013: 18543: 18505: 18108: 17828: 17547: 17453: 17139: 16902: 16836: 16800: 16653:

Methods in Field Theory – Les Houches Summer School in Theoretical Physics.

16254: 16167: 16116:

Stairs, Ingrid H. (2003), "Testing General Relativity with Pulsar Timing",

16101: 15810: 15477: 15266: 14891: 14808: 14715: 14508: 14479: 14082: 14003:

Mannheim, Philip D. (2006), "Alternatives to Dark Matter and Dark Energy",

13903: 13803: 13764: 13640: 13521: 13408:

Komar, Arthur (1959), "Covariant Conservation Laws in General Relativity",

13220: 13149: 12818: 12774: 12767: 12675: 12520: 12452: 12398: 12128:

Goroff, Marc H.; Sagnotti, Augusto (1985), "Quantum gravity at two loops",

12121: 12065: 11831: 11824: 11768: 11730: 11652: 11596: 10916: 10702: 10549: 10431: 10251: 10075: 10039: 9978: 9463: 9335: 9189: 9150: 7769: 7263: 6450: 6038:; the most recent and most accurate Sirius B measurements are published in 4784: 4749: 4717: 4544: 4535:

covariance, Einstein's theory is not sufficient by itself to determine the

4497: 4474: 3712: 3690: 3644:. More recently, it has been measured for test masses aboard the satellite 3579: 3571: 3308: 3051: 2906: 2863: 2859:, spacetime tells matter how to move; matter tells spacetime how to curve. 2787: 2756:

in the circular motion. The third term represents the relativistic effect.

1398: 1306: 1277:

9.8 m/s (the acceleration due to gravity at the surface of the Earth).

1167: 1030: 648: 628: 19160: 18269: 18021: 12799: 12080:

Goenner, Hubert F. M. (2004), "On the History of Unified Field Theories",

9225: 3254: 1579:–momentum corresponds to the statement that the energy–momentum tensor is 21392: 21366: 21288: 20977: 20916: 20466: 20150: 19972: 19543: 18912: 18881: 18428: 17761: 17304: 16883: 16865: 16130: 16076: 16029: 16012: 15979: 15848: 15749: 15646: 15544: 15348: 15157: 14549: 14422: 14376: 14259:(1918), "On the Energy of the Gravitational Field in Einstein's Theory", 14226: 14017: 13980: 13884: 13603: 13496: 13446: 13280: 12634:

Heusler, Markus (1998), "Stationary Black Holes: Uniqueness and Beyond",

11571: 11367:"Über das Relativitätsprinzip und die aus demselben gezogene Folgerungen" 11056: 10974: 10855: 10748: 10665: 10524: 10406: 10119: 9892: 9672: 9615: 9316: 8585: 8377:

An overview of quantum theory can be found in standard textbooks such as

7295: 6242: 4818:

Observation of gravitational waves from binary black hole merger GW150914

4804: 4675: 4188: 4138: 3723: 3707: 3602: 3598: 3575: 3437:{\displaystyle \sigma ={\frac {24\pi ^{3}L^{2}}{T^{2}c^{2}(1-e^{2})}}\ ,} 3304: 2981: 2969: 2953: 2836:, which describe the relation between the geometry of a four-dimensional 2288: 1505: 1318: 1034: 1022: 983:, general relativity continues to be the simplest theory consistent with 960: 848: 538: 508: 37:

For a more accessible and less technical introduction to this topic, see

18237: 18100: 17120: 16781: 16235: 16148: 16093: 15458: 15247: 14872: 13621: 13201: 12748: 12656: 12102: 11940: 11805: 11711: 11633: 11258:(1973), "Survey of general relativity theory", in Israel, Werner (ed.), 11185:

Dediu, Adrian-Horia; Magdalena, Luis; Martín-Vide, Carlos, eds. (2015).

10908: 10683: 10232: 10067: 9959: 9131: 8896:; for a general review of experimental tests of general relativity, see 8600:. The theory is successful in that one mode will always correspond to a 7662:

is simply postulated, but not derived from an underlying physical theory

1166:—to match that observational presumption. By 1929, however, the work of 871:, providing a unified description of gravity as a geometric property of 21273: 20787: 20777: 20471: 19757: 19123: 18689: 18533: 18528: 17644: 17414: 17280: 17088:(2006), "The Confrontation between General Relativity and Experiment", 16688: 16620: 16548: 16432: 16314: 15949: 15629: 15225:

Reula, Oscar A. (1998), "Hyperbolic Methods for Einstein's Equations",

14147: 13820: 13777: 13091: 12876: 12568: 12528: 11681: 11611:

Font, José A. (2003), "Numerical Hydrodynamics in General Relativity",

11397:

Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin

11223: 10822: 10307: 9560: 8969:

Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity

8439:; an accessible introduction to black hole evaporation can be found in 7372: 6196:, pp. 202–204; in fact, Einstein published one such derivation as 5498:

An in-depth comparison between the two symmetry groups can be found in

5151:. Einstein's condemnation would prove to be premature, cf. the section 4932: 4582: 4501: 4203: 3853: 3834: 3660: 3243: 3180: 2852: 2849: 2105: 1580: 1413: 1337: 748: 310: 236: 168:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }} 83: 18259: 18003: 17950: 17910: 17102: 16763: 16725:

Quantum field theory in curved spacetime and black hole thermodynamics

16604: 16578: 16525: 16472: 16464:

Particle and Nuclear Astrophysics and Cosmology in the Next Millennium

16366: 15932: 15892: 15709: 15498: 15430: 15407: 14854: 14285: 14188: 13965: 13948: 13931: 13866: 13721: 13679: 13014: 12922: 12862:

Ibanez, L. E. (2000), "The second string (phenomenology) revolution",

12162: 11992: 11895: 11751: 11441: 11159: 11016: 10598: 10464:

Buchert, Thomas (2008), "Dark Energy from Structure—A Status Report",

9941: 9713: 9492: 9356: 9298: 9100: 9017:(1962), "The dynamics of general relativity", in Witten, Louis (ed.), 8564:

An accessible introduction at the undergraduate level can be found in

3163:. On 11 February 2016, the Advanced LIGO team announced that they had 21248: 21199: 20638: 19762: 19744: 19548: 19525: 18520: 18221: 16828: 16265: 15096: 14621: 14514:'Subtle is the Lord ...' The Science and life of Albert Einstein 13391: 12285: 10446:

Bruhat, Yvonne (1962), "The Cauchy Problem", in Witten, Louis (ed.),

10356:(1961), "Mach's Principle and a Relativistic Theory of Gravitation", 9651:

Einstein's Unfinished Symphony: Listening to the Sounds of Space-Time

9409: 8952: 8073:, ch. 2; a more extensive treatment of this solution can be found in 7655: 7287: 6550:

for a brief introduction to the methods of numerical relativity, and

4848: – Hypothetical FTL transportation by warping space (warp drive) 4101: 3818: 3632: 3055: 2929:

The core concept of general-relativistic model-building is that of a

2845: 1482: 1353: 1261: 968: 936: 906: 888: 883: 319: 79: 17309:

Space, Time, and Gravity: the Theory of the Big Bang and Black Holes

15112: 14722:(1965), "A measurement of excess antenna temperature at 4080 Mc/s", 12171:

Gowdy, Robert H. (1971), "Gravitational Waves in Closed Universes",

12056: 9455: 7520:, sec. 4.6, and simulations of large-scale structure formation, see 6478:"Gravitational waves detected 100 years after Einstein's prediction" 5597: 5562: 5239: 5235: 1494: 1389:. From this, one can deduce that spacetime is curved. The resulting 1229:

has noted that at multiple levels, general relativity exhibits what

21278: 21263: 19819: 19629: 19622: 19063: 18413: 18129: 16046: 15996: 15561: 15174: 14825:

Comptes rendus hebdomadaires des séances de l'Académie des sciences

14744: 14573: 14439: 14393: 14074: 12854: 12277: 11588: 10991: 10734:(1999), "Astrophysical evidence for the existence of black holes", 9909: 9107:

Ashby, Neil (2003), "Relativity in the Global Positioning System",

8601: 7832: 7791:

Sachs, R. (1962). "Asymptotic symmetries in gravitational theory".

7659: 6698: 6425: 5450:. The simple thought experiment in question was first described in 4753: 4578: 3783: 3756:

Artist's impression of the space-borne gravitational wave detector

3284:, but akin to an ellipse that rotates on its focus, resulting in a 3103: 3059: 1584: 1564: 1424: 1369: 1333: 1314: 1272: 1179: 1159: 972: 932: 898: 688: 498: 338: 19830: 18083: 16851:

Wambsganss, Joachim (1998), "Gravitational Lensing in Astronomy",

16173: 14791: 14672:(1969), "Gravitational collapse: the role of general relativity", 14552:(1966), "Primordial Helium abundance and primordial fireball II", 13340: 12968: 10478: 10178: 9242: 9040: 7709: 7067: 5863: 5839: 5243: 3904: 3231: 3031: 1336:), can be used to define the geometry of space, as well as a time 1109:

into his relativistic framework. In 1907, beginning with a simple

62:

GW150914 as seen by a nearby observer, during 0.33 s of its final

20972: 20934: 19191: 17243:

The Einstein Theory of Relativity: A Trip to the Fourth Dimension

16304:

Thiemann, Thomas (2007), "Loop Quantum Gravity: An Inside View",

14994:

Pound, R. V.; Rebka, G. A. (1960), "Apparent weight of photons",

14128:

Stellar Structure and Evolution (Lecture notes for Astronomy 606)

13771:, Contemporary Concepts in Physics, vol. 5, pp. 1–362, 11538:, Project Report: NASA, Stanford University and Lockheed Martin, 11413:"Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie" 6942:

For overviews of gravitational lensing and its applications, see

6241:, ch. 8, sec. 7; for signals actively sent back by space probes ( 6200:. Such calculations tacitly assume that the geometry of space is 4439: 4167:

showed that solutions to Einstein's equations exist that contain

3775: 3339:, expressed in radians per revolution, is approximately given by 3281: 1556: 1290: 1254: 1106: 18253:

Carroll, Sean M. (1997). "Lecture Notes on General Relativity".

17015:"Elements of the past: Big Bang Nucleosynthesis and observation" 13480: 13228: 13126:"Of pots and holes: Einstein's bumpy road to general relativity" 7230:. To a distant observer, some of these jets even appear to move 6649: 6285:"Näherungsweise Integration der Feldgleichungen der Gravitation" 5596:, sec. 1.4 for the experimental evidence, see once more section 4265:. This implies the conclusion that General Relativity (GR) does 3080: 2889:

As it is constructed using tensors, general relativity exhibits

1943:. Matching the theory's prediction to observational results for 1348:), there is a universality of free fall (also known as the weak 21298: 20890: 18743: 18151: 17833:

A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics

14901:

A Relativist's Toolkit. The Mathematics of Black-Hole Mechanics

14777:

Peskin, Michael E. (2007), "Dark Matter and Particle Physics",

14457:; Snyder, H. (1939), "On continued gravitational contraction", 8865: 7544: 7051: 6254: 4740:

of space in addition to the usual three. In what is called the

4712: 3731: 3664: 3583: 3559: 2871: 1944: 1218: 1026: 902: 894: 18286:"Introduction to Differential Geometry and General Relativity" 14639:(1965), "Gravitational collapse and spacetime singularities", 14407: 7435: 4752:

form part of a hypothesized eleven-dimensional model known as

4430:

Even more remarkably, there is a general set of laws known as

1178:

used these solutions to formulate the earliest version of the

19411: 18304: 17607:

Spacetime and Geometry: An Introduction to General Relativity

17050:, Scientific American Library, San Francisco: W. H. Freeman, 16749:

Wald, Robert M. (2001), "The Thermodynamics of Black Holes",

16308:, Lecture Notes in Physics, vol. 721, pp. 185–263, 11986:, Lecture Notes in Physics, vol. 721, pp. 105–120, 10584:

Carlip, Steven (2001), "Quantum Gravity: a Progress Report",

10383: 10301:, Lecture Notes in Physics, vol. 738, pp. 393–424, 9813: 8163: 7404: 6230: 5757:, ch. 7 or, in fact, any other textbook on general relativity 4680: 4602:, such as a more precise formulation of the hoop conjecture. 4459: 4159: 3799: 3667:

satellites, confirming the relativistic prediction. Also the

3269: 2576: 1947: 1879:{\displaystyle R_{\mu \nu }={R^{\alpha }}_{\mu \alpha \nu }.} 1409: 1298: 872: 18300:

The Feynman Lectures on Physics Vol. II Ch. 42: Curved Space

16360:. Lecture Notes in Physics. Vol. 631. pp. 41–135. 13265: 11934:, Lecture Notes in Physics, vol. 702, pp. 45–111, 11417:

Sitzungsberichte der Preußischen Akademie der Wissenschaften

10716:, Cambridge University Press, pp. 294–369 and 860–863, 9876: 8126:", are resolved into finite peaks beyond that approximation. 7658:

function that is crucial to determining the dynamics of the

7497: 7489: 7480:

and, for precision measurements by satellite observatories,

7286:, observations of stellar dynamics in the center of our own 6771:; for the initial evidence for gravitational radiation, see 3806:, is currently under development, with a precursor mission ( 2866:

gravitational potential of classical physics by a symmetric

18724: 17398:

Relativity, Gravitation and Cosmology: a Basic Introduction

17006:

Proceedings of "Radio Pulsars," Chania, Crete, August, 2002

14930:

String Theory Vol. I: An Introduction to the Bosonic String

13440:. Lecture Notes in Physics. Vol. 648. pp. 33–54. 11276:

Ehlers, Jürgen; Falco, Emilio E.; Schneider, Peter (1992),

11209:(1996), "M-Theory (the Theory Formerly Known as Strings)", 10146:

Ripples on a Cosmic Sea. The Search for Gravitational Waves

9927:(2002), "Numerical Approaches to Spacetime Singularities", 9160:(1986), "New variables for classical and quantum gravity", 8869: 7388: 6860:; further updates will be available on the mission website 6621: 5704:

Relativity, Gravitation and Cosmology: a Basic Introduction

4910: – Tensor index notation for tensor-based calculations 4617: 4104:

and homogeneous solutions of these enhanced equations, the

3779: 1393:

is a geometric formulation of Newtonian gravity using only

1302: 1062: 21538: 20859: 18159: – produced by the numerical relativity group at the 17262:(2001), "Gravitational radiation", in Murdin, Paul (ed.), 14823:

Poincaré, M. H. (1905), "Sur la dynamique de l'électron",

14048: 13536:

Kraus, Ute (1998), "Light Deflection Near Neutron Stars",

13390:

Kochanek, C.S.; Falco, E.E.; Impey, C.; Lehar, J. (2007),

11529:

Everitt, C. W. F.; Parkinson, Bradford; Kahn, Bob (2007),

10841:(2005), "The Lithium Content of the Galactic Halo Stars", 8966: 8191:, pp. 302–305. For numerical results, see the review 7481: 7258:; for simulating accretion and the formation of jets, cf. 6633: 5051:, ch. 9. The publication featuring the field equations is 5039:, pp. 110ff. Einstein's original papers are found in 4892:

Introduction to the mathematics of general relativity

4423:, and location at a specified time. This is stated by the 4318:. In consequence, an exploration of all light worldlines ( 4291: 3150: 2913:, as expressed in the equivalence principle, spacetime is 1939:

is the energy–momentum tensor. All tensors are written in

1029:, can escape from them. Black holes are the end-state for 17458:

Gravity: an Introduction to Einstein's General Relativity

17373:

Exploring Black Holes: Introduction to General Relativity

17330:

Geons, Black Holes, & Quantum Foam: a life in physics

16519:

Townsend, Paul K. (1997). "Black Holes (Lecture notes)".

16060: 15963: 15780: 15142: 11510: 11184: 10958: 7957:

The laws of black hole mechanics were first described in

7521: 7458: 7419: 7392: 7299: 7183: 6853: 6679: 6660: 6145: 5561:, ch. 12. For the experimental evidence, cf. the section 5259:

and references therein; the historical development is in

3612: 3113: 1385:

mathematically speaking, the Newtonian connection is not

855:

in 1915 and is the current description of gravitation in

18145: 17286:

Black Holes and Time Warps: Einstein's Outrageous Legacy

16446:

Black Holes and Time Warps: Einstein's Outrageous Legacy

15401:(2000). "Notes for a brief history of quantum gravity". 14599: 12499:(1966), "The occurrence of singularities in cosmology", 12287:

Quantum Gravitation – The Feynman Path Integral Approach

10509: 9025: 8885: 7747: 7376: 7255: 5862:

For background information and a list of solutions, cf.

5486:; for accounts of precision experiments, cf. part IV of 4783:. Space is represented by a web-like structure called a 3828: 3722:. It is used to detect the presence and distribution of 1423:

of (special) relativistic mechanics. In the language of

1289:'s motion can be described as a combination of free (or 18163:, with an elementary introduction to general relativity 16534:

Townsend, Paul K. (1996). "Four Lectures on M-Theory".

14174:

Narayan, Ramesh (2006), "Black holes in astrophysics",

13665:

Lehner, Luis (2001), "Numerical Relativity: A review",

13389: 12034:

Gnedin, Nickolay Y. (2005), "Digitizing the Universe",

10006: 9387: 8301: 6987: 6826: 6366:

Castelvecchi, Davide; Witze, Witze (11 February 2016).

6067:

experiment; an overview of experiments can be found in

4280: 3752: 1174:

in 1922, which do not require a cosmological constant.

82:

itself is distorted and dragged around by the rotating

15529: 11932:

Special Relativity—Will it Survive the Next 101 Years?

11296:

Special Relativity—Will it Survive the Next 101 Years?

8105:; a closer look at the singularity itself is taken in 7694: 7291: 4807:, twistor models or the path integral based models of 3964: 1976: 1639: 1376:. Space, in this construction, still has the ordinary 990:

Reconciliation of general relativity with the laws of

21515: 20103:

Die Grundlagen der Einsteinschen Relativitäts-Theorie

13070:(1971), "Event Horizons and Gravitational Collapse", 11885:(1996). "Partial Differential Equations of Physics". 10162: 9666: 8931:; Herman, R. C. (1948), "Evolution of the universe", 8881: 7227: 4986: 4903:

Problems with Einstein's general theory of relativity

4075: 3945: 3922: 3528: 3506: 3481: 3456: 3348: 3325: 3193: 3183:, arbitrarily strong gravitational waves do not obey 2759: 2591: 2433: 2372: 2352: 2297: 2273: 2124: 2071: 2048: 2024: 1956: 1915: 1895: 1831: 1772: 1739: 1709: 1604: 931:. These predictions concern the passage of time, the 111: 17346: 16956:

The Quantum Theory of Fields II: Modern Applications

16806: 16010: 15484:

Schäfer, Gerhard (2004), "Gravitomagnetic Effects",

14948:

String Theory Vol. II: Superstring Theory and Beyond

14089:

It's About Time. Understanding Einstein's Relativity

12832:(1975), "Discovery of a pulsar in a binary system", 12284:

Hamber, Herbert W. (2009), Hamber, Herbert W (ed.),

12255: 11528: 11275: 10930: 9600: 9523: 8361:, or Penrose's quasi-local energy–momentum based on 7958: 7642: 6975: 6943: 6912: 6857: 6554:

for the connection with gravitational wave astronomy

6039: 6035: 4562: 3022: 2804: 2401: 2398:

whose motion is described by the geodesic equation.

1301:

acting on a body is equal to that body's (inertial)

17862:"Die Grundlage der allgemeinen Relativitätstheorie" 17389: 16408: 16358:

Quantum Gravity: From Theory to Experimental Search

14838:"The Motion of Point Particles in Curved Spacetime" 14142: 11041: 10729: 10045: 9341: 9005: 8705: 8514: 8282: 8270: 8220: 8216: 7516:, ch. 18, evidence from gravitational lensing, cf. 7318: 7283: 7171: 6798: 6682:"Strong-Field Gravity Tests with the Double Pulsar" 6246: 5995: 4605: 3718:Gravitational lensing has developed into a tool of 3659:where, for any object entering a zone known as the 3651:Near a rotating mass, there are gravitomagnetic or 979:radiation. Despite the introduction of a number of 17496: 17371:Taylor, Edwin F.; Wheeler, John Archibald (2000), 16975: 16953: 16462:Thorne, Kip S. (1995), "Gravitational radiation", 15334:Roulet, E.; Mollerach, S. (1997), "Microlensing", 15295: 14756: 14086: 12226: 11485:International Studies in the Philosophy of Science 9986: 8621: 8609: 7961:; a more pedagogical presentation can be found in 4999:, Scotland: School of Mathematics and Statistics, 4888: – Debate about credit for general relativity 4091: 4057: 3928: 3534: 3512: 3487: 3462: 3436: 3331: 3209: 2823: 2729: 2564: 2378: 2358: 2322: 2279: 2256: 2093: 2054: 2030: 2010: 1962: 1931: 1901: 1878: 1807: 1755: 1725: 1688: 167: 58:Slow motion computer simulation of the black hole 20036:Investigations on the Theory of Brownian Movement 15734: 14453: 14219: 12624: 11689:Frauendiener, Jörg (2004), "Conformal Infinity", 9675:(1984), "Theory of extragalactic radio sources", 8536:In particular, a perturbative technique known as 8353:Such quasi-local mass–energy definitions are the 7501: 7247: 7009: 6963: 6365: 6361: 6359: 5451: 5364:"Gravity and Me: The force that shapes our lives" 4539:of the metric tensor. It must be combined with a 3558:Orbital decay for PSR J0737−3039: time shift (in 1131:and published by Grossmann and Einstein in 1913. 1076:Widely acknowledged as a theory of extraordinary 21589: 17692:Gravity: Newtonian, Post-Newtonian, Relativistic 17400:, Oxford and New York: Oxford University Press, 15615:(2007), "String Theory: Progress and Problems", 13178: 13110:, Cambridge University Press, pp. 199–276, 12587:, Cambridge University Press, pp. 631–651, 11294:Ehlers, Jürgen; Lämmerzahl, Claus, eds. (2006), 11293: 10836: 10714:General Relativity, an Einstein Centenary Survey 10448:Gravitation: An Introduction to Current Research 10265:, Cambridge University Press, pp. 277–329, 9283: 9019:Gravitation: An Introduction to Current Research 8888:; for a review of work on compact binaries, see 7917: 7741: 7439: 6508: 5706:. Oxford and New York: Oxford University Press. 5676: 5600:. Choosing a different connection with non-zero 5487: 4914:Timeline of gravitational physics and relativity 2815:Physical theories modified by general relativity 2323:{\displaystyle \Gamma ^{\mu }{}_{\alpha \beta }} 1267: 1150:objects were taken, eventually resulting in the 17581:(Second ed.), Cambridge University Press, 16978:The Quantum Theory of Fields III: Supersymmetry 15416:Rovelli, Carlo (1998), "Loop Quantum Gravity", 15333: 14751: 14296:"What was Einstein's principle of equivalence?" 11556: 11087: 10886: 9984: 9433: 8487: 8485: 8401: 7819: 7473: 7431: 7368: 7307: 7104: 6998: 6908: 5598:Gravitational time dilation and frequency shift 5563:Gravitational time dilation and frequency shift 5318: 5240:Gravitational time dilation and frequency shift 5236:Orbital effects and the relativity of direction 4899: – Predecessor to the theory of relativity 3737: 3232:Orbital effects and the relativity of direction 3032:Gravitational time dilation and frequency shift 20043:Relativity: The Special and the General Theory 19485:Degenerate Higher-Order Scalar-Tensor theories 18141:Max Planck Institute for Gravitational Physics 17891:English translation at Einstein Papers Project 17186:Relativity: The Special and the General Theory 17069:Theory and experiment in gravitational physics 17025:Max Planck Institute for Gravitational Physics 16995: 15679: 15660: 14343: 13806:(2006), "Towards inflation in string theory", 13646: 13438:Astrophysics, Clocks and Fundamental Constants 12413: 12359: 12127: 11864:Max Planck Institute for Gravitational Physics 11423:English translation at Einstein Papers Project 11403:English translation at Einstein Papers Project 11381:English translation at Einstein Papers Project 11153:, Singapore: World Scientific, p. 12024, 9705:Encyclopedia of Mathematical Physics, Volume 2 9074:"Relativity and the Global Positioning System" 8549: 6889: 6877: 6814: 6764: 6637: 6368:"Einstein's gravitational waves found at last" 6356: 6250: 6141: 6124: 6112: 6068: 6060: 6056: 6023: 5971: 5557:, pp. 17ff; a derivation can be found in 5094: 5090: 4959: 4530:Initial value formulation (general relativity) 3798:to detect gravitational waves in the 10 to 10 2968:(which opens up the intriguing possibility of 2874:, the latter reduces to the former in certain 1404: 1255:From classical mechanics to general relativity 1248:Relativity: The Special and the General Theory 1233:has termed a "strangeness in the proportion" ( 1158:. In 1917, Einstein applied his theory to the 909:are present. The relation is specified by the 30:For the graduate textbook by Robert Wald, see 20875: 20537: 20523: 19846: 19176: 18320: 18148:, the official website of the GEO600 project. 17984:"The quest to understand the Pioneer anomaly" 17726:, W. H. Freeman, Princeton University Press, 16176:Exact Solutions of Einstein's Field Equations 15820:(1964), "Fourth test of general relativity", 15278:Relativity. Special, General and Cosmological 15191:The Genesis of General Relativity (4 Volumes) 14714: 14361: 13396:, Harvard-Smithsonian Center for Astrophysics 11658: 11427: 11311: 11090:"LISA Technology—Concepts, Status, Prospects" 11088:Danzmann, Karsten; Rüdiger, Albrecht (2003), 10800: 10779: 10630:Carroll, Bradley W.; Ostlie, Dale A. (1996), 10629: 10392:(2003), "Precision Cosmology? Not Just Yet", 10296: 10139: 10117: 9702: 8420: 8232: 8102: 8090: 8089:, ch. 4; for a more extensive treatment, cf. 8074: 8004:; a more thorough derivation can be found in 7945: 7671: 7625:; a pedagogical introduction can be found in 7477: 7427: 7211: 7029: 6856:; a first post-flight evaluation is given in 6205: 5878: 5701: 5295: 5244:Light deflection and gravitational time delay 4724:One attempt to overcome these limitations is 4692:the extra dimensions posited by string theory 4337:Aware of the importance of causal structure, 4106:Friedmann–Lemaître–Robertson–Walker solutions 3912:The current models of cosmology are based on 3771:are currently in operation, most notably the 3674: 3081:Light deflection and gravitational time delay 3007:. Approximate solutions may also be found by 1427:: where gravity can be neglected, physics is 810: 16596:Encyclopedia of Mathematical Physics, Vol. 2 14303:Studies in History and Philosophy of Science 13051:Canonical Gravity: From Classical to Quantum 12600: 11782: 11688: 11005: 10089: 9477: 8927: 8893: 8482: 8204: 8054: 7933: 7909: 7873: 7857: 7849: 7469: 7454: 7329: 7148: 6410: 6022:; a list of further experiments is given in 5903: 5577:, sec. 1.13; for an elementary account, see 4987:O'Connor, J.J.; Robertson, E.F. (May 1996). 4933:"GW150914: LIGO Detects Gravitational Waves" 2011:{\textstyle \kappa ={\frac {8\pi G}{c^{4}}}} 939:, and the propagation of light, and include 19190: 16935:The Quantum Theory of Fields I: Foundations 16623:(1976), "Notes on Black Hole Evaporation", 15023: 14945: 14924: 13769:Particle Physics and Inflationary Cosmology 12824: 12155: 10564:Caldwell, Robert R. (2004), "Dark Energy", 8990:Mathematical Methods of Classical Mechanics 8573: 8569: 8255: 8243:, ch. 10; an online review can be found in 8164:Belinskii, Khalatnikov & Lifschitz 1971 7418:; for a recent account of predictions, see 6768: 6323: 6282: 6019: 5348:The following exposition re-traces that of 5322:Relativity – The Special and General Theory 5043:, volumes 4 and 6. An early key article is 4779:, this leads to a promising model known as 3319:In general relativity the perihelion shift 20882: 20868: 20530: 20516: 19853: 19839: 19183: 19169: 18327: 18313: 17264:Encyclopedia of Astronomy and Astrophysics 17208:: CS1 maint: location missing publisher ( 16850: 16178:(2 ed.), Cambridge University Press, 15617:Progress of Theoretical Physics Supplement 14993: 14963: 12725: 12699: 12240:Superstring theory. Volume 1: Introduction 11371:Jahrbuch der Radioaktivität und Elektronik 10348: 9777: 9738: 8700:; more extensive accounts can be found in 8568:; more complete overviews can be found in 8382: 7989: 7985: 7825: 7784: 7498:Kamionkowski, Kosowsky & Stebbins 1997 7040: 6947: 6896:, sec. 9.7; for a more recent review, see 6192:, sec. 1.16; for the historical examples, 6015: 6011: 5726: 5361: 4996:History Topics: Mathematical Physics Index 4863:Derivations of the Lorentz transformations 4160:Exotic solutions: time travel, warp drives 3916:, which include the cosmological constant 2287:is a scalar parameter of motion (e.g. the 1808:{\displaystyle R=g^{\mu \nu }R_{\mu \nu }} 1021:implications, including the prediction of 817: 803: 18258: 18082: 18020: 18002: 17949: 17927: 17909: 17673:, Cambridge: Cambridge University Press, 17481:, Cambridge: Cambridge University Press, 17129: 17119: 17101: 16892: 16882: 16864: 16790: 16780: 16762: 16603: 16577: 16547: 16524: 16471: 16365: 16313: 16266:"Binary pulsars and relativistic gravity" 16244: 16234: 16157: 16147: 16129: 16075: 16028: 15978: 15931: 15891: 15748: 15708: 15628: 15543: 15497: 15467: 15457: 15447: 15429: 15406: 15347: 15256: 15246: 15156: 15045: 15015: 14985: 14881: 14871: 14853: 14790: 14743: 14660: 14478: 14421: 14375: 14284: 14278: 14261:Verhandl. Koninkl. Ned. Akad. Wetenschap. 14255: 14225: 14187: 14073: 14016: 13947: 13937: 13893: 13883: 13865: 13819: 13776: 13720: 13678: 13630: 13620: 13602: 13588: 13495: 13445: 13339: 13329: 13311: 13279: 13226: 13210: 13200: 13013: 12967: 12921: 12875: 12853: 12808: 12798: 12757: 12747: 12665: 12655: 12350: 12303: 12161: 12111: 12101: 12055: 11991: 11939: 11894: 11851: 11814: 11804: 11750: 11736: 11720: 11710: 11680: 11642: 11632: 11570: 11440: 11345: 11222: 11158: 11124: 11055: 11015: 10973: 10854: 10747: 10692: 10682: 10664: 10597: 10523: 10477: 10405: 10306: 10257: 10241: 10231: 10177: 9968: 9958: 9940: 9891: 9712: 9648: 9632: 9614: 9491: 9355: 9325: 9315: 9297: 9241: 9140: 9130: 9039: 8861: 8837: 8171: 7888:; for more recent numerical studies, see 7831: 7708: 7476:, ch. 11; for the initial detection, see 7450: 7198: 7066: 7025: 6861: 6830: 6752: 6697: 6624:measurements of planetary positions; see 6424: 6137: 5867: 5180: 5176: 5102: 5035:; an accessible overview can be found in 4925: 4041: 3977: 1685: 1652: 1002:with the three non-gravitational forces: 998:. It is not yet known how gravity can be 27:Theory of gravitation as curved spacetime 21239:Covariance and contravariance of vectors 20477:Emergency Committee of Atomic Scientists 18208:Series of lectures on General Relativity 17770:. Chicago: University of Chicago Press. 17653:The Classical Theory of Fields (4th ed.) 17311:, Chicago: University of Chicago Press, 17224:, Chicago: University of Chicago Press, 17145: 16973: 16951: 16931: 16908: 16593: 16564: 16533: 16518: 16355: 16303: 16212: 15110: 14835: 14822: 14779:Journal of the Physical Society of Japan 14344:Ohanian, Hans C.; Ruffini, Remo (1994), 14234: 14002: 13909: 11407: 11387: 11361: 11148: 10563: 10209: 9231: 9195: 9156: 8889: 8780: 8709: 8697: 8685: 8681: 8645: 8633: 8565: 8541: 8525: 8502:A timeline and overview can be found in 8440: 8397: 8381:; a more elementary account is given in 8366: 8341: 8086: 8070: 8021: 7626: 7582: 7578: 7562: 7422:; an accessible account can be found in 7340: 6893: 6873: 6810: 6406: 6404: 6238: 6197: 6177: 5975: 5890: 5827: 5767: 5754: 5738: 5730: 5644: 5640: 5624: 5114: 5098: 5052: 5044: 4975: 4813: 4787:, evolving over time in discrete steps. 4764: 4720:of the type used in loop quantum gravity 4711: 4679: 4624:Quantum field theory in curved spacetime 4618:Quantum field theory in curved spacetime 4393: 4290: 4244:, M. G. van der Burg, A. W. Metzner and 4214: 3903: 3879: 3838: 3751: 3689: 3679: 3553: 3253: 3249: 3149: 3102: 3041: 2420:revolving around a massive central body 1571:for gravity relates this tensor and the 1408: 1271: 45: 19789:Gravitational interaction of antimatter 18189:Einstein's General Theory of Relativity 17810:The Large Scale Structure of Space-time 17628:Einstein's General Theory of Relativity 17437:Einstein's General theory of Relativity 17042: 16650: 15816: 15611: 15483: 15415: 15397: 15379: 15293: 15272: 14898: 14759:An Introduction to Quantum Field Theory 14691: 14668: 14635: 14579: 14548: 14530: 14486: 14173: 14106: 13647:Landau, L. D.; Lifshitz, E. M. (1975), 13123: 12681: 12633: 12608:The large scale structure of space-time 12582: 12542: 12495: 12079: 11981: 11929: 11911: 11433:Theoretical and Observational Cosmology 10647: 10463: 9985:Bergström, Lars; Goobar, Ariel (2003), 9537:"The Four Laws of Black Hole Mechanics" 8799: 8784: 8769: 8757: 8701: 8545: 8503: 8432: 8378: 8187:; a textbook-level account is given in 8184: 8170:. An accessible exposition is given by 8151: 8140: 8034: 8001: 7974: 7970: 7941: 7937: 7586: 7574: 7548: 7532: 7517: 7513: 7415: 7356: 7303: 6897: 6794: 6661:Dediu, Magdalena & Martín-Vide 2015 6579: 6519: 6234: 6217: 6189: 5991: 5967: 5866:; a more recent review can be found in 5815: 5778: 5734: 5578: 5574: 5510: 5499: 5483: 5475: 5427: 5283: 5260: 5028: 4141:, the nature of which remains unclear. 2832:theory of gravitation. At its core are 1531: 971:, thus leading to the discovery of the 14: 21590: 20728:Atomic, molecular, and optical physics 20221:Albert Einstein World Award of Science 18032:from the original on 24 September 2015 17245:, Philadelphia: Paul Dry Books, Inc., 16497: 16461: 16443: 16431: 16414:Annales de l'Institut Henri Poincaré A 16260: 16115: 15917: 15878: 15698: 15593: 15575: 15315: 15053: 14776: 14332:from the original on 22 September 2006 14293: 14124: 14081: 13979: 13710: 13664: 13578:Annales de l'Institut Henri Poincaré A 13571: 13556:Relativity, Astrophysics and Cosmology 13553: 13435: 13371: 13105: 13066: 12861: 12773: 12549:Communications in Mathematical Physics 12283: 12033: 11881: 11482: 11260:Relativity, Astrophysics and Cosmology 11254: 10785:The Mathematical Theory of Black Holes 10708: 10632:An Introduction to Modern Astrophysics 10583: 10445: 10278: 9923: 9574: 9415: 8984: 8971:, World Scientific, pp. 353–355, 8849: 8795: 8746: 8724: 8660: 8597: 8553: 8544:, ch. 17, 18, fails in this case; cf. 8491: 8476: 8464: 8405: 8393: 8296:; for a pedagogical introduction, see 8259: 8239:; for a pedagogical introduction, see 8236: 8219:; for a pedagogical introduction, see 8192: 8167: 8106: 7962: 7929: 7901: 7889: 7885: 7630: 7536: 7472:, for a pedagogical introduction, see 7215: 7187: 7159: 7137: 7021: 6959: 6852:A mission description can be found in 6783: 6772: 6748: 6744: 6563: 6551: 6547: 6495: 6472: 6470: 6468: 6258: 6193: 6165: 6100: 6096: 5915: 5851: 5664: 5652: 5636: 5620: 5616: 5593: 5582: 5558: 5554: 5542: 5538: 5526: 5479: 5462: 5447: 5439: 5399: 5387: 5349: 5307: 5222: 5218: 5203: 5131: 4523: 4353:) are used to derive general results. 4341:and others developed what is known as 4295:Penrose–Carter diagram of an infinite 3613:Geodetic precession and frame-dragging 3238:Two-body problem in general relativity 3139: 3134:parameterized post-Newtonian formalism 2862:While general relativity replaces the 2408:Two-body problem in general relativity 20863: 20511: 19834: 19794:Physics in the medieval Islamic world 19717:(2+1)-dimensional topological gravity 19213:Newton's law of universal gravitation 19164: 18308: 17793: 17745:General Relativity for Mathematicians 17595: 17012: 16665:"On Particle Creation by Black Holes" 16619: 16191: 15847: 15224: 13802: 13763: 13535: 13407: 13045: 12999: 12953: 12907: 12467: 12321: 12198: 12170: 11914:Special Relativity: A First Encounter 11830: 11393:"Die Feldgleichungen der Gravitation" 10803:Journal of Astrophysics and Astronomy 10787:, New York: Oxford University Press, 10651:(2001), "The Cosmological Constant", 10281:The Early Universe. Facts and Fiction 9418:How is Quantum Field Theory Possible? 9106: 9071: 8720: 8293: 8244: 8038: 7940:, ch. 9 and 10 for a derivation, and 7790: 7683: 7602: 7598: 7577:; an accessible overview is given in 7423: 7267: 6931: 6927: 6916: 6827:Bertotti, Ciufolini & Bender 1987 6675: 6673: 6535: 6531: 6401: 6085: 6081: 5970:, pp. 24–26 vs. pp. 236–237 and 5514: 5443: 5148: 5135: 5134:; an accessible overview is given in 4876: – Concept in general relativity 4870: – Paradox in special relativity 4636:leading to the possibility that they 4492:have "ragged edges"—regions known as 4196:exact solutions in general relativity 3829:Black holes and other compact objects 3165:directly detected gravitational waves 2901:. It thus satisfies a more stringent 2412:In general relativity, the effective 1431:as in special relativity rather than 1201:"), and in 1919 an expedition led by 925:Newton's law of universal gravitation 921:Newton's law of universal gravitation 869:Newton's law of universal gravitation 19539:Asymptotic safety in quantum gravity 19265: 17578:A First Course in General Relativity 17084: 17063: 17048:A Journey Into Gravity and Spacetime 16748: 16722: 16702: 16659: 16199:, North-Holland Publishing Company, 15854:Big Bang: The Origin of the Universe 15578:A first course in general relativity 15206: 15188: 15074: 14507: 13851: 13649:The Classical Theory of Fields, v. 2 13254:, Stanford University News Service, 13244: 12700:Hey, Tony; Walters, Patrick (2003), 11610: 11205: 10730:Celotti, Annalisa; Miller, John C.; 9579:, World Scientific, pp. 24–34, 8897: 8812: 8735: 8649: 8451: 8436: 8416: 8325: 8313: 8312:For a pedagogical introduction, see 8297: 8240: 8188: 8122:of many wave equations, namely the " 8058: 8042: 8009: 8005: 7966: 7913: 7869: 7853: 7344: 7259: 7251: 7250:or, for more recent numerical work, 7235: 6841: 6629: 6625: 6608: 6604: 6591: 6567: 6270: 5979: 5955: 5951: 5939: 5927: 5802: 5790: 5750: 5604:leads to a modified theory known as 5415: 5411: 5271: 5191: 5164: 5118: 5056: 5048: 5036: 5032: 5024: 4955: 4953: 4854: – Proposed theories of gravity 4281:Causal structure and global geometry 4209: 4153: 4133:by a form of energy with an unusual 3810:) having launched in December 2015. 3706:, or partial rings called arcs. The 3648:to a precision of better than 0.3%. 3307:to a body's gravity (encoded in the 3091:Kepler problem in general relativity 19860: 18899:Tolman–Oppenheimer–Volkoff equation 18852:Friedmann–Lemaître–Robertson–Walker 17850: 17503:, Oxford: Oxford University Press, 15685:Sitzungsber. Preuss. Akad. D. Wiss. 15666:Sitzungsber. Preuss. Akad. D. Wiss. 13002:Central European Journal of Physics 9989:Cosmology and Particle Astrophysics 9742:(1973), "Black Holes and Entropy", 8302:Ashtekar & Magnon-Ashtekar 1979 7948:as overviews of more recent results 7426:; compare with the observations in 7228:Begelman, Blandford & Rees 1984 6465: 4886:General relativity priority dispute 4383:, the relevant length scale is the 3566:According to general relativity, a 2958:Friedmann–Lemaître–Robertson–Walker 1197:without any arbitrary parameters (" 24: 21102:Tensors in curvilinear coordinates 20250:Albert Einstein: Creator and Rebel 19963:Einstein–Infeld–Hoffmann equations 19933:Einstein relation (kinetic theory) 19291:Gibbons–Hawking–York boundary term 18270:"Understanding General Relativity" 17521:(1st ed.). Berlin: Springer. 17479:Introduction to General Relativity 17168: 15516:10.1023/B:GERG.0000046180.97877.32 15486:General Relativity and Gravitation 15384:, Boston: Walter de Gruyter GmbH, 15298:Introduction to Special Relativity 13713:General Relativity and Gravitation 13261:from the original on 23 April 2007 13072:General Relativity and Gravitation 12545:"Particle Creation by Black Holes" 11854:"Of singularities and breadmaking" 11318:General Relativity and Gravitation 11010:, World Scientific, p. 4090, 10466:General Relativity and Gravitation 9993:(2nd ed.), Wiley & Sons, 7959:Bardeen, Carter & Hawking 1973 6976:Walsh, Carswell & Weymann 1979 6944:Ehlers, Falco & Schneider 1992 6913:Ciufolini, Pavlis & Peron 2006 6858:Everitt, Parkinson & Kahn 2007 6670: 6620:The most precise measurements are 6036:Greenstein, Oke & Shipman 1971 4858:Contributors to general relativity 4852:Alternatives to general relativity 4647: 3994: 3923: 3764: 2917:, and the laws of physics exhibit 2772:alternatives to general relativity 2766:Alternatives to general relativity 2760:Alternatives to general relativity 2299: 2170: 1443:, and with high-energy phenomena. 1264:derivation of general relativity. 935:of space, the motion of bodies in 128: 39:Introduction to general relativity 25: 21624: 20111:The Einstein Theory of Relativity 19406:Modified Newtonian dynamics, MOND 19322:Classical theories of gravitation 18669:Hamilton–Jacobi–Einstein equation 18123: 17655:, London: Butterworth-Heinemann, 17609:, San Francisco: Addison-Wesley, 17499:Introducing Einstein's Relativity 17460:, San Francisco: Addison-Wesley, 17347:Beginning undergraduate textbooks 16536:High Energy Physics and Cosmology 16439:, W. H. Freeman, pp. 231–258 16306:Approaches to Fundamental Physics 11984:Approaches to Fundamental Physics 11435:, vol. 541, pp. 1–116, 8283:Arnowitt, Deser & Misner 1962 8271:Misner, Thorne & Wheeler 1973 8221:Misner, Thorne & Wheeler 1973 8217:Arnowitt, Deser & Misner 1962 7284:Celotti, Miller & Sciama 1999 7172:Celotti, Miller & Sciama 1999 6799:Misner, Thorne & Wheeler 1973 6482:NSF – National Science Foundation 6247:Bertotti, Iess & Tortora 2003 5996:Misner, Thorne & Wheeler 1973 5362:Al-Khalili, Jim (26 March 2021). 4950: 4897:Nordström's theory of gravitation 4826: 4759:Another approach starts with the 4563:Global and quasi-local quantities 4406:black hole) and the axisymmetric 4225:The spacetime symmetry group for 4120:echo" from the early cosmos, the 3671:probe around Mars has been used. 3023:Consequences of Einstein's theory 2924: 2811:Mathematics of general relativity 2805:Definition and basic applications 2402:Total force in general relativity 2040:Newtonian constant of gravitation 1542:Mathematics of general relativity 1134:The Einstein field equations are 1094:Classical theories of gravitation 21573: 21561: 21549: 21537: 21525: 20490: 20489: 20226:Einstein Prize for Laser Science 19147: 19146: 17390:Advanced undergraduate textbooks 17173: 16017:Astrophysical Journal Supplement 14583:Principles of physical cosmology 13314:One hundred authors for Einstein 11545:from the original on 9 June 2007 9634:10.1111/j.1365-2966.2005.09359.x 8902: 8874: 8854: 8842: 8831: 8818: 8805: 8789: 8774: 8763: 8751: 8740: 8729: 8714: 8708:as well as in the lecture notes 8690: 8666: 8654: 8639: 8627: 8622:Green, Schwarz & Witten 1987 8615: 8610:Green, Schwarz & Witten 1987 8578: 8558: 8530: 8519: 8508: 8496: 8470: 8457: 8445: 8425: 8410: 8404:; a more accessible overview is 8387: 8371: 8365:methods; cf. the review article 8347: 8335: 8319: 8306: 8287: 8276: 8264: 8249: 8226: 8210: 8198: 8176: 8166:; for a more recent review, see 8156: 8145: 8129: 8112: 8096: 8080: 8064: 8048: 8027: 8015: 7994: 7979: 7965:; for a more recent review, see 7951: 7922: 7895: 7879: 7863: 7843: 7688: 7677: 7665: 7648: 7636: 7607: 7601:; for a more recent review, see 7591: 7568: 7554: 7526: 7506: 7463: 7444: 7409: 7397: 7381: 7362: 7350: 7334: 7323: 7312: 7272: 7240: 7220: 7204: 7192: 7177: 7165: 7153: 7142: 7131: 7109: 7098: 7045: 7034: 7015: 7003: 6992: 6980: 6969: 6952: 6936: 6921: 6902: 6883: 6867: 6846: 6835: 6820: 6804: 6788: 6767:; for the pulsar discovery, see 4703:effective (quantum) field theory 4640:over time. As briefly mentioned 4606:Relationship with quantum theory 4480: 4185:chronology protection conjecture 3562:), tracked over 16 years (2021). 3549: 2931:solution of Einstein's equations 2909:are the same for all observers. 2840:representing spacetime, and the 2748:. The first term represents the 1215:golden age of general relativity 1156:electrically charged black holes 784: 783: 770: 100: 20849:Timeline of physics discoveries 20271:Einstein: His Life and Universe 19228:History of gravitational theory 17151:A First Course in String Theory 16727:, University of Chicago Press, 16709:, University of Chicago Press, 14755:; Schroeder, Daniel V. (1995), 13558:, D. Reidel, pp. 237–288, 13334:, vol. 0709, p. 685, 13316:, Wiley-VCH, pp. 178–181, 10384:Bridle, Sarah L.; Lahav, Ofer; 9390:Journal of Mathematical Physics 8920: 8706:Ashtekar & Lewandowski 2004 7973:. For the Penrose process, see 6777: 6758: 6738: 6643: 6614: 6597: 6585: 6573: 6557: 6540: 6525: 6513: 6501: 6488: 6324:Einstein, A (31 January 1918). 6317: 6276: 6264: 6223: 6210: 6183: 6171: 6150: 6130: 6118: 6105: 6090: 6074: 6045: 6029: 6001: 5985: 5961: 5945: 5933: 5920: 5909: 5896: 5884: 5872: 5856: 5844: 5832: 5820: 5808: 5796: 5784: 5772: 5760: 5744: 5720: 5695: 5670: 5658: 5630: 5610: 5587: 5568: 5548: 5532: 5520: 5504: 5492: 5468: 5456: 5433: 5421: 5405: 5393: 5381: 5355: 5342: 5312: 5301: 5289: 5277: 5265: 5249: 5228: 5212: 5197: 5185: 5170: 5158: 5141: 5124: 3520:is the speed of light in vacuum 2903:general principle of relativity 2824:Definition and basic properties 2094:{\displaystyle R_{\mu \nu }=0.} 1763:and the metric. In particular, 1154:, which is now associated with 19893:Mass–energy equivalence (E=mc) 19534:Causal dynamical triangulation 19223:Poisson's equation for gravity 18476:Mass–energy equivalence (E=mc) 18334: 18200:recorded 22 September 2008 at 17982:Nieto, Michael Martin (2006), 17835:. Cambridge University Press. 17813:. Cambridge University Press. 17694:. Cambridge University Press. 17421:, Princeton University Press, 17222:General Relativity from A to B 17153:, Cambridge University Press, 17071:, Cambridge University Press, 16982:, Cambridge University Press, 16960:, Cambridge University Press, 16938:, Cambridge University Press, 16598:, Elsevier, pp. 189–195, 16504:. Cambridge University Press. 16197:Relativity: The Special Theory 15598:, Cambridge University Press, 15580:, Cambridge University Press, 14950:, Cambridge University Press, 14932:, Cambridge University Press, 14903:, Cambridge University Press, 14836:Poisson, Eric (27 May 2004a). 14586:, Princeton University Press, 14535:, Cambridge University Press, 14348:, W. W. Norton & Company, 14242:, Cambridge University Press, 14093:, Princeton University Press, 12704:, Cambridge University Press, 12686:, Cambridge University Press, 12684:Black Hole Uniqueness Theorems 12611:, Cambridge University Press, 12305:11858/00-001M-0000-0013-471D-A 12242:, Cambridge University Press, 11347:11858/00-001M-0000-0013-5AB5-4 11126:11858/00-001M-0000-0013-5233-E 10196:10.1088/0264-9381/25/16/165003 10126:, Cambridge University Press, 10124:Quantum Fields in Curved Space 10112:10.1146/annurev.astro.36.1.599 9871:10.3367/ufnr.0102.197011d.0463 8825: 7613:More precisely, these are the 7420:Coc, Vangioni‐Flam et al. 2004 6443:10.1103/PhysRevLett.116.061102 5256: 5152: 5108: 5084: 5071: 5062: 5018: 4980: 4969: 4880:Einstein's thought experiments 4672:Causal dynamical triangulation 4410:(used to describe a rotating, 3852:, neutron stars of around 1.4 3422: 3403: 2608: 2602: 2450: 2444: 2414:gravitational potential energy 1499:Einstein equivalence principle 1465:in the image). These sets are 1358:Einstein's elevator experiment 915:partial differential equations 13: 1: 21155:Exterior covariant derivative 21087:Tensor (intrinsic definition) 17556:A General Relativity Workbook 17435:Gron, O.; Hervik, S. (2007), 17266:, Institute of Physics Pub., 16655:, vol. 77, North Holland 15803:10.1103/PhysRevLett.92.121101 15366:10.1016/S0370-1573(96)00020-8 15059:Field Theory: A Modern Primer 13986:What Remains To Be Discovered 12910:Classical and Quantum Gravity 11262:, D. Reidel, pp. 1–125, 10542:10.1103/PhysRevLett.90.241101 10166:Classical and Quantum Gravity 10092:Annu. Rev. Astron. Astrophys. 9847:; original paper in Russian: 7502:Seljak & Zaldarriaga 1997 7432:Bania, Rood & Balser 2002 7308:Falcke, Melia & Agol 2000 7248:Oppenheimer & Snyder 1939 7210:For the basic mechanism, see 7186:and the accompanying summary 7085:10.1088/0264-9381/27/8/084013 7055:Classical and Quantum Gravity 7010:Narayan & Bartelmann 1997 6964:Narayan & Bartelmann 1997 6958:For a simple derivation, see 5452:Heckmann & Schücking 1959 5325:. Read Books Ltd. p. 4. 5130:Hubble's original article is 4919: 4742:second superstring revolution 4641: 4589:at null infinity. Just as in 4454:. This radiation is known as 4425:black hole uniqueness theorem 4391:, is not a physical barrier. 4306:can reach any other location 2346:is used for repeated indices 1699:On the left-hand side is the 1268:Geometry of Newtonian gravity 1090:History of general relativity 1061:by the physics collaboration 21180:Raising and lowering indices 18218:General Relativity Tutorials 17558:, University Science Books, 17419:General Theory of Relativity 17090:Living Reviews in Relativity 16853:Living Reviews in Relativity 16751:Living Reviews in Relativity 16448:, W W Norton & Company, 16332:10.1007/978-3-540-71117-9_10 16215:Living Reviews in Relativity 16118:Living Reviews in Relativity 15418:Living Reviews in Relativity 15380:Rovelli, Carlo, ed. (2015), 15227:Living Reviews in Relativity 14946:Polchinski, Joseph (1998b), 14842:Living Reviews in Relativity 14323:10.1016/0039-3681(85)90002-0 13854:Living Reviews in Relativity 13591:Living Reviews in Relativity 13181:Living Reviews in Relativity 12728:Living Reviews in Relativity 12636:Living Reviews in Relativity 12445:10.1126/science.177.4044.168 12391:10.1126/science.177.4044.166 12221:10.1016/0003-4916(74)90384-4 12150:10.1016/0370-2693(85)91470-4 12082:Living Reviews in Relativity 11785:Living Reviews in Relativity 11691:Living Reviews in Relativity 11613:Living Reviews in Relativity 11515:, Springer, pp. 52–82, 10953:10.1016/j.newast.2006.02.001 10843:Astronomy & Astrophysics 10766:10.1088/0264-9381/16/12A/301 10653:Living Reviews in Relativity 10325:10.1007/978-3-540-74353-8_11 10212:Living Reviews in Relativity 9929:Living Reviews in Relativity 9286:Living Reviews in Relativity 9110:Living Reviews in Relativity 8911:Living Reviews in Relativity 7918:Ashtekar & Krishnan 2004 7440:Charbonnel & Primas 2005 7246:For stellar end states, cf. 6509:Jaranowski & Królak 2005 6283:Einstein, A (22 June 1916). 5766:At least approximately, cf. 5488:Ehlers & Lämmerzahl 2006 4964:general theory of relativity 4664:Canonical general relativity 4517:cosmic censorship hypothesis 4434:, which is analogous to the 4363:Horizon (general relativity) 4092:{\displaystyle g_{\mu \nu }} 3893: 3860:with a few million to a few 3769:gravitational wave detectors 3748:Gravitational-wave astronomy 3738:Gravitational-wave astronomy 3715:events" have been observed. 3015:and its generalization, the 2579:can then be obtained as its 1932:{\displaystyle T_{\mu \nu }} 1756:{\displaystyle R_{\mu \nu }} 1726:{\displaystyle G_{\mu \nu }} 1526:locally inertial coordinates 1297:, which states that the net 1207:solar eclipse of 29 May 1919 1191:anomalous perihelion advance 838:Einstein's theory of gravity 834:general theory of relativity 18:General theory of relativity 7: 21418:Gluon field strength tensor 20889: 20813:Quantum information science 20216:Albert Einstein Peace Prize 19993:Unsuccessful investigations 19396:Infinite derivative gravity 18491:Relativistic Doppler effect 18230:"Reflections on relativity" 18062:10.1103/PhysRevLett.20.1265 17968:10.1088/0264-9381/22/10/048 17626:; Hervik, Sigbjørn (2007), 17542:, Oxford University Press, 17536:Møller, Christian (1955) , 17353:Callahan, James J. (2000), 17283:; Hawking, Stephen (1994). 16384:10.1007/978-3-540-45230-0_3 15884:Lectures on Quantum Gravity 15767:10.1103/PhysRevLett.78.2054 15594:Schutz, Bernard F. (2003), 15576:Schutz, Bernard F. (1985), 15302:, Clarendon Press, Oxford, 15280:, Oxford University Press, 14517:, Oxford University Press, 13697:10.1088/0264-9381/18/17/202 13464:10.1007/978-3-540-40991-5_3 13376:, Oxford University Press, 13358:10.1016/j.shpsa.2012.07.010 13245:Kahn, Bob (14 April 2007), 12940:10.1088/0264-9381/23/17/N01 12629:, vol. 53, p. 489 12010:10.1007/978-3-540-71117-9_6 11916:, Oxford University Press, 11459:10.1007/978-94-011-4455-1_1 11117:10.1088/0264-9381/20/10/301 10781:Chandrasekhar, Subrahmanyan 10032:10.1103/PhysRevLett.58.1062 9420:, Oxford University Press, 9374:10.1088/0264-9381/21/15/R01 9182:10.1103/PhysRevLett.57.2244 8515:'t Hooft & Veltman 1974 8402:Peskin & Schroeder 1995 8162:The conjecture was made in 8037:, sec. 12.4. Unruh effect: 7474:Bergström & Goobar 2003 7369:Bergström & Goobar 2003 7105:Danzmann & Rüdiger 2003 6999:Roulet & Mollerach 1997 6909:Ciufolini & Pavlis 2004 4839: 4356: 4122:cosmic background radiation 3268:In general relativity, the 3038:Gravitational time dilation 2972:in curved spacetimes), the 2950:Reissner–Nordström solution 2104:In general relativity, the 1405:Relativistic generalization 1152:Reissner–Nordström solution 1116:Prussian Academy of Science 977:cosmic microwave background 965:tests of general relativity 941:gravitational time dilation 913:, a system of second-order 893:is directly related to the 281:Gravitational time dilation 10: 21629: 21229:Cartan formalism (physics) 21049:Penrose graphical notation 20644:Classical electromagnetism 20143:Picasso at the Lapin Agile 20081:Russell–Einstein Manifesto 19953:Bose–Einstein correlations 19584:Unified-field-theoric and 18962:In computational physics: 18486:Relativity of simultaneity 17332:, New York: W. W. Norton, 17289:. New York: W. W. Norton. 15886:, Springer, p. 9009, 15842:10.1103/PhysRevLett.13.789 15596:Gravity from the ground up 15294:Rindler, Wolfgang (1991), 15207:Renn, Jürgen, ed. (2005), 15189:Renn, Jürgen, ed. (2007), 15047:10.1103/PhysRevLett.13.539 14580:Peebles, P. J. E. (1993), 14035:10.1016/j.ppnp.2005.08.001 13940:AIP Conference Proceedings 13838:10.1088/1742-6596/24/1/018 13739:10.1142/9789812776556_0010 13651:, Elsevier Science, Ltd., 13540:, Vieweg, pp. 66–81, 12894:10.1088/0264-9381/17/5/321 12193:10.1103/PhysRevLett.27.826 11912:Giulini, Domenico (2005), 11074:10.1103/PhysRevD.74.063006 10873:10.1051/0004-6361:20042491 10616:10.1088/0034-4885/64/8/301 10143:; McNamara, Geoff (1997), 9707:, Elsevier, p. 2041, 9649:Bartusiak, Marcia (2000), 9510:10.1103/PhysRevD.69.082005 9260:10.1142/9789812834300_0008 9058:10.1103/PhysRevD.77.064035 8798:, pp. 52–59, 98–122; 8550:Goroff & Sagnotti 1985 7727:10.1103/PhysRevD.79.124017 7117:"LISA pathfinder overview" 6890:Ohanian & Ruffini 1994 6878:Ohanian & Ruffini 1994 6815:Ohanian & Ruffini 1994 6765:Weisberg & Taylor 2003 6716:10.1103/PhysRevX.11.041050 6638:Ohanian & Ruffini 1994 6251:Ohanian & Ruffini 1994 6142:Ohanian & Ruffini 1994 6125:Ohanian & Ruffini 1994 6113:Ohanian & Ruffini 1994 6069:Ohanian & Ruffini 1994 6061:Hafele & Keating 1972b 6057:Hafele & Keating 1972a 6024:Ohanian & Ruffini 1994 5972:Ohanian & Ruffini 1994 4960:Landau & Lifshitz 1975 4767:above), the result is the 4728:, a quantum theory not of 4657: 4651: 4621: 4581:) or suitable symmetries ( 4569:Mass in general relativity 4566: 4527: 4484: 4360: 4284: 4218: 4114:primordial nucleosynthesis 3914:Einstein's field equations 3897: 3832: 3741: 3683: 3675:Astrophysical applications 3631:: the axis direction of a 3616: 3597:, using the binary pulsar 3261: 3235: 3143: 3084: 3035: 2976:(a model universe that is 2838:pseudo-Riemannian manifold 2808: 2763: 2750:force of Newtonian gravity 2405: 1596:Einstein's field equations 1535: 1227:Subrahmanyan Chandrasekhar 1087: 1083: 401:Mathisson–Papapetrou–Dixon 242:Pseudo-Riemannian manifold 36: 29: 21435: 21375: 21324: 21317: 21209: 21140: 21077: 21021: 20968: 20915: 20908: 20901:Glossary of tensor theory 20897: 20821: 20758: 20686: 20602: 20574: 20546: 20485: 20399: 20308: 20239: 20193: 20090: 20051:The Meaning of Relativity 20016: 19868: 19807: 19739: 19727:Jackiw–Teitelboim gravity 19697: 19583: 19562: 19506:Canonical quantum gravity 19501:Euclidean quantum gravity 19493: 19340: 19314: 19303: 19236: 19205: 19198: 19144: 18976: 18841: 18813: 18799:Lense–Thirring precession 18682: 18631: 18593: 18572: 18561: 18519: 18463: 18447: 18389: 18381:Doubly special relativity 18353: 18342: 17929:10.1088/1367-2630/7/1/204 16974:Weinberg, Steven (2000), 16952:Weinberg, Steven (1996), 16932:Weinberg, Steven (1995), 16572:, World Scientific: 180, 16290:10.1103/RevModPhys.66.711 15967:Astrophys. J. Suppl. Ser. 15017:10.1103/PhysRevLett.4.337 14987:10.1103/PhysRevLett.3.439 14674:Rivista del Nuovo Cimento 14662:10.1103/PhysRevLett.14.57 14531:Peacock, John A. (1999), 14346:Gravitation and Spacetime 14240:Introduction to Cosmology 14206:10.1088/1367-2630/7/1/199 13538:Relativistic Astrophysics 13032:10.2478/s11534-009-0117-6 12986:10.1007/s11214-008-9478-1 12490:10.1103/RevModPhys.36.938 12352:10.1103/RevModPhys.21.447 12296:10.1007/978-3-540-85293-3 11852:Garfinkle, David (2007), 11497:10.1080/02698590120118800 11241:10.1142/S0217751X96002583 10578:10.1088/2058-7058/17/5/36 10496:10.1007/s10714-007-0554-8 9880:Astrophys. J. Suppl. Ser. 9851:Uspekhi Fizicheskikh Nauk 9844:10.1080/00018737000101171 9697:10.1103/RevModPhys.56.255 9603:Mon. Not. R. Astron. Soc. 9416:Auyang, Sunny Y. (1995), 9021:, Wiley, pp. 227–265 8760:, ch. 33 and refs therein 8421:Birrell & Davies 1984 8223:, § 21.4–§ 21.7 8103:Ellis & Van Elst 1999 7946:Beig & Chruściel 2006 7478:Penzias & Wilson 1965 7428:Olive & Skillman 2004 7212:Carroll & Ostlie 1996 7030:Blair & McNamara 1997 6380:10.1038/nature.2016.19361 6326:"Über Gravitationswellen" 6206:Ehlers & Rindler 1997 6063:, and culminating in the 6053:Hafele–Keating experiment 6040:Barstow, Bond et al. 2005 5838:Introductory chapters of 5677:Weinberg, Steven (1972). 5001:University of St. Andrews 4989:"General relativity]" 4547:in other field theories. 4310:before light sent out at 4221:Bondi–Metzner–Sachs group 4100:is the spacetime metric. 3773:interferometric detectors 3071:Global Positioning System 2880:weak gravitational fields 32:General Relativity (book) 21485:Gregorio Ricci-Curbastro 21357:Riemann curvature tensor 21064:Van der Waerden notation 20750:Condensed matter physics 20442:Albert Einstein Archives 20372:Bernhard Caesar Einstein 20119:Relics: Einstein's Brain 20067:The Evolution of Physics 19948:Bose–Einstein statistics 19943:Bose–Einstein condensate 19923:Einstein field equations 19603:Superfluid vacuum theory 19427:Nonsymmetric gravitation 19276:Post-Newtonian formalism 18659:Post-Newtonian formalism 18649:Einstein field equations 18585:Mathematical formulation 18409:Hyperbolic orthogonality 18214:(introductory/advanced). 17886:10.1002/andp.19163540702 17690:; Poisson, Eric (2014). 17554:Moore, Thomas A (2012), 17539:The Theory of Relativity 17328:; Ford, Kenneth (1998), 16723:Wald, Robert M. (1994), 16703:Wald, Robert M. (1984), 15137:10.1002/andp.19163550905 14294:Norton, John D. (1985), 14131:, University of Maryland 14107:Messiah, Albert (1999), 13298:10.1103/PhysRevD.55.7368 13124:Janssen, Michel (2005), 13108:300 Years of Gravitation 12702:The new quantum universe 12682:Heusler, Markus (1996), 12585:300 Years of Gravitation 10279:Börner, Gerhard (1993), 10263:300 Years of Gravitation 9218:10.1103/PhysRevD.36.1587 8894:Futamase & Itoh 2006 8205:Hawking & Ellis 1973 8055:Hawking & Ellis 1973 7934:Hawking & Ellis 1973 7910:Hawking & Ellis 1973 7874:Hawking & Ellis 1973 7858:Hawking & Ellis 1973 7813:10.1103/PhysRev.128.2851 7470:Alpher & Herman 1948 7330:Barack & Cutler 2004 7149:Cutler & Thorne 2002 5904:Hawking & Ellis 1973 5319:Albert Einstein (2011). 5246:, and references therein 4801:dynamical triangulations 4543:, which is analogous to 3929:{\displaystyle \Lambda } 3815:electromagnetic spectrum 3301:post-Newtonian formalism 3210:{\displaystyle 10^{-21}} 3017:post-Newtonian expansion 2919:local Lorentz invariance 2828:General relativity is a 1889:On the right-hand side, 1820:Riemann curvature tensor 1538:Einstein field equations 1344:and its successors (see 1047:supermassive black holes 911:Einstein field equations 406:Hamilton–Jacobi–Einstein 386:Einstein field equations 209:Mathematical formulation 21455:Elwin Bruno Christoffel 21388:Angular momentum tensor 21059:Tetrad (index notation) 21029:Abstract index notation 20452:Einstein Papers Project 19968:Einstein–de Haas effect 19768:Mechanical explanations 19637:Heterotic string theory 19593:Noncommutative geometry 19512:Wheeler–DeWitt equation 19238:General relativity (GR) 19218:Gauss's law for gravity 19192:Theories of gravitation 18370:Galilean transformation 18361:Principle of relativity 18212:Institut Henri Poincaré 18157:NCSA Spacetime Wrinkles 17669:Stephani, Hans (1990), 17519:Einstein in Matrix Form 17495:d'Inverno, Ray (1992), 17008:, ASP Conference Series 16645:10.1103/PhysRevD.14.870 16444:Thorne, Kip S. (1994), 15703:, IUCAA, p. 6088, 15193:, Dordrecht: Springer, 14966:Physical Review Letters 14641:Physical Review Letters 14005:Prog. Part. Nucl. Phys. 13514:10.1126/science.1132305 13430:10.1103/PhysRev.113.934 13230:Gravity Probe B Website 13227:Kahn, Bob (1996–2012), 12627:Encyclopedia of Physics 12543:Hawking, S. W. (1975), 12290:, Springer Publishing, 11958:10.1007/3-540-34523-X_4 11338:10.1023/A:1018843001842 10865:2005A&A...442..961C 10424:10.1126/science.1082158 10378:10.1103/PhysRev.124.925 10104:1998ARA&A..36..599B 10012:Physical Review Letters 9800:10.1103/PhysRevD.9.3292 9764:10.1103/PhysRevD.7.2333 9667:Begelman, Mitchell C.; 8008:. A review is given in 7597:A good introduction is 6769:Hulse & Taylor 1975 6413:Physical Review Letters 6249:; for an overview, see 6020:Pound & Snider 1964 5978:and the description in 5826:section 5 in ch. 12 of 5101:(later complemented in 4874:Einstein–Hilbert action 4769:Wheeler–deWitt equation 4494:spacetime singularities 4324:Penrose–Carter diagrams 3858:supermassive black hole 3720:observational astronomy 3332:{\displaystyle \sigma } 3087:Schwarzschild geodesics 2388:Newton's laws of motion 2359:{\displaystyle \alpha } 1963:{\displaystyle \kappa } 1941:abstract index notation 1902:{\displaystyle \kappa } 1473:or conformal geometry. 1374:gravitational potential 21269:Levi-Civita connection 20834:Nobel Prize in Physics 20696:Relativistic mechanics 20384:Thomas Martin Einstein 20292:Introducing Relativity 20264:Einstein for Beginners 20167:Einstein and Eddington 19958:Einstein–Cartan theory 19609:Logarithmic BEC vacuum 19554:Rainbow gravity theory 19432:Scalar–tensor theories 19206:Newtonian gravity (NG) 18455:Lorentz transformation 17718:John Archibald Wheeler 17630:, New York: Springer, 17517:Ludyk, Günter (2013). 17396:Cheng, Ta-Pei (2005), 17357:, New York: Springer, 15318:Active galactic nuclei 14899:Poisson, Eric (2004), 14809:10.1143/JPSJ.76.111017 14480:10.1103/PhysRev.56.455 14455:Oppenheimer, J. Robert 14176:New Journal of Physics 14111:, Dover Publications, 13572:Künzle, H. P. (1972), 13393:CASTLES Survey Website 13372:Kenyon, I. R. (1990), 13150:10.1002/andp.200410130 12779:Proc. Natl. Acad. Sci. 12521:10.1098/rspa.1966.0221 11769:10.1002/andp.200510173 10450:, Wiley, p. 130, 10299:Inflationary Cosmology 8431:For Hawking radiation 8383:Hey & Walters 2003 8135:Namely when there are 8120:eikonal approximations 7912:, pp. 312–320 or 7770:10.1098/rspa.1962.0161 7041:Hough & Rowan 2000 6016:Pound & Rebka 1960 6012:Pound & Rebka 1959 6008:Pound–Rebka experiment 5727:Brans & Dicke 1961 5702:Cheng, Ta-Pei (2005). 5606:Einstein–Cartan theory 4819: 4761:canonical quantization 4721: 4693: 4630:quantum field theories 4460:quantum theory section 4436:laws of thermodynamics 4399: 4299: 4237:, the Poincaré group. 4169:closed timelike curves 4093: 4059: 3930: 3909: 3844: 3760: 3698: 3563: 3536: 3514: 3489: 3464: 3438: 3333: 3280:; the orbit is not an 3259: 3211: 3155: 3108: 3047: 2946:Schwarzschild solution 2899:background independent 2857:John Archibald Wheeler 2848:, orbital motion, and 2799:Einstein–Cartan theory 2731: 2566: 2380: 2379:{\displaystyle \beta } 2360: 2340:Levi-Civita connection 2334:(sometimes called the 2324: 2281: 2258: 2095: 2056: 2032: 2012: 1964: 1933: 1903: 1880: 1809: 1757: 1727: 1690: 1577:conservation of energy 1551:, which includes both 1549:energy–momentum tensor 1522:Levi-Civita connection 1416: 1278: 1121:non-Euclidean geometry 1039:active galactic nuclei 1017:Einstein's theory has 949:gravitational redshift 276:Gravitational redshift 169: 87: 21603:Concepts in astronomy 21495:Jan Arnoldus Schouten 21450:Augustin-Louis Cauchy 20930:Differential geometry 20839:Philosophy of physics 20457:Einstein refrigerator 20447:Einstein's Blackboard 20326:(second wife; cousin) 20257:Einstein and Religion 20206:Albert Einstein Medal 20201:Albert Einstein Award 20059:The World as I See It 19998:Wave–particle duality 19978:Bohr–Einstein debates 19938:Cosmological constant 19918:Equivalence principle 19908:Einstein coefficients 19686:Twistor string theory 19665:Type II string theory 19658:Bosonic string theory 19598:Semiclassical gravity 19563:Unified-field-theoric 19348:Poincaré gauge theory 18923:Weyl−Lewis−Papapetrou 18664:Raychaudhuri equation 18603:Equivalence principle 18210:given in 2006 at the 17477:; Tod, K. P. (1991), 17013:Weiss, Achim (2006), 15532:Astrophysical Journal 15211:, Berlin: Wiley-VCH, 15145:Astrophysical Journal 15111:Reissner, H. (1916), 14724:Astrophysical Journal 14554:Astrophysical Journal 14410:Astrophysical Journal 14364:Astrophysical Journal 14125:Miller, Cole (2002), 14054:Astrophysical Journal 13233:, Stanford University 13047:Isham, Christopher J. 12800:10.1073/pnas.15.3.168 12601:Hawking, Stephen W.; 12258:Astrophysical Journal 11559:Astrophysical Journal 10962:Astrophysical Journal 10386:Ostriker, Jeremiah P. 8137:trapped null surfaces 7928:For first steps, cf. 7654:More concretely, the 7629:, sec. 6.4, see also 7451:Lahav & Suto 2004 7300:Remillard et al. 2006 6071:, table 4.1 on p. 186 6026:, table 4.1 on p. 186 4817: 4715: 4707:non-renormalizability 4688:, one of the ways of 4683: 4487:Spacetime singularity 4397: 4347:Raychaudhuri equation 4294: 4271:soft graviton theorem 4215:Asymptotic symmetries 4177:traversable wormholes 4094: 4060: 3931: 3907: 3842: 3755: 3693: 3686:Gravitational lensing 3680:Gravitational lensing 3557: 3537: 3515: 3490: 3465: 3439: 3334: 3257: 3250:Precession of apsides 3212: 3161:electromagnetic waves 3153: 3106: 3045: 3009:perturbation theories 2997:numerical integration 2732: 2575:A conservative total 2567: 2416:of an object of mass 2381: 2361: 2325: 2282: 2259: 2096: 2057: 2033: 2013: 1965: 1934: 1904: 1881: 1810: 1758: 1728: 1691: 1589:covariant derivatives 1412: 1368:which depends on the 1350:equivalence principle 1285:is the notion that a 1275: 1164:cosmological constant 1051:gravitational lensing 945:gravitational lensing 886:. In particular, the 564:Weyl−Lewis−Papapetrou 519:Kerr–Newman–de Sitter 339:Einstein–Rosen bridge 271:Gravitational lensing 227:Equivalence principle 170: 76:gravitational lensing 57: 21470:Carl Friedrich Gauss 21403:stress–energy tensor 21398:Cauchy stress tensor 21150:Covariant derivative 21112:Antisymmetric tensor 21044:Multi-index notation 20798:Mathematical physics 20336:Hans Albert Einstein 20285:I Am Albert Einstein 20231:Einstein Prize (APS) 19903:Photoelectric effect 19876:Theory of relativity 19753:Aristotelian physics 19722:Gauss–Bonnet gravity 19672:Little string theory 19651:Type 0 string theory 19644:Type I string theory 19519:Loop quantum gravity 19446:Scalar–tensor–vector 19419:Tensor–vector–scalar 19374:Gauge theory gravity 19332:Theory of everything 18964:Numerical relativity 18805:pulsar timing arrays 17938:Class. Quantum Grav. 16884:10.12942/lrr-1998-12 15920:Int. J. Theor. Phys. 15647:10.1143/PTPS.170.214 15316:Robson, Ian (1996), 14533:Cosmological Physics 13885:10.12942/lrr-1998-13 13667:Class. Quantum Grav. 12864:Class. Quantum Grav. 11278:Gravitational lenses 11211:Int. J. Mod. Phys. A 11144:on 26 September 2007 11097:Class. Quantum Grav. 10736:Class. Quantum Grav. 9740:Bekenstein, Jacob D. 9344:Class. Quantum Grav. 9317:10.12942/lrr-2004-10 9072:Ashby, Neil (2002), 8882:Blanchet et al. 2008 7541:elementary particles 7522:Springel et al. 2005 7459:Springel et al. 2005 7280:Eddington luminosity 7184:Springel et al. 2005 7162:, lectures 19 and 21 6988:Kochanek et al. 2007 6751:, pp. 317–321, 6634:Anderson et al. 1992 5864:Stephani et al. 2003 5840:Stephani et al. 2003 4781:loop quantum gravity 4773:Schrödinger equation 4771:(an analogue of the 4668:Loop quantum gravity 4541:coordinate condition 4509:singularity theorems 4432:black hole mechanics 4385:Schwarzschild radius 4371:Black hole mechanics 4275:quantum field theory 4202:present examples of 4073: 3943: 3920: 3792:pulsar timing arrays 3669:Mars Global Surveyor 3657:rotating black holes 3582:, one of which is a 3544:orbital eccentricity 3526: 3504: 3479: 3454: 3346: 3323: 3191: 3185:linear superposition 3001:numerical relativity 2990:Maldacena conjecture 2986:anti-de Sitter space 2834:Einstein's equations 2589: 2431: 2370: 2350: 2344:summation convention 2295: 2271: 2122: 2069: 2046: 2022: 1974: 1954: 1913: 1893: 1829: 1770: 1737: 1707: 1602: 1532:Einstein's equations 1391:Newton–Cartan theory 1148:electrically charged 1144:Schwarzschild metric 981:alternative theories 832:, also known as the 494:Einstein–Rosen waves 220:Fundamental concepts 109: 21347:Nonmetricity tensor 21202:(2nd-order tensors) 21170:Hodge star operator 21160:Exterior derivative 21009:Transport phenomena 20994:Continuum mechanics 20950:Multilinear algebra 20773:Atmospheric physics 20612:Classical mechanics 20540:branches of physics 19988:Thought experiments 19570:Kaluza–Klein theory 18856:Friedmann equations 18750:Hulse–Taylor binary 18712:Gravitational waves 18608:Riemannian geometry 18434:Proper acceleration 18419:Maxwell's equations 18365:Galilean relativity 18240:on 18 December 2015 18202:Stanford University 18135:1 June 2014 at the 18101:10.1038/nature06896 18093:2008Natur.452..851V 18054:1968PhRvL..20.1265S 18022:10.1051/epn:2006604 18013:2006ENews..37f..30N 17960:2005CQGra..22S.487L 17920:2005NJPh....7..204F 17878:1916AnP...354..769E 17747:, Springer-Verlag, 17649:Lifshitz, Evgeny F. 17121:10.12942/lrr-2006-3 17112:2006LRR.....9....3W 16996:Weisberg, Joel M.; 16875:1998LRR.....1...12W 16821:1979Natur.279..381W 16782:10.12942/lrr-2001-6 16773:2001LRR.....4....6W 16733:1994qftc.book.....W 16681:1975CMaPh..45....9W 16669:Commun. 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Phys. 16637:1976PhRvD..14..870U 16614:2006gr.qc.....6062T 16588:2000mmp..conf..180T 16558:1997hepcbconf..385T 16482:1995pnac.conf..160T 16437:Magic without Magic 16426:1974AIHPA..20...69T 16376:2003LNP...631...41T 16324:2007LNP...721..185T 16282:1994RvMP...66..711T 16236:10.12942/lrr-2004-4 16227:2004LRR.....7....4S 16149:10.12942/lrr-2003-5 16140:2003LRR.....6....5S 16094:10.1038/nature03597 16086:2005Natur.435..629S 16039:2007ApJS..170..377S 15989:2003ApJS..148..175S 15942:1997IJTP...36.2759S 15902:2003gr.qc.....9009S 15863:2004biba.book.....S 15834:1964PhRvL..13..789S 15795:2004PhRvL..92l1101S 15759:1997PhRvL..78.2054S 15719:1998gr.qc.....6088S 15693:1916skpa.conf..424S 15681:Schwarzschild, Karl 15674:1916SPAW.......189S 15662:Schwarzschild, Karl 15639:2007PThPS.170..214S 15554:2003ApJ...596.1015S 15508:2004GReGr..36.2223S 15459:10.12942/lrr-1998-1 15440:1998LRR.....1....1R 15358:1997PhR...279...67R 15248:10.12942/lrr-1998-3 15239:1998LRR.....1....3R 15167:2006ApJ...646..407R 15129:1916AnP...355..106R 15089:1966Natur.211..468R 15038:1964PhRvL..13..539P 15008:1960PhRvL...4..337P 14978:1959PhRvL...3..439P 14909:2004rtmb.book.....P 14873:10.12942/lrr-2004-6 14864:2004LRR.....7....6P 14801:2007JPSJ...76k1017P 14736:1965ApJ...142..419P 14698:The Road to Reality 14686:1969NCimR...1..252P 14653:1965PhRvL..14...57P 14614:1991Natur.352..769P 14566:1966ApJ...146..542P 14471:1939PhRv...56..455O 14432:2001ApJ...552..718O 14386:2004ApJ...617...29O 14315:1985SHPSA..16..203N 14273:1918KNAB...20.1238N 14236:Narlikar, Jayant V. 14198:2005NJPh....7..199N 14066:1994ApJ...420..439M 14027:2006PrPNP..56..340M 13958:2006AIPC..841..129M 13924:1972JMP....13..874L 13876:1998LRR.....1...13L 13830:2005JPhCS..24..151L 13808:J. 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R. 12561:1975CMaPh..43..199H 12513:1966RSPSA.294..511H 12497:Hawking, Stephen W. 12482:1964RvMP...36..938H 12437:1972Sci...177..168H 12383:1972Sci...177..166H 12343:1949RvMP...21..447G 12270:1971ApJ...169..563G 12213:1974AnPhy..83..203G 12185:1971PhRvL..27..826G 12142:1985PhLB..160...81G 12103:10.12942/lrr-2004-2 12094:2004LRR.....7....2G 12048:2005Natur.435..572G 12002:2007LNP...721..105G 11950:2006math.ph...2018G 11905:1996gere.conf...19G 11806:10.12942/lrr-2007-2 11797:2007LRR....10....2F 11761:2006AnP...518...84F 11712:10.12942/lrr-2004-1 11703:2004LRR.....7....1F 11673:1952AcMa...88..141F 11634:10.12942/lrr-2003-4 11625:2003LRR.....6....4F 11581:2000ApJ...528L..13F 11451:1999ASIC..541....1E 11330:1997GReGr..29..519E 11233:1996IJMPA..11.5623D 11169:1995gr.qc....12024D 11109:2003CQGra..20S...1D 11066:2006PhRvD..74f3006D 11026:2002gr.qc.....4090C 10984:2004ApJ...600..544C 10945:2006NewA...11..527C 10909:10.1038/nature03007 10901:2004Natur.431..958C 10815:1984JApA....5....3C 10758:1999CQGra..16A...3C 10684:10.12942/lrr-2001-1 10675:2001LRR.....4....1C 10608:2001RPPh...64..885C 10534:2003PhRvL..90x1101B 10488:2008GReGr..40..467B 10416:2003Sci...299.1532B 10400:(5612): 1532–1533, 10390:Steinhardt, Paul J. 10370:1961PhRv..124..925B 10317:2007LNP...738..393B 10233:10.12942/lrr-2006-4 10224:2006LRR.....9....4B 10188:2008CQGra..25p5003B 10068:10.1038/nature01997 10060:2003Natur.425..374B 10024:1987PhRvL..58.1062B 9960:10.12942/lrr-2002-1 9951:2002LRR.....5....1B 9902:2003ApJS..148....1B 9863:1970UsFiN.102..463B 9836:1970AdPhy..19..525B 9824:Advances in Physics 9792:1974PhRvD...9.3292B 9756:1973PhRvD...7.2333B 9723:2005gr.qc.....2041B 9689:1984RvMP...56..255B 9669:Blandford, Roger D. 9625:2005MNRAS.362.1134B 9585:2005grg..conf.....F 9553:1973CMaPh..31..161B 9502:2004PhRvD..69h2005B 9448:2002Natur.415...54B 9402:1979JMP....20..793A 9366:2004CQGra..21R..53A 9308:2004LRR.....7...10A 9252:2008mgm..conf..126A 9210:1987PhRvD..36.1587A 9174:1986PhRvL..57.2244A 9132:10.12942/lrr-2003-1 9123:2003LRR.....6....1A 9093:2002PhT....55e..41A 9050:2008PhRvD..77f4035A 8945:1948Natur.162..774A 8802:, sec. 34.1, ch. 30 8024:, sec. 4.4.4, 4.4.5 7872:, sec. 9.2–9.4 and 7805:1962PhRv..128.2851S 7762:1962RSPSA.269...21B 7719:2009PhRvD..79l4017F 7643:Spergel et al. 2007 7490:Bennett et al. 2003 7436:O'Meara et al. 2001 7393:Spergel et al. 2003 7377:Peebles et al. 1991 7292:Schödel et al. 2003 7077:2010CQGra..27h4013H 6854:Everitt et al. 2001 6708:2021PhRvX..11d1050K 6636:; for an overview, 6484:. 11 February 2016. 6435:2016PhRvL.116f1102A 6338:1918SPAW.......154E 6332:(part 1): 154–167. 6299:1916SPAW.......688E 6293:(part 1): 688–696. 6245:measurements), cf. 6146:Shapiro et al. 2004 5733:, sec. 3 in ch. 7, 5095:Schwarzschild 1916b 5091:Schwarzschild 1916a 4765:evolution equations 4734:unified description 4686:Calabi–Yau manifold 4612:solid-state physics 4524:Evolution equations 4254:gravitational waves 4250:asymptotic symmetry 3884:black hole binaries 3880:Gravitational waves 3796:millisecond pulsars 3642:lunar laser ranging 3629:geodetic precession 3619:Geodetic precession 3314:orders of magnitude 3140:Gravitational waves 3005:naked singularities 2962:de Sitter universes 2935:Riemannian manifold 2332:Christoffel symbols 1471:conformal structure 1283:classical mechanics 1211:theoretical physics 1125:Riemannian geometry 1103:gravitational waves 1055:gravitational waves 1049:. It also predicts 1043:stellar black holes 1041:are believed to be 448:Kaluza–Klein theory 334:Minkowski spacetime 286:Gravitational waves 21598:General relativity 21480:Tullio Levi-Civita 21423:Metric tensor (GR) 21337:Levi-Civita symbol 21190:Tensor contraction 21004:General relativity 20940:Euclidean geometry 20829:History of physics 20437:Things named after 20299:Subtle is the Lord 20122:(1994 documentary) 20114:(1923 documentary) 20106:(1922 documentary) 20003:Gravitational wave 19886:General relativity 19881:Special relativity 19815:Gravitational wave 19698:Generalisations / 19586:quantum-mechanical 19494:Quantum-mechanical 19307:general relativity 19281:Linearized gravity 18905:Reissner–Nordström 18823:Brans–Dicke theory 18654:Linearized gravity 18481:Length contraction 18399:Frame of reference 18376:Special relativity 17866:Annalen der Physik 17794:Specialists' books 17767:General Relativity 17596:Graduate textbooks 17375:, Addison Wesley, 17260:Schutz, Bernard F. 17031:on 8 February 2007 16706:General Relativity 16689:10.1007/BF01609863 15950:10.1007/BF02435709 15117:Annalen der Physik 15061:, Addison-Wesley, 14926:Polchinski, Joseph 14763:, Addison-Wesley, 14753:Peskin, Michael E. 14144:Misner, Charles W. 13374:General Relativity 13130:Annalen der Physik 13092:10.1007/BF02450518 12569:10.1007/BF02345020 11887:General Relativity 11739:Annalen der Physik 11682:10.1007/BF02392131 10823:10.1007/BF02714967 10634:, Addison-Wesley, 9925:Berger, Beverly K. 9816:Khalatnikov, I. M. 9814:Belinskii, V. A.; 9561:10.1007/BF01645742 9015:Misner, Charles W. 8696:For a review, see 8606:messenger particle 8233:Fourès-Bruhat 1952 8091:Chandrasekhar 1983 8075:Chandrasekhar 1983 7672:Brandenberger 2008 7482:Mather et al. 1994 7405:Bridle et al. 2003 7347:, pp. 285–288 7226:For a review, see 6880:, pp. 469–471 6650:Kramer et al. 2006 6640:, pp. 406–407 6594:, pp. 177–181 6570:, pp. 253–254 6566:, pp. 48–49, 6273:, sec. 7.1 and 7.2 6127:, pp. 164–172 6051:Starting with the 5994:, pp. 24–26; 5982:, pp. 196–198 5942:, sec. 4.1 and 4.2 5879:Chandrasekhar 1983 5649:Bianchi identities 5296:Chandrasekhar 1984 5167:, pp. 253–254 5007:on 4 February 2015 4820: 4777:Ashtekar variables 4722: 4694: 4400: 4332:spacetime diagrams 4300: 4297:Minkowski universe 4227:special relativity 4146:inflationary phase 4089: 4055: 3968: 3926: 3910: 3900:Physical cosmology 3882:, above). Merging 3845: 3761: 3744:Gravitational wave 3699: 3638:parallel transport 3564: 3532: 3510: 3485: 3460: 3434: 3329: 3264:Apsidal precession 3260: 3219:Fourier decomposed 3207: 3156: 3146:Gravitational wave 3109: 3095:Gravitational lens 3048: 3013:linearized gravity 2905:, namely that the 2895:coordinate systems 2891:general covariance 2780:Brans–Dicke theory 2776:Whitehead's theory 2727: 2562: 2376: 2356: 2320: 2277: 2254: 2091: 2052: 2028: 2008: 1960: 1929: 1909:is a constant and 1899: 1876: 1805: 1753: 1723: 1686: 1643: 1417: 1378:Euclidean geometry 1305:multiplied by its 1279: 1245:In the preface to 1140:Karl Schwarzschild 1111:thought experiment 1071:expanding universe 1057:, which have been 953:Shapiro time delay 865:special relativity 830:General relativity 777:Physics portal 549:Oppenheimer–Snyder 489:Reissner–Nordström 381:Linearized gravity 329:Spacetime diagrams 232:Special relativity 165: 94:General relativity 88: 21513: 21512: 21475:Hermann Grassmann 21431: 21430: 21383:Moment of inertia 21244:Differential form 21219:Affine connection 21034:Einstein notation 21017: 21016: 20945:Exterior calculus 20925:Coordinate system 20857: 20856: 20844:Physics education 20793:Materials science 20760:Interdisciplinary 20718:Quantum mechanics 20505: 20504: 20407:Awards and honors 20390:Siegbert Einstein 20278:Einstein's Cosmos 19828: 19827: 19799:Theory of impetus 19735: 19734: 19707:Liouville gravity 19451:Conformal gravity 19379:Composite gravity 19369:Bimetric theories 19299: 19298: 19158: 19157: 18972: 18971: 18951:Ozsváth–Schücking 18557: 18556: 18539:Minkowski diagram 18496:Thomas precession 18439:Relativistic mass 18077:(7189): 851–853, 18048:(22): 1265–1269, 17944:(10): S487–S492, 17842:978-0-521-53780-3 17820:978-0-521-09906-6 17710:Charles W. Misner 17701:978-1-107-03286-6 17680:978-0-521-37941-0 17662:978-0-7506-2768-9 17637:978-0-387-69199-2 17616:978-0-8053-8732-2 17588:978-0-521-88705-2 17565:978-1-891389-82-5 17528:978-3-642-35797-8 17510:978-0-19-859686-8 17488:978-0-521-33943-8 17467:978-0-8053-8662-2 17446:978-0-387-69199-2 17428:978-0-691-01146-2 17407:978-0-19-852957-6 17382:978-0-201-38423-9 17364:978-0-387-98641-8 17339:978-0-393-31991-0 17318:978-0-226-87029-8 17273:978-1-56159-268-5 17252:978-1-58988-044-3 17231:978-0-226-28864-2 17196:978-3-528-06059-6 17160:978-0-521-83143-7 17086:Will, Clifford M. 17078:978-0-521-43973-2 17065:Will, Clifford M. 17057:978-0-7167-6034-4 16998:Taylor, Joseph H. 16989:978-0-521-66000-6 16967:978-0-521-55002-4 16945:978-0-521-55001-7 16925:978-0-471-92567-5 16742:978-0-226-87027-4 16716:978-0-226-87033-5 16511:978-0-521-82081-3 16491:978-0-521-36853-7 16455:978-0-393-31276-8 16393:978-3-540-40810-9 16341:978-3-540-71115-5 16262:Taylor, Joseph H. 16206:978-0-7204-0064-9 16185:978-0-521-46136-8 16070:(7042): 629–636, 15926:(12): 2759–2781, 15911:978-0-387-23995-8 15880:Sorkin, Rafael D. 15872:978-0-00-715251-3 15857:, Fourth Estate, 15818:Shapiro, Irwin I. 15743:(11): 2054–2057, 15728:978-81-900378-3-9 15605:978-0-521-45506-0 15587:978-0-521-27703-7 15492:(10): 2223–2235, 15391:978-3-11-034042-6 15327:978-0-471-95853-6 15309:978-0-19-853952-0 15287:978-0-19-850836-6 15274:Rindler, Wolfgang 15218:978-3-527-40571-8 15200:978-1-4020-3999-7 15083:(5048): 468–470, 15068:978-0-201-54611-8 14957:978-0-521-63304-8 14939:978-0-521-63303-1 14918:978-0-521-83091-1 14770:978-0-201-50397-5 14708:978-0-679-45443-4 14608:(6338): 769–776, 14593:978-0-691-01933-8 14550:Peebles, P. 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Math. Phys. 9427:978-0-19-509345-2 9269:978-981-283-426-3 9168:(18): 2244–2247, 9101:10.1063/1.1485583 9028:Physical Review D 9007:Arnowitt, Richard 8999:978-3-540-96890-0 8978:978-981-02-0950-6 8939:(4124): 774–775, 8608:of gravity, e.g. 7906:apparent horizons 7850:Frauendiener 2004 7697:Physical Review D 7455:Bertschinger 1998 7319:Dalal et al. 2006 7302:for an overview, 7256:Buras et al. 2003 7232:faster than light 6686:Physical Review X 6233:and Mercury, cf. 6162:WKB approximation 5893:, ch. 4, sec. 3.3 5713:978-0-19-852957-6 5688:978-0-471-92567-5 5338:Extract of page 4 5332:978-1-4474-9358-7 5080:English translate 4962:, p. 228 "...the 4868:Ehrenfest paradox 4809:quantum cosmology 4634:Hawking radiation 4591:classical physics 4456:Hawking radiation 4448:thermal radiation 4351:energy conditions 4263:supertranslations 4210:Advanced concepts 4135:equation of state 4039: 3999: 3972: 3873:relativistic jets 3850:stellar evolution 3782:(two detectors), 3535:{\displaystyle e} 3513:{\displaystyle c} 3488:{\displaystyle T} 3463:{\displaystyle L} 3430: 3426: 3294:Urbain Le Verrier 2974:Taub–NUT solution 2754:centrifugal force 2725: 2672: 2640: 2581:negative gradient 2560: 2510: 2475: 2392:Einstein notation 2336:affine connection 2280:{\displaystyle s} 2243: 2216: 2164: 2113:geodesic equation 2055:{\displaystyle c} 2031:{\displaystyle G} 2006: 1647: 1518:pseudo-Riemannian 1433:Galilei invariant 1429:Lorentz invariant 1346:Eötvös experiment 1187:Newtonian gravity 1059:observed directly 985:experimental data 929:classical physics 827: 826: 460: 459: 346: 345: 55: 16:(Redirected from 21620: 21578: 21577: 21566: 21565: 21554: 21553: 21552: 21542: 21541: 21530: 21529: 21528: 21521: 21490:Bernhard Riemann 21322: 21321: 21165:Exterior product 21132:Two-point tensor 21117:Symmetric tensor 20999:Electromagnetism 20913: 20912: 20884: 20877: 20870: 20861: 20860: 20783:Chemical physics 20723:Particle physics 20649:Classical optics 20532: 20525: 20518: 20509: 20508: 20493: 20492: 20392:(distant cousin) 20386:(great-grandson) 20348:Hermann Einstein 20330:Lieserl Einstein 20008:Tea leaf paradox 19855: 19848: 19841: 19832: 19831: 19783: 19781:Entropic gravity 19776: 19700:extensions of GR 19688: 19674: 19667: 19660: 19653: 19646: 19639: 19632: 19625: 19611: 19528: 19521: 19514: 19475:Geometrodynamics 19464: 19440: 19421: 19414: 19363: 19356: 19312: 19311: 19203: 19202: 19185: 19178: 19171: 19162: 19161: 19150: 19149: 18933:van Stockum dust 18705:Two-body problem 18623:Mach's principle 18570: 18569: 18511:Terrell rotation 18351: 18350: 18329: 18322: 18315: 18306: 18305: 18296: 18294: 18292: 18280: 18278: 18276: 18264: 18262: 18249: 18247: 18245: 18236:. Archived from 18198:Leonard Susskind 18190: 18146:GEO600 home page 18119: 18086: 18064: 18042:Phys. Rev. Lett. 18033: 18031: 18024: 18006: 17991:Europhysics News 17988: 17978: 17953: 17932: 17931: 17913: 17888: 17858:Einstein, Albert 17851:Journal articles 17846: 17824: 17801:Hawking, Stephen 17789: 17757: 17743:; H. Wu (1977), 17736: 17705: 17683: 17665: 17640: 17619: 17603:Carroll, Sean M. 17591: 17568: 17550: 17532: 17513: 17502: 17491: 17470: 17454:Hartle, James B. 17449: 17431: 17410: 17385: 17367: 17342: 17321: 17300: 17276: 17255: 17234: 17213: 17207: 17199: 17163: 17147:Zwiebach, Barton 17142: 17133: 17123: 17105: 17081: 17060: 17044:Wheeler, John A. 17039: 17038: 17036: 17027:, archived from 17009: 16992: 16981: 16970: 16959: 16948: 16928: 16910:Weinberg, Steven 16905: 16896: 16886: 16868: 16866:astro-ph/9812021 16847: 16829:10.1038/279381a0 16803: 16794: 16784: 16766: 16745: 16719: 16699: 16656: 16647: 16616: 16607: 16590: 16581: 16566:Traschen, Jennie 16561: 16551: 16530: 16528: 16515: 16494: 16475: 16458: 16440: 16428: 16410:'t Hooft, Gerard 16405: 16369: 16352: 16317: 16300: 16257: 16248: 16238: 16209: 16188: 16170: 16161: 16151: 16133: 16131:astro-ph/0307536 16112: 16079: 16077:astro-ph/0504097 16057: 16032: 16030:astro-ph/0603449 16011:Spergel, D. N.; 16007: 15982: 15980:astro-ph/0302209 15960: 15935: 15914: 15895: 15875: 15844: 15822:Phys. Rev. Lett. 15813: 15783:Phys. Rev. Lett. 15777: 15752: 15750:astro-ph/9609169 15737:Phys. Rev. Lett. 15731: 15712: 15695: 15676: 15657: 15632: 15613:Schwarz, John H. 15608: 15590: 15572: 15547: 15545:astro-ph/0306214 15538:(2): 1015–1034, 15526: 15501: 15480: 15471: 15461: 15451: 15433: 15412: 15410: 15394: 15376: 15351: 15349:astro-ph/9603119 15330: 15312: 15301: 15290: 15269: 15260: 15250: 15221: 15203: 15185: 15160: 15158:astro-ph/0509758 15139: 15107: 15097:10.1038/211468a0 15071: 15050: 15049: 15026:Phys. Rev. Lett. 15020: 15019: 14996:Phys. Rev. Lett. 14990: 14989: 14960: 14942: 14921: 14895: 14885: 14875: 14857: 14832: 14819: 14794: 14773: 14762: 14748: 14747: 14711: 14688: 14665: 14664: 14632: 14622:10.1038/352769a0 14596: 14576: 14545: 14527: 14504: 14483: 14482: 14450: 14425: 14423:astro-ph/0011179 14404: 14379: 14377:astro-ph/0405588 14358: 14340: 14339: 14337: 14331: 14300: 14290: 14288: 14275: 14252: 14231: 14229: 14227:astro-ph/9606001 14216: 14191: 14170: 14152:Wheeler, John A. 14139: 14138: 14136: 14121: 14103: 14092: 14083:Mermin, N. David 14078: 14077: 14045: 14020: 14018:astro-ph/0505266 13999: 13976: 13951: 13934: 13906: 13897: 13887: 13869: 13848: 13823: 13799: 13780: 13760: 13724: 13707: 13682: 13661: 13643: 13634: 13624: 13606: 13604:astro-ph/0310642 13585: 13568: 13550: 13532: 13499: 13497:astro-ph/0609417 13490:(5796): 97–102, 13477: 13449: 13447:astro-ph/0405178 13432: 13404: 13403: 13401: 13386: 13368: 13343: 13326: 13308: 13283: 13281:astro-ph/9611125 13262: 13260: 13253: 13241: 13240: 13238: 13223: 13214: 13204: 13175: 13174: 13172: 13166: 13160:, archived from 13120: 13102: 13063: 13042: 13017: 12996: 12971: 12962:(1–4): 363–381, 12950: 12925: 12904: 12879: 12870:(5): 1117–1128, 12858: 12857: 12821: 12812: 12802: 12770: 12761: 12751: 12722: 12696: 12678: 12669: 12659: 12630: 12621: 12597: 12579: 12539: 12492: 12464: 12410: 12356: 12354: 12318: 12307: 12280: 12252: 12223: 12195: 12173:Phys. Rev. Lett. 12167: 12165: 12152: 12124: 12115: 12105: 12076: 12059: 12030: 11995: 11978: 11943: 11926: 11908: 11898: 11878: 11877: 11875: 11866:, archived from 11848: 11838:, Viking Press, 11827: 11818: 11808: 11779: 11754: 11733: 11724: 11714: 11685: 11684: 11661:Acta Mathematica 11655: 11646: 11636: 11607: 11574: 11572:astro-ph/9912263 11553: 11552: 11550: 11544: 11537: 11525: 11507: 11479: 11444: 11420: 11409:Einstein, Albert 11400: 11389:Einstein, Albert 11378: 11363:Einstein, Albert 11358: 11349: 11312:Ehlers, Jürgen; 11308: 11290: 11272: 11251: 11226: 11202: 11181: 11162: 11145: 11143: 11137:, archived from 11128: 11094: 11084: 11059: 11057:astro-ph/0601275 11038: 11019: 11002: 10977: 10975:astro-ph/0309480 10955: 10927: 10883: 10858: 10856:astro-ph/0505247 10837:Charbonnel, C.; 10833: 10797: 10776: 10751: 10749:astro-ph/9912186 10726: 10705: 10696: 10686: 10668: 10666:astro-ph/0004075 10649:Carroll, Sean M. 10644: 10626: 10601: 10586:Rep. Prog. Phys. 10580: 10560: 10527: 10525:astro-ph/0303171 10512:Phys. Rev. Lett. 10506: 10481: 10472:(2–3): 467–527, 10460: 10442: 10409: 10407:astro-ph/0303180 10380: 10345: 10310: 10293: 10275: 10259:Blandford, R. D. 10254: 10245: 10235: 10206: 10181: 10159: 10136: 10118:Birrell, N. D.; 10114: 10086: 10042: 10003: 9992: 9981: 9972: 9962: 9944: 9920: 9895: 9893:astro-ph/0302207 9873: 9846: 9820:Lifschitz, E. M. 9810: 9774: 9750:(8): 2333–2346, 9735: 9716: 9699: 9663: 9645: 9636: 9618: 9616:astro-ph/0506600 9609:(4): 1134–1142, 9597: 9571: 9520: 9495: 9474: 9430: 9412: 9410:10.1063/1.524151 9384: 9359: 9350:(15): R53–R152, 9338: 9329: 9319: 9301: 9280: 9245: 9228: 9204:(6): 1587–1602, 9192: 9162:Phys. Rev. Lett. 9153: 9144: 9134: 9103: 9078: 9068: 9043: 9022: 9002: 8981: 8963: 8953:10.1038/162774b0 8915: 8906: 8900: 8886:Arun et al. 2008 8878: 8872: 8858: 8852: 8846: 8840: 8835: 8829: 8822: 8816: 8809: 8803: 8793: 8787: 8778: 8772: 8767: 8761: 8755: 8749: 8744: 8738: 8733: 8727: 8718: 8712: 8694: 8688: 8670: 8664: 8658: 8652: 8643: 8637: 8631: 8625: 8619: 8613: 8582: 8576: 8574:Polchinski 1998b 8570:Polchinski 1998a 8562: 8556: 8534: 8528: 8523: 8517: 8512: 8506: 8500: 8494: 8489: 8480: 8474: 8468: 8461: 8455: 8449: 8443: 8429: 8423: 8414: 8408: 8391: 8385: 8375: 8369: 8351: 8345: 8339: 8333: 8323: 8317: 8310: 8304: 8291: 8285: 8280: 8274: 8268: 8262: 8256:Gourgoulhon 2007 8253: 8247: 8230: 8224: 8214: 8208: 8202: 8196: 8180: 8174: 8160: 8154: 8149: 8143: 8133: 8127: 8116: 8110: 8100: 8094: 8084: 8078: 8068: 8062: 8052: 8046: 8031: 8025: 8019: 8013: 7998: 7992: 7983: 7977: 7955: 7949: 7926: 7920: 7899: 7893: 7883: 7877: 7867: 7861: 7847: 7841: 7840: 7835: 7823: 7817: 7816: 7799:(6): 2851–2864. 7788: 7782: 7781: 7745: 7739: 7738: 7712: 7692: 7686: 7681: 7675: 7669: 7663: 7652: 7646: 7640: 7634: 7623:monopole problem 7615:flatness problem 7611: 7605: 7595: 7589: 7572: 7566: 7558: 7552: 7530: 7524: 7510: 7504: 7467: 7461: 7448: 7442: 7413: 7407: 7401: 7395: 7385: 7379: 7366: 7360: 7354: 7348: 7338: 7332: 7327: 7321: 7316: 7310: 7276: 7270: 7244: 7238: 7224: 7218: 7208: 7202: 7196: 7190: 7181: 7175: 7169: 7163: 7157: 7151: 7146: 7140: 7135: 7129: 7128: 7126: 7124: 7113: 7107: 7102: 7096: 7095: 7070: 7049: 7043: 7038: 7032: 7019: 7013: 7007: 7001: 6996: 6990: 6984: 6978: 6973: 6967: 6956: 6950: 6940: 6934: 6925: 6919: 6906: 6900: 6887: 6881: 6871: 6865: 6850: 6844: 6839: 6833: 6824: 6818: 6808: 6802: 6792: 6786: 6781: 6775: 6762: 6756: 6755:, pp. 70–86 6742: 6736: 6735: 6701: 6677: 6668: 6658: 6652: 6647: 6641: 6618: 6612: 6601: 6595: 6589: 6583: 6577: 6571: 6561: 6555: 6544: 6538: 6529: 6523: 6517: 6511: 6505: 6499: 6492: 6486: 6485: 6474: 6463: 6462: 6428: 6408: 6399: 6398: 6396: 6394: 6363: 6354: 6353: 6351: 6349: 6344:on 21 March 2019 6340:. Archived from 6321: 6315: 6314: 6312: 6310: 6305:on 21 March 2019 6301:. Archived from 6280: 6274: 6268: 6262: 6227: 6221: 6214: 6208: 6187: 6181: 6175: 6169: 6154: 6148: 6134: 6128: 6122: 6116: 6109: 6103: 6094: 6088: 6078: 6072: 6049: 6043: 6033: 6027: 6005: 5999: 5989: 5983: 5965: 5959: 5949: 5943: 5937: 5931: 5924: 5918: 5913: 5907: 5900: 5894: 5888: 5882: 5876: 5870: 5860: 5854: 5848: 5842: 5836: 5830: 5824: 5818: 5812: 5806: 5800: 5794: 5788: 5782: 5776: 5770: 5764: 5758: 5748: 5742: 5737:, sec. 7.2, and 5724: 5718: 5717: 5699: 5693: 5692: 5674: 5668: 5662: 5656: 5634: 5628: 5614: 5608: 5591: 5585: 5572: 5566: 5552: 5546: 5536: 5530: 5524: 5518: 5508: 5502: 5496: 5490: 5472: 5466: 5460: 5454: 5437: 5431: 5425: 5419: 5409: 5403: 5397: 5391: 5385: 5379: 5378: 5376: 5374: 5359: 5353: 5346: 5340: 5336: 5316: 5310: 5305: 5299: 5293: 5287: 5281: 5275: 5269: 5263: 5253: 5247: 5232: 5226: 5221:, ch. 7.8–7.10, 5216: 5210: 5201: 5195: 5189: 5183: 5174: 5168: 5162: 5156: 5145: 5139: 5128: 5122: 5112: 5106: 5088: 5082: 5075: 5069: 5066: 5060: 5041:Digital Einstein 5022: 5016: 5015: 5014: 5012: 5003:, archived from 4992: 4984: 4978: 4973: 4967: 4957: 4948: 4947: 4945: 4943: 4929: 4846:Alcubierre drive 4738:extra dimensions 4684:Projection of a 4600:isolated systems 4467:particle horizon 4421:angular momentum 4287:Causal structure 4200:Alcubierre drive 4130:particle physics 4098: 4096: 4095: 4090: 4088: 4087: 4064: 4062: 4061: 4056: 4054: 4053: 4040: 4038: 4037: 4028: 4017: 4012: 4011: 3997: 3990: 3989: 3973: 3963: 3958: 3957: 3935: 3933: 3932: 3927: 3708:earliest example 3574:or for ordinary 3541: 3539: 3538: 3533: 3519: 3517: 3516: 3511: 3494: 3492: 3491: 3486: 3469: 3467: 3466: 3461: 3443: 3441: 3440: 3435: 3428: 3427: 3425: 3421: 3420: 3402: 3401: 3392: 3391: 3381: 3380: 3379: 3370: 3369: 3356: 3338: 3336: 3335: 3330: 3216: 3214: 3213: 3208: 3206: 3205: 2746:angular momentum 2736: 2734: 2733: 2728: 2726: 2724: 2723: 2722: 2713: 2712: 2699: 2698: 2697: 2678: 2673: 2671: 2670: 2669: 2656: 2655: 2646: 2641: 2639: 2638: 2629: 2618: 2601: 2600: 2571: 2569: 2568: 2563: 2561: 2559: 2558: 2557: 2548: 2547: 2534: 2533: 2532: 2516: 2511: 2509: 2508: 2507: 2491: 2490: 2481: 2476: 2471: 2460: 2443: 2442: 2385: 2383: 2382: 2377: 2365: 2363: 2362: 2357: 2338:coefficients or 2329: 2327: 2326: 2321: 2319: 2318: 2310: 2307: 2306: 2286: 2284: 2283: 2278: 2263: 2261: 2260: 2255: 2244: 2242: 2234: 2233: 2232: 2219: 2217: 2215: 2207: 2206: 2205: 2192: 2190: 2189: 2181: 2178: 2177: 2165: 2163: 2162: 2161: 2148: 2147: 2146: 2137: 2136: 2126: 2100: 2098: 2097: 2092: 2084: 2083: 2061: 2059: 2058: 2053: 2037: 2035: 2034: 2029: 2017: 2015: 2014: 2009: 2007: 2005: 2004: 1995: 1984: 1969: 1967: 1966: 1961: 1938: 1936: 1935: 1930: 1928: 1927: 1908: 1906: 1905: 1900: 1885: 1883: 1882: 1877: 1872: 1871: 1860: 1859: 1858: 1844: 1843: 1814: 1812: 1811: 1806: 1804: 1803: 1791: 1790: 1762: 1760: 1759: 1754: 1752: 1751: 1732: 1730: 1729: 1724: 1722: 1721: 1695: 1693: 1692: 1687: 1684: 1683: 1665: 1664: 1648: 1638: 1633: 1632: 1617: 1616: 1510:Minkowski metric 1491:preferred frames 1464: 1460: 1456: 1452: 1380:. However, space 1330:electromagnetism 1323:curved spacetime 1311:reference frames 1129:Marcel Grossmann 881:four-dimensional 819: 812: 805: 792: 787: 786: 779: 775: 774: 559:van Stockum dust 544:Robertson–Walker 370: 369: 260: 259: 174: 172: 171: 166: 164: 163: 151: 143: 142: 124: 123: 104: 90: 89: 56: 21: 21628: 21627: 21623: 21622: 21621: 21619: 21618: 21617: 21613:1915 in science 21608:Albert Einstein 21588: 21587: 21584: 21572: 21560: 21550: 21548: 21536: 21526: 21524: 21516: 21514: 21509: 21460:Albert Einstein 21427: 21408:Einstein tensor 21371: 21352:Ricci curvature 21332:Kronecker delta 21318:Notable tensors 21313: 21234:Connection form 21211: 21205: 21136: 21122:Tensor operator 21079: 21073: 21013: 20989:Computer vision 20982: 20964: 20960:Tensor calculus 20904: 20893: 20888: 20858: 20853: 20817: 20803:Medical physics 20754: 20713:Nuclear physics 20682: 20676:Non-equilibrium 20598: 20570: 20542: 20536: 20506: 20501: 20481: 20432:Religious views 20427:Political views 20395: 20380:(granddaughter) 20378:Evelyn Einstein 20366:Robert Einstein 20360:Eduard Einstein 20304: 20241: 20235: 20189: 20159:Einstein's Gift 20093: 20086: 20027:Annus mirabilis 20012: 19983:Teleparallelism 19928:Einstein radius 19898:Brownian motion 19864: 19862:Albert Einstein 19859: 19829: 19824: 19803: 19779: 19772: 19743: 19741: 19731: 19712:Lovelock theory 19699: 19693: 19684: 19670: 19663: 19656: 19649: 19642: 19635: 19628: 19621: 19607: 19585: 19579: 19558: 19524: 19517: 19510: 19489: 19480:Induced gravity 19460: 19456:Scalar theories 19436: 19417: 19410: 19401:Massive gravity 19361:Teleparallelism 19359: 19354:Einstein–Cartan 19352: 19336: 19327:Quantum gravity 19306: 19305:Alternatives to 19295: 19261:Exact solutions 19232: 19194: 19189: 19159: 19154: 19140: 18968: 18872:BKL singularity 18862:Lemaître–Tolman 18837: 18833:Quantum gravity 18815: 18809: 18795:geodetic effect 18769:(together with 18739:LISA Pathfinder 18678: 18627: 18613:Penrose diagram 18595: 18589: 18564: 18553: 18549:Minkowski space 18515: 18459: 18443: 18391: 18385: 18345: 18338: 18333: 18290: 18288: 18284:Waner, Stefan. 18283: 18274: 18272: 18267: 18252: 18243: 18241: 18227: 18188: 18183: 18152:LIGO Laboratory 18137:Wayback Machine 18130:Einstein Online 18126: 18067: 18036: 18029: 17986: 17981: 17935: 17895: 17856: 17853: 17843: 17827: 17821: 17799: 17796: 17778: 17762:Wald, Robert M. 17760: 17755: 17739: 17734: 17708: 17702: 17686: 17681: 17668: 17663: 17643: 17638: 17622: 17617: 17601: 17598: 17589: 17571: 17566: 17553: 17535: 17529: 17516: 17511: 17494: 17489: 17475:Hughston, L. P. 17473: 17468: 17452: 17447: 17434: 17429: 17413: 17408: 17395: 17392: 17383: 17370: 17365: 17352: 17349: 17340: 17324: 17319: 17305:Wald, Robert M. 17303: 17297: 17279: 17274: 17258: 17253: 17239:Lieber, Lillian 17237: 17232: 17216: 17201: 17200: 17197: 17179: 17176: 17171: 17169:Further reading 17166: 17161: 17079: 17058: 17034: 17032: 17020:Einstein Online 17002:Thorsett, S. E. 16990: 16968: 16946: 16926: 16815:(5712): 381–4, 16743: 16717: 16661:Wald, Robert M. 16512: 16492: 16456: 16394: 16342: 16270:Rev. Mod. Phys. 16207: 16186: 15912: 15873: 15828:(26): 789–791, 15729: 15606: 15588: 15392: 15336:Physics Reports 15328: 15310: 15288: 15219: 15201: 15069: 15032:(18): 539–540, 14958: 14940: 14919: 14771: 14709: 14701:, A. A. Knopf, 14594: 14543: 14525: 14502: 14488:Overbye, Dennis 14459:Physical Review 14356: 14335: 14333: 14329: 14298: 14250: 14168: 14148:Thorne, Kip. S. 14134: 14132: 14119: 14101: 13997: 13797: 13749: 13673:(17): R25–R86, 13659: 13566: 13548: 13474: 13399: 13397: 13384: 13324: 13258: 13251: 13236: 13234: 13170: 13168: 13164: 13118: 13061: 12956:Space Sci. Rev. 12720: 12694: 12619: 12595: 12470:Rev. Mod. Phys. 12316: 12250: 12179:(12): 826–829, 12057:10.1038/435572a 12020: 11968: 11941:math-ph/0602018 11924: 11873: 11871: 11859:Einstein Online 11846: 11745:(1–2): 84–108, 11548: 11546: 11542: 11535: 11523: 11469: 11306: 11288: 11270: 11199: 11179: 11141: 11092: 11036: 10795: 10742:(12A): A3–A21, 10724: 10710:Carter, Brandon 10642: 10458: 10358:Physical Review 10335: 10291: 10273: 10157: 10134: 10008:Bertotti, Bruno 10001: 9857:(11): 463–500, 9848: 9830:(80): 525–573, 9733: 9677:Rev. Mod. Phys. 9673:Rees, Martin J. 9661: 9595: 9456:10.1038/415054a 9442:(6867): 54–57, 9428: 9270: 9236:, p. 126, 9158:Ashtekar, Abhay 9076: 9000: 8979: 8923: 8918: 8908:See, e.g., the 8907: 8903: 8879: 8875: 8859: 8855: 8847: 8843: 8836: 8832: 8823: 8819: 8813:Quantum gravity 8810: 8806: 8794: 8790: 8779: 8775: 8768: 8764: 8756: 8752: 8745: 8741: 8734: 8730: 8719: 8715: 8695: 8691: 8678:magnetic fields 8671: 8667: 8659: 8655: 8644: 8640: 8632: 8628: 8620: 8616: 8612:, sec. 2.3, 5.3 8583: 8579: 8563: 8559: 8538:renormalization 8535: 8531: 8524: 8520: 8513: 8509: 8501: 8497: 8490: 8483: 8475: 8471: 8462: 8458: 8450: 8446: 8430: 8426: 8415: 8411: 8392: 8388: 8376: 8372: 8352: 8348: 8340: 8336: 8324: 8320: 8311: 8307: 8292: 8288: 8281: 8277: 8269: 8265: 8254: 8250: 8231: 8227: 8215: 8211: 8203: 8199: 8181: 8177: 8161: 8157: 8150: 8146: 8134: 8130: 8117: 8113: 8101: 8097: 8085: 8081: 8069: 8065: 8053: 8049: 8032: 8028: 8020: 8016: 7999: 7995: 7990:Bekenstein 1974 7986:Bekenstein 1973 7984: 7980: 7956: 7952: 7927: 7923: 7900: 7896: 7884: 7880: 7868: 7864: 7860:, sec. 6.8, 6.9 7848: 7844: 7824: 7820: 7793:Physical Review 7789: 7785: 7756:(1336): 21–52. 7746: 7742: 7693: 7689: 7682: 7678: 7670: 7666: 7653: 7649: 7641: 7637: 7619:horizon problem 7612: 7608: 7596: 7592: 7573: 7569: 7559: 7555: 7531: 7527: 7511: 7507: 7468: 7464: 7449: 7445: 7414: 7410: 7402: 7398: 7386: 7382: 7367: 7363: 7355: 7351: 7339: 7335: 7328: 7324: 7317: 7313: 7277: 7273: 7245: 7241: 7225: 7221: 7209: 7205: 7197: 7193: 7182: 7178: 7170: 7166: 7158: 7154: 7147: 7143: 7136: 7132: 7122: 7120: 7115: 7114: 7110: 7103: 7099: 7050: 7046: 7039: 7035: 7020: 7016: 7008: 7004: 6997: 6993: 6985: 6981: 6974: 6970: 6957: 6953: 6948:Wambsganss 1998 6941: 6937: 6926: 6922: 6907: 6903: 6888: 6884: 6872: 6868: 6851: 6847: 6840: 6836: 6825: 6821: 6809: 6805: 6797:, § 14.5, 6793: 6789: 6782: 6778: 6763: 6759: 6743: 6739: 6678: 6671: 6659: 6655: 6648: 6644: 6619: 6615: 6607:, sec. 3.5 and 6602: 6598: 6590: 6586: 6578: 6574: 6562: 6558: 6545: 6541: 6530: 6526: 6518: 6514: 6506: 6502: 6493: 6489: 6476: 6475: 6466: 6409: 6402: 6392: 6390: 6364: 6357: 6347: 6345: 6322: 6318: 6308: 6306: 6281: 6277: 6269: 6265: 6228: 6224: 6215: 6211: 6188: 6184: 6176: 6172: 6155: 6151: 6135: 6131: 6123: 6119: 6110: 6106: 6095: 6091: 6079: 6075: 6065:Gravity Probe A 6050: 6046: 6034: 6030: 6006: 6002: 5990: 5986: 5966: 5962: 5950: 5946: 5938: 5934: 5925: 5921: 5914: 5910: 5901: 5897: 5889: 5885: 5877: 5873: 5861: 5857: 5849: 5845: 5837: 5833: 5825: 5821: 5813: 5809: 5801: 5797: 5789: 5785: 5777: 5773: 5765: 5761: 5749: 5745: 5725: 5721: 5714: 5700: 5696: 5689: 5675: 5671: 5663: 5659: 5635: 5631: 5615: 5611: 5592: 5588: 5573: 5569: 5553: 5549: 5537: 5533: 5525: 5521: 5509: 5505: 5497: 5493: 5473: 5469: 5461: 5457: 5438: 5434: 5426: 5422: 5410: 5406: 5398: 5394: 5386: 5382: 5372: 5370: 5360: 5356: 5347: 5343: 5333: 5317: 5313: 5306: 5302: 5294: 5290: 5282: 5278: 5270: 5266: 5254: 5250: 5233: 5229: 5217: 5213: 5202: 5198: 5190: 5186: 5175: 5171: 5163: 5159: 5147:As reported in 5146: 5142: 5129: 5125: 5113: 5109: 5089: 5085: 5076: 5072: 5067: 5063: 5027:, ch. 9 to 15, 5023: 5019: 5010: 5008: 4993: 4985: 4981: 4974: 4970: 4958: 4951: 4941: 4939: 4937:Black-holes.org 4931: 4930: 4926: 4922: 4842: 4829: 4730:point particles 4678: 4656: 4654:Quantum gravity 4650: 4648:Quantum gravity 4626: 4620: 4608: 4571: 4565: 4551:example is the 4532: 4526: 4489: 4483: 4475:Unruh radiation 4444:Penrose process 4417:linear momentum 4381:hoop conjecture 4373: 4367:No hair theorem 4361:Main articles: 4359: 4343:global geometry 4289: 4283: 4248:addressed this 4246:Rainer K. Sachs 4223: 4217: 4212: 4181:Stephen Hawking 4173:Tipler cylinder 4162: 4154:quantum gravity 4080: 4076: 4074: 4071: 4070: 4046: 4042: 4033: 4029: 4018: 4016: 4004: 4000: 3982: 3978: 3962: 3950: 3946: 3944: 3941: 3940: 3921: 3918: 3917: 3902: 3896: 3888:standard candle 3837: 3831: 3808:LISA Pathfinder 3750: 3742:Main articles: 3740: 3728:Hubble constant 3688: 3682: 3677: 3646:Gravity Probe B 3625: 3617:Main articles: 3615: 3552: 3527: 3524: 3523: 3505: 3502: 3501: 3480: 3477: 3476: 3472:semi-major axis 3455: 3452: 3451: 3416: 3412: 3397: 3393: 3387: 3383: 3382: 3375: 3371: 3365: 3361: 3357: 3355: 3347: 3344: 3343: 3324: 3321: 3320: 3266: 3252: 3240: 3234: 3226:Gowdy universes 3198: 3194: 3192: 3189: 3188: 3171:of black holes 3148: 3142: 3101: 3085:Main articles: 3083: 3040: 3034: 3025: 2942:exact solutions 2927: 2907:laws of physics 2842:energy–momentum 2826: 2817: 2807: 2784:teleparallelism 2768: 2762: 2718: 2714: 2708: 2704: 2700: 2693: 2689: 2679: 2677: 2665: 2661: 2657: 2651: 2647: 2645: 2634: 2630: 2619: 2617: 2596: 2592: 2590: 2587: 2586: 2553: 2549: 2543: 2539: 2535: 2528: 2524: 2517: 2515: 2503: 2499: 2492: 2486: 2482: 2480: 2461: 2459: 2438: 2434: 2432: 2429: 2428: 2410: 2404: 2371: 2368: 2367: 2351: 2348: 2347: 2311: 2309: 2308: 2302: 2298: 2296: 2293: 2292: 2272: 2269: 2268: 2235: 2228: 2224: 2220: 2218: 2208: 2201: 2197: 2193: 2191: 2182: 2180: 2179: 2173: 2169: 2157: 2153: 2149: 2142: 2138: 2132: 2128: 2127: 2125: 2123: 2120: 2119: 2076: 2072: 2070: 2067: 2066: 2047: 2044: 2043: 2023: 2020: 2019: 2000: 1996: 1985: 1983: 1975: 1972: 1971: 1970:is found to be 1955: 1952: 1951: 1920: 1916: 1914: 1911: 1910: 1894: 1891: 1890: 1861: 1854: 1850: 1849: 1848: 1836: 1832: 1830: 1827: 1826: 1796: 1792: 1783: 1779: 1771: 1768: 1767: 1744: 1740: 1738: 1735: 1734: 1714: 1710: 1708: 1705: 1704: 1701:Einstein tensor 1697: 1676: 1672: 1657: 1653: 1637: 1625: 1621: 1609: 1605: 1603: 1600: 1599: 1544: 1536:Main articles: 1534: 1479:inertial frames 1462: 1458: 1454: 1450: 1407: 1281:At the base of 1270: 1257: 1096: 1088:Main articles: 1086: 1012:electromagnetic 996:quantum gravity 992:quantum physics 853:Albert Einstein 823: 782: 769: 768: 761: 760: 584: 583: 574: 573: 529:Lemaître–Tolman 474: 473: 462: 461: 453:Quantum gravity 440:Advanced theory 367: 366: 365: 348: 347: 296:Geodetic effect 257: 256: 247: 246: 222: 221: 205: 175: 156: 152: 147: 135: 131: 116: 112: 110: 107: 106: 46: 42: 35: 28: 23: 22: 15: 12: 11: 5: 21626: 21616: 21615: 21610: 21605: 21600: 21583: 21582: 21570: 21558: 21546: 21534: 21511: 21510: 21508: 21507: 21502: 21500:Woldemar Voigt 21497: 21492: 21487: 21482: 21477: 21472: 21467: 21465:Leonhard Euler 21462: 21457: 21452: 21447: 21441: 21439: 21437:Mathematicians 21433: 21432: 21429: 21428: 21426: 21425: 21420: 21415: 21410: 21405: 21400: 21395: 21390: 21385: 21379: 21377: 21373: 21372: 21370: 21369: 21364: 21362:Torsion tensor 21359: 21354: 21349: 21344: 21339: 21334: 21328: 21326: 21319: 21315: 21314: 21312: 21311: 21306: 21301: 21296: 21291: 21286: 21281: 21276: 21271: 21266: 21261: 21256: 21251: 21246: 21241: 21236: 21231: 21226: 21221: 21215: 21213: 21207: 21206: 21204: 21203: 21197: 21195:Tensor product 21192: 21187: 21185:Symmetrization 21182: 21177: 21175:Lie derivative 21172: 21167: 21162: 21157: 21152: 21146: 21144: 21138: 21137: 21135: 21134: 21129: 21124: 21119: 21114: 21109: 21104: 21099: 21097:Tensor density 21094: 21089: 21083: 21081: 21075: 21074: 21072: 21071: 21069:Voigt notation 21066: 21061: 21056: 21054:Ricci calculus 21051: 21046: 21041: 21039:Index notation 21036: 21031: 21025: 21023: 21019: 21018: 21015: 21014: 21012: 21011: 21006: 21001: 20996: 20991: 20985: 20983: 20981: 20980: 20975: 20969: 20966: 20965: 20963: 20962: 20957: 20955:Tensor algebra 20952: 20947: 20942: 20937: 20935:Dyadic algebra 20932: 20927: 20921: 20919: 20910: 20906: 20905: 20898: 20895: 20894: 20887: 20886: 20879: 20872: 20864: 20855: 20854: 20852: 20851: 20846: 20841: 20836: 20831: 20825: 20823: 20819: 20818: 20816: 20815: 20810: 20805: 20800: 20795: 20790: 20785: 20780: 20775: 20770: 20764: 20762: 20756: 20755: 20753: 20752: 20747: 20746: 20745: 20740: 20735: 20725: 20720: 20715: 20710: 20709: 20708: 20703: 20692: 20690: 20684: 20683: 20681: 20680: 20679: 20678: 20673: 20666:Thermodynamics 20663: 20662: 20661: 20656: 20646: 20641: 20636: 20635: 20634: 20629: 20624: 20619: 20608: 20606: 20600: 20599: 20597: 20596: 20595: 20594: 20584: 20578: 20576: 20572: 20571: 20569: 20568: 20567: 20566: 20556: 20550: 20548: 20544: 20543: 20535: 20534: 20527: 20520: 20512: 20503: 20502: 20500: 20499: 20486: 20483: 20482: 20480: 20479: 20474: 20469: 20464: 20459: 20454: 20449: 20444: 20439: 20434: 20429: 20424: 20419: 20414: 20409: 20403: 20401: 20397: 20396: 20394: 20393: 20387: 20381: 20375: 20369: 20363: 20357: 20351: 20345: 20339: 20333: 20327: 20321: 20314: 20312: 20306: 20305: 20303: 20302: 20295: 20288: 20281: 20274: 20267: 20260: 20253: 20245: 20243: 20237: 20236: 20234: 20233: 20228: 20223: 20218: 20213: 20208: 20203: 20197: 20195: 20191: 20190: 20188: 20187: 20179: 20171: 20170:(2008 TV film) 20163: 20155: 20147: 20139: 20135:Young Einstein 20131: 20127:Insignificance 20123: 20115: 20107: 20098: 20096: 20088: 20087: 20085: 20084: 20078: 20075:Why Socialism? 20071: 20063: 20055: 20047: 20039: 20032: 20022: 20020: 20014: 20013: 20011: 20010: 20005: 20000: 19995: 19990: 19985: 19980: 19975: 19970: 19965: 19960: 19955: 19950: 19945: 19940: 19935: 19930: 19925: 19920: 19915: 19913:Einstein solid 19910: 19905: 19900: 19895: 19890: 19889: 19888: 19883: 19872: 19870: 19866: 19865: 19858: 19857: 19850: 19843: 19835: 19826: 19825: 19823: 19822: 19817: 19811: 19809: 19808:Related topics 19805: 19804: 19802: 19801: 19796: 19791: 19786: 19785: 19784: 19777: 19765: 19760: 19755: 19749: 19747: 19737: 19736: 19733: 19732: 19730: 19729: 19724: 19719: 19714: 19709: 19703: 19701: 19695: 19694: 19692: 19691: 19690: 19689: 19680:Twistor theory 19677: 19676: 19675: 19668: 19661: 19654: 19647: 19640: 19633: 19626: 19614: 19613: 19612: 19600: 19595: 19589: 19587: 19581: 19580: 19578: 19577: 19572: 19566: 19564: 19560: 19559: 19557: 19556: 19551: 19546: 19541: 19536: 19531: 19530: 19529: 19522: 19515: 19503: 19497: 19495: 19491: 19490: 19488: 19487: 19482: 19477: 19472: 19467: 19466: 19465: 19453: 19448: 19443: 19442: 19441: 19429: 19424: 19423: 19422: 19415: 19403: 19398: 19393: 19381: 19376: 19371: 19366: 19365: 19364: 19357: 19344: 19342: 19338: 19337: 19335: 19334: 19329: 19324: 19318: 19316: 19309: 19301: 19300: 19297: 19296: 19294: 19293: 19288: 19283: 19278: 19273: 19268: 19263: 19258: 19253: 19248: 19242: 19240: 19234: 19233: 19231: 19230: 19225: 19220: 19215: 19209: 19207: 19200: 19196: 19195: 19188: 19187: 19180: 19173: 19165: 19156: 19155: 19145: 19142: 19141: 19139: 19138: 19131: 19126: 19121: 19116: 19111: 19106: 19101: 19096: 19091: 19086: 19081: 19076: 19071: 19066: 19061: 19059:Choquet-Bruhat 19056: 19051: 19046: 19041: 19036: 19031: 19026: 19021: 19016: 19011: 19006: 19001: 18996: 18991: 18986: 18980: 18978: 18974: 18973: 18970: 18969: 18967: 18966: 18959: 18958: 18953: 18948: 18941: 18940: 18935: 18930: 18925: 18920: 18911:Axisymmetric: 18908: 18907: 18902: 18896: 18885: 18884: 18879: 18874: 18869: 18864: 18859: 18850:Cosmological: 18847: 18845: 18839: 18838: 18836: 18835: 18830: 18825: 18819: 18817: 18811: 18810: 18808: 18807: 18802: 18791:frame-dragging 18788: 18783: 18778: 18775:Einstein rings 18771:Einstein cross 18764: 18753: 18752: 18747: 18741: 18736: 18731: 18718: 18708: 18707: 18702: 18697: 18692: 18686: 18684: 18680: 18679: 18677: 18676: 18674:Ernst equation 18671: 18666: 18661: 18656: 18651: 18646: 18644:BSSN formalism 18641: 18635: 18633: 18629: 18628: 18626: 18625: 18620: 18615: 18610: 18605: 18599: 18597: 18591: 18590: 18588: 18587: 18582: 18576: 18574: 18567: 18559: 18558: 18555: 18554: 18552: 18551: 18546: 18541: 18536: 18531: 18525: 18523: 18517: 18516: 18514: 18513: 18508: 18503: 18501:Ladder paradox 18498: 18493: 18488: 18483: 18478: 18473: 18467: 18465: 18461: 18460: 18458: 18457: 18451: 18449: 18445: 18444: 18442: 18441: 18436: 18431: 18426: 18421: 18416: 18411: 18406: 18404:Speed of light 18401: 18395: 18393: 18387: 18386: 18384: 18383: 18378: 18373: 18367: 18357: 18355: 18348: 18340: 18339: 18332: 18331: 18324: 18317: 18309: 18303: 18302: 18297: 18281: 18265: 18250: 18228:Brown, Kevin. 18225: 18215: 18205: 18182: 18181: 18178: 18175: 18171: 18165: 18164: 18154: 18149: 18143: 18125: 18124:External links 18122: 18121: 18120: 18065: 18038:Shapiro, I. I. 18034: 17979: 17933: 17893: 17872:(7): 769–822, 17852: 17849: 17848: 17847: 17841: 17825: 17819: 17795: 17792: 17791: 17790: 17776: 17758: 17753: 17737: 17732: 17706: 17700: 17688:Will, Clifford 17684: 17679: 17666: 17661: 17645:Landau, Lev D. 17641: 17636: 17620: 17615: 17597: 17594: 17593: 17592: 17587: 17569: 17564: 17551: 17533: 17527: 17514: 17509: 17492: 17487: 17471: 17466: 17450: 17445: 17432: 17427: 17411: 17406: 17391: 17388: 17387: 17386: 17381: 17368: 17363: 17348: 17345: 17344: 17343: 17338: 17322: 17317: 17301: 17295: 17277: 17272: 17256: 17251: 17235: 17230: 17214: 17195: 17175: 17172: 17170: 17167: 17165: 17164: 17159: 17143: 17082: 17077: 17061: 17056: 17040: 17010: 16993: 16988: 16971: 16966: 16949: 16944: 16929: 16924: 16918:, John Wiley, 16906: 16848: 16804: 16746: 16741: 16720: 16715: 16700: 16657: 16648: 16631:(4): 870–892, 16617: 16591: 16562: 16549:hep-th/9612121 16531: 16516: 16510: 16495: 16490: 16459: 16454: 16441: 16433:Thorne, Kip S. 16429: 16406: 16392: 16353: 16340: 16315:hep-th/0608210 16301: 16276:(3): 711–719, 16258: 16210: 16205: 16189: 16184: 16171: 16113: 16058: 16047:10.1086/513700 16023:(2): 377–408, 16008: 15997:10.1086/377226 15973:(1): 175–194, 15961: 15915: 15910: 15876: 15871: 15845: 15814: 15789:(12): 121101, 15778: 15732: 15727: 15696: 15677: 15658: 15630:hep-th/0702219 15609: 15604: 15591: 15586: 15573: 15562:10.1086/378122 15527: 15481: 15449:10.1.1.90.7036 15413: 15399:Rovelli, Carlo 15395: 15390: 15377: 15331: 15326: 15320:, John Wiley, 15313: 15308: 15291: 15286: 15270: 15222: 15217: 15204: 15199: 15186: 15175:10.1086/504862 15151:(1): 407–419, 15140: 15123:(9): 106–120, 15108: 15072: 15067: 15055:Ramond, Pierre 15051: 15021: 15002:(7): 337–341, 14991: 14972:(9): 439–441, 14961: 14956: 14943: 14938: 14922: 14917: 14896: 14833: 14820: 14785:(11): 111017, 14774: 14769: 14749: 14745:10.1086/148307 14716:Penzias, A. A. 14712: 14707: 14693:Penrose, Roger 14689: 14670:Penrose, Roger 14666: 14637:Penrose, Roger 14633: 14597: 14592: 14577: 14574:10.1086/148918 14546: 14541: 14528: 14523: 14505: 14500: 14484: 14465:(5): 455–459, 14451: 14440:10.1086/320579 14416:(2): 718–730, 14405: 14394:10.1086/425170 14359: 14354: 14341: 14309:(3): 203–246, 14291: 14276: 14253: 14248: 14232: 14217: 14171: 14166: 14140: 14122: 14117: 14104: 14099: 14079: 14075:10.1086/173574 14046: 14011:(2): 340–445, 14000: 13995: 13977: 13935: 13918:(6): 874–876, 13912:J. Math. Phys. 13907: 13849: 13821:hep-th/0503195 13814:(1): 151–160, 13800: 13795: 13778:hep-th/0503203 13761: 13747: 13708: 13662: 13657: 13644: 13586: 13569: 13564: 13551: 13546: 13533: 13478: 13472: 13433: 13416:(3): 934–936, 13405: 13387: 13382: 13369: 13327: 13322: 13309: 13263: 13242: 13224: 13176: 13121: 13116: 13103: 13068:Israel, Werner 13064: 13059: 13043: 13008:(3): 509–513, 12997: 12951: 12905: 12877:hep-ph/9911499 12859: 12855:10.1086/181708 12822: 12785:(3): 168–173, 12771: 12723: 12718: 12697: 12692: 12679: 12631: 12622: 12617: 12598: 12593: 12580: 12555:(3): 199–220, 12540: 12493: 12476:(4): 938–965, 12465: 12419:Keating, R. E. 12411: 12365:Keating, R. E. 12357: 12337:(3): 447–450. 12331:Rev. Mod. Phys 12319: 12314: 12281: 12278:10.1086/151174 12253: 12248: 12232:Schwarz, J. H. 12224: 12207:(1): 203–241, 12196: 12168: 12153: 12136:(1–3): 81–86, 12125: 12077: 12031: 12018: 11979: 11966: 11927: 11922: 11909: 11883:Geroch, Robert 11879: 11849: 11844: 11828: 11780: 11734: 11686: 11667:(1): 141–225, 11656: 11608: 11589:10.1086/312423 11565:(1): L13–L16, 11554: 11526: 11521: 11508: 11480: 11467: 11425: 11405: 11384: 11383: 11359: 11324:(4): 519–529, 11309: 11304: 11291: 11286: 11273: 11268: 11256:Ehlers, Jürgen 11252: 11224:hep-th/9608117 11203: 11197: 11182: 11177: 11146: 11085: 11039: 11034: 11003: 10992:10.1086/380121 10968:(2): 544–552, 10956: 10939:(8): 527–550, 10928: 10884: 10849:(3): 961–992, 10834: 10798: 10793: 10777: 10727: 10722: 10706: 10645: 10640: 10627: 10592:(8): 885–942, 10581: 10561: 10518:(24): 241101, 10507: 10461: 10456: 10443: 10381: 10364:(3): 925–935, 10346: 10333: 10308:hep-th/0701111 10294: 10289: 10276: 10271: 10255: 10207: 10172:(16): 165003, 10160: 10155: 10137: 10132: 10115: 10098:(1): 599–654, 10087: 10043: 10004: 9999: 9982: 9921: 9910:10.1086/377253 9874: 9811: 9775: 9736: 9731: 9700: 9683:(2): 255–351, 9664: 9659: 9646: 9598: 9593: 9572: 9547:(2): 161–170, 9533:Hawking, S. W. 9525:Bardeen, J. M. 9521: 9475: 9431: 9426: 9413: 9396:(5): 793–800, 9385: 9339: 9281: 9268: 9229: 9193: 9154: 9104: 9069: 9023: 9011:Deser, Stanley 9003: 8998: 8982: 8977: 8964: 8924: 8922: 8919: 8917: 8916: 8901: 8873: 8862:Bartusiak 2000 8853: 8841: 8838:Friedrich 2005 8830: 8817: 8804: 8788: 8773: 8762: 8750: 8739: 8728: 8713: 8689: 8665: 8653: 8638: 8626: 8614: 8577: 8557: 8529: 8518: 8507: 8495: 8481: 8469: 8456: 8444: 8424: 8409: 8386: 8370: 8355:Hawking energy 8346: 8334: 8318: 8305: 8286: 8275: 8263: 8248: 8225: 8209: 8197: 8175: 8172:Garfinkle 2007 8155: 8144: 8128: 8111: 8095: 8079: 8063: 8047: 8033:Horizons: cf. 8026: 8014: 7993: 7978: 7950: 7936:, sec. 9.3 or 7921: 7894: 7878: 7862: 7842: 7818: 7783: 7740: 7703:(12): 124017. 7687: 7676: 7664: 7647: 7635: 7606: 7590: 7567: 7553: 7547:matter"), cf. 7525: 7505: 7462: 7443: 7408: 7396: 7380: 7361: 7349: 7333: 7322: 7311: 7271: 7239: 7219: 7203: 7199:Blandford 1987 7191: 7176: 7164: 7152: 7141: 7130: 7108: 7097: 7044: 7033: 7026:Bartusiak 2000 7014: 7002: 6991: 6979: 6968: 6962:, ch. 23; cf. 6951: 6935: 6920: 6901: 6882: 6876:, sec. 4.2.1, 6866: 6862:Kahn 1996–2012 6845: 6834: 6831:Nordtvedt 2003 6819: 6803: 6787: 6776: 6757: 6753:Bartusiak 2000 6737: 6669: 6653: 6642: 6613: 6596: 6584: 6572: 6556: 6539: 6524: 6512: 6500: 6487: 6464: 6400: 6355: 6316: 6275: 6263: 6222: 6209: 6182: 6170: 6149: 6138:Kennefick 2005 6129: 6117: 6104: 6089: 6073: 6044: 6028: 6000: 5984: 5960: 5944: 5932: 5919: 5908: 5895: 5883: 5871: 5868:MacCallum 2006 5855: 5843: 5831: 5819: 5807: 5795: 5783: 5771: 5759: 5743: 5741:, respectively 5719: 5712: 5694: 5687: 5681:. John Wiley. 5669: 5657: 5629: 5619:, p. 16, 5609: 5586: 5567: 5547: 5531: 5519: 5503: 5491: 5467: 5465:, pp. 10f 5455: 5432: 5420: 5404: 5392: 5380: 5354: 5341: 5331: 5311: 5300: 5288: 5276: 5264: 5248: 5227: 5211: 5196: 5184: 5181:Kennefick 2007 5177:Kennefick 2005 5169: 5157: 5140: 5123: 5107: 5103:Nordström 1918 5083: 5070: 5061: 5017: 4979: 4968: 4949: 4923: 4921: 4918: 4917: 4916: 4911: 4908:Ricci calculus 4905: 4900: 4894: 4889: 4883: 4877: 4871: 4865: 4860: 4855: 4849: 4841: 4838: 4828: 4827:Current status 4825: 4797:Regge calculus 4652:Main article: 4649: 4646: 4622:Main article: 4619: 4616: 4607: 4604: 4567:Main article: 4564: 4561: 4537:time evolution 4528:Main article: 4525: 4522: 4513:BKL conjecture 4485:Main article: 4482: 4479: 4358: 4355: 4320:null geodesics 4285:Main article: 4282: 4279: 4231:Poincaré group 4219:Main article: 4216: 4213: 4211: 4208: 4161: 4158: 4086: 4083: 4079: 4066: 4065: 4052: 4049: 4045: 4036: 4032: 4027: 4024: 4021: 4015: 4010: 4007: 4003: 3996: 3993: 3988: 3985: 3981: 3976: 3971: 3967: 3961: 3956: 3953: 3949: 3925: 3898:Main article: 3895: 3892: 3833:Main article: 3830: 3827: 3739: 3736: 3695:Einstein cross 3684:Main article: 3681: 3678: 3676: 3673: 3653:frame-dragging 3623:Frame dragging 3614: 3611: 3607:PSR J0737−3039 3551: 3548: 3547: 3546: 3531: 3521: 3509: 3499: 3497:orbital period 3484: 3474: 3459: 3445: 3444: 3433: 3424: 3419: 3415: 3411: 3408: 3405: 3400: 3396: 3390: 3386: 3378: 3374: 3368: 3364: 3360: 3354: 3351: 3328: 3274:center of mass 3262:Main article: 3251: 3248: 3236:Main article: 3233: 3230: 3204: 3201: 3197: 3144:Main article: 3141: 3138: 3082: 3079: 3075:binary pulsars 3036:Main article: 3033: 3030: 3024: 3021: 2966:Gödel universe 2926: 2925:Model-building 2923: 2876:limiting cases 2825: 2822: 2806: 2803: 2764:Main article: 2761: 2758: 2738: 2737: 2721: 2717: 2711: 2707: 2703: 2696: 2692: 2688: 2685: 2682: 2676: 2668: 2664: 2660: 2654: 2650: 2644: 2637: 2633: 2628: 2625: 2622: 2616: 2613: 2610: 2607: 2604: 2599: 2595: 2573: 2572: 2556: 2552: 2546: 2542: 2538: 2531: 2527: 2523: 2520: 2514: 2506: 2502: 2498: 2495: 2489: 2485: 2479: 2474: 2470: 2467: 2464: 2458: 2455: 2452: 2449: 2446: 2441: 2437: 2403: 2400: 2375: 2355: 2317: 2314: 2305: 2301: 2276: 2265: 2264: 2253: 2250: 2247: 2241: 2238: 2231: 2227: 2223: 2214: 2211: 2204: 2200: 2196: 2188: 2185: 2176: 2172: 2168: 2160: 2156: 2152: 2145: 2141: 2135: 2131: 2102: 2101: 2090: 2087: 2082: 2079: 2075: 2051: 2027: 2003: 1999: 1994: 1991: 1988: 1982: 1979: 1959: 1926: 1923: 1919: 1898: 1887: 1886: 1875: 1870: 1867: 1864: 1857: 1853: 1847: 1842: 1839: 1835: 1816: 1815: 1802: 1799: 1795: 1789: 1786: 1782: 1778: 1775: 1750: 1747: 1743: 1720: 1717: 1713: 1682: 1679: 1675: 1671: 1668: 1663: 1660: 1656: 1651: 1646: 1642: 1636: 1631: 1628: 1624: 1620: 1615: 1612: 1608: 1594: 1587:counterparts, 1569:field equation 1533: 1530: 1441:speed of light 1437:Poincaré group 1406: 1403: 1269: 1266: 1256: 1253: 1193:of the planet 1099:Henri Poincaré 1085: 1082: 963:. So far, all 951:of light, the 857:modern physics 825: 824: 822: 821: 814: 807: 799: 796: 795: 794: 793: 780: 763: 762: 759: 758: 751: 746: 741: 736: 731: 726: 721: 716: 711: 706: 701: 696: 691: 686: 681: 676: 671: 666: 661: 656: 651: 646: 641: 636: 631: 626: 621: 616: 611: 606: 601: 596: 591: 585: 581: 580: 579: 576: 575: 572: 571: 566: 561: 556: 551: 546: 541: 536: 531: 526: 521: 516: 511: 506: 501: 496: 491: 486: 475: 469: 468: 467: 464: 463: 458: 457: 456: 455: 450: 442: 441: 437: 436: 435: 434: 432:Post-Newtonian 429: 424: 416: 415: 411: 410: 409: 408: 403: 398: 393: 388: 383: 375: 374: 368: 364: 363: 360: 356: 355: 354: 353: 350: 349: 344: 343: 342: 341: 336: 331: 323: 322: 316: 315: 314: 313: 308: 303: 298: 293: 291:Frame-dragging 288: 283: 278: 273: 268: 266:Kepler problem 258: 254: 253: 252: 249: 248: 245: 244: 239: 234: 229: 223: 219: 218: 217: 214: 213: 212: 211: 206: 204: 203: 198: 193: 187: 185: 177: 176: 162: 159: 155: 150: 146: 141: 138: 134: 130: 127: 122: 119: 115: 105: 97: 96: 26: 9: 6: 4: 3: 2: 21625: 21614: 21611: 21609: 21606: 21604: 21601: 21599: 21596: 21595: 21593: 21586: 21581: 21576: 21571: 21569: 21564: 21559: 21557: 21547: 21545: 21540: 21535: 21533: 21523: 21522: 21519: 21506: 21503: 21501: 21498: 21496: 21493: 21491: 21488: 21486: 21483: 21481: 21478: 21476: 21473: 21471: 21468: 21466: 21463: 21461: 21458: 21456: 21453: 21451: 21448: 21446: 21443: 21442: 21440: 21438: 21434: 21424: 21421: 21419: 21416: 21414: 21411: 21409: 21406: 21404: 21401: 21399: 21396: 21394: 21391: 21389: 21386: 21384: 21381: 21380: 21378: 21374: 21368: 21365: 21363: 21360: 21358: 21355: 21353: 21350: 21348: 21345: 21343: 21342:Metric tensor 21340: 21338: 21335: 21333: 21330: 21329: 21327: 21323: 21320: 21316: 21310: 21307: 21305: 21302: 21300: 21297: 21295: 21292: 21290: 21287: 21285: 21282: 21280: 21277: 21275: 21272: 21270: 21267: 21265: 21262: 21260: 21257: 21255: 21254:Exterior form 21252: 21250: 21247: 21245: 21242: 21240: 21237: 21235: 21232: 21230: 21227: 21225: 21222: 21220: 21217: 21216: 21214: 21208: 21201: 21198: 21196: 21193: 21191: 21188: 21186: 21183: 21181: 21178: 21176: 21173: 21171: 21168: 21166: 21163: 21161: 21158: 21156: 21153: 21151: 21148: 21147: 21145: 21143: 21139: 21133: 21130: 21128: 21127:Tensor bundle 21125: 21123: 21120: 21118: 21115: 21113: 21110: 21108: 21105: 21103: 21100: 21098: 21095: 21093: 21090: 21088: 21085: 21084: 21082: 21076: 21070: 21067: 21065: 21062: 21060: 21057: 21055: 21052: 21050: 21047: 21045: 21042: 21040: 21037: 21035: 21032: 21030: 21027: 21026: 21024: 21020: 21010: 21007: 21005: 21002: 21000: 20997: 20995: 20992: 20990: 20987: 20986: 20984: 20979: 20976: 20974: 20971: 20970: 20967: 20961: 20958: 20956: 20953: 20951: 20948: 20946: 20943: 20941: 20938: 20936: 20933: 20931: 20928: 20926: 20923: 20922: 20920: 20918: 20914: 20911: 20907: 20903: 20902: 20896: 20892: 20885: 20880: 20878: 20873: 20871: 20866: 20865: 20862: 20850: 20847: 20845: 20842: 20840: 20837: 20835: 20832: 20830: 20827: 20826: 20824: 20820: 20814: 20811: 20809: 20808:Ocean physics 20806: 20804: 20801: 20799: 20796: 20794: 20791: 20789: 20786: 20784: 20781: 20779: 20776: 20774: 20771: 20769: 20766: 20765: 20763: 20761: 20757: 20751: 20748: 20744: 20743:Modern optics 20741: 20739: 20736: 20734: 20731: 20730: 20729: 20726: 20724: 20721: 20719: 20716: 20714: 20711: 20707: 20704: 20702: 20699: 20698: 20697: 20694: 20693: 20691: 20689: 20685: 20677: 20674: 20672: 20669: 20668: 20667: 20664: 20660: 20657: 20655: 20652: 20651: 20650: 20647: 20645: 20642: 20640: 20637: 20633: 20630: 20628: 20625: 20623: 20620: 20618: 20615: 20614: 20613: 20610: 20609: 20607: 20605: 20601: 20593: 20592:Computational 20590: 20589: 20588: 20585: 20583: 20580: 20579: 20577: 20573: 20565: 20562: 20561: 20560: 20557: 20555: 20552: 20551: 20549: 20545: 20541: 20533: 20528: 20526: 20521: 20519: 20514: 20513: 20510: 20498: 20497: 20488: 20487: 20484: 20478: 20475: 20473: 20470: 20468: 20465: 20463: 20460: 20458: 20455: 20453: 20450: 20448: 20445: 20443: 20440: 20438: 20435: 20433: 20430: 20428: 20425: 20423: 20420: 20418: 20415: 20413: 20410: 20408: 20405: 20404: 20402: 20398: 20391: 20388: 20385: 20382: 20379: 20376: 20373: 20370: 20367: 20364: 20361: 20358: 20355: 20354:Maja Einstein 20352: 20349: 20346: 20343: 20340: 20337: 20334: 20331: 20328: 20325: 20324:Elsa Einstein 20322: 20319: 20316: 20315: 20313: 20311: 20307: 20301: 20300: 20296: 20294: 20293: 20289: 20287: 20286: 20282: 20280: 20279: 20275: 20273: 20272: 20268: 20266: 20265: 20261: 20259: 20258: 20254: 20252: 20251: 20247: 20246: 20244: 20238: 20232: 20229: 20227: 20224: 20222: 20219: 20217: 20214: 20212: 20211:Kalinga Prize 20209: 20207: 20204: 20202: 20199: 20198: 20196: 20192: 20185: 20184: 20180: 20178:(2017 series) 20177: 20176: 20172: 20169: 20168: 20164: 20161: 20160: 20156: 20153: 20152: 20148: 20145: 20144: 20140: 20137: 20136: 20132: 20129: 20128: 20124: 20121: 20120: 20116: 20113: 20112: 20108: 20105: 20104: 20100: 20099: 20097: 20095: 20089: 20082: 20079: 20076: 20072: 20069: 20068: 20064: 20061: 20060: 20056: 20053: 20052: 20048: 20045: 20044: 20040: 20037: 20033: 20030: 20028: 20024: 20023: 20021: 20019: 20015: 20009: 20006: 20004: 20001: 19999: 19996: 19994: 19991: 19989: 19986: 19984: 19981: 19979: 19976: 19974: 19971: 19969: 19966: 19964: 19961: 19959: 19956: 19954: 19951: 19949: 19946: 19944: 19941: 19939: 19936: 19934: 19931: 19929: 19926: 19924: 19921: 19919: 19916: 19914: 19911: 19909: 19906: 19904: 19901: 19899: 19896: 19894: 19891: 19887: 19884: 19882: 19879: 19878: 19877: 19874: 19873: 19871: 19867: 19863: 19856: 19851: 19849: 19844: 19842: 19837: 19836: 19833: 19821: 19818: 19816: 19813: 19812: 19810: 19806: 19800: 19797: 19795: 19792: 19790: 19787: 19782: 19778: 19775: 19774:Fatio–Le Sage 19771: 19770: 19769: 19766: 19764: 19761: 19759: 19756: 19754: 19751: 19750: 19748: 19746: 19740:Pre-Newtonian 19738: 19728: 19725: 19723: 19720: 19718: 19715: 19713: 19710: 19708: 19705: 19704: 19702: 19696: 19687: 19683: 19682: 19681: 19678: 19673: 19669: 19666: 19662: 19659: 19655: 19652: 19648: 19645: 19641: 19638: 19634: 19631: 19627: 19624: 19620: 19619: 19618: 19617:String theory 19615: 19610: 19606: 19605: 19604: 19601: 19599: 19596: 19594: 19591: 19590: 19588: 19582: 19576: 19573: 19571: 19568: 19567: 19565: 19561: 19555: 19552: 19550: 19547: 19545: 19542: 19540: 19537: 19535: 19532: 19527: 19523: 19520: 19516: 19513: 19509: 19508: 19507: 19504: 19502: 19499: 19498: 19496: 19492: 19486: 19483: 19481: 19478: 19476: 19473: 19471: 19468: 19463: 19459: 19458: 19457: 19454: 19452: 19449: 19447: 19444: 19439: 19435: 19434: 19433: 19430: 19428: 19425: 19420: 19416: 19413: 19409: 19408: 19407: 19404: 19402: 19399: 19397: 19394: 19392: 19390: 19386: 19382: 19380: 19377: 19375: 19372: 19370: 19367: 19362: 19358: 19355: 19351: 19350: 19349: 19346: 19345: 19343: 19339: 19333: 19330: 19328: 19325: 19323: 19320: 19319: 19317: 19313: 19310: 19308: 19302: 19292: 19289: 19287: 19286:ADM formalism 19284: 19282: 19279: 19277: 19274: 19272: 19269: 19267: 19264: 19262: 19259: 19257: 19254: 19252: 19249: 19247: 19244: 19243: 19241: 19239: 19235: 19229: 19226: 19224: 19221: 19219: 19216: 19214: 19211: 19210: 19208: 19204: 19201: 19197: 19193: 19186: 19181: 19179: 19174: 19172: 19167: 19166: 19163: 19153: 19143: 19137: 19136: 19132: 19130: 19127: 19125: 19122: 19120: 19117: 19115: 19112: 19110: 19107: 19105: 19102: 19100: 19097: 19095: 19092: 19090: 19087: 19085: 19082: 19080: 19077: 19075: 19072: 19070: 19067: 19065: 19062: 19060: 19057: 19055: 19052: 19050: 19047: 19045: 19044:Chandrasekhar 19042: 19040: 19037: 19035: 19032: 19030: 19027: 19025: 19022: 19020: 19017: 19015: 19012: 19010: 19007: 19005: 19004:Schwarzschild 19002: 19000: 18997: 18995: 18992: 18990: 18987: 18985: 18982: 18981: 18979: 18975: 18965: 18961: 18960: 18957: 18954: 18952: 18949: 18947: 18943: 18942: 18939: 18936: 18934: 18931: 18929: 18926: 18924: 18921: 18918: 18914: 18910: 18909: 18906: 18903: 18900: 18897: 18895: 18891: 18890:Schwarzschild 18887: 18886: 18883: 18880: 18878: 18875: 18873: 18870: 18868: 18865: 18863: 18860: 18857: 18853: 18849: 18848: 18846: 18844: 18840: 18834: 18831: 18829: 18826: 18824: 18821: 18820: 18818: 18812: 18806: 18803: 18800: 18796: 18792: 18789: 18787: 18786:Shapiro delay 18784: 18782: 18779: 18776: 18772: 18768: 18765: 18762: 18758: 18755: 18754: 18751: 18748: 18745: 18742: 18740: 18737: 18735: 18732: 18730: 18729:collaboration 18726: 18722: 18719: 18717: 18713: 18710: 18709: 18706: 18703: 18701: 18698: 18696: 18695:Event horizon 18693: 18691: 18688: 18687: 18685: 18681: 18675: 18672: 18670: 18667: 18665: 18662: 18660: 18657: 18655: 18652: 18650: 18647: 18645: 18642: 18640: 18639:ADM formalism 18637: 18636: 18634: 18630: 18624: 18621: 18619: 18616: 18614: 18611: 18609: 18606: 18604: 18601: 18600: 18598: 18592: 18586: 18583: 18581: 18578: 18577: 18575: 18571: 18568: 18566: 18560: 18550: 18547: 18545: 18544:Biquaternions 18542: 18540: 18537: 18535: 18532: 18530: 18527: 18526: 18524: 18522: 18518: 18512: 18509: 18507: 18504: 18502: 18499: 18497: 18494: 18492: 18489: 18487: 18484: 18482: 18479: 18477: 18474: 18472: 18471:Time dilation 18469: 18468: 18466: 18462: 18456: 18453: 18452: 18450: 18446: 18440: 18437: 18435: 18432: 18430: 18427: 18425: 18424:Proper length 18422: 18420: 18417: 18415: 18412: 18410: 18407: 18405: 18402: 18400: 18397: 18396: 18394: 18388: 18382: 18379: 18377: 18374: 18371: 18368: 18366: 18362: 18359: 18358: 18356: 18352: 18349: 18347: 18341: 18337: 18330: 18325: 18323: 18318: 18316: 18311: 18310: 18307: 18301: 18298: 18287: 18282: 18271: 18266: 18261: 18260:gr-qc/9712019 18256: 18251: 18239: 18235: 18234:Mathpages.com 18231: 18226: 18223: 18219: 18216: 18213: 18209: 18206: 18203: 18199: 18195: 18191: 18186: 18185: 18184: 18179: 18176: 18173: 18172: 18169: 18168: 18162: 18158: 18155: 18153: 18150: 18147: 18144: 18142: 18138: 18134: 18131: 18128: 18127: 18118: 18114: 18110: 18106: 18102: 18098: 18094: 18090: 18085: 18080: 18076: 18072: 18066: 18063: 18059: 18055: 18051: 18047: 18043: 18039: 18035: 18028: 18023: 18018: 18014: 18010: 18005: 18004:gr-qc/0702017 18000: 17996: 17992: 17985: 17980: 17977: 17973: 17969: 17965: 17961: 17957: 17952: 17951:gr-qc/0411071 17947: 17943: 17939: 17934: 17930: 17925: 17921: 17917: 17912: 17911:gr-qc/0501041 17907: 17903: 17899: 17894: 17892: 17887: 17883: 17879: 17875: 17871: 17867: 17863: 17859: 17855: 17854: 17844: 17838: 17834: 17830: 17829:Poisson, Eric 17826: 17822: 17816: 17812: 17811: 17806: 17805:Ellis, George 17802: 17798: 17797: 17787: 17783: 17779: 17777:0-226-87032-4 17773: 17769: 17768: 17763: 17759: 17756: 17750: 17746: 17742: 17738: 17735: 17733:0-7167-0344-0 17729: 17725: 17724: 17719: 17715: 17714:Kip S. Thorne 17711: 17707: 17703: 17697: 17693: 17689: 17685: 17682: 17676: 17672: 17667: 17664: 17658: 17654: 17650: 17646: 17642: 17639: 17633: 17629: 17625: 17621: 17618: 17612: 17608: 17604: 17600: 17599: 17590: 17584: 17580: 17579: 17574: 17573:Schutz, B. F. 17570: 17567: 17561: 17557: 17552: 17549: 17545: 17541: 17540: 17534: 17530: 17524: 17520: 17515: 17512: 17506: 17501: 17500: 17493: 17490: 17484: 17480: 17476: 17472: 17469: 17463: 17459: 17455: 17451: 17448: 17442: 17438: 17433: 17430: 17424: 17420: 17416: 17412: 17409: 17403: 17399: 17394: 17393: 17384: 17378: 17374: 17369: 17366: 17360: 17356: 17351: 17350: 17341: 17335: 17331: 17327: 17326:Wheeler, John 17323: 17320: 17314: 17310: 17306: 17302: 17298: 17296:0-393-03505-0 17292: 17288: 17287: 17282: 17278: 17275: 17269: 17265: 17261: 17257: 17254: 17248: 17244: 17240: 17236: 17233: 17227: 17223: 17219: 17215: 17211: 17205: 17198: 17192: 17188: 17187: 17182: 17178: 17177: 17174:Popular books 17162: 17156: 17152: 17148: 17144: 17141: 17137: 17132: 17127: 17122: 17117: 17113: 17109: 17104: 17103:gr-qc/0510072 17099: 17095: 17091: 17087: 17083: 17080: 17074: 17070: 17066: 17062: 17059: 17053: 17049: 17045: 17041: 17030: 17026: 17022: 17021: 17016: 17011: 17007: 17003: 16999: 16994: 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14741: 14737: 14733: 14729: 14725: 14721: 14720:Wilson, R. W. 14717: 14713: 14710: 14704: 14700: 14699: 14694: 14690: 14687: 14683: 14679: 14675: 14671: 14667: 14663: 14658: 14654: 14650: 14646: 14642: 14638: 14634: 14631: 14627: 14623: 14619: 14615: 14611: 14607: 14603: 14598: 14595: 14589: 14585: 14584: 14578: 14575: 14571: 14567: 14563: 14559: 14555: 14551: 14547: 14544: 14538: 14534: 14529: 14526: 14520: 14516: 14515: 14510: 14509:Pais, Abraham 14506: 14503: 14497: 14493: 14489: 14485: 14481: 14476: 14472: 14468: 14464: 14460: 14456: 14452: 14449: 14445: 14441: 14437: 14433: 14429: 14424: 14419: 14415: 14411: 14406: 14403: 14399: 14395: 14391: 14387: 14383: 14378: 14373: 14369: 14365: 14360: 14357: 14351: 14347: 14342: 14328: 14324: 14320: 14316: 14312: 14308: 14304: 14297: 14292: 14287: 14286:gr-qc/0301024 14282: 14277: 14274: 14270: 14267:: 1238–1245, 14266: 14262: 14258: 14254: 14251: 14245: 14241: 14237: 14233: 14228: 14223: 14218: 14215: 14211: 14207: 14203: 14199: 14195: 14190: 14189:gr-qc/0506078 14185: 14181: 14177: 14172: 14169: 14163: 14159: 14158: 14153: 14149: 14145: 14141: 14130: 14129: 14123: 14120: 14114: 14110: 14105: 14102: 14096: 14091: 14090: 14084: 14080: 14076: 14071: 14067: 14063: 14059: 14055: 14051: 14050:Mather, J. C. 14047: 14044: 14040: 14036: 14032: 14028: 14024: 14019: 14014: 14010: 14006: 14001: 13998: 13992: 13989:, Macmillan, 13988: 13987: 13982: 13978: 13975: 13971: 13967: 13963: 13959: 13955: 13950: 13949:gr-qc/0601102 13945: 13941: 13936: 13933: 13929: 13925: 13921: 13917: 13913: 13908: 13905: 13901: 13896: 13891: 13886: 13881: 13877: 13873: 13868: 13867:gr-qc/9805049 13863: 13859: 13855: 13850: 13847: 13843: 13839: 13835: 13831: 13827: 13822: 13817: 13813: 13809: 13805: 13804:Linde, Andrei 13801: 13798: 13792: 13788: 13784: 13779: 13774: 13770: 13766: 13765:Linde, Andrei 13762: 13758: 13754: 13750: 13744: 13740: 13736: 13732: 13728: 13723: 13722:gr-qc/0202055 13718: 13714: 13709: 13706: 13702: 13698: 13694: 13690: 13686: 13681: 13680:gr-qc/0106072 13676: 13672: 13668: 13663: 13660: 13654: 13650: 13645: 13642: 13638: 13633: 13628: 13623: 13618: 13614: 13610: 13605: 13600: 13596: 13592: 13587: 13583: 13579: 13575: 13570: 13567: 13561: 13557: 13552: 13549: 13543: 13539: 13534: 13531: 13527: 13523: 13519: 13515: 13511: 13507: 13503: 13498: 13493: 13489: 13485: 13479: 13475: 13469: 13465: 13461: 13457: 13453: 13448: 13443: 13439: 13434: 13431: 13427: 13423: 13419: 13415: 13411: 13406: 13395: 13394: 13388: 13385: 13379: 13375: 13370: 13367: 13363: 13359: 13355: 13351: 13347: 13342: 13337: 13333: 13328: 13325: 13319: 13315: 13310: 13307: 13303: 13299: 13295: 13291: 13287: 13282: 13277: 13273: 13269: 13264: 13257: 13250: 13249: 13243: 13232: 13231: 13225: 13222: 13218: 13213: 13208: 13203: 13198: 13194: 13190: 13186: 13182: 13177: 13163: 13159: 13155: 13151: 13147: 13143: 13139: 13136:(S1): 58–85, 13135: 13131: 13127: 13122: 13119: 13113: 13109: 13104: 13101: 13097: 13093: 13089: 13085: 13081: 13077: 13073: 13069: 13065: 13062: 13056: 13052: 13048: 13044: 13041: 13037: 13033: 13029: 13025: 13021: 13016: 13015:gr-qc/0701146 13011: 13007: 13003: 12998: 12995: 12991: 12987: 12983: 12979: 12975: 12970: 12965: 12961: 12957: 12952: 12949: 12945: 12941: 12937: 12933: 12929: 12924: 12923:gr-qc/0606092 12919: 12915: 12911: 12906: 12903: 12899: 12895: 12891: 12887: 12883: 12878: 12873: 12869: 12865: 12860: 12856: 12851: 12847: 12843: 12839: 12835: 12834:Astrophys. J. 12831: 12827: 12823: 12820: 12816: 12811: 12806: 12801: 12796: 12792: 12788: 12784: 12780: 12776: 12775:Hubble, Edwin 12772: 12769: 12765: 12760: 12755: 12750: 12745: 12741: 12737: 12733: 12729: 12724: 12721: 12715: 12711: 12707: 12703: 12698: 12695: 12689: 12685: 12680: 12677: 12673: 12668: 12663: 12658: 12653: 12649: 12645: 12641: 12637: 12632: 12628: 12623: 12620: 12614: 12610: 12609: 12604: 12599: 12596: 12590: 12586: 12581: 12578: 12574: 12570: 12566: 12562: 12558: 12554: 12550: 12546: 12541: 12538: 12534: 12530: 12526: 12522: 12518: 12514: 12510: 12506: 12502: 12498: 12494: 12491: 12487: 12483: 12479: 12475: 12471: 12466: 12462: 12458: 12454: 12450: 12446: 12442: 12438: 12434: 12430: 12426: 12425: 12420: 12416: 12415:Hafele, J. C. 12412: 12408: 12404: 12400: 12396: 12392: 12388: 12384: 12380: 12376: 12372: 12371: 12366: 12362: 12361:Hafele, J. C. 12358: 12353: 12348: 12344: 12340: 12336: 12332: 12328: 12324: 12320: 12317: 12311: 12306: 12301: 12297: 12293: 12289: 12288: 12282: 12279: 12275: 12271: 12267: 12263: 12259: 12254: 12251: 12245: 12241: 12237: 12233: 12229: 12225: 12222: 12218: 12214: 12210: 12206: 12202: 12197: 12194: 12190: 12186: 12182: 12178: 12174: 12169: 12164: 12163:gr-qc/0703035 12159: 12154: 12151: 12147: 12143: 12139: 12135: 12131: 12126: 12123: 12119: 12114: 12109: 12104: 12099: 12095: 12091: 12087: 12083: 12078: 12075: 12071: 12067: 12063: 12058: 12053: 12049: 12045: 12041: 12037: 12032: 12029: 12025: 12021: 12015: 12011: 12007: 12003: 11999: 11994: 11993:gr-qc/0603087 11989: 11985: 11980: 11977: 11973: 11969: 11963: 11959: 11955: 11951: 11947: 11942: 11937: 11933: 11928: 11925: 11919: 11915: 11910: 11906: 11902: 11897: 11896:gr-qc/9602055 11892: 11888: 11884: 11880: 11869: 11865: 11861: 11860: 11855: 11850: 11847: 11841: 11837: 11836:My World Line 11833: 11832:Gamow, George 11829: 11826: 11822: 11817: 11812: 11807: 11802: 11798: 11794: 11790: 11786: 11781: 11778: 11774: 11770: 11766: 11762: 11758: 11753: 11752:gr-qc/0508016 11748: 11744: 11740: 11735: 11732: 11728: 11723: 11718: 11713: 11708: 11704: 11700: 11696: 11692: 11687: 11683: 11678: 11674: 11670: 11666: 11662: 11657: 11654: 11650: 11645: 11640: 11635: 11630: 11626: 11622: 11618: 11614: 11609: 11606: 11602: 11598: 11594: 11590: 11586: 11582: 11578: 11573: 11568: 11564: 11560: 11555: 11541: 11534: 11533: 11527: 11524: 11518: 11514: 11509: 11506: 11502: 11498: 11494: 11490: 11486: 11481: 11478: 11474: 11470: 11464: 11460: 11456: 11452: 11448: 11443: 11442:gr-qc/9812046 11438: 11434: 11430: 11426: 11424: 11418: 11414: 11410: 11406: 11404: 11398: 11394: 11390: 11386: 11385: 11382: 11376: 11372: 11368: 11364: 11360: 11357: 11353: 11348: 11343: 11339: 11335: 11331: 11327: 11323: 11319: 11315: 11310: 11307: 11301: 11297: 11292: 11289: 11283: 11279: 11274: 11271: 11265: 11261: 11257: 11253: 11250: 11246: 11242: 11238: 11234: 11230: 11225: 11220: 11216: 11212: 11208: 11207:Duff, Michael 11204: 11200: 11194: 11190: 11189: 11183: 11180: 11174: 11170: 11166: 11161: 11160:gr-qc/9512024 11156: 11152: 11147: 11140: 11136: 11132: 11127: 11122: 11118: 11114: 11110: 11106: 11103:(10): S1–S9, 11102: 11098: 11091: 11086: 11083: 11079: 11075: 11071: 11067: 11063: 11058: 11053: 11050:(6): 063006, 11049: 11045: 11040: 11037: 11031: 11027: 11023: 11018: 11017:gr-qc/0204090 11013: 11009: 11004: 11001: 10997: 10993: 10989: 10985: 10981: 10976: 10971: 10967: 10963: 10957: 10954: 10950: 10946: 10942: 10938: 10934: 10929: 10926: 10922: 10918: 10914: 10910: 10906: 10902: 10898: 10894: 10890: 10885: 10882: 10878: 10874: 10870: 10866: 10862: 10857: 10852: 10848: 10844: 10840: 10835: 10832: 10828: 10824: 10820: 10816: 10812: 10808: 10804: 10799: 10796: 10790: 10786: 10782: 10778: 10775: 10771: 10767: 10763: 10759: 10755: 10750: 10745: 10741: 10737: 10733: 10728: 10725: 10719: 10715: 10711: 10707: 10704: 10700: 10695: 10690: 10685: 10680: 10676: 10672: 10667: 10662: 10658: 10654: 10650: 10646: 10643: 10637: 10633: 10628: 10625: 10621: 10617: 10613: 10609: 10605: 10600: 10599:gr-qc/0108040 10595: 10591: 10587: 10582: 10579: 10575: 10571: 10567: 10566:Physics World 10562: 10559: 10555: 10551: 10547: 10543: 10539: 10535: 10531: 10526: 10521: 10517: 10513: 10508: 10505: 10501: 10497: 10493: 10489: 10485: 10480: 10475: 10471: 10467: 10462: 10459: 10453: 10449: 10444: 10441: 10437: 10433: 10429: 10425: 10421: 10417: 10413: 10408: 10403: 10399: 10395: 10391: 10387: 10382: 10379: 10375: 10371: 10367: 10363: 10359: 10355: 10351: 10347: 10344: 10340: 10336: 10330: 10326: 10322: 10318: 10314: 10309: 10304: 10300: 10295: 10292: 10286: 10282: 10277: 10274: 10268: 10264: 10260: 10256: 10253: 10249: 10244: 10239: 10234: 10229: 10225: 10221: 10217: 10213: 10208: 10205: 10201: 10197: 10193: 10189: 10185: 10180: 10175: 10171: 10167: 10161: 10158: 10152: 10148: 10147: 10142: 10138: 10135: 10129: 10125: 10121: 10120:Davies, P. C. 10116: 10113: 10109: 10105: 10101: 10097: 10093: 10088: 10085: 10081: 10077: 10073: 10069: 10065: 10061: 10057: 10053: 10049: 10044: 10041: 10037: 10033: 10029: 10025: 10021: 10017: 10013: 10009: 10005: 10002: 9996: 9991: 9990: 9983: 9980: 9976: 9971: 9966: 9961: 9956: 9952: 9948: 9943: 9942:gr-qc/0201056 9938: 9934: 9930: 9926: 9922: 9919: 9915: 9911: 9907: 9903: 9899: 9894: 9889: 9885: 9881: 9875: 9872: 9868: 9864: 9860: 9856: 9852: 9845: 9841: 9837: 9833: 9829: 9825: 9821: 9817: 9812: 9809: 9805: 9801: 9797: 9793: 9789: 9785: 9781: 9776: 9773: 9769: 9765: 9761: 9757: 9753: 9749: 9745: 9741: 9737: 9734: 9728: 9724: 9720: 9715: 9714:gr-qc/0502041 9710: 9706: 9701: 9698: 9694: 9690: 9686: 9682: 9678: 9674: 9670: 9665: 9662: 9656: 9652: 9647: 9644: 9640: 9635: 9630: 9626: 9622: 9617: 9612: 9608: 9604: 9599: 9596: 9590: 9586: 9582: 9578: 9573: 9570: 9566: 9562: 9558: 9554: 9550: 9546: 9542: 9538: 9534: 9530: 9526: 9522: 9519: 9515: 9511: 9507: 9503: 9499: 9494: 9493:gr-qc/0310125 9489: 9486:(8): 082005, 9485: 9481: 9476: 9473: 9469: 9465: 9461: 9457: 9453: 9449: 9445: 9441: 9437: 9432: 9429: 9423: 9419: 9414: 9411: 9407: 9403: 9399: 9395: 9391: 9386: 9383: 9379: 9375: 9371: 9367: 9363: 9358: 9357:gr-qc/0404018 9353: 9349: 9345: 9340: 9337: 9333: 9328: 9323: 9318: 9313: 9309: 9305: 9300: 9299:gr-qc/0407042 9295: 9291: 9287: 9282: 9279: 9275: 9271: 9265: 9261: 9257: 9253: 9249: 9244: 9239: 9235: 9230: 9227: 9223: 9219: 9215: 9211: 9207: 9203: 9199: 9194: 9191: 9187: 9183: 9179: 9175: 9171: 9167: 9163: 9159: 9155: 9152: 9148: 9143: 9138: 9133: 9128: 9124: 9120: 9116: 9112: 9111: 9105: 9102: 9098: 9094: 9090: 9086: 9082: 9081:Physics Today 9075: 9070: 9067: 9063: 9059: 9055: 9051: 9047: 9042: 9037: 9034:(6): 064035, 9033: 9029: 9024: 9020: 9016: 9012: 9008: 9004: 9001: 8995: 8991: 8987: 8986:Arnold, V. I. 8983: 8980: 8974: 8970: 8965: 8962: 8958: 8954: 8950: 8946: 8942: 8938: 8934: 8930: 8929:Alpher, R. A. 8926: 8925: 8913: 8912: 8905: 8899: 8895: 8891: 8890:Blanchet 2006 8887: 8883: 8877: 8871: 8867: 8863: 8857: 8851: 8845: 8839: 8834: 8827: 8821: 8814: 8808: 8801: 8797: 8792: 8786: 8782: 8781:Ashtekar 2007 8777: 8771: 8766: 8759: 8754: 8748: 8743: 8737: 8732: 8726: 8722: 8717: 8711: 8710:Thiemann 2003 8707: 8703: 8699: 8698:Thiemann 2007 8693: 8687: 8686:Ashtekar 1987 8683: 8682:Ashtekar 1986 8679: 8675: 8669: 8662: 8657: 8651: 8647: 8646:Townsend 1996 8642: 8635: 8634:Weinberg 2000 8630: 8623: 8618: 8611: 8607: 8603: 8599: 8595: 8591: 8587: 8581: 8575: 8571: 8567: 8566:Zwiebach 2004 8561: 8555: 8551: 8547: 8543: 8542:Weinberg 1996 8539: 8533: 8527: 8526:Donoghue 1995 8522: 8516: 8511: 8505: 8499: 8493: 8488: 8486: 8479:, p. 407 8478: 8473: 8466: 8460: 8453: 8448: 8442: 8441:Traschen 2000 8438: 8434: 8428: 8422: 8418: 8413: 8407: 8403: 8399: 8398:Weinberg 1995 8395: 8390: 8384: 8380: 8374: 8368: 8367:Szabados 2004 8364: 8360: 8359:Geroch energy 8356: 8350: 8343: 8342:Townsend 1997 8338: 8331: 8330:negative mass 8327: 8322: 8315: 8309: 8303: 8299: 8295: 8290: 8284: 8279: 8273:, § 20.4 8272: 8267: 8261: 8257: 8252: 8246: 8242: 8238: 8234: 8229: 8222: 8218: 8213: 8206: 8201: 8194: 8190: 8186: 8179: 8173: 8169: 8165: 8159: 8153: 8148: 8142: 8138: 8132: 8125: 8121: 8115: 8108: 8104: 8099: 8092: 8088: 8087:Townsend 1997 8083: 8076: 8072: 8071:Townsend 1997 8067: 8060: 8056: 8051: 8044: 8040: 8036: 8030: 8023: 8022:Narlikar 1993 8018: 8011: 8007: 8003: 7997: 7991: 7987: 7982: 7976: 7972: 7968: 7964: 7960: 7954: 7947: 7943: 7939: 7935: 7931: 7925: 7919: 7915: 7911: 7907: 7903: 7898: 7891: 7887: 7882: 7875: 7871: 7866: 7859: 7856:, sec. 11.1, 7855: 7851: 7846: 7839: 7834: 7829: 7822: 7814: 7810: 7806: 7802: 7798: 7794: 7787: 7779: 7775: 7771: 7767: 7763: 7759: 7755: 7751: 7744: 7736: 7732: 7728: 7724: 7720: 7716: 7711: 7706: 7702: 7698: 7691: 7685: 7680: 7673: 7668: 7661: 7657: 7651: 7644: 7639: 7632: 7628: 7627:Narlikar 1993 7624: 7620: 7616: 7610: 7604: 7600: 7594: 7588: 7584: 7583:Mannheim 2006 7580: 7579:Caldwell 2004 7576: 7571: 7564: 7563:Mannheim 2006 7557: 7550: 7546: 7542: 7538: 7534: 7529: 7523: 7519: 7515: 7509: 7503: 7499: 7495: 7491: 7487: 7483: 7479: 7475: 7471: 7466: 7460: 7456: 7452: 7447: 7441: 7437: 7433: 7429: 7425: 7421: 7417: 7412: 7406: 7400: 7394: 7390: 7384: 7378: 7374: 7370: 7365: 7358: 7353: 7346: 7342: 7341:Einstein 1917 7337: 7331: 7326: 7320: 7315: 7309: 7305: 7301: 7297: 7293: 7289: 7285: 7281: 7275: 7269: 7265: 7264:X-ray pulsars 7261: 7257: 7253: 7249: 7243: 7237: 7233: 7229: 7223: 7217: 7213: 7207: 7200: 7195: 7189: 7185: 7180: 7173: 7168: 7161: 7156: 7150: 7145: 7139: 7134: 7118: 7112: 7106: 7101: 7094: 7090: 7086: 7082: 7078: 7074: 7069: 7064: 7061:(8): 084013, 7060: 7056: 7048: 7042: 7037: 7031: 7027: 7023: 7018: 7011: 7006: 7000: 6995: 6989: 6983: 6977: 6972: 6965: 6961: 6955: 6949: 6945: 6939: 6933: 6929: 6924: 6918: 6914: 6910: 6905: 6899: 6895: 6894:Weinberg 1972 6891: 6886: 6879: 6875: 6874:Townsend 1997 6870: 6863: 6859: 6855: 6849: 6843: 6838: 6832: 6828: 6823: 6816: 6812: 6811:Weinberg 1972 6807: 6801:, § 11.4 6800: 6796: 6791: 6785: 6780: 6774: 6770: 6766: 6761: 6754: 6750: 6746: 6741: 6733: 6729: 6725: 6721: 6717: 6713: 6709: 6705: 6700: 6695: 6692:(4): 041050. 6691: 6687: 6683: 6676: 6674: 6666: 6662: 6657: 6651: 6646: 6639: 6635: 6631: 6627: 6623: 6617: 6610: 6606: 6600: 6593: 6588: 6581: 6576: 6569: 6565: 6560: 6553: 6549: 6543: 6537: 6533: 6528: 6521: 6516: 6510: 6504: 6497: 6491: 6483: 6479: 6473: 6471: 6469: 6460: 6456: 6452: 6448: 6444: 6440: 6436: 6432: 6427: 6422: 6419:(6): 061102. 6418: 6414: 6407: 6405: 6389: 6385: 6381: 6377: 6373: 6369: 6362: 6360: 6343: 6339: 6335: 6331: 6327: 6320: 6304: 6300: 6296: 6292: 6291: 6286: 6279: 6272: 6267: 6260: 6256: 6252: 6248: 6244: 6240: 6239:Weinberg 1972 6236: 6232: 6226: 6219: 6213: 6207: 6203: 6199: 6198:Einstein 1907 6195: 6191: 6186: 6179: 6178:Blanchet 2006 6174: 6167: 6163: 6159: 6153: 6147: 6143: 6139: 6133: 6126: 6121: 6114: 6108: 6102: 6098: 6093: 6087: 6083: 6077: 6070: 6066: 6062: 6058: 6054: 6048: 6041: 6037: 6032: 6025: 6021: 6017: 6013: 6009: 6004: 5998:, § 38.5 5997: 5993: 5988: 5981: 5977: 5976:Einstein 1907 5973: 5969: 5964: 5957: 5953: 5948: 5941: 5936: 5929: 5926:For instance 5923: 5917: 5912: 5905: 5899: 5892: 5891:Narlikar 1993 5887: 5880: 5875: 5869: 5865: 5859: 5853: 5847: 5841: 5835: 5829: 5828:Weinberg 1972 5823: 5817: 5811: 5804: 5799: 5792: 5787: 5780: 5775: 5769: 5768:Poisson 2004a 5763: 5756: 5755:Weinberg 1972 5752: 5747: 5740: 5739:Trautman 2006 5736: 5732: 5731:Weinberg 1972 5728: 5723: 5715: 5709: 5705: 5698: 5690: 5684: 5680: 5673: 5666: 5661: 5654: 5650: 5646: 5645:Lovelock 1972 5642: 5641:Weinberg 1972 5638: 5633: 5626: 5625:Weinberg 1972 5622: 5618: 5613: 5607: 5603: 5599: 5595: 5590: 5584: 5580: 5576: 5571: 5564: 5560: 5556: 5551: 5544: 5540: 5535: 5528: 5523: 5517:, ch. 1 and 2 5516: 5512: 5507: 5501: 5495: 5489: 5485: 5481: 5477: 5471: 5464: 5459: 5453: 5449: 5445: 5441: 5436: 5429: 5424: 5417: 5413: 5408: 5402:, pp. 5f 5401: 5396: 5389: 5384: 5369: 5368:www.bbc.co.uk 5365: 5358: 5351: 5345: 5339: 5334: 5328: 5324: 5323: 5315: 5309: 5304: 5297: 5292: 5285: 5280: 5273: 5268: 5262: 5258: 5252: 5245: 5241: 5237: 5231: 5224: 5220: 5215: 5209: 5205: 5200: 5193: 5188: 5182: 5178: 5173: 5166: 5161: 5154: 5150: 5144: 5137: 5133: 5127: 5120: 5116: 5115:Einstein 1917 5111: 5104: 5100: 5099:Reissner 1916 5096: 5092: 5087: 5081: 5074: 5065: 5058: 5054: 5053:Einstein 1915 5050: 5046: 5045:Einstein 1907 5042: 5038: 5034: 5030: 5026: 5021: 5006: 5002: 4998: 4997: 4990: 4983: 4977: 4976:Poincaré 1905 4972: 4965: 4961: 4956: 4954: 4938: 4934: 4928: 4924: 4915: 4912: 4909: 4906: 4904: 4901: 4898: 4895: 4893: 4890: 4887: 4884: 4881: 4878: 4875: 4872: 4869: 4866: 4864: 4861: 4859: 4856: 4853: 4850: 4847: 4844: 4843: 4837: 4833: 4824: 4816: 4812: 4810: 4806: 4802: 4798: 4795:approach and 4794: 4793:Path Integral 4788: 4786: 4782: 4778: 4774: 4770: 4766: 4762: 4757: 4755: 4751: 4747: 4746:supersymmetry 4743: 4739: 4735: 4731: 4727: 4726:string theory 4719: 4714: 4710: 4708: 4704: 4698: 4691: 4690:compactifying 4687: 4682: 4677: 4673: 4669: 4665: 4661: 4660:String theory 4655: 4645: 4643: 4639: 4635: 4631: 4625: 4615: 4613: 4603: 4601: 4596: 4592: 4588: 4584: 4580: 4575: 4570: 4560: 4558: 4554: 4553:ADM formalism 4548: 4546: 4542: 4538: 4531: 4521: 4518: 4514: 4510: 4505: 4503: 4499: 4495: 4488: 4481:Singularities 4478: 4476: 4472: 4471:Rindler space 4468: 4463: 4461: 4457: 4453: 4449: 4445: 4441: 4437: 4433: 4428: 4426: 4422: 4418: 4413: 4409: 4408:Kerr solution 4405: 4396: 4392: 4390: 4386: 4382: 4378: 4372: 4368: 4364: 4354: 4352: 4348: 4344: 4340: 4339:Roger Penrose 4335: 4333: 4329: 4325: 4321: 4317: 4313: 4309: 4305: 4298: 4293: 4288: 4278: 4276: 4272: 4268: 4264: 4259: 4255: 4251: 4247: 4243: 4242:Hermann Bondi 4238: 4236: 4232: 4228: 4222: 4207: 4205: 4201: 4197: 4192: 4190: 4186: 4182: 4178: 4174: 4170: 4166: 4157: 4155: 4151: 4147: 4142: 4140: 4136: 4131: 4125: 4123: 4119: 4115: 4111: 4107: 4103: 4099: 4084: 4081: 4077: 4050: 4047: 4043: 4034: 4030: 4025: 4022: 4019: 4013: 4008: 4005: 4001: 3991: 3986: 3983: 3979: 3974: 3969: 3965: 3959: 3954: 3951: 3947: 3939: 3938: 3937: 3915: 3906: 3901: 3891: 3889: 3885: 3881: 3876: 3874: 3870: 3865: 3863: 3859: 3855: 3851: 3841: 3836: 3826: 3824: 3823:cosmic string 3820: 3816: 3811: 3809: 3805: 3801: 3797: 3793: 3789: 3785: 3781: 3777: 3774: 3770: 3766: 3765:Orbital decay 3759: 3754: 3749: 3745: 3735: 3733: 3729: 3725: 3721: 3716: 3714: 3709: 3705: 3704:Einstein ring 3696: 3692: 3687: 3672: 3670: 3666: 3662: 3658: 3654: 3649: 3647: 3643: 3639: 3634: 3630: 3624: 3620: 3610: 3608: 3604: 3600: 3596: 3592: 3587: 3585: 3581: 3580:neutron stars 3577: 3573: 3569: 3568:binary system 3561: 3556: 3550:Orbital decay 3545: 3529: 3522: 3507: 3500: 3498: 3482: 3475: 3473: 3457: 3450: 3449: 3448: 3431: 3417: 3413: 3409: 3406: 3398: 3394: 3388: 3384: 3376: 3372: 3366: 3362: 3358: 3352: 3349: 3342: 3341: 3340: 3326: 3317: 3315: 3310: 3306: 3302: 3297: 3295: 3291: 3290:test particle 3287: 3283: 3279: 3275: 3271: 3265: 3256: 3247: 3245: 3239: 3229: 3227: 3222: 3220: 3202: 3199: 3195: 3186: 3182: 3176: 3174: 3170: 3166: 3162: 3152: 3147: 3137: 3135: 3129: 3126: 3122: 3121:null geodesic 3117: 3115: 3105: 3100: 3099:Shapiro delay 3096: 3092: 3088: 3078: 3076: 3072: 3068: 3067:atomic clocks 3063: 3061: 3057: 3053: 3044: 3039: 3029: 3020: 3018: 3014: 3010: 3006: 3002: 2998: 2993: 2991: 2987: 2983: 2979: 2975: 2971: 2967: 2963: 2959: 2955: 2951: 2947: 2943: 2938: 2936: 2932: 2922: 2920: 2916: 2912: 2908: 2904: 2900: 2896: 2892: 2887: 2885: 2881: 2877: 2873: 2869: 2865: 2860: 2858: 2854: 2851: 2847: 2843: 2839: 2835: 2831: 2821: 2816: 2812: 2802: 2800: 2796: 2794: 2790: 2785: 2781: 2777: 2773: 2767: 2757: 2755: 2751: 2747: 2743: 2719: 2715: 2709: 2705: 2701: 2694: 2690: 2686: 2683: 2680: 2674: 2666: 2662: 2658: 2652: 2648: 2642: 2635: 2631: 2626: 2623: 2620: 2614: 2611: 2605: 2597: 2593: 2585: 2584: 2583: 2582: 2578: 2554: 2550: 2544: 2540: 2536: 2529: 2525: 2521: 2518: 2512: 2504: 2500: 2496: 2493: 2487: 2483: 2477: 2472: 2468: 2465: 2462: 2456: 2453: 2447: 2439: 2435: 2427: 2426: 2425: 2423: 2419: 2415: 2409: 2399: 2397: 2396:test particle 2393: 2389: 2373: 2353: 2345: 2341: 2337: 2333: 2315: 2312: 2303: 2290: 2274: 2251: 2248: 2245: 2239: 2236: 2229: 2225: 2221: 2212: 2209: 2202: 2198: 2194: 2186: 2183: 2174: 2166: 2158: 2154: 2150: 2143: 2139: 2133: 2129: 2118: 2117: 2116: 2114: 2109: 2107: 2088: 2085: 2080: 2077: 2073: 2065: 2064: 2063: 2049: 2041: 2025: 2001: 1997: 1992: 1989: 1986: 1980: 1977: 1957: 1949: 1946: 1942: 1924: 1921: 1917: 1896: 1873: 1868: 1865: 1862: 1855: 1851: 1845: 1840: 1837: 1833: 1825: 1824: 1823: 1821: 1800: 1797: 1793: 1787: 1784: 1780: 1776: 1773: 1766: 1765: 1764: 1748: 1745: 1741: 1718: 1715: 1711: 1702: 1696: 1680: 1677: 1673: 1669: 1666: 1661: 1658: 1654: 1649: 1644: 1640: 1634: 1629: 1626: 1622: 1618: 1613: 1610: 1606: 1597: 1593: 1590: 1586: 1582: 1578: 1574: 1570: 1566: 1562: 1558: 1555:and momentum 1554: 1550: 1543: 1539: 1529: 1527: 1523: 1519: 1515: 1514:metric tensor 1511: 1507: 1502: 1500: 1496: 1492: 1486: 1484: 1480: 1474: 1472: 1468: 1449: 1444: 1442: 1438: 1434: 1430: 1426: 1422: 1421:limiting case 1415: 1411: 1402: 1400: 1399:tidal effects 1396: 1392: 1388: 1383: 1379: 1375: 1371: 1367: 1361: 1359: 1355: 1351: 1347: 1343: 1339: 1335: 1331: 1326: 1324: 1320: 1316: 1312: 1308: 1304: 1300: 1296: 1295:law of motion 1292: 1288: 1284: 1274: 1265: 1263: 1260:results in a 1252: 1250: 1249: 1243: 1240: 1236: 1232: 1231:Francis Bacon 1228: 1223: 1220: 1216: 1212: 1208: 1204: 1200: 1199:fudge factors 1196: 1192: 1188: 1183: 1181: 1177: 1173: 1169: 1165: 1161: 1157: 1153: 1149: 1145: 1141: 1137: 1132: 1130: 1126: 1122: 1117: 1112: 1108: 1104: 1100: 1095: 1091: 1081: 1079: 1074: 1072: 1069:models of an 1068: 1064: 1060: 1056: 1052: 1048: 1044: 1040: 1036: 1032: 1031:massive stars 1028: 1024: 1020: 1019:astrophysical 1015: 1013: 1009: 1005: 1001: 997: 993: 988: 986: 982: 978: 974: 970: 966: 962: 958: 957:singularities 954: 950: 946: 942: 938: 934: 930: 926: 922: 918: 916: 912: 908: 904: 900: 896: 892: 890: 885: 882: 878: 874: 870: 866: 862: 858: 854: 851:published by 850: 846: 843: 839: 835: 831: 820: 815: 813: 808: 806: 801: 800: 798: 797: 791: 781: 778: 773: 767: 766: 765: 764: 757: 756: 752: 750: 747: 745: 742: 740: 737: 735: 732: 730: 727: 725: 722: 720: 717: 715: 712: 710: 707: 705: 702: 700: 697: 695: 694:Chandrasekhar 692: 690: 687: 685: 682: 680: 677: 675: 672: 670: 667: 665: 662: 660: 657: 655: 652: 650: 647: 645: 642: 640: 637: 635: 632: 630: 627: 625: 622: 620: 617: 615: 612: 610: 609:Schwarzschild 607: 605: 602: 600: 597: 595: 592: 590: 587: 586: 578: 577: 570: 569:Hartle–Thorne 567: 565: 562: 560: 557: 555: 552: 550: 547: 545: 542: 540: 537: 535: 532: 530: 527: 525: 522: 520: 517: 515: 512: 510: 507: 505: 502: 500: 497: 495: 492: 490: 487: 484: 480: 479:Schwarzschild 477: 476: 472: 466: 465: 454: 451: 449: 446: 445: 444: 443: 439: 438: 433: 430: 428: 425: 423: 420: 419: 418: 417: 413: 412: 407: 404: 402: 399: 397: 394: 392: 389: 387: 384: 382: 379: 378: 377: 376: 372: 371: 361: 358: 357: 352: 351: 340: 337: 335: 332: 330: 327: 326: 325: 324: 321: 318: 317: 312: 309: 307: 304: 302: 301:Event horizon 299: 297: 294: 292: 289: 287: 284: 282: 279: 277: 274: 272: 269: 267: 264: 263: 262: 261: 251: 250: 243: 240: 238: 235: 233: 230: 228: 225: 224: 216: 215: 210: 207: 202: 199: 197: 194: 192: 189: 188: 186: 184: 181: 180: 179: 178: 160: 157: 153: 148: 144: 139: 136: 132: 125: 120: 117: 113: 103: 99: 98: 95: 92: 91: 85: 81: 77: 73: 69: 65: 61: 60:binary system 44: 40: 33: 19: 21585: 21505:Hermann Weyl 21309:Vector space 21294:Pseudotensor 21259:Fiber bundle 21212:abstractions 21107:Mixed tensor 21092:Tensor field 21003: 20899: 20768:Astrophysics 20705: 20582:Experimental 20494: 20462:Einsteinhaus 20342:Pauline Koch 20320:(first wife) 20318:Mileva Marić 20297: 20290: 20283: 20276: 20269: 20262: 20255: 20248: 20181: 20173: 20165: 20157: 20149: 20141: 20133: 20125: 20117: 20109: 20101: 20065: 20057: 20049: 20041: 20026: 19885: 19742:theories and 19575:Supergravity 19388: 19384: 19246:Introduction 19237: 19134: 18828:Kaluza–Klein 18580:Introduction 18562: 18506:Twin paradox 18289:. Retrieved 18273:. Retrieved 18268:Moor, Rafi. 18242:. Retrieved 18238:the original 18233: 18196:(lecture by 18170: 18167: 18166: 18074: 18070: 18045: 18041: 17997:(6): 30–34, 17994: 17990: 17941: 17937: 17901: 17898:New J. Phys. 17897: 17869: 17865: 17832: 17809: 17766: 17744: 17722: 17691: 17670: 17652: 17627: 17624:Grøn, Øyvind 17606: 17577: 17555: 17538: 17518: 17498: 17478: 17457: 17439:, Springer, 17436: 17418: 17397: 17372: 17354: 17329: 17308: 17285: 17263: 17242: 17221: 17185: 17181:Einstein, A. 17150: 17093: 17089: 17068: 17047: 17033:, retrieved 17029:the original 17018: 17005: 16977: 16955: 16934: 16914: 16856: 16852: 16812: 16808: 16754: 16750: 16724: 16705: 16672: 16668: 16652: 16628: 16625:Phys. Rev. D 16624: 16621:Unruh, W. G. 16595: 16569: 16539: 16535: 16500: 16463: 16445: 16436: 16417: 16413: 16357: 16305: 16273: 16269: 16218: 16214: 16196: 16193:Synge, J. L. 16175: 16121: 16117: 16067: 16063: 16020: 16016: 15970: 15966: 15923: 15919: 15883: 15853: 15849:Singh, Simon 15825: 15821: 15786: 15782: 15740: 15736: 15700: 15684: 15665: 15620: 15616: 15595: 15577: 15535: 15531: 15489: 15485: 15421: 15417: 15381: 15339: 15335: 15317: 15297: 15277: 15230: 15226: 15208: 15190: 15148: 15144: 15120: 15116: 15080: 15076: 15058: 15029: 15025: 14999: 14995: 14969: 14965: 14947: 14929: 14900: 14845: 14841: 14828: 14824: 14782: 14778: 14758: 14727: 14723: 14697: 14677: 14673: 14647:(3): 57–59, 14644: 14640: 14605: 14601: 14582: 14557: 14553: 14532: 14513: 14494:, Back Bay, 14491: 14462: 14458: 14413: 14409: 14370:(1): 29–49, 14367: 14363: 14345: 14334:, retrieved 14306: 14302: 14264: 14260: 14239: 14179: 14175: 14156: 14133:, retrieved 14127: 14108: 14088: 14057: 14053: 14008: 14004: 13985: 13981:Maddox, John 13939: 13915: 13911: 13857: 13853: 13811: 13807: 13768: 13712: 13670: 13666: 13648: 13594: 13590: 13581: 13577: 13555: 13537: 13487: 13483: 13437: 13413: 13409: 13398:, retrieved 13392: 13373: 13331: 13313: 13271: 13268:Phys. Rev. D 13267: 13247: 13235:, retrieved 13229: 13184: 13180: 13169:, retrieved 13162:the original 13133: 13129: 13107: 13078:(1): 53–59, 13075: 13071: 13053:, Springer, 13050: 13005: 13001: 12959: 12955: 12913: 12909: 12867: 12863: 12837: 12833: 12782: 12778: 12731: 12727: 12701: 12683: 12639: 12635: 12626: 12607: 12584: 12552: 12548: 12504: 12500: 12473: 12469: 12428: 12422: 12374: 12368: 12334: 12330: 12286: 12261: 12257: 12239: 12228:Green, M. B. 12204: 12200: 12176: 12172: 12133: 12129: 12085: 12081: 12039: 12035: 11983: 11931: 11913: 11886: 11872:, retrieved 11868:the original 11857: 11835: 11788: 11784: 11742: 11738: 11694: 11690: 11664: 11660: 11616: 11612: 11562: 11558: 11547:, retrieved 11531: 11512: 11491:(1): 27–37, 11488: 11484: 11432: 11416: 11396: 11374: 11370: 11321: 11317: 11298:, Springer, 11295: 11280:, Springer, 11277: 11259: 11214: 11210: 11191:. Springer. 11187: 11150: 11139:the original 11100: 11096: 11047: 11044:Phys. Rev. D 11043: 11007: 10965: 10961: 10936: 10932: 10892: 10888: 10846: 10842: 10806: 10802: 10784: 10739: 10735: 10713: 10656: 10652: 10631: 10589: 10585: 10572:(5): 37–42, 10569: 10565: 10515: 10511: 10469: 10465: 10447: 10397: 10393: 10361: 10357: 10354:Dicke, R. H. 10350:Brans, C. H. 10298: 10283:, Springer, 10280: 10262: 10215: 10211: 10169: 10165: 10145: 10141:Blair, David 10123: 10095: 10091: 10051: 10047: 10015: 10011: 9988: 9932: 9928: 9883: 9879: 9854: 9850: 9827: 9823: 9783: 9780:Phys. Rev. D 9779: 9747: 9744:Phys. Rev. D 9743: 9704: 9680: 9676: 9650: 9606: 9602: 9576: 9544: 9540: 9483: 9480:Phys. Rev. D 9479: 9439: 9435: 9417: 9393: 9389: 9347: 9343: 9289: 9285: 9233: 9201: 9198:Phys. Rev. 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Retrieved 7111: 7100: 7058: 7054: 7047: 7036: 7017: 7005: 6994: 6982: 6971: 6954: 6938: 6923: 6904: 6898:Schäfer 2004 6892:, sec. 4.7, 6885: 6869: 6848: 6837: 6822: 6813:, sec. 9.6, 6806: 6795:Penrose 2004 6790: 6779: 6760: 6740: 6689: 6685: 6656: 6645: 6632:, sec. 3.5, 6616: 6599: 6587: 6580:Rindler 2001 6575: 6559: 6542: 6527: 6520:Rindler 2001 6515: 6507:For example 6503: 6490: 6481: 6416: 6412: 6391:. Retrieved 6371: 6346:. Retrieved 6342:the original 6329: 6319: 6307:. Retrieved 6303:the original 6288: 6278: 6266: 6235:Shapiro 1964 6225: 6220:, sec. 11.11 6218:Rindler 2001 6212: 6190:Rindler 2001 6185: 6173: 6152: 6132: 6120: 6107: 6092: 6076: 6047: 6031: 6003: 5992:Rindler 2001 5987: 5968:Rindler 2001 5963: 5954:, sec. 3.2, 5947: 5935: 5922: 5911: 5898: 5886: 5874: 5858: 5846: 5834: 5822: 5816:Giulini 2007 5810: 5798: 5786: 5781:, p. xi 5779:Wheeler 1990 5774: 5762: 5746: 5735:Goenner 2004 5722: 5703: 5697: 5678: 5672: 5660: 5632: 5623:, sec. 7.2, 5612: 5589: 5579:Wheeler 1990 5575:Rindler 2001 5570: 5550: 5541:, sec. 1.4, 5534: 5522: 5511:Rindler 1991 5506: 5500:Giulini 2006 5494: 5484:Rindler 1991 5476:Giulini 2005 5470: 5458: 5442:, sec. 1.2, 5435: 5428:Wheeler 1990 5423: 5414:, sec. 2.4, 5407: 5395: 5383: 5371:. Retrieved 5367: 5357: 5344: 5321: 5314: 5303: 5291: 5284:Rovelli 2015 5279: 5267: 5261:Overbye 1999 5251: 5230: 5214: 5199: 5187: 5172: 5160: 5143: 5126: 5110: 5086: 5073: 5064: 5029:Janssen 2005 5020: 5009:, retrieved 5005:the original 4995: 4982: 4971: 4963: 4940:. Retrieved 4936: 4927: 4834: 4830: 4821: 4789: 4785:spin network 4758: 4750:supergravity 4723: 4718:spin network 4699: 4695: 4627: 4609: 4594: 4576: 4572: 4549: 4545:gauge fixing 4533: 4506: 4498:Ricci scalar 4490: 4464: 4452:Planck's law 4429: 4401: 4388: 4374: 4336: 4328:compactified 4315: 4311: 4307: 4303: 4301: 4266: 4262: 4257: 4239: 4234: 4224: 4193: 4163: 4143: 4126: 4069: 4067: 3911: 3877: 3866: 3854:solar masses 3846: 3812: 3762: 3717: 3713:microlensing 3700: 3650: 3626: 3588: 3576:double stars 3572:Solar System 3565: 3446: 3318: 3309:nonlinearity 3298: 3267: 3241: 3223: 3177: 3157: 3130: 3118: 3110: 3064: 3052:gravity well 3049: 3026: 2994: 2939: 2928: 2888: 2861: 2853:trajectories 2827: 2818: 2792: 2788: 2769: 2741: 2739: 2574: 2424:is given by 2421: 2417: 2411: 2266: 2110: 2103: 1888: 1817: 1698: 1598: 1595: 1573:Ricci tensor 1545: 1503: 1487: 1475: 1445: 1418: 1381: 1362: 1357: 1327: 1307:acceleration 1280: 1258: 1246: 1244: 1238: 1234: 1224: 1184: 1133: 1097: 1075: 1067:cosmological 1035:Microquasars 1016: 989: 919: 901:of whatever 891:of spacetime 887: 867:and refines 863:generalizes 837: 833: 829: 828: 754: 714:Raychaudhuri 183:Introduction 93: 43: 21568:Mathematics 21556:Outer space 21445:Élie Cartan 21393:Spin tensor 21367:Weyl tensor 21325:Mathematics 21289:Multivector 21080:definitions 20978:Engineering 20917:Mathematics 20671:Statistical 20587:Theoretical 20564:Engineering 20467:Einsteinium 20240:Books about 20186:(2023 film) 20183:Oppenheimer 20162:(2003 play) 20154:(1994 film) 20146:(1993 play) 20138:(1988 film) 20130:(1985 film) 19973:EPR paradox 19544:Causal sets 19438:Brans–Dicke 19256:Mathematics 18917:Kerr–Newman 18888:Spherical: 18757:Other tests 18700:Singularity 18632:Formulation 18594:Fundamental 18448:Formulation 18429:Proper time 18390:Fundamental 17723:Gravitation 17415:Dirac, Paul 17281:Thorne, Kip 17035:24 February 16675:(3): 9–34, 15687:: 424–434, 15668:: 189–196, 15623:: 214–226, 14831:: 1504–1508 14730:: 419–421, 14680:: 252–276, 14560:: 542–552, 14157:Gravitation 14060:: 439–444, 12840:: L51–L55, 12323:Gödel, Kurt 12130:Phys. Lett. 10933:New Astron. 10809:(1): 3–11, 10149:, Perseus, 9886:(1): 1–27, 9653:, Berkley, 8850:Lehner 2002 8796:Maddox 1998 8747:Sorkin 2005 8725:Sorkin 1997 8661:Kuchař 1973 8598:Ibanez 2000 8592:and other) 8554:Hamber 2009 8492:Hamber 2009 8477:Schutz 2003 8465:Carlip 2001 8406:Auyang 1995 8394:Ramond 1990 8316:, sec. 11.2 8260:Lehner 2001 8237:Bruhat 1962 8193:Berger 2002 8168:Berger 2002 8107:Börner 1993 7963:Carter 1979 7944:as well as 7930:Israel 1971 7902:Israel 1987 7890:Berger 2002 7886:Thorne 1972 7631:Börner 1993 7537:Peskin 2007 7373:light-years 7216:Robson 1996 7188:Gnedin 2005 7160:Miller 2002 7138:Thorne 1995 7022:Barish 2005 6960:Schutz 2003 6784:Kramer 2004 6773:Taylor 1994 6749:Schutz 2003 6745:Stairs 2003 6582:, sec. 11.9 6564:Schutz 2003 6552:Seidel 1998 6548:Lehner 2002 6496:Schutz 1985 6393:11 February 6372:Nature News 6348:12 February 6309:12 February 6259:Stairs 2003 6243:transponder 6194:Israel 1987 6166:Ehlers 1973 6101:Kramer 2004 6097:Stairs 2003 5916:Lehner 2002 5881:, ch. 3,5,6 5852:Geroch 1996 5665:Kenyon 1990 5653:Schutz 1985 5637:Ehlers 1973 5621:Kenyon 1990 5617:Ehlers 1973 5594:Ehlers 1973 5583:Norton 1985 5559:Mermin 2005 5555:Ehlers 1973 5543:Schutz 1985 5539:Ehlers 1973 5527:Ehlers 1973 5513:, sec. 22, 5480:Mermin 2005 5463:Ehlers 1973 5448:Künzle 1972 5440:Ehlers 1973 5400:Ehlers 1973 5388:Arnold 1989 5350:Ehlers 1973 5308:Engler 2002 5223:Thorne 1994 5219:Israel 1987 5204:Thorne 2003 5132:Hubble 1929 5059:, ch. 11–15 4805:causal sets 4676:Causal sets 4595:quasi-local 4504:) as well. 4189:time travel 4183:introduced 4150:singularity 4139:dark energy 4137:, known as 3804:eLISA / NGO 3724:dark matter 3603:Nobel Prize 3305:self-energy 3056:blueshifted 2982:anisotropic 2978:homogeneous 2970:time travel 2954:Kerr metric 2915:Minkowskian 2289:proper time 1559:as well as 1506:proper time 1319:world lines 1317:, straight 1023:black holes 961:black holes 849:gravitation 729:van Stockum 659:Oppenheimer 514:Kerr–Newman 306:Singularity 84:black holes 21592:Categories 21274:Linear map 21142:Operations 20788:Geophysics 20778:Biophysics 20622:Analytical 20575:Approaches 20472:Max Talmey 20374:(grandson) 20332:(daughter) 20092:In popular 19758:CGHS model 19745:toy models 19069:Zel'dovich 18977:Scientists 18956:Alcubierre 18763:of Mercury 18761:precession 18690:Black hole 18573:Background 18565:relativity 18534:World line 18529:Light cone 18354:Background 18346:relativity 18336:Relativity 17904:(1): 204, 17754:1461299055 17741:R.K. Sachs 17218:Geroch, R. 17189:, Berlin, 14182:(1): 199, 13410:Phys. Rev. 12236:Witten, E. 10839:Primas, F. 9529:Carter, B. 8721:Isham 1994 8624:, sec. 4.2 8294:Komar 1959 8245:Reula 1998 8207:, sec. 7.1 8195:, sec. 2.1 8109:, sec. 1.2 8061:, sec. 9.1 8039:Unruh 1976 7892:, sec. 2.1 7833:1703.05448 7684:Gödel 1949 7645:, sec. 5,6 7633:, sec. 9.1 7621:, and the 7603:Linde 2006 7599:Linde 2005 7535:, ch. 12, 7424:Weiss 2006 7391:data, see 7387:E.g. with 7268:Kraus 1998 7012:, sec. 3.7 6932:Iorio 2010 6928:Iorio 2006 6917:Iorio 2009 6817:, sec. 7.8 6699:2112.06795 6663:, p. 6611:, sec. 7.3 6536:Gowdy 1974 6532:Gowdy 1971 6426:1602.03837 6261:, sec. 4.4 6180:, sec. 1.3 6158:Lagrangian 6115:, sec. 4.2 6086:Ashby 2003 6082:Ashby 2002 5930:, sec. 4.4 5805:, sec. 4.1 5793:, sec. 4.4 5667:, sec. 7.4 5655:, sec. 8.3 5627:, sec. 2.8 5545:, sec. 5.1 5529:, sec. 2.3 5515:Synge 1972 5444:Havas 1964 5206:, p. 5149:Gamow 1970 5136:Singh 2004 5011:4 February 4920:References 4658:See also: 4587:Bondi mass 4583:Komar mass 4502:Big Crunch 4462:, below). 4412:stationary 4204:warp drive 4165:Kurt Gödel 4156:, below). 3835:Black hole 3794:are using 3790:. Various 3661:ergosphere 3599:PSR1913+16 3286:rose curve 3244:precession 3181:non-linear 3125:invariance 3060:redshifted 2884:slow speed 2850:spacecraft 2809:See also: 2770:There are 2406:See also: 2106:world line 1581:divergence 1414:Light cone 1387:integrable 1366:connection 1338:coordinate 861:relativity 859:. General 582:Scientists 414:Formalisms 362:Formalisms 311:Black hole 237:World line 21532:Astronomy 21413:EM tensor 21249:Dimension 21200:Transpose 20738:Molecular 20639:Acoustics 20632:Continuum 20627:Celestial 20617:Newtonian 20604:Classical 20547:Divisions 19763:RST model 19549:DGP model 19526:Spin foam 19470:Whitehead 19462:Nordström 19391:) gravity 19341:Classical 19315:Paradigms 19266:Resources 19039:Robertson 19024:Friedmann 19019:Eddington 19009:de Sitter 18843:Solutions 18721:detectors 18716:astronomy 18683:Phenomena 18618:Geodesics 18521:Spacetime 18464:Phenomena 18222:John Baez 18180:Tutorials 18084:0809.1280 17976:119476595 17889:See also 16859:(1): 12, 16697:120950657 16420:(1): 69, 16402:119151491 16350:119572847 16298:120534048 15444:CiteSeerX 15374:262685723 14928:(1998a), 14792:0707.1536 13860:(1): 13, 13846:250677699 13584:: 337–362 13400:21 August 13366:119203172 13341:0709.0685 13171:28 August 13100:119645546 12969:0809.1373 12948:118233440 12537:120730123 11605:119433133 11505:120160056 11477:122994560 11421:See also 11401:See also 11399:: 844–847 11379:See also 11356:118162303 11135:250836327 10881:119340132 10831:120910934 10624:118923209 10479:0707.2153 10440:119368762 10179:0802.1249 9808:123043135 9772:122636624 9382:119175535 9292:(1): 10, 9278:119663169 9243:0705.2222 9041:0711.0302 8898:Will 2006 8826:Cosmology 8736:Loll 1998 8650:Duff 1996 8452:Wald 2001 8437:Wald 1975 8417:Wald 1994 8326:Wald 1984 8314:Wald 1984 8298:Wald 1984 8241:Wald 1984 8189:Wald 1984 8059:Wald 1984 8043:Wald 2001 8010:Wald 2001 8006:Wald 1975 7967:Wald 2001 7914:Wald 1984 7870:Wald 1984 7854:Wald 1984 7778:120125096 7710:0904.0141 7656:potential 7345:Pais 1982 7288:Milky Way 7260:Font 2003 7252:Font 2003 7236:Rees 1966 7068:0911.5206 6842:Kahn 2007 6732:245124502 6724:2160-3308 6630:Will 2006 6628:, ch. 5, 6626:Will 1993 6609:Will 1993 6605:Will 2006 6592:Will 1993 6568:Pais 1982 6459:124959784 6388:182916902 6271:Will 1993 6202:Euclidean 5980:Pais 1982 5956:Will 1993 5952:Will 2006 5940:Will 1993 5928:Wald 1984 5803:Wald 1984 5791:Wald 1984 5753:, ch. 4, 5751:Wald 1984 5416:Will 2006 5412:Will 1993 5272:Wald 1984 5257:Cosmology 5234:Sections 5225:, ch. 3–9 5192:Pais 1982 5165:Pais 1982 5153:Cosmology 5138:, ch. 2–4 5121:, ch. 15e 5119:Pais 1982 5057:Pais 1982 5049:Pais 1982 5037:Renn 2005 5033:Renn 2007 5025:Pais 1982 4748:known as 4638:evaporate 4628:Ordinary 4458:(cf. the 4102:Isotropic 4085:ν 4082:μ 4051:ν 4048:μ 4023:π 4009:ν 4006:μ 3995:Λ 3987:ν 3984:μ 3960:− 3955:ν 3952:μ 3924:Λ 3894:Cosmology 3869:Accretion 3819:supernova 3633:gyroscope 3410:− 3363:π 3350:σ 3327:σ 3200:− 2846:free-fall 2795:) gravity 2675:− 2615:− 2513:− 2457:− 2374:β 2354:α 2316:β 2313:α 2304:μ 2300:Γ 2230:β 2203:α 2187:β 2184:α 2175:μ 2171:Γ 2144:μ 2081:ν 2078:μ 1990:π 1978:κ 1958:κ 1945:planetary 1925:ν 1922:μ 1897:κ 1869:ν 1866:α 1863:μ 1856:α 1841:ν 1838:μ 1801:ν 1798:μ 1788:ν 1785:μ 1749:ν 1746:μ 1719:ν 1716:μ 1681:ν 1678:μ 1670:κ 1662:ν 1659:μ 1635:− 1630:ν 1627:μ 1619:≡ 1614:ν 1611:μ 1557:densities 1483:time-like 1395:covariant 1354:test body 1315:geodesics 1262:heuristic 1203:Eddington 1172:Friedmann 1136:nonlinear 1123:, called 969:cosmology 937:free fall 907:radiation 889:curvature 884:spacetime 842:geometric 840:, is the 836:, and as 674:Robertson 639:Friedmann 634:Eddington 624:Nordström 614:de Sitter 471:Solutions 396:Geodesics 391:Friedmann 373:Equations 359:Equations 320:Spacetime 255:Phenomena 161:ν 158:μ 149:κ 140:ν 137:μ 129:Λ 121:ν 118:μ 80:spacetime 21279:Manifold 21264:Geodesic 21022:Notation 20496:Category 20422:Memorial 20368:(cousin) 20356:(sister) 20350:(father) 20344:(mother) 20242:Einstein 20077:" (1949) 20038:" (1905) 19820:Graviton 19630:F-theory 19623:M-theory 19199:Standard 19152:Category 19029:Lemaître 18994:Einstein 18984:Poincaré 18944:Others: 18928:Taub–NUT 18894:interior 18816:theories 18814:Advanced 18781:redshift 18596:concepts 18414:Rapidity 18392:concepts 18177:Lectures 18133:Archived 18109:18421348 18027:archived 17860:(1916), 17831:(2007). 17807:(1975). 17786:10018614 17764:(1984). 17720:(1973), 17651:(1980), 17605:(2004), 17575:(2009), 17456:(2003), 17417:(1996), 17307:(1992), 17241:(2008), 17220:(1981), 17204:citation 17183:(1916), 17149:(2004), 17140:28179873 17096:(1): 3, 17067:(1993), 17046:(1990), 17004:(eds.), 16912:(1972), 16903:28937183 16837:16068158 16801:28163633 16757:(1): 6, 16663:(1975), 16264:(1994), 16255:28179865 16221:(1): 4, 16195:(1972), 16168:28163640 16124:(1): 5, 16102:15931216 16013:Bean, R. 16005:10794058 15851:(2004), 15811:15089661 15775:30795875 15655:16762545 15570:17719367 15524:14255129 15478:28937180 15424:(1): 1, 15276:(2001), 15267:28191833 15233:(1): 3, 15183:14949527 15105:41065207 15057:(1990), 14892:28179866 14848:(1). 6. 14817:16276112 14695:(2004), 14511:(1982), 14490:(1999), 14448:14164537 14402:15187664 14327:archived 14238:(1993), 14214:17986323 14154:(1973), 14085:(2005), 14043:14024934 13983:(1998), 13974:13096531 13904:28191826 13767:(2005), 13641:28163643 13597:(1): 8, 13522:16973838 13306:14018215 13256:archived 13237:20 April 13221:28163647 13187:(1): 3, 13158:10641693 13040:16052420 12994:15698399 12902:15707877 12819:16577160 12768:28179855 12734:(1): 3, 12676:28937184 12642:(1): 6, 12605:(1973), 12577:55539246 12461:37376002 12453:17779918 12407:10067969 12399:17779917 12325:(1949). 12238:(1987), 12122:28179864 12088:(1): 2, 12066:15931201 12028:14772226 11976:15948765 11874:3 August 11834:(1970), 11825:28179819 11791:(1): 2, 11777:37236624 11731:28179863 11697:(1): 1, 11653:29104452 11619:(1): 4, 11597:10587484 11549:5 August 11540:archived 11411:(1917), 11391:(1915), 11365:(1907), 11249:17432791 11082:10008243 11000:16276658 10917:15496915 10783:(1983), 10774:17677758 10703:28179856 10659:(1): 1, 10558:27632148 10550:12857181 10504:17281664 10432:12624255 10343:18752698 10252:28179874 10218:(1): 4, 10204:54608927 10122:(1984), 10076:14508481 10040:10034329 9979:28179859 9935:(1): 1, 9569:54690354 9535:(1973), 9518:21565397 9464:11780112 9336:28163644 9190:10033673 9151:28163638 9117:(1): 1, 9066:55825202 8988:(1989), 8914:journal. 8824:section 8811:section 8674:electric 8663:, sec. 3 8636:, ch. 31 8602:graviton 8590:electric 8467:, sec. 2 8124:caustics 7735:59575856 7674:, sec. 2 7660:inflaton 7565:, sec. 9 7551:, ch. 12 7545:baryonic 7282:"), see 7174:, sec. 3 7123:23 April 7093:56073764 6966:, sec. 3 6522:, ch. 13 6451:26918975 6168:, sec. 5 6160:using a 5418:, sec. 2 5352:, sec. 1 5255:Section 5194:, ch. 16 4942:18 April 4840:See also 4754:M-theory 4579:ADM mass 4377:horizons 4357:Horizons 4258:a priori 4240:In 1962 4198:such as 3784:TAMA 300 3732:galaxies 3011:such as 2952:and the 2018:, where 1585:manifold 1565:pressure 1467:observer 1425:symmetry 1370:gradient 1334:friction 1291:inertial 1180:Big Bang 1176:Lemaître 1160:universe 973:Big Bang 933:geometry 899:momentum 790:Category 654:Lemaître 619:Reissner 604:Poincaré 589:Einstein 534:Taub–NUT 499:Wormhole 483:interior 196:Timeline 72:ringdown 64:inspiral 21580:Physics 21518:Portals 21376:Physics 21210:Related 20973:Physics 20891:Tensors 20822:Related 20706:General 20701:Special 20559:Applied 20400:Related 20094:culture 19869:Physics 19251:History 19094:Hawking 19089:Penrose 19074:Novikov 19054:Wheeler 18999:Hilbert 18989:Lorentz 18946:pp-wave 18767:lensing 18563:General 18344:Special 18291:5 April 18275:11 July 18194:YouTube 18174:Courses 18117:4412396 18089:Bibcode 18050:Bibcode 18009:Bibcode 17956:Bibcode 17916:Bibcode 17874:Bibcode 17548:7644624 17131:5256066 17108:Bibcode 16894:5567250 16871:Bibcode 16845:2142707 16817:Bibcode 16792:5253844 16769:Bibcode 16729:Bibcode 16677:Bibcode 16633:Bibcode 16610:Bibcode 16584:Bibcode 16554:Bibcode 16542:: 385. 16478:Bibcode 16466:: 160, 16422:Bibcode 16372:Bibcode 16320:Bibcode 16278:Bibcode 16246:5255888 16223:Bibcode 16159:5253800 16136:Bibcode 16110:4383030 16082:Bibcode 16055:1386346 16035:Bibcode 15985:Bibcode 15958:4803804 15938:Bibcode 15898:Bibcode 15859:Bibcode 15830:Bibcode 15791:Bibcode 15755:Bibcode 15715:Bibcode 15689:Bibcode 15670:Bibcode 15635:Bibcode 15550:Bibcode 15504:Bibcode 15469:5567241 15436:Bibcode 15354:Bibcode 15258:5253804 15235:Bibcode 15163:Bibcode 15125:Bibcode 15085:Bibcode 15034:Bibcode 15004:Bibcode 14974:Bibcode 14905:Bibcode 14883:5256043 14860:Bibcode 14797:Bibcode 14732:Bibcode 14682:Bibcode 14649:Bibcode 14630:4337502 14610:Bibcode 14562:Bibcode 14467:Bibcode 14428:Bibcode 14382:Bibcode 14336:11 June 14311:Bibcode 14269:Bibcode 14194:Bibcode 14135:25 July 14062:Bibcode 14023:Bibcode 13954:Bibcode 13920:Bibcode 13895:5253799 13872:Bibcode 13826:Bibcode 13783:Bibcode 13757:9145148 13727:Bibcode 13705:9715975 13685:Bibcode 13632:5253994 13609:Bibcode 13530:6674714 13502:Bibcode 13484:Science 13452:Bibcode 13418:Bibcode 13346:Bibcode 13286:Bibcode 13212:5253919 13189:Bibcode 13138:Bibcode 13080:Bibcode 13020:Bibcode 12974:Bibcode 12928:Bibcode 12882:Bibcode 12842:Bibcode 12787:Bibcode 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