Einstein field equations

1154: 757: 2320: 87: 1975: 7722:. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as 10773: 9913: 1167: 770: 7557: 982: 1287: 2315:{\displaystyle {\begin{aligned}g_{\mu \nu }&=\times \operatorname {diag} (-1,+1,+1,+1)\\{R^{\mu }}_{\alpha \beta \gamma }&=\times \left(\Gamma _{\alpha \gamma ,\beta }^{\mu }-\Gamma _{\alpha \beta ,\gamma }^{\mu }+\Gamma _{\sigma \beta }^{\mu }\Gamma _{\gamma \alpha }^{\sigma }-\Gamma _{\sigma \gamma }^{\mu }\Gamma _{\beta \alpha }^{\sigma }\right)\\G_{\mu \nu }&=\times \kappa T_{\mu \nu }\end{aligned}}} 7287: 7276: 8078: 6368: 7928: 6210: 9457: 7091: 3678: 3552: 7552:{\displaystyle {\begin{aligned}{F^{\alpha \beta }}_{;\beta }&=0\\F_{}&={\tfrac {1}{3}}\left(F_{\alpha \beta ;\gamma }+F_{\beta \gamma ;\alpha }+F_{\gamma \alpha ;\beta }\right)={\tfrac {1}{3}}\left(F_{\alpha \beta ,\gamma }+F_{\beta \gamma ,\alpha }+F_{\gamma \alpha ,\beta }\right)=0.\end{aligned}}} 7748:

One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum. In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright,

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The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of

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The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign. This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity.

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Despite the EFE as written containing the inverse of the metric tensor, they can be arranged in a form that contains the metric tensor in polynomial form and without its inverse. First, the determinant of the metric in 4 dimensions can be written

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in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived. However, there are global solutions of the equation that may lack a globally defined potential.

4275: 4948: 4589: 4710: 5180: 3148: 6436: 3250: 5758: 3909: 4936: 8333:, the stress–energy tensor on the right side of the equation must be written with each component in units of energy density (i.e., energy per volume, equivalently pressure). In Einstein's original publication, the choice is 5614: 3353: 5356: 4403: 6857: 4408:

which expresses the local conservation of stress–energy. This conservation law is a physical requirement. With his field equations Einstein ensured that general relativity is consistent with this conservation condition.

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Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side and incorporated as part of the stress–energy tensor:

158: 1530: 5329: 1281: 6710: 4717: 5891: 5753: 1980: 7271:{\displaystyle G^{\alpha \beta }+\Lambda g^{\alpha \beta }={\frac {\kappa }{\mu _{0}}}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+{\tfrac {1}{4}}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right).} 4088: 1924:

The equations are more complex than they appear. Given a specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor

8073:{\displaystyle g^{\alpha \kappa }={\frac {{\tfrac {1}{6}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }}{\det(g)}}\,.} 7767:

The nonlinearity of the EFE makes finding exact solutions difficult. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the

6647: 2781: 1954:, since both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic 899:

of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear

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Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace

4327: 5670: 6699: 6363:{\displaystyle R_{00}=\Gamma _{00,\rho }^{\rho }-\Gamma _{\rho 0,0}^{\rho }+\Gamma _{\rho \lambda }^{\rho }\Gamma _{00}^{\lambda }-\Gamma _{0\lambda }^{\rho }\Gamma _{\rho 0}^{\lambda }.} 7923:{\displaystyle \det(g)={\tfrac {1}{24}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\alpha \kappa }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }} 2440: 4193: 6205:{\displaystyle K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)\approx K\left(\rho c^{4}-{\tfrac {1}{2}}\left(-\rho c^{2}\right)\left(-c^{2}\right)\right)={\tfrac {1}{2}}K\rho c^{4}\,.} 4493: 4602: 5090: 3416: 3071: 6780: 1728: 7086:{\displaystyle T^{\alpha \beta }=\,-{\frac {1}{\mu _{0}}}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+{\tfrac {1}{4}}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right)} 3180: 2919: 2437:

Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative:

1952: 1606: 1563: 1394: 1360: 1326: 4851: 5518: 3284: 1762: 4334: 3673:{\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }-{R^{\gamma }}_{\beta \gamma \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0} 3547:{\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }+{R^{\gamma }}_{\beta \varepsilon \gamma ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0} 1418: 64: 6787: 1442: 1906:

dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when

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Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in

1587: 5185: 4021: 3899:{\displaystyle g^{\beta \delta }\left(R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }\right)=0} 2942: 9828: 1055: 3052:

The inclusion of this term does not create inconsistencies. For many years the cosmological constant was almost universally assumed to be zero. More recent

9853: 5255: 801: 4488: 8766: 5677: 9439: 5078:{\displaystyle {\frac {d^{2}x^{\alpha }}{d\tau ^{2}}}=-\Gamma _{\beta \gamma }^{\alpha }{\frac {dx^{\beta }}{d\tau }}{\frac {dx^{\gamma }}{d\tau }}\,.} 9237: 180: 3018:

was absent from the version in which he originally published them. Einstein then included the term with the cosmological constant to allow for a

8228: 6581:{\displaystyle \Phi _{,ii}\approx \Gamma _{00,i}^{i}\approx R_{00}=K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)\approx {\tfrac {1}{2}}K\rho c^{4}} 9464: 5879:{\displaystyle T_{\mu \nu }\approx \operatorname {diag} \left(T_{00},0,0,0\right)\approx \operatorname {diag} \left(\rho c^{4},0,0,0\right)\,.} 6591: 4009:{\displaystyle {R^{\delta }}_{\delta ;\varepsilon }-{R^{\delta }}_{\varepsilon ;\delta }+{R^{\gamma \delta }}_{\delta \varepsilon ;\gamma }=0} 93: 9781: 1457: 8797: 2569: 1218: 385: 5496:{\displaystyle \Phi _{,i}\approx \Gamma _{00}^{i}={\tfrac {1}{2}}g^{i\alpha }\left(g_{\alpha 0,0}+g_{0\alpha ,0}-g_{00,\alpha }\right)\,.} 10523: 6379: 10758: 9858: 9384: 6914: 1896: 1110: 738: 9214: 10547: 10293: 9414: 9274: 8134: 7709: 4284: 548: 390: 8974:

Kohli, Ikjyot Singh; Haslam, Michael C. (2013). "Dynamical systems approach to a Bianchi type I viscous magnetohydrodynamic model".

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to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives. The

5619: 9898: 6935: 1085: 794: 503: 7779:. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the 6652: 10810: 9787: 3771:{\displaystyle R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0} 2762:{\displaystyle R_{\mu \nu }-{\frac {2}{D-2}}\Lambda g_{\mu \nu }=\kappa \left(T_{\mu \nu }-{\frac {1}{D-2}}Tg_{\mu \nu }\right).} 8888:

Hsu, L.; Wainwright, J (1986). "Self-similar spatially homogeneous cosmologies: orthogonal perfect fluid and vacuum solutions".

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Substituting this expression of the inverse of the metric into the equations then multiplying both sides by a suitable power of

10329: 10283: 9642: 7281: 3057: 416: 250: 8690: 10467: 9224: 9116: 8776: 8304: 8119: 7727: 3362: 934: 455: 4839:{\displaystyle {\begin{aligned}\Phi _{,ii}&=4\pi G\rho \\{\frac {d^{2}x^{i}}{dt^{2}}}&=-\Phi _{,i}\,.\end{aligned}}} 9951: 6862: 6009:{\displaystyle T=g^{\alpha \beta }T_{\alpha \beta }\approx g^{00}T_{00}\approx -{\frac {1}{c^{2}}}\rho c^{4}=-\rho c^{2}\,} 950: 9202: 4417:

The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example,

3367: 1593:. This is a symmetric second-degree tensor that depends on only the metric tensor and its first and second derivatives. 1199: 787: 10204: 9848: 9671: 9513: 9354: 9168: 7715: 6745: 1363: 1050: 896: 167: 17: 10575: 10209: 9532: 9409: 9142: 8667: 8621: 8551: 8433: 8401: 8368: 8277: 8139: 8094:

from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields.

1874: 1130: 193: 9647: 4177:{\displaystyle \left({R^{\gamma }}_{\varepsilon }-{\tfrac {1}{2}}{g^{\gamma }}_{\varepsilon }R\right)_{;\gamma }=0} 10529: 1153: 756: 473: 86: 10242: 9692: 2882:{\displaystyle R_{\mu \nu }-\Lambda g_{\mu \nu }=\kappa \left(T_{\mu \nu }-{\frac {1}{2}}T\,g_{\mu \nu }\right).} 908: 380: 5252:

are negligible. Applying these simplifying assumptions to the spatial components of the geodesic equation gives

10345: 9502: 9374: 7726:. Even so, there are several cases where the field equations have been solved completely, and those are called 6713:

A Swiss commemorative coin from 1979, showing the vacuum field equations with zero cosmological constant (top).

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To see how the latter reduces to the former, we assume that the test particle's velocity is approximately zero

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In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form

919: 7564: 4433:, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is 10800: 10776: 10518: 10423: 10353: 9267: 9195: 8129: 7692:{\displaystyle F_{\alpha \beta }=A_{\alpha ;\beta }-A_{\beta ;\alpha }=A_{\alpha ,\beta }-A_{\beta ,\alpha }} 5248:

and that the metric and its derivatives are approximately static and that the squares of deviations from the

1955: 1870: 900: 467: 175: 10486: 513: 10340: 9480: 9369: 9364: 9082: 8567:

Efstathiou, G.; Sutherland, W. J.; Maddox, S. J. (1990). "The cosmological constant and cold dark matter".

8124: 7723: 533: 4434: 10805: 10381: 10115: 10100: 9637: 9596: 9540: 9389: 9314: 9190: 8109: 945:. Further simplification is achieved in approximating the spacetime as having only small deviations from 774: 265: 185: 67: 9185: 7783:, ignoring higher-power terms. This linearization procedure can be used to investigate the phenomena of 10668: 10110: 10063: 9564: 9379: 8425: 4454: 3556:

The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten:

1970:(MTW). The authors analyzed conventions that exist and classified these according to three signs ( ): 678: 226: 6725: 1706: 10698: 10324: 10005: 9472: 9419: 9399: 8104: 4462: 1696:{\displaystyle \kappa ={\frac {8\pi G}{c^{4}}}\approx 2.07665(5)\times 10^{-43}\,{\textrm {N}}^{-1},} 942: 478: 290: 2633:

and substituting this in the original EFE, one gets the following equivalent "trace-reversed" form:

10795: 10723: 10033: 9944: 9917: 9868: 9863: 9843: 9833: 9793: 9488: 9260: 4458: 2533:{\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }-\Lambda g_{\mu \nu }=-\kappa T_{\mu \nu }.} 1397: 869: 835: 713: 703: 553: 2894: 1927: 1852:{\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }.} 1538: 1369: 1335: 1301: 10452: 10374: 9994: 9985: 9873: 8091: 7784: 6742:, the vacuum field equations, also known as 'Einstein vacuum equations' (EVE), can be written as 3279:

General relativity is consistent with the local conservation of energy and momentum expressed as

1192: 698: 432: 6724:

is zero in the region under consideration, then the field equations are also referred to as the

10678: 10541: 10447: 10405: 10079: 10043: 9805: 9713: 9685: 9588: 9447: 9098: 6870: 6709: 4418: 4270:{\displaystyle \left(R^{\gamma \delta }-{\tfrac {1}{2}}g^{\gamma \delta }R\right)_{;\gamma }=0} 1738: 1025: 888: 668: 653: 498: 260: 8294: 4584:{\displaystyle \nabla ^{2}\Phi \left({\vec {x}},t\right)=4\pi G\rho \left({\vec {x}},t\right)} 2563: 2430:, Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) and Peacock (1999) are 1403: 33: 10683: 10288: 10227: 9878: 9838: 9678: 9627: 9622: 9548: 9359: 9339: 9329: 8114: 7098: 6927: 4705:{\displaystyle {\ddot {\vec {x}}}(t)={\vec {g}}=-\nabla \Phi \left({\vec {x}}(t),t\right)\,.} 3012: 2934: 1427: 1421: 1120: 1020: 903:

when used in this way. The solutions of the EFE are the components of the metric tensor. The

876: 643: 211: 5175:{\displaystyle {\frac {dx^{\beta }}{d\tau }}\approx \left({\frac {dt}{d\tau }},0,0,0\right)} 10588: 10514: 10358: 9960: 9757: 9706: 9699: 9652: 9324: 9297: 9211:— An explanation of Einstein's field equation, its derivation, and some of its consequences 9046: 8993: 8940: 8897: 8854: 8806: 8795:(1977). "Solutions of the Maxwell and Yang–Mills equations associated with Hopf fibrings". 8731: 8578: 8472: 8187: 8144: 7560: 3143:{\displaystyle T_{\mu \nu }^{\mathrm {(vac)} }=-{\frac {\Lambda }{\kappa }}g_{\mu \nu }\,.} 930: 463: 7753:. New solutions have been discovered using these methods by LeBlanc and Kohli and Haslam. 7586:

that in a coordinate chart it is possible to introduce an electromagnetic field potential

8: 10663: 10658: 10648: 10480: 10429: 10247: 10232: 10058: 9989: 9937: 7768: 7749:

self-similar solutions to the Einstein field equations are fixed points of the resulting

7579: 6898: 4480:, which is the gravitational potential in joules per kilogram of the gravitational field 1873:, which dictates how freely falling matter moves through spacetime, form the core of the 954: 938: 658: 628: 623: 375: 10653: 9050: 8997: 8944: 8901: 8858: 8810: 8735: 8582: 8476: 8191: 3064:

is needed. The cosmological constant is negligible at the scale of a galaxy or smaller.

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The study of exact solutions of Einstein's field equations is one of the activities of

3245:{\displaystyle \rho _{\mathrm {vac} }=-p_{\mathrm {vac} }={\frac {\Lambda }{\kappa }},} 3034: 2415: 1967: 1889: 1878: 1572: 1185: 1171: 1135: 997: 992: 892: 815: 708: 593: 518: 365: 270: 255: 216: 78: 8203: 4018:

The definitions of the Ricci curvature tensor and the scalar curvature then show that

3049:"that the introduction of the cosmological term was the biggest blunder of his life". 1892:

reduce the number of independent equations from 10 to 6, leaving the metric with four

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in the limit of a weak gravitational field and velocities that are much less than the

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to the distribution of mass–energy, momentum and stress, that is, they determine the

538: 313: 10608: 8909: 8751: 8598: 7583: 4931:{\displaystyle R_{\mu \nu }=K\left(T_{\mu \nu }-{\tfrac {1}{2}}Tg_{\mu \nu }\right)} 929:

Exact solutions for the EFE can only be found under simplifying assumptions such as

588: 10643: 10633: 10580: 10557: 10135: 9769: 9751: 9429: 9066: 9054: 9001: 8948: 8905: 8862: 8814: 8792: 8739: 8698: 8586: 8569: 8490: 8480: 8195: 7780: 7750: 5249: 4422: 2922: 1885: 1590: 884: 880: 663: 618: 598: 543: 10703: 9238:

The Einstein field equation on the wall of the Museum Boerhaave in downtown Leiden

8653: 8246: 5609:{\displaystyle 2\Phi _{,i}\approx g^{ij}\left(-g_{00,j}\right)\approx -g_{00,i}\,} 3348:{\displaystyle \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0.} 2324:

The third sign above is related to the choice of convention for the Ricci tensor:

683: 10718: 10693: 10618: 10613: 10496: 10457: 10419: 10363: 10237: 10173: 9799: 9731: 9496: 9404: 9349: 9319: 9283: 9152: 9126: 8842: 8265: 8224: 7776: 6866: 4398:{\displaystyle \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0} 4278: 3019: 2547: 1566: 1448: 1329: 1030: 946: 861: 842: 718: 693: 578: 573: 437: 318: 280: 10501: 9739: 6852:{\displaystyle R_{\mu \nu }={\frac {\Lambda }{{\frac {D}{2}}-1}}g_{\mu \nu }\,.} 488: 10743: 10415: 10399: 10395: 10298: 10268: 10125: 10028: 9556: 9334: 9090: 9005: 8347:, in which case the stress–energy tensor components have units of mass density. 8149: 5353:

have been divided out. This will reduce to its Newtonian counterpart, provided

4430: 4426: 1746: 1158: 923: 761: 728: 723: 411: 275: 9227:

Physics Today November 2015, History of the Development of the Field Equations

9058: 8931:

LeBlanc, V. G. (1997). "Asymptotic states of magnetic Bianchi I cosmologies".

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

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Einstein's General Theory of Relativity: With Modern Applications in Cosmology

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is the simplest example of a vacuum solution. Nontrivial examples include the

937:

are most often studied since they model many gravitational phenomena, such as

10789: 10733: 10713: 10708: 10623: 10491: 10319: 10263: 10095: 10048: 9775: 9745: 9632: 9242: 9108: 9094: 8199: 6874: 3157: 3026:

any desired steady state solution described by this equation is unstable, and

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The sign of the cosmological term would change in both these versions if the

688: 603: 583: 508: 406: 285: 8299:(illustrated ed.). Springer Science & Business Media. p. 180. 648: 10753: 10673: 10638: 10168: 10130: 9883: 9763: 8659: 8649: 8504: 8378: 6884: 4442: 3779: 3422:

gives, using the fact that the metric tensor is covariantly constant, i.e.

3153: 3046: 3030: 1893: 1125: 633: 613: 9027:

Katanaev, M. O. (2006). "Polynomial form of the Hilbert–Einstein action".

2405:{\displaystyle R_{\mu \nu }=\times \times {R^{\alpha }}_{\mu \alpha \nu }} 918:

As well as implying local energy–momentum conservation, the EFE reduce to

10537: 10506: 10053: 9888: 9572: 9394: 8726: 5241:{\displaystyle {\frac {d}{dt}}\left({\frac {dt}{d\tau }}\right)\approx 0} 4078:{\displaystyle R_{;\varepsilon }-2{R^{\gamma }}_{\varepsilon ;\gamma }=0} 1105: 523: 493: 8485: 8358: 5674:

Turning to the Einstein equations, we only need the time-time component

3004:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }\,,} 10748: 10314: 10158: 10153: 9893: 8866: 8818: 8394:

Dreams of a Final Theory: the search for the fundamental laws of nature

8233:

Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin

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appearing in the EFE is determined by making these two approximations.

733: 295: 221: 153:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }} 9041: 8467: 8451:"Theorems on Existence and Global Dynamics for the Einstein Equations" 8319:

With the choice of the Einstein gravitational constant as given here,

4476:

Newtonian gravitation can be written as the theory of a scalar field,

4277:

which by the symmetry of the bracketed term and the definition of the

1525:{\displaystyle G_{\mu \nu }=R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu },} 10145: 8590: 7772: 7719: 5324:{\displaystyle {\frac {d^{2}x^{i}}{dt^{2}}}\approx -\Gamma _{00}^{i}} 4596: 3053: 1276:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }} 1015: 981: 973: 854: 831: 304: 8712:

Turner, Michael S. (May 2001). "Making Sense of the New Cosmology".

8511: 10688: 10038: 9246: 7934:; and the inverse of the metric in 4 dimensions can be written as: 7742: 6880: 865: 673: 483: 323: 9252: 8988: 8419: 5748:{\displaystyle R_{00}=K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)} 1286: 1899:, which correspond to the freedom to choose a coordinate system. 1080: 904: 6901:

and manifolds with a Ricci tensor proportional to the metric as

6784:

In the case of nonzero cosmological constant, the equations are

1862:

In standard units, each term on the left has units of 1/length.

10368: 7572: 5515:

and the time (0) derivatives to be zero. So this simplifies to

1750: 1291: 1212:

The Einstein field equations (EFE) may be written in the form:

846: 8270:

Spacetime and Geometry – An Introduction to General Relativity

9929: 9249:, Einstein field equation on the side of a train in Bolivia. 10349: 8566: 7093:

is used, then the Einstein field equations are called the

911:) in the resulting geometry are then calculated using the 9225:

Arch and scaffold: How Einstein found his field equations

9218: 8612:

Collins, P. D. B.; Martin, A. D.; Squires, E. J. (1989).

8357:

Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975).

8208: 1966:

The above form of the EFE is the standard established by

9247:"The impact of general relativity on the Atacama Desert" 6642:{\displaystyle {\tfrac {1}{2}}K\rho c^{4}=4\pi G\rho \,} 7104:, taken to be zero in conventional relativity theory): 6861:

The solutions to the vacuum field equations are called

6588:

which reduces to the Newtonian field equation provided

9205:— A simple introduction to Einstein's Field Equations. 7961: 7819: 7455: 7367: 7210: 7028: 6596: 6551: 6518: 6171: 6102: 6045: 5755:

the low speed and static field assumptions imply that

5716: 5395: 4896: 4220: 4121: 3274: 1888:. Each tensor has 10 independent components. The four 953:. These equations are used to study phenomena such as 9525:

Die Grundlagen der Einsteinschen Relativitäts-Theorie

8691:"Was Einstein's 'biggest blunder' a stellar success?" 7940: 7802: 7599: 7290: 7110: 6943: 6790: 6748: 6655: 6594: 6439: 6382: 6220: 6022: 5894: 5761: 5680: 5622: 5521: 5359: 5258: 5188: 5093: 4951: 4854: 4720: 4605: 4496: 4337: 4287: 4196: 4091: 4024: 3912: 3791: 3686: 3564: 3438: 3370: 3287: 3183: 3074: 2945: 2897: 2784: 2639: 2616:{\displaystyle R-{\frac {D}{2}}R+D\Lambda =\kappa T,} 2572: 2443: 2330: 1978: 1930: 1765: 1709: 1609: 1575: 1541: 1460: 1430: 1406: 1372: 1338: 1304: 1221: 96: 36: 30:"Einstein equation" redirects here. For the equation 8611: 8523: 8174:"The Foundation of the General Theory of Relativity" 9089: 8630: 8517: 8252: 6424:{\displaystyle R_{00}\approx \Gamma _{00,i}^{i}\,.} 868:and stress within that spacetime (expressed by the 9156: 9130: 8356: 8072: 7922: 7714:The solutions of the Einstein field equations are 7691: 7551: 7270: 7085: 6851: 6774: 6693: 6641: 6580: 6423: 6362: 6204: 6008: 5878: 5747: 5664: 5608: 5495: 5323: 5240: 5174: 5077: 4930: 4838: 4704: 4583: 4397: 4321: 4269: 4176: 4077: 4008: 3898: 3770: 3672: 3546: 3410: 3347: 3244: 3142: 3003: 2913: 2881: 2761: 2615: 2532: 2426:, Peebles (1980) and Efstathiou et al. (1990) are 2404: 2314: 1946: 1851: 1722: 1695: 1581: 1557: 1524: 1436: 1412: 1388: 1354: 1320: 1275: 152: 58: 9458:Investigations on the Theory of Brownian Movement 10787: 8415: 8413: 8051: 7803: 6372:Our simplifying assumptions make the squares of 3358:Derivation of local energy–momentum conservation 4448: 4322:{\displaystyle {G^{\alpha \beta }}_{;\beta }=0} 1884:The EFE is a tensor equation relating a set of 9465:Relativity: The Special and the General Theory 8840: 8655:My World Line : An Informal Autobiography 5665:{\displaystyle g_{00}\approx -c^{2}-2\Phi \,.} 9945: 9268: 8887: 8422:Exact Solutions of Einstein's Field Equations 8410: 6908: 6376:disappear together with the time derivatives 3020:universe that is not expanding or contracting 1193: 795: 8843:"A class of homogeneous cosmological models" 8798:International Journal of Theoretical Physics 7741:and to different models of evolution of the 7582:is zero. From the latter, it follows by the 6739: 6694:{\displaystyle K={\frac {8\pi G}{c^{4}}}\,.} 2928: 9215:Video Lecture on Einstein's Field Equations 8973: 9952: 9938: 9275: 9261: 8292: 3060:, and to explain this a positive value of 2557: 1200: 1186: 802: 788: 9040: 8987: 8725: 8544:The Large-scale Structure of the Universe 8494: 8484: 8466: 8066: 7710:Solutions of the Einstein field equations 6960: 6845: 6768: 6704: 6687: 6638: 6417: 6198: 6005: 5872: 5658: 5605: 5489: 5071: 4828: 4698: 3136: 2997: 2857: 1672: 907:trajectories of particles and radiation ( 9899:Emergency Committee of Atomic Scientists 9125: 9026: 8791: 8771:. Oxford University Press. p. 164. 8529: 8391: 8223: 8171: 7567:. The first equation asserts that the 4- 6708: 6214:From the definition of the Ricci tensor 4281:, gives, after relabelling the indices, 3022:. This effort was unsuccessful because: 2921:in the expression on the right with the 2627:is the spacetime dimension. Solving for 1444:is the Einstein gravitational constant. 1285: 9151: 8930: 8841:Ellis, G. F. R.; MacCallum, M. (1969). 8697:. University of Toronto. Archived from 8636: 8541: 8448: 8293:Grøn, Øyvind; Hervik, Sigbjorn (2007). 8264: 6915:Maxwell's equations in curved spacetime 6433:Combining the above equations together 4331:Using the EFE, this immediately gives, 2925:without significant loss of accuracy). 14: 10788: 9643:Albert Einstein World Award of Science 9221:Physics Professor Edmund Bertschinger. 8711: 8705: 8363:(2d ed.). New York: McGraw-Hill. 8258: 8217: 3177:that are fixed constants and given by 3058:accelerating expansion of the universe 9933: 9256: 8764: 8648: 8642: 8542:Peebles, Phillip James Edwin (1980). 8385: 8229:"Die Feldgleichungen der Gravitation" 8167: 8165: 8120:Exact solutions in general relativity 7756: 4473:Derivation of Newton's law of gravity 3785:Next, contract again with the metric 2554:metric sign convention adopted here. 1086:Newton's law of universal gravitation 27:Field-equations in general relativity 8688: 8682: 8396:. Vintage Press. pp. 107, 233. 8272:. Addison Wesley. pp. 151–159. 6936:electromagnetic stress–energy tensor 4595:is the mass density. The orbit of a 1756:The EFE can thus also be written as 960: 10524:Tolman–Oppenheimer–Volkoff equation 10477:Friedmann–Lemaître–Robertson–Walker 9282: 8518:Misner, Thorne & Wheeler (1973) 8253:Misner, Thorne & Wheeler (1973) 7284:are also applicable in free space: 3411:{\displaystyle R_{\alpha \beta }=0} 3275:Conservation of energy and momentum 1869:These equations, together with the 879:are related to the distribution of 24: 9672:Albert Einstein: Creator and Rebel 9385:Einstein–Infeld–Hoffmann equations 9355:Einstein relation (kinetic theory) 9231: 9209:The Meaning of Einstein's Equation 8360:Introduction to general relativity 8162: 7790: 7127: 6809: 6460: 6441: 6397: 6340: 6322: 6304: 6286: 6259: 6235: 5655: 5526: 5377: 5361: 5307: 5000: 4816: 4726: 4658: 4655: 4507: 4498: 4339: 3289: 3231: 3220: 3217: 3214: 3196: 3193: 3190: 3115: 3098: 3095: 3092: 2962: 2801: 2674: 2598: 2566:of both sides of the EFE one gets 2489: 2231: 2213: 2192: 2174: 2147: 2120: 1961: 1811: 1407: 1238: 1051:Introduction to general relativity 113: 25: 10822: 10294:Hamilton–Jacobi–Einstein equation 9533:The Einstein Theory of Relativity 9178: 8140:Mathematics of general relativity 8135:Hamilton–Jacobi–Einstein equation 7578:is zero, and the second that its 7559:where the semicolon represents a 6775:{\displaystyle R_{\mu \nu }=0\,.} 4714:In tensor notation, these become 1886:symmetric 4 × 4 tensors 1739:Newtonian constant of gravitation 1131:Mathematics of general relativity 1056:Mathematics of general relativity 10772: 10771: 9912: 9911: 9648:Einstein Prize for Laser Science 7737:. It leads to the prediction of 2939:In the Einstein field equations 2564:trace with respect to the metric 1723:{\displaystyle {\textrm {m/J}},} 1166: 1165: 1152: 980: 841:The equations were published by 769: 768: 755: 85: 9693:Einstein: His Life and Universe 9020: 8967: 8924: 8881: 8834: 8785: 8758: 8605: 8560: 8535: 8442: 4412: 1598:Einstein gravitational constant 10811:Partial differential equations 10101:Mass–energy equivalence (E=mc) 9959: 9315:Mass–energy equivalence (E=mc) 9203:Caltech Tutorial on Relativity 9163:. Cambridge University Press. 8614:Particle Physics and Cosmology 8546:. Princeton University Press. 8350: 8313: 8286: 8060: 8054: 7812: 7806: 7354: 7339: 6919:If the energy–momentum tensor 6740:trace-reversed field equations 6717:If the energy–momentum tensor 5616:which is satisfied by letting 4684: 4678: 4672: 4643: 4631: 4625: 4614: 4564: 4521: 3397: 3382: 3101: 3089: 2371: 2362: 2356: 2347: 2286: 2277: 2108: 2099: 2060: 2024: 2012: 2003: 1956:partial differential equations 1653: 1647: 901:partial differential equations 13: 1: 9076: 8130:History of general relativity 7724:post-Newtonian approximations 3363:differential Bianchi identity 2422:, whereas Weinberg (1972) is 9083:General relativity resources 8689:Wahl, Nicolle (2005-11-22). 8125:General relativity resources 7703: 4449:The correspondence principle 3778:using the definition of the 3033:showed that our universe is 2914:{\displaystyle g_{\mu \nu }} 2550:is used rather than the MTW 1947:{\displaystyle g_{\mu \nu }} 1558:{\displaystyle R_{\mu \nu }} 1389:{\displaystyle T_{\mu \nu }} 1355:{\displaystyle g_{\mu \nu }} 1321:{\displaystyle G_{\mu \nu }} 875:Analogously to the way that 816:general theory of relativity 7: 10116:Relativistic Doppler effect 9638:Albert Einstein Peace Prize 9415:Unsuccessful investigations 9191:Encyclopedia of Mathematics 8097: 7282:covariant Maxwell equations 3269: 3056:observations have shown an 2778:dimensions this reduces to 2416:Misner, Thorne, and Wheeler 1968:Misner, Thorne, and Wheeler 1917:is everywhere zero) define 1081:Introduction to gravitation 920:Newton's law of gravitation 266:Gravitational time dilation 10: 10827: 10587:In computational physics: 10111:Relativity of simultaneity 9565:Picasso at the Lapin Agile 9503:Russell–Einstein Manifesto 9375:Bose–Einstein correlations 9006:10.1103/physrevd.88.063518 8953:10.1088/0264-9381/14/8/025 8426:Cambridge University Press 7760: 7707: 7563:, and the brackets denote 7095:Einstein–Maxwell equations 6912: 6909:Einstein–Maxwell equations 4085:which can be rewritten as 2932: 386:Mathisson–Papapetrou–Dixon 227:Pseudo-Riemannian manifold 29: 10769: 10601: 10466: 10438: 10424:Lense–Thirring precession 10307: 10256: 10218: 10197: 10186: 10144: 10088: 10072: 10014: 10006:Doubly special relativity 9978: 9967: 9907: 9821: 9730: 9661: 9615: 9512: 9473:The Meaning of Relativity 9438: 9290: 9137:. John Wiley & Sons. 9133:Gravitation and Cosmology 9059:10.1007/s10714-006-0310-5 8910:10.1088/0264-9381/3/6/011 8744:10.1142/S0217751X02013113 8449:Rendall, Alan D. (2005). 8392:Weinberg, Steven (1993). 8172:Einstein, Albert (1916). 8105:Conformastatic spacetimes 4463:slow-motion approximation 4441:, which is linear in the 4186:A final contraction with 3252:where it is assumed that 2929:The cosmological constant 1111:Derivations of relativity 864:) with the local energy, 845:in 1915 in the form of a 830:) relate the geometry of 10284:Post-Newtonian formalism 10274:Einstein field equations 10210:Mathematical formulation 10034:Hyperbolic orthogonality 9864:Albert Einstein Archives 9794:Bernhard Caesar Einstein 9541:Relics: Einstein's Brain 9489:The Evolution of Physics 9370:Bose–Einstein statistics 9365:Bose–Einstein condensate 9345:Einstein field equations 8461:(1). Article number: 6. 8200:10.1002/andp.19163540702 8156: 4465:. In fact, the constant 4459:weak-field approximation 3152:This tensor describes a 3041:Einstein then abandoned 3011:the term containing the 1875:mathematical formulation 1413:{\displaystyle \Lambda } 1061:Einstein field equations 849:which related the local 820:Einstein field equations 391:Hamilton–Jacobi–Einstein 371:Einstein field equations 194:Mathematical formulation 59:{\displaystyle E=mc^{2}} 9995:Galilean transformation 9986:Principle of relativity 9874:Einstein Papers Project 9390:Einstein–de Haas effect 9099:Wheeler, John Archibald 8110:Einstein–Hilbert action 8092:Einstein-Hilbert action 7785:gravitational radiation 4489:Gauss's law for gravity 4455:Newton's law of gravity 3680:which is equivalent to 3168:and isotropic pressure 2558:Equivalent formulations 2418:classify themselves as 2414:With these definitions 1437:{\displaystyle \kappa } 1026:Lorentz transformations 834:to the distribution of 68:Mass–energy equivalence 10080:Lorentz transformation 9806:Thomas Martin Einstein 9714:Introducing Relativity 9686:Einstein for Beginners 9589:Einstein and Eddington 9380:Einstein–Cartan theory 8765:Brown, Harvey (2005). 8074: 7924: 7693: 7553: 7272: 7087: 6871:Schwarzschild solution 6853: 6776: 6726:vacuum field equations 6714: 6705:Vacuum field equations 6695: 6643: 6582: 6425: 6364: 6206: 6010: 5880: 5749: 5666: 5610: 5505:Our assumptions force 5497: 5325: 5242: 5176: 5079: 4932: 4840: 4706: 4585: 4435:Schrödinger's equation 4399: 4323: 4271: 4178: 4079: 4010: 3900: 3772: 3674: 3548: 3412: 3349: 3246: 3144: 3005: 2915: 2883: 2763: 2617: 2534: 2406: 2316: 1948: 1853: 1724: 1697: 1583: 1567:Ricci curvature tensor 1559: 1526: 1438: 1414: 1390: 1356: 1322: 1295: 1277: 877:electromagnetic fields 261:Gravitational redshift 154: 60: 10548:Weyl−Lewis−Papapetrou 10289:Raychaudhuri equation 10228:Equivalence principle 9879:Einstein refrigerator 9869:Einstein's Blackboard 9748:(second wife; cousin) 9679:Einstein and Religion 9628:Albert Einstein Medal 9623:Albert Einstein Award 9481:The World as I See It 9420:Wave–particle duality 9400:Bohr–Einstein debates 9360:Cosmological constant 9340:Equivalence principle 9330:Einstein coefficients 8227:(November 25, 1915). 8115:Equivalence principle 8075: 7925: 7775:approximates that of 7771:is very weak and the 7694: 7554: 7273: 7099:cosmological constant 7088: 6928:electromagnetic field 6897:, are referred to as 6854: 6777: 6712: 6696: 6644: 6583: 6426: 6365: 6207: 6011: 5881: 5750: 5667: 5611: 5498: 5331:where two factors of 5326: 5243: 5177: 5080: 4933: 4841: 4707: 4586: 4400: 4324: 4272: 4179: 4080: 4011: 3901: 3773: 3675: 3549: 3413: 3350: 3262:is defined as above. 3247: 3145: 3013:cosmological constant 3006: 2935:Cosmological constant 2916: 2884: 2764: 2618: 2535: 2407: 2317: 1949: 1854: 1725: 1698: 1584: 1560: 1527: 1439: 1422:cosmological constant 1415: 1391: 1357: 1323: 1289: 1278: 1121:Differential geometry 1021:Equivalence principle 933:. Special classes of 891:, the EFE relate the 549:Weyl−Lewis−Papapetrou 504:Kerr–Newman–de Sitter 324:Einstein–Rosen bridge 256:Gravitational lensing 212:Equivalence principle 155: 61: 10801:Equations of physics 10589:Numerical relativity 10430:pulsar timing arrays 9758:Hans Albert Einstein 9707:I Am Albert Einstein 9653:Einstein Prize (APS) 9325:Photoelectric effect 9298:Theory of relativity 9186:"Einstein equations" 9159:Cosmological Physics 8714:Int. J. Mod. Phys. A 8145:Numerical relativity 7938: 7800: 7597: 7561:covariant derivative 7288: 7108: 6941: 6899:Ricci-flat manifolds 6788: 6746: 6653: 6649:which will occur if 6592: 6437: 6380: 6218: 6020: 5892: 5759: 5678: 5620: 5519: 5357: 5256: 5186: 5091: 4949: 4852: 4718: 4603: 4494: 4335: 4285: 4194: 4089: 4022: 3910: 3789: 3684: 3562: 3436: 3368: 3285: 3181: 3072: 2943: 2895: 2782: 2637: 2570: 2441: 2328: 1976: 1928: 1763: 1707: 1607: 1573: 1539: 1458: 1428: 1404: 1398:stress–energy tensor 1370: 1336: 1302: 1219: 1099:Relevant mathematics 939:rotating black holes 870:stress–energy tensor 828:Einstein's equations 479:Einstein–Rosen waves 205:Fundamental concepts 94: 34: 10481:Friedmann equations 10375:Hulse–Taylor binary 10337:Gravitational waves 10233:Riemannian geometry 10059:Proper acceleration 10044:Maxwell's equations 9990:Galilean relativity 9410:Thought experiments 9051:2006GReGr..38.1233K 8998:2013PhRvD..88f3518K 8945:1997CQGra..14.2281L 8933:Class. Quantum Grav 8902:1986CQGra...3.1105H 8890:Class. Quantum Grav 8859:1969CMaPh..12..108E 8811:1977IJTP...16..561T 8768:Physical Relativity 8736:2002IJMPA..17S.180T 8616:. New York: Wiley. 8583:1990Natur.348..705E 8486:10.12942/lrr-2005-6 8477:2005LRR.....8....6R 8455:Living Rev. Relativ 8192:1916AnP...354..769E 7769:gravitational field 7580:exterior derivative 7565:anti-symmetrization 6479: 6416: 6356: 6338: 6317: 6302: 6281: 6254: 5390: 5320: 5016: 4938:for some constant, 4599:particle satisfies 4419:Maxwell's equations 3106: 2247: 2229: 2208: 2190: 2169: 2142: 968:Part of a series on 955:gravitational waves 889:Maxwell's equations 433:Kaluza–Klein theory 319:Minkowski spacetime 271:Gravitational waves 10806:General relativity 10530:Reissner–Nordström 10448:Brans–Dicke theory 10279:Linearized gravity 10106:Length contraction 10024:Frame of reference 10001:Special relativity 9859:Things named after 9721:Subtle is the Lord 9544:(1994 documentary) 9536:(1923 documentary) 9528:(1922 documentary) 9425:Gravitational wave 9308:General relativity 9303:Special relativity 9091:Misner, Charles W. 8867:10.1007/BF01645908 8819:10.1007/BF01811088 8652:(April 28, 1970). 8179:Annalen der Physik 8070: 7970: 7932:Levi-Civita symbol 7920: 7828: 7763:Linearized gravity 7757:The linearized EFE 7689: 7549: 7547: 7464: 7376: 7280:Additionally, the 7268: 7219: 7083: 7037: 6903:Einstein manifolds 6849: 6772: 6715: 6691: 6639: 6605: 6578: 6560: 6527: 6459: 6421: 6396: 6360: 6339: 6321: 6303: 6285: 6258: 6234: 6202: 6180: 6111: 6054: 6006: 5876: 5745: 5725: 5662: 5606: 5493: 5404: 5376: 5321: 5306: 5238: 5172: 5075: 4999: 4928: 4905: 4836: 4834: 4702: 4581: 4474: 4457:by using both the 4453:The EFE reduce to 4425:are linear in the 4395: 4319: 4267: 4229: 4174: 4130: 4075: 4006: 3896: 3768: 3670: 3544: 3408: 3359: 3345: 3256:has SI unit m and 3242: 3140: 3075: 3001: 2911: 2879: 2759: 2613: 2530: 2402: 2312: 2310: 2230: 2212: 2191: 2173: 2146: 2119: 1944: 1919:Einstein manifolds 1897:degrees of freedom 1890:Bianchi identities 1879:general relativity 1849: 1720: 1693: 1579: 1555: 1522: 1434: 1410: 1386: 1352: 1318: 1296: 1273: 1159:Physics portal 1136:Spacetime topology 1116:Spacetime diagrams 1044:General relativity 1016:Spacetime manifold 1009:Spacetime concepts 998:General relativity 993:Special relativity 943:expanding universe 893:spacetime geometry 860:(expressed by the 762:Physics portal 534:Oppenheimer–Snyder 474:Reissner–Nordström 366:Linearized gravity 314:Spacetime diagrams 217:Special relativity 150: 79:General relativity 56: 18:Einstein Equations 10783: 10782: 10597: 10596: 10576:Ozsváth–Schücking 10182: 10181: 10164:Minkowski diagram 10121:Thomas precession 10064:Relativistic mass 9927: 9926: 9829:Awards and honors 9812:Siegbert Einstein 9700:Einstein's Cosmos 9118:978-0-7167-0344-0 9107:. San Francisco: 8793:Trautman, Andrzej 8778:978-0-19-927583-0 8306:978-0-387-69200-5 8064: 7969: 7827: 7463: 7375: 7218: 7161: 7036: 6979: 6883:with a vanishing 6830: 6821: 6685: 6604: 6559: 6526: 6179: 6110: 6053: 5971: 5724: 5403: 5298: 5226: 5202: 5147: 5119: 5069: 5042: 4991: 4944:geodesic equation 4904: 4803: 4675: 4646: 4622: 4617: 4567: 4524: 4472: 4439:quantum mechanics 4228: 4129: 3357: 3237: 3121: 2852: 2733: 2672: 2587: 2468: 1871:geodesic equation 1790: 1714: 1678: 1639: 1582:{\displaystyle R} 1501: 1290:EFE on a wall in 1210: 1209: 1074:Classical gravity 961:Mathematical form 949:, leading to the 913:geodesic equation 812: 811: 445: 444: 331: 330: 16:(Redirected from 10818: 10775: 10774: 10558:van Stockum dust 10330:Two-body problem 10248:Mach's principle 10195: 10194: 10136:Terrell rotation 9976: 9975: 9954: 9947: 9940: 9931: 9930: 9915: 9914: 9814:(distant cousin) 9808:(great-grandson) 9770:Hermann Einstein 9752:Lieserl Einstein 9430:Tea leaf paradox 9277: 9270: 9263: 9254: 9253: 9199: 9174: 9162: 9153:Peacock, John A. 9148: 9136: 9127:Weinberg, Steven 9122: 9071: 9070: 9044: 9035:(8): 1233–1240. 9024: 9018: 9017: 8991: 8971: 8965: 8964: 8928: 8922: 8921: 8896:(6): 1105–1124. 8885: 8879: 8878: 8847:Comm. Math. Phys 8838: 8832: 8830: 8789: 8783: 8782: 8762: 8756: 8755: 8729: 8727:astro-ph/0202008 8709: 8703: 8702: 8686: 8680: 8679: 8677: 8676: 8646: 8640: 8634: 8628: 8627: 8609: 8603: 8602: 8591:10.1038/348705a0 8564: 8558: 8557: 8539: 8533: 8527: 8521: 8520:, p. 501ff. 8515: 8509: 8508: 8498: 8488: 8470: 8446: 8440: 8439: 8417: 8408: 8407: 8389: 8383: 8382: 8354: 8348: 8346: 8332: 8317: 8311: 8310: 8290: 8284: 8283: 8262: 8256: 8250: 8244: 8243: 8241: 8240: 8225:Einstein, Albert 8221: 8215: 8214: 8212: 8202:. Archived from 8169: 8089: 8079: 8077: 8076: 8071: 8065: 8063: 8049: 8048: 8047: 8035: 8034: 8022: 8021: 8009: 8008: 7990: 7989: 7971: 7962: 7958: 7953: 7952: 7929: 7927: 7926: 7921: 7919: 7918: 7906: 7905: 7893: 7892: 7880: 7879: 7867: 7866: 7848: 7847: 7829: 7820: 7781:Minkowski metric 7751:dynamical system 7698: 7696: 7695: 7690: 7688: 7687: 7669: 7668: 7650: 7649: 7631: 7630: 7612: 7611: 7592: 7577: 7558: 7556: 7555: 7550: 7548: 7538: 7534: 7533: 7532: 7511: 7510: 7489: 7488: 7465: 7456: 7450: 7446: 7445: 7444: 7423: 7422: 7401: 7400: 7377: 7368: 7358: 7357: 7319: 7318: 7310: 7309: 7308: 7277: 7275: 7274: 7269: 7264: 7260: 7259: 7258: 7246: 7245: 7233: 7232: 7220: 7211: 7205: 7204: 7199: 7198: 7197: 7186: 7185: 7180: 7179: 7178: 7162: 7160: 7159: 7147: 7142: 7141: 7123: 7122: 7103: 7092: 7090: 7089: 7084: 7082: 7078: 7077: 7076: 7064: 7063: 7051: 7050: 7038: 7029: 7023: 7022: 7017: 7016: 7015: 7004: 7003: 6998: 6997: 6996: 6980: 6978: 6977: 6965: 6956: 6955: 6925: 6896: 6863:vacuum solutions 6858: 6856: 6855: 6850: 6844: 6843: 6831: 6829: 6822: 6814: 6808: 6803: 6802: 6781: 6779: 6778: 6773: 6761: 6760: 6737: 6723: 6700: 6698: 6697: 6692: 6686: 6684: 6683: 6674: 6663: 6648: 6646: 6645: 6640: 6622: 6621: 6606: 6597: 6587: 6585: 6584: 6579: 6577: 6576: 6561: 6552: 6546: 6542: 6541: 6540: 6528: 6519: 6513: 6512: 6492: 6491: 6478: 6473: 6455: 6454: 6430: 6428: 6427: 6422: 6415: 6410: 6392: 6391: 6375: 6369: 6367: 6366: 6361: 6355: 6350: 6337: 6332: 6316: 6311: 6301: 6296: 6280: 6275: 6253: 6248: 6230: 6229: 6211: 6209: 6208: 6203: 6197: 6196: 6181: 6172: 6166: 6162: 6161: 6157: 6156: 6155: 6138: 6134: 6133: 6132: 6112: 6103: 6097: 6096: 6073: 6069: 6068: 6067: 6055: 6046: 6040: 6039: 6015: 6013: 6012: 6007: 6004: 6003: 5985: 5984: 5972: 5970: 5969: 5957: 5949: 5948: 5939: 5938: 5926: 5925: 5913: 5912: 5885: 5883: 5882: 5877: 5871: 5867: 5848: 5847: 5821: 5817: 5798: 5797: 5774: 5773: 5754: 5752: 5751: 5746: 5744: 5740: 5739: 5738: 5726: 5717: 5711: 5710: 5690: 5689: 5671: 5669: 5668: 5663: 5648: 5647: 5632: 5631: 5615: 5613: 5612: 5607: 5604: 5603: 5582: 5578: 5577: 5576: 5553: 5552: 5537: 5536: 5514: 5502: 5500: 5499: 5494: 5488: 5484: 5483: 5482: 5464: 5463: 5442: 5441: 5418: 5417: 5405: 5396: 5389: 5384: 5372: 5371: 5352: 5351: 5349: 5348: 5343: 5340: 5330: 5328: 5327: 5322: 5319: 5314: 5299: 5297: 5296: 5295: 5282: 5281: 5280: 5271: 5270: 5260: 5250:Minkowski metric 5247: 5245: 5244: 5239: 5231: 5227: 5225: 5217: 5209: 5203: 5201: 5190: 5181: 5179: 5178: 5173: 5171: 5167: 5148: 5146: 5138: 5130: 5120: 5118: 5110: 5109: 5108: 5095: 5084: 5082: 5081: 5076: 5070: 5068: 5060: 5059: 5058: 5045: 5043: 5041: 5033: 5032: 5031: 5018: 5015: 5010: 4992: 4990: 4989: 4988: 4975: 4974: 4973: 4964: 4963: 4953: 4941: 4937: 4935: 4934: 4929: 4927: 4923: 4922: 4921: 4906: 4897: 4891: 4890: 4867: 4866: 4845: 4843: 4842: 4837: 4835: 4827: 4826: 4804: 4802: 4801: 4800: 4787: 4786: 4785: 4776: 4775: 4765: 4740: 4739: 4711: 4709: 4708: 4703: 4697: 4693: 4677: 4676: 4668: 4648: 4647: 4639: 4624: 4623: 4618: 4610: 4608: 4594: 4590: 4588: 4587: 4582: 4580: 4576: 4569: 4568: 4560: 4537: 4533: 4526: 4525: 4517: 4506: 4505: 4486: 4479: 4468: 4423:electromagnetism 4404: 4402: 4401: 4396: 4388: 4387: 4379: 4378: 4377: 4360: 4359: 4347: 4346: 4328: 4326: 4325: 4320: 4312: 4311: 4303: 4302: 4301: 4276: 4274: 4273: 4268: 4260: 4259: 4251: 4247: 4243: 4242: 4230: 4221: 4215: 4214: 4189: 4183: 4181: 4180: 4175: 4167: 4166: 4158: 4154: 4150: 4149: 4144: 4143: 4142: 4131: 4122: 4116: 4115: 4110: 4109: 4108: 4084: 4082: 4081: 4076: 4068: 4067: 4056: 4055: 4054: 4037: 4036: 4015: 4013: 4012: 4007: 3999: 3998: 3984: 3983: 3982: 3965: 3964: 3953: 3952: 3951: 3937: 3936: 3925: 3924: 3923: 3905: 3903: 3902: 3897: 3889: 3885: 3884: 3883: 3866: 3865: 3864: 3850: 3849: 3828: 3827: 3804: 3803: 3777: 3775: 3774: 3769: 3761: 3760: 3743: 3742: 3741: 3727: 3726: 3705: 3704: 3679: 3677: 3676: 3671: 3663: 3662: 3645: 3644: 3643: 3629: 3628: 3611: 3610: 3609: 3595: 3594: 3577: 3576: 3575: 3553: 3551: 3550: 3545: 3537: 3536: 3519: 3518: 3517: 3503: 3502: 3485: 3484: 3483: 3469: 3468: 3451: 3450: 3449: 3431: 3421: 3417: 3415: 3414: 3409: 3401: 3400: 3361:Contracting the 3354: 3352: 3351: 3346: 3338: 3337: 3329: 3328: 3327: 3310: 3309: 3297: 3296: 3261: 3255: 3251: 3249: 3248: 3243: 3238: 3230: 3225: 3224: 3223: 3201: 3200: 3199: 3176: 3167: 3149: 3147: 3146: 3141: 3135: 3134: 3122: 3114: 3105: 3104: 3086: 3063: 3044: 3029:observations by 3017: 3010: 3008: 3007: 3002: 2996: 2995: 2977: 2976: 2958: 2957: 2923:Minkowski metric 2920: 2918: 2917: 2912: 2910: 2909: 2888: 2886: 2885: 2880: 2875: 2871: 2870: 2869: 2853: 2845: 2840: 2839: 2816: 2815: 2797: 2796: 2777: 2768: 2766: 2765: 2760: 2755: 2751: 2750: 2749: 2734: 2732: 2718: 2713: 2712: 2689: 2688: 2673: 2671: 2657: 2652: 2651: 2632: 2626: 2622: 2620: 2619: 2614: 2588: 2580: 2553: 2545: 2539: 2537: 2536: 2531: 2526: 2525: 2504: 2503: 2485: 2484: 2469: 2461: 2456: 2455: 2433: 2429: 2425: 2421: 2411: 2409: 2408: 2403: 2401: 2400: 2389: 2388: 2387: 2343: 2342: 2321: 2319: 2318: 2313: 2311: 2307: 2306: 2269: 2268: 2252: 2248: 2246: 2241: 2228: 2223: 2207: 2202: 2189: 2184: 2168: 2163: 2141: 2136: 2091: 2090: 2079: 2078: 2077: 1995: 1994: 1953: 1951: 1950: 1945: 1943: 1942: 1916: 1905: 1858: 1856: 1855: 1850: 1845: 1844: 1826: 1825: 1807: 1806: 1791: 1783: 1778: 1777: 1744: 1736: 1729: 1727: 1726: 1721: 1716: 1715: 1712: 1702: 1700: 1699: 1694: 1689: 1688: 1680: 1679: 1676: 1671: 1670: 1640: 1638: 1637: 1628: 1617: 1591:scalar curvature 1588: 1586: 1585: 1580: 1564: 1562: 1561: 1556: 1554: 1553: 1531: 1529: 1528: 1523: 1518: 1517: 1502: 1494: 1489: 1488: 1473: 1472: 1443: 1441: 1440: 1435: 1419: 1417: 1416: 1411: 1395: 1393: 1392: 1387: 1385: 1384: 1361: 1359: 1358: 1353: 1351: 1350: 1327: 1325: 1324: 1319: 1317: 1316: 1282: 1280: 1279: 1274: 1272: 1271: 1253: 1252: 1234: 1233: 1202: 1195: 1188: 1174: 1169: 1168: 1161: 1157: 1156: 1126:Curved spacetime 984: 965: 964: 858: 857: 826:; also known as 804: 797: 790: 777: 772: 771: 764: 760: 759: 544:van Stockum dust 529:Robertson–Walker 355: 354: 245: 244: 159: 157: 156: 151: 149: 148: 136: 128: 127: 109: 108: 89: 75: 74: 65: 63: 62: 57: 55: 54: 21: 10826: 10825: 10821: 10820: 10819: 10817: 10816: 10815: 10796:Albert Einstein 10786: 10785: 10784: 10779: 10765: 10593: 10497:BKL singularity 10487:Lemaître–Tolman 10462: 10458:Quantum gravity 10440: 10434: 10420:geodetic effect 10394:(together with 10364:LISA Pathfinder 10303: 10252: 10238:Penrose diagram 10220: 10214: 10189: 10178: 10174:Minkowski space 10140: 10084: 10068: 10016: 10010: 9970: 9963: 9958: 9928: 9923: 9903: 9854:Religious views 9849:Political views 9817: 9802:(granddaughter) 9800:Evelyn Einstein 9788:Robert Einstein 9782:Eduard Einstein 9726: 9663: 9657: 9611: 9581:Einstein's Gift 9515: 9508: 9449:Annus mirabilis 9434: 9405:Teleparallelism 9350:Einstein radius 9320:Brownian motion 9286: 9284:Albert Einstein 9281: 9234: 9232:External images 9184: 9181: 9171: 9145: 9119: 9079: 9074: 9025: 9021: 8972: 8968: 8929: 8925: 8886: 8882: 8839: 8835: 8790: 8786: 8779: 8763: 8759: 8720:(S1): 180–196. 8710: 8706: 8687: 8683: 8674: 8672: 8670: 8647: 8643: 8635: 8631: 8624: 8610: 8606: 8565: 8561: 8554: 8540: 8536: 8530:Weinberg (1972) 8528: 8524: 8516: 8512: 8447: 8443: 8436: 8418: 8411: 8404: 8390: 8386: 8371: 8355: 8351: 8334: 8320: 8318: 8314: 8307: 8291: 8287: 8280: 8263: 8259: 8255:, p. 916 . 8251: 8247: 8238: 8236: 8222: 8218: 8206: 8170: 8163: 8159: 8154: 8100: 8083: 8050: 8040: 8036: 8027: 8023: 8014: 8010: 7995: 7991: 7976: 7972: 7960: 7959: 7957: 7945: 7941: 7939: 7936: 7935: 7911: 7907: 7898: 7894: 7885: 7881: 7872: 7868: 7853: 7849: 7834: 7830: 7818: 7801: 7798: 7797: 7793: 7791:Polynomial form 7777:Minkowski space 7765: 7759: 7728:exact solutions 7712: 7706: 7677: 7673: 7658: 7654: 7639: 7635: 7620: 7616: 7604: 7600: 7598: 7595: 7594: 7591: 7587: 7575: 7546: 7545: 7519: 7515: 7497: 7493: 7475: 7471: 7470: 7466: 7454: 7431: 7427: 7409: 7405: 7387: 7383: 7382: 7378: 7366: 7359: 7338: 7334: 7331: 7330: 7320: 7311: 7301: 7297: 7296: 7295: 7291: 7289: 7286: 7285: 7251: 7247: 7238: 7234: 7225: 7221: 7209: 7200: 7193: 7189: 7188: 7187: 7181: 7174: 7170: 7169: 7168: 7167: 7163: 7155: 7151: 7146: 7134: 7130: 7115: 7111: 7109: 7106: 7105: 7101: 7069: 7065: 7056: 7052: 7043: 7039: 7027: 7018: 7011: 7007: 7006: 7005: 6999: 6992: 6988: 6987: 6986: 6985: 6981: 6973: 6969: 6964: 6948: 6944: 6942: 6939: 6938: 6924: 6920: 6917: 6911: 6893: 6888: 6867:Minkowski space 6836: 6832: 6813: 6812: 6807: 6795: 6791: 6789: 6786: 6785: 6753: 6749: 6747: 6744: 6743: 6734: 6729: 6722: 6718: 6707: 6702: 6679: 6675: 6664: 6662: 6654: 6651: 6650: 6617: 6613: 6595: 6593: 6590: 6589: 6572: 6568: 6550: 6536: 6532: 6517: 6508: 6504: 6503: 6499: 6487: 6483: 6474: 6463: 6444: 6440: 6438: 6435: 6434: 6411: 6400: 6387: 6383: 6381: 6378: 6377: 6373: 6351: 6343: 6333: 6325: 6312: 6307: 6297: 6289: 6276: 6262: 6249: 6238: 6225: 6221: 6219: 6216: 6215: 6192: 6188: 6170: 6151: 6147: 6143: 6139: 6128: 6124: 6117: 6113: 6101: 6092: 6088: 6084: 6080: 6063: 6059: 6044: 6035: 6031: 6030: 6026: 6021: 6018: 6017: 5999: 5995: 5980: 5976: 5965: 5961: 5956: 5944: 5940: 5934: 5930: 5918: 5914: 5905: 5901: 5893: 5890: 5889: 5843: 5839: 5835: 5831: 5793: 5789: 5788: 5784: 5766: 5762: 5760: 5757: 5756: 5734: 5730: 5715: 5706: 5702: 5701: 5697: 5685: 5681: 5679: 5676: 5675: 5643: 5639: 5627: 5623: 5621: 5618: 5617: 5593: 5589: 5566: 5562: 5558: 5554: 5545: 5541: 5529: 5525: 5520: 5517: 5516: 5506: 5472: 5468: 5450: 5446: 5428: 5424: 5423: 5419: 5410: 5406: 5394: 5385: 5380: 5364: 5360: 5358: 5355: 5354: 5344: 5341: 5336: 5335: 5333: 5332: 5315: 5310: 5291: 5287: 5283: 5276: 5272: 5266: 5262: 5261: 5259: 5257: 5254: 5253: 5218: 5210: 5208: 5204: 5194: 5189: 5187: 5184: 5183: 5139: 5131: 5129: 5128: 5124: 5111: 5104: 5100: 5096: 5094: 5092: 5089: 5088: 5061: 5054: 5050: 5046: 5044: 5034: 5027: 5023: 5019: 5017: 5011: 5003: 4984: 4980: 4976: 4969: 4965: 4959: 4955: 4954: 4952: 4950: 4947: 4946: 4939: 4914: 4910: 4895: 4883: 4879: 4878: 4874: 4859: 4855: 4853: 4850: 4849: 4833: 4832: 4819: 4815: 4805: 4796: 4792: 4788: 4781: 4777: 4771: 4767: 4766: 4764: 4761: 4760: 4741: 4729: 4725: 4721: 4719: 4716: 4715: 4667: 4666: 4665: 4661: 4638: 4637: 4609: 4607: 4606: 4604: 4601: 4600: 4592: 4559: 4558: 4557: 4553: 4516: 4515: 4514: 4510: 4501: 4497: 4495: 4492: 4491: 4481: 4477: 4466: 4451: 4431:magnetic fields 4415: 4406: 4380: 4370: 4366: 4365: 4364: 4352: 4348: 4342: 4338: 4336: 4333: 4332: 4304: 4294: 4290: 4289: 4288: 4286: 4283: 4282: 4279:Einstein tensor 4252: 4235: 4231: 4219: 4207: 4203: 4202: 4198: 4197: 4195: 4192: 4191: 4187: 4159: 4145: 4138: 4134: 4133: 4132: 4120: 4111: 4104: 4100: 4099: 4098: 4097: 4093: 4092: 4090: 4087: 4086: 4057: 4050: 4046: 4045: 4044: 4029: 4025: 4023: 4020: 4019: 3985: 3975: 3971: 3970: 3969: 3954: 3947: 3943: 3942: 3941: 3926: 3919: 3915: 3914: 3913: 3911: 3908: 3907: 3867: 3860: 3856: 3855: 3854: 3836: 3832: 3814: 3810: 3809: 3805: 3796: 3792: 3790: 3787: 3786: 3744: 3737: 3733: 3732: 3731: 3713: 3709: 3691: 3687: 3685: 3682: 3681: 3646: 3639: 3635: 3634: 3633: 3612: 3605: 3601: 3600: 3599: 3578: 3571: 3567: 3566: 3565: 3563: 3560: 3559: 3520: 3513: 3509: 3508: 3507: 3486: 3479: 3475: 3474: 3473: 3452: 3445: 3441: 3440: 3439: 3437: 3434: 3433: 3428: 3423: 3419: 3375: 3371: 3369: 3366: 3365: 3330: 3320: 3316: 3315: 3314: 3302: 3298: 3292: 3288: 3286: 3283: 3282: 3277: 3272: 3257: 3253: 3229: 3213: 3212: 3208: 3189: 3188: 3184: 3182: 3179: 3178: 3175: 3169: 3166: 3160: 3127: 3123: 3113: 3088: 3087: 3079: 3073: 3070: 3069: 3061: 3045:, remarking to 3042: 3015: 2988: 2984: 2969: 2965: 2950: 2946: 2944: 2941: 2940: 2937: 2931: 2902: 2898: 2896: 2893: 2892: 2862: 2858: 2844: 2832: 2828: 2827: 2823: 2808: 2804: 2789: 2785: 2783: 2780: 2779: 2772: 2742: 2738: 2722: 2717: 2705: 2701: 2700: 2696: 2681: 2677: 2661: 2656: 2644: 2640: 2638: 2635: 2634: 2628: 2624: 2579: 2571: 2568: 2567: 2560: 2551: 2548:sign convention 2543: 2518: 2514: 2496: 2492: 2477: 2473: 2460: 2448: 2444: 2442: 2439: 2438: 2431: 2427: 2423: 2419: 2390: 2383: 2379: 2378: 2377: 2335: 2331: 2329: 2326: 2325: 2309: 2308: 2299: 2295: 2270: 2261: 2257: 2254: 2253: 2242: 2234: 2224: 2216: 2203: 2195: 2185: 2177: 2164: 2150: 2137: 2123: 2118: 2114: 2092: 2080: 2073: 2069: 2068: 2067: 2064: 2063: 1996: 1987: 1983: 1979: 1977: 1974: 1973: 1964: 1962:Sign convention 1935: 1931: 1929: 1926: 1925: 1915: 1907: 1903: 1837: 1833: 1818: 1814: 1799: 1795: 1782: 1770: 1766: 1764: 1761: 1760: 1742: 1734: 1711: 1710: 1708: 1705: 1704: 1681: 1675: 1674: 1673: 1663: 1659: 1633: 1629: 1618: 1616: 1608: 1605: 1604: 1574: 1571: 1570: 1546: 1542: 1540: 1537: 1536: 1510: 1506: 1493: 1481: 1477: 1465: 1461: 1459: 1456: 1455: 1449:Einstein tensor 1429: 1426: 1425: 1405: 1402: 1401: 1377: 1373: 1371: 1368: 1367: 1343: 1339: 1337: 1334: 1333: 1330:Einstein tensor 1309: 1305: 1303: 1300: 1299: 1264: 1260: 1245: 1241: 1226: 1222: 1220: 1217: 1216: 1206: 1177: 1164: 1151: 1150: 1142: 1141: 1140: 1100: 1092: 1091: 1090: 1075: 1067: 1066: 1065: 1045: 1037: 1036: 1035: 1031:Minkowski space 1010: 1002: 963: 935:exact solutions 862:Einstein tensor 852: 851: 847:tensor equation 843:Albert Einstein 808: 767: 754: 753: 746: 745: 569: 568: 559: 558: 514:Lemaître–Tolman 459: 458: 447: 446: 438:Quantum gravity 425:Advanced theory 352: 351: 350: 333: 332: 281:Geodetic effect 242: 241: 232: 231: 207: 206: 190: 160: 141: 137: 132: 120: 116: 101: 97: 95: 92: 91: 71: 50: 46: 35: 32: 31: 28: 23: 22: 15: 12: 11: 5: 10824: 10814: 10813: 10808: 10803: 10798: 10781: 10780: 10770: 10767: 10766: 10764: 10763: 10756: 10751: 10746: 10741: 10736: 10731: 10726: 10721: 10716: 10711: 10706: 10701: 10696: 10691: 10686: 10684:Choquet-Bruhat 10681: 10676: 10671: 10666: 10661: 10656: 10651: 10646: 10641: 10636: 10631: 10626: 10621: 10616: 10611: 10605: 10603: 10599: 10598: 10595: 10594: 10592: 10591: 10584: 10583: 10578: 10573: 10566: 10565: 10560: 10555: 10550: 10545: 10536:Axisymmetric: 10533: 10532: 10527: 10521: 10510: 10509: 10504: 10499: 10494: 10489: 10484: 10475:Cosmological: 10472: 10470: 10464: 10463: 10461: 10460: 10455: 10450: 10444: 10442: 10436: 10435: 10433: 10432: 10427: 10416:frame-dragging 10413: 10408: 10403: 10400:Einstein rings 10396:Einstein cross 10389: 10378: 10377: 10372: 10366: 10361: 10356: 10343: 10333: 10332: 10327: 10322: 10317: 10311: 10309: 10305: 10304: 10302: 10301: 10299:Ernst equation 10296: 10291: 10286: 10281: 10276: 10271: 10269:BSSN formalism 10266: 10260: 10258: 10254: 10253: 10251: 10250: 10245: 10240: 10235: 10230: 10224: 10222: 10216: 10215: 10213: 10212: 10207: 10201: 10199: 10192: 10184: 10183: 10180: 10179: 10177: 10176: 10171: 10166: 10161: 10156: 10150: 10148: 10142: 10141: 10139: 10138: 10133: 10128: 10126:Ladder paradox 10123: 10118: 10113: 10108: 10103: 10098: 10092: 10090: 10086: 10085: 10083: 10082: 10076: 10074: 10070: 10069: 10067: 10066: 10061: 10056: 10051: 10046: 10041: 10036: 10031: 10029:Speed of light 10026: 10020: 10018: 10012: 10011: 10009: 10008: 10003: 9998: 9992: 9982: 9980: 9973: 9965: 9964: 9957: 9956: 9949: 9942: 9934: 9925: 9924: 9922: 9921: 9908: 9905: 9904: 9902: 9901: 9896: 9891: 9886: 9881: 9876: 9871: 9866: 9861: 9856: 9851: 9846: 9841: 9836: 9831: 9825: 9823: 9819: 9818: 9816: 9815: 9809: 9803: 9797: 9791: 9785: 9779: 9773: 9767: 9761: 9755: 9749: 9743: 9736: 9734: 9728: 9727: 9725: 9724: 9717: 9710: 9703: 9696: 9689: 9682: 9675: 9667: 9665: 9659: 9658: 9656: 9655: 9650: 9645: 9640: 9635: 9630: 9625: 9619: 9617: 9613: 9612: 9610: 9609: 9601: 9593: 9592:(2008 TV film) 9585: 9577: 9569: 9561: 9557:Young Einstein 9553: 9549:Insignificance 9545: 9537: 9529: 9520: 9518: 9510: 9509: 9507: 9506: 9500: 9497:Why Socialism? 9493: 9485: 9477: 9469: 9461: 9454: 9444: 9442: 9436: 9435: 9433: 9432: 9427: 9422: 9417: 9412: 9407: 9402: 9397: 9392: 9387: 9382: 9377: 9372: 9367: 9362: 9357: 9352: 9347: 9342: 9337: 9335:Einstein solid 9332: 9327: 9322: 9317: 9312: 9311: 9310: 9305: 9294: 9292: 9288: 9287: 9280: 9279: 9272: 9265: 9257: 9251: 9250: 9240: 9233: 9230: 9229: 9228: 9222: 9212: 9206: 9200: 9180: 9179:External links 9177: 9176: 9175: 9170:978-0521410724 9169: 9149: 9143: 9123: 9117: 9095:Thorne, Kip S. 9078: 9075: 9073: 9072: 9029:Gen. Rel. Grav 9019: 8966: 8923: 8880: 8853:(2): 108–141. 8833: 8805:(9): 561–565. 8784: 8777: 8757: 8704: 8701:on 2007-03-07. 8681: 8668: 8641: 8637:Peacock (1999) 8629: 8622: 8604: 8559: 8552: 8534: 8522: 8510: 8441: 8434: 8409: 8402: 8384: 8369: 8349: 8312: 8305: 8285: 8278: 8257: 8245: 8216: 8213:on 2012-02-06. 8160: 8158: 8155: 8153: 8152: 8150:Ricci calculus 8147: 8142: 8137: 8132: 8127: 8122: 8117: 8112: 8107: 8101: 8099: 8096: 8069: 8062: 8059: 8056: 8053: 8046: 8043: 8039: 8033: 8030: 8026: 8020: 8017: 8013: 8007: 8004: 8001: 7998: 7994: 7988: 7985: 7982: 7979: 7975: 7968: 7965: 7956: 7951: 7948: 7944: 7917: 7914: 7910: 7904: 7901: 7897: 7891: 7888: 7884: 7878: 7875: 7871: 7865: 7862: 7859: 7856: 7852: 7846: 7843: 7840: 7837: 7833: 7826: 7823: 7817: 7814: 7811: 7808: 7805: 7792: 7789: 7761:Main article: 7758: 7755: 7708:Main article: 7705: 7702: 7686: 7683: 7680: 7676: 7672: 7667: 7664: 7661: 7657: 7653: 7648: 7645: 7642: 7638: 7634: 7629: 7626: 7623: 7619: 7615: 7610: 7607: 7603: 7589: 7584:Poincaré lemma 7544: 7541: 7537: 7531: 7528: 7525: 7522: 7518: 7514: 7509: 7506: 7503: 7500: 7496: 7492: 7487: 7484: 7481: 7478: 7474: 7469: 7462: 7459: 7453: 7449: 7443: 7440: 7437: 7434: 7430: 7426: 7421: 7418: 7415: 7412: 7408: 7404: 7399: 7396: 7393: 7390: 7386: 7381: 7374: 7371: 7365: 7362: 7360: 7356: 7353: 7350: 7347: 7344: 7341: 7337: 7333: 7332: 7329: 7326: 7323: 7321: 7317: 7314: 7307: 7304: 7300: 7294: 7293: 7267: 7263: 7257: 7254: 7250: 7244: 7241: 7237: 7231: 7228: 7224: 7217: 7214: 7208: 7203: 7196: 7192: 7184: 7177: 7173: 7166: 7158: 7154: 7150: 7145: 7140: 7137: 7133: 7129: 7126: 7121: 7118: 7114: 7081: 7075: 7072: 7068: 7062: 7059: 7055: 7049: 7046: 7042: 7035: 7032: 7026: 7021: 7014: 7010: 7002: 6995: 6991: 6984: 6976: 6972: 6968: 6963: 6959: 6954: 6951: 6947: 6934:, i.e. if the 6926:is that of an 6922: 6910: 6907: 6891: 6848: 6842: 6839: 6835: 6828: 6825: 6820: 6817: 6811: 6806: 6801: 6798: 6794: 6771: 6767: 6764: 6759: 6756: 6752: 6732: 6720: 6706: 6703: 6690: 6682: 6678: 6673: 6670: 6667: 6661: 6658: 6637: 6634: 6631: 6628: 6625: 6620: 6616: 6612: 6609: 6603: 6600: 6575: 6571: 6567: 6564: 6558: 6555: 6549: 6545: 6539: 6535: 6531: 6525: 6522: 6516: 6511: 6507: 6502: 6498: 6495: 6490: 6486: 6482: 6477: 6472: 6469: 6466: 6462: 6458: 6453: 6450: 6447: 6443: 6420: 6414: 6409: 6406: 6403: 6399: 6395: 6390: 6386: 6359: 6354: 6349: 6346: 6342: 6336: 6331: 6328: 6324: 6320: 6315: 6310: 6306: 6300: 6295: 6292: 6288: 6284: 6279: 6274: 6271: 6268: 6265: 6261: 6257: 6252: 6247: 6244: 6241: 6237: 6233: 6228: 6224: 6201: 6195: 6191: 6187: 6184: 6178: 6175: 6169: 6165: 6160: 6154: 6150: 6146: 6142: 6137: 6131: 6127: 6123: 6120: 6116: 6109: 6106: 6100: 6095: 6091: 6087: 6083: 6079: 6076: 6072: 6066: 6062: 6058: 6052: 6049: 6043: 6038: 6034: 6029: 6025: 6002: 5998: 5994: 5991: 5988: 5983: 5979: 5975: 5968: 5964: 5960: 5955: 5952: 5947: 5943: 5937: 5933: 5929: 5924: 5921: 5917: 5911: 5908: 5904: 5900: 5897: 5875: 5870: 5866: 5863: 5860: 5857: 5854: 5851: 5846: 5842: 5838: 5834: 5830: 5827: 5824: 5820: 5816: 5813: 5810: 5807: 5804: 5801: 5796: 5792: 5787: 5783: 5780: 5777: 5772: 5769: 5765: 5743: 5737: 5733: 5729: 5723: 5720: 5714: 5709: 5705: 5700: 5696: 5693: 5688: 5684: 5661: 5657: 5654: 5651: 5646: 5642: 5638: 5635: 5630: 5626: 5602: 5599: 5596: 5592: 5588: 5585: 5581: 5575: 5572: 5569: 5565: 5561: 5557: 5551: 5548: 5544: 5540: 5535: 5532: 5528: 5524: 5492: 5487: 5481: 5478: 5475: 5471: 5467: 5462: 5459: 5456: 5453: 5449: 5445: 5440: 5437: 5434: 5431: 5427: 5422: 5416: 5413: 5409: 5402: 5399: 5393: 5388: 5383: 5379: 5375: 5370: 5367: 5363: 5318: 5313: 5309: 5305: 5302: 5294: 5290: 5286: 5279: 5275: 5269: 5265: 5237: 5234: 5230: 5224: 5221: 5216: 5213: 5207: 5200: 5197: 5193: 5170: 5166: 5163: 5160: 5157: 5154: 5151: 5145: 5142: 5137: 5134: 5127: 5123: 5117: 5114: 5107: 5103: 5099: 5074: 5067: 5064: 5057: 5053: 5049: 5040: 5037: 5030: 5026: 5022: 5014: 5009: 5006: 5002: 4998: 4995: 4987: 4983: 4979: 4972: 4968: 4962: 4958: 4926: 4920: 4917: 4913: 4909: 4903: 4900: 4894: 4889: 4886: 4882: 4877: 4873: 4870: 4865: 4862: 4858: 4831: 4825: 4822: 4818: 4814: 4811: 4808: 4806: 4799: 4795: 4791: 4784: 4780: 4774: 4770: 4763: 4762: 4759: 4756: 4753: 4750: 4747: 4744: 4742: 4738: 4735: 4732: 4728: 4724: 4723: 4701: 4696: 4692: 4689: 4686: 4683: 4680: 4674: 4671: 4664: 4660: 4657: 4654: 4651: 4645: 4642: 4636: 4633: 4630: 4627: 4621: 4616: 4613: 4579: 4575: 4572: 4566: 4563: 4556: 4552: 4549: 4546: 4543: 4540: 4536: 4532: 4529: 4523: 4520: 4513: 4509: 4504: 4500: 4471: 4450: 4447: 4414: 4411: 4394: 4391: 4386: 4383: 4376: 4373: 4369: 4363: 4358: 4355: 4351: 4345: 4341: 4318: 4315: 4310: 4307: 4300: 4297: 4293: 4266: 4263: 4258: 4255: 4250: 4246: 4241: 4238: 4234: 4227: 4224: 4218: 4213: 4210: 4206: 4201: 4173: 4170: 4165: 4162: 4157: 4153: 4148: 4141: 4137: 4128: 4125: 4119: 4114: 4107: 4103: 4096: 4074: 4071: 4066: 4063: 4060: 4053: 4049: 4043: 4040: 4035: 4032: 4028: 4005: 4002: 3997: 3994: 3991: 3988: 3981: 3978: 3974: 3968: 3963: 3960: 3957: 3950: 3946: 3940: 3935: 3932: 3929: 3922: 3918: 3895: 3892: 3888: 3882: 3879: 3876: 3873: 3870: 3863: 3859: 3853: 3848: 3845: 3842: 3839: 3835: 3831: 3826: 3823: 3820: 3817: 3813: 3808: 3802: 3799: 3795: 3767: 3764: 3759: 3756: 3753: 3750: 3747: 3740: 3736: 3730: 3725: 3722: 3719: 3716: 3712: 3708: 3703: 3700: 3697: 3694: 3690: 3669: 3666: 3661: 3658: 3655: 3652: 3649: 3642: 3638: 3632: 3627: 3624: 3621: 3618: 3615: 3608: 3604: 3598: 3593: 3590: 3587: 3584: 3581: 3574: 3570: 3543: 3540: 3535: 3532: 3529: 3526: 3523: 3516: 3512: 3506: 3501: 3498: 3495: 3492: 3489: 3482: 3478: 3472: 3467: 3464: 3461: 3458: 3455: 3448: 3444: 3426: 3407: 3404: 3399: 3396: 3393: 3390: 3387: 3384: 3381: 3378: 3374: 3356: 3344: 3341: 3336: 3333: 3326: 3323: 3319: 3313: 3308: 3305: 3301: 3295: 3291: 3276: 3273: 3271: 3268: 3241: 3236: 3233: 3228: 3222: 3219: 3216: 3211: 3207: 3204: 3198: 3195: 3192: 3187: 3173: 3164: 3158:energy density 3139: 3133: 3130: 3126: 3120: 3117: 3112: 3109: 3103: 3100: 3097: 3094: 3091: 3085: 3082: 3078: 3039: 3038: 3027: 3000: 2994: 2991: 2987: 2983: 2980: 2975: 2972: 2968: 2964: 2961: 2956: 2953: 2949: 2933:Main article: 2930: 2927: 2908: 2905: 2901: 2878: 2874: 2868: 2865: 2861: 2856: 2851: 2848: 2843: 2838: 2835: 2831: 2826: 2822: 2819: 2814: 2811: 2807: 2803: 2800: 2795: 2792: 2788: 2758: 2754: 2748: 2745: 2741: 2737: 2731: 2728: 2725: 2721: 2716: 2711: 2708: 2704: 2699: 2695: 2692: 2687: 2684: 2680: 2676: 2670: 2667: 2664: 2660: 2655: 2650: 2647: 2643: 2612: 2609: 2606: 2603: 2600: 2597: 2594: 2591: 2586: 2583: 2578: 2575: 2559: 2556: 2529: 2524: 2521: 2517: 2513: 2510: 2507: 2502: 2499: 2495: 2491: 2488: 2483: 2480: 2476: 2472: 2467: 2464: 2459: 2454: 2451: 2447: 2399: 2396: 2393: 2386: 2382: 2376: 2373: 2370: 2367: 2364: 2361: 2358: 2355: 2352: 2349: 2346: 2341: 2338: 2334: 2305: 2302: 2298: 2294: 2291: 2288: 2285: 2282: 2279: 2276: 2273: 2271: 2267: 2264: 2260: 2256: 2255: 2251: 2245: 2240: 2237: 2233: 2227: 2222: 2219: 2215: 2211: 2206: 2201: 2198: 2194: 2188: 2183: 2180: 2176: 2172: 2167: 2162: 2159: 2156: 2153: 2149: 2145: 2140: 2135: 2132: 2129: 2126: 2122: 2117: 2113: 2110: 2107: 2104: 2101: 2098: 2095: 2093: 2089: 2086: 2083: 2076: 2072: 2066: 2065: 2062: 2059: 2056: 2053: 2050: 2047: 2044: 2041: 2038: 2035: 2032: 2029: 2026: 2023: 2020: 2017: 2014: 2011: 2008: 2005: 2002: 1999: 1997: 1993: 1990: 1986: 1982: 1981: 1963: 1960: 1941: 1938: 1934: 1911: 1860: 1859: 1848: 1843: 1840: 1836: 1832: 1829: 1824: 1821: 1817: 1813: 1810: 1805: 1802: 1798: 1794: 1789: 1786: 1781: 1776: 1773: 1769: 1747:speed of light 1731: 1730: 1719: 1692: 1687: 1684: 1669: 1666: 1662: 1658: 1655: 1652: 1649: 1646: 1643: 1636: 1632: 1627: 1624: 1621: 1615: 1612: 1600:is defined as 1578: 1552: 1549: 1545: 1533: 1532: 1521: 1516: 1513: 1509: 1505: 1500: 1497: 1492: 1487: 1484: 1480: 1476: 1471: 1468: 1464: 1451:is defined as 1433: 1409: 1383: 1380: 1376: 1349: 1346: 1342: 1315: 1312: 1308: 1284: 1283: 1270: 1267: 1263: 1259: 1256: 1251: 1248: 1244: 1240: 1237: 1232: 1229: 1225: 1208: 1207: 1205: 1204: 1197: 1190: 1182: 1179: 1178: 1176: 1175: 1162: 1147: 1144: 1143: 1139: 1138: 1133: 1128: 1123: 1118: 1113: 1108: 1102: 1101: 1098: 1097: 1094: 1093: 1089: 1088: 1083: 1077: 1076: 1073: 1072: 1069: 1068: 1064: 1063: 1058: 1053: 1047: 1046: 1043: 1042: 1039: 1038: 1034: 1033: 1028: 1023: 1018: 1012: 1011: 1008: 1007: 1004: 1003: 1001: 1000: 995: 989: 986: 985: 977: 976: 970: 969: 962: 959: 951:linearized EFE 947:flat spacetime 924:speed of light 810: 809: 807: 806: 799: 792: 784: 781: 780: 779: 778: 765: 748: 747: 744: 743: 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: 586: 581: 576: 570: 566: 565: 564: 561: 560: 557: 556: 551: 546: 541: 536: 531: 526: 521: 516: 511: 506: 501: 496: 491: 486: 481: 476: 471: 460: 454: 453: 452: 449: 448: 443: 442: 441: 440: 435: 427: 426: 422: 421: 420: 419: 417:Post-Newtonian 414: 409: 401: 400: 396: 395: 394: 393: 388: 383: 378: 373: 368: 360: 359: 353: 349: 348: 345: 341: 340: 339: 338: 335: 334: 329: 328: 327: 326: 321: 316: 308: 307: 301: 300: 299: 298: 293: 288: 283: 278: 276:Frame-dragging 273: 268: 263: 258: 253: 251:Kepler problem 243: 239: 238: 237: 234: 233: 230: 229: 224: 219: 214: 208: 204: 203: 202: 199: 198: 197: 196: 191: 189: 188: 183: 178: 172: 170: 162: 161: 147: 144: 140: 135: 131: 126: 123: 119: 115: 112: 107: 104: 100: 90: 82: 81: 53: 49: 45: 42: 39: 26: 9: 6: 4: 3: 2: 10823: 10812: 10809: 10807: 10804: 10802: 10799: 10797: 10794: 10793: 10791: 10778: 10768: 10762: 10761: 10757: 10755: 10752: 10750: 10747: 10745: 10742: 10740: 10737: 10735: 10732: 10730: 10727: 10725: 10722: 10720: 10717: 10715: 10712: 10710: 10707: 10705: 10702: 10700: 10697: 10695: 10692: 10690: 10687: 10685: 10682: 10680: 10677: 10675: 10672: 10670: 10669:Chandrasekhar 10667: 10665: 10662: 10660: 10657: 10655: 10652: 10650: 10647: 10645: 10642: 10640: 10637: 10635: 10632: 10630: 10629:Schwarzschild 10627: 10625: 10622: 10620: 10617: 10615: 10612: 10610: 10607: 10606: 10604: 10600: 10590: 10586: 10585: 10582: 10579: 10577: 10574: 10572: 10568: 10567: 10564: 10561: 10559: 10556: 10554: 10551: 10549: 10546: 10543: 10539: 10535: 10534: 10531: 10528: 10525: 10522: 10520: 10516: 10515:Schwarzschild 10512: 10511: 10508: 10505: 10503: 10500: 10498: 10495: 10493: 10490: 10488: 10485: 10482: 10478: 10474: 10473: 10471: 10469: 10465: 10459: 10456: 10454: 10451: 10449: 10446: 10445: 10443: 10437: 10431: 10428: 10425: 10421: 10417: 10414: 10412: 10411:Shapiro delay 10409: 10407: 10404: 10401: 10397: 10393: 10390: 10387: 10383: 10380: 10379: 10376: 10373: 10370: 10367: 10365: 10362: 10360: 10357: 10355: 10354:collaboration 10351: 10347: 10344: 10342: 10338: 10335: 10334: 10331: 10328: 10326: 10323: 10321: 10320:Event horizon 10318: 10316: 10313: 10312: 10310: 10306: 10300: 10297: 10295: 10292: 10290: 10287: 10285: 10282: 10280: 10277: 10275: 10272: 10270: 10267: 10265: 10264:ADM formalism 10262: 10261: 10259: 10255: 10249: 10246: 10244: 10241: 10239: 10236: 10234: 10231: 10229: 10226: 10225: 10223: 10217: 10211: 10208: 10206: 10203: 10202: 10200: 10196: 10193: 10191: 10185: 10175: 10172: 10170: 10169:Biquaternions 10167: 10165: 10162: 10160: 10157: 10155: 10152: 10151: 10149: 10147: 10143: 10137: 10134: 10132: 10129: 10127: 10124: 10122: 10119: 10117: 10114: 10112: 10109: 10107: 10104: 10102: 10099: 10097: 10096:Time dilation 10094: 10093: 10091: 10087: 10081: 10078: 10077: 10075: 10071: 10065: 10062: 10060: 10057: 10055: 10052: 10050: 10049:Proper length 10047: 10045: 10042: 10040: 10037: 10035: 10032: 10030: 10027: 10025: 10022: 10021: 10019: 10013: 10007: 10004: 10002: 9999: 9996: 9993: 9991: 9987: 9984: 9983: 9981: 9977: 9974: 9972: 9966: 9962: 9955: 9950: 9948: 9943: 9941: 9936: 9935: 9932: 9920: 9919: 9910: 9909: 9906: 9900: 9897: 9895: 9892: 9890: 9887: 9885: 9882: 9880: 9877: 9875: 9872: 9870: 9867: 9865: 9862: 9860: 9857: 9855: 9852: 9850: 9847: 9845: 9842: 9840: 9837: 9835: 9832: 9830: 9827: 9826: 9824: 9820: 9813: 9810: 9807: 9804: 9801: 9798: 9795: 9792: 9789: 9786: 9783: 9780: 9777: 9776:Maja Einstein 9774: 9771: 9768: 9765: 9762: 9759: 9756: 9753: 9750: 9747: 9746:Elsa Einstein 9744: 9741: 9738: 9737: 9735: 9733: 9729: 9723: 9722: 9718: 9716: 9715: 9711: 9709: 9708: 9704: 9702: 9701: 9697: 9695: 9694: 9690: 9688: 9687: 9683: 9681: 9680: 9676: 9674: 9673: 9669: 9668: 9666: 9660: 9654: 9651: 9649: 9646: 9644: 9641: 9639: 9636: 9634: 9633:Kalinga Prize 9631: 9629: 9626: 9624: 9621: 9620: 9618: 9614: 9607: 9606: 9602: 9600:(2017 series) 9599: 9598: 9594: 9591: 9590: 9586: 9583: 9582: 9578: 9575: 9574: 9570: 9567: 9566: 9562: 9559: 9558: 9554: 9551: 9550: 9546: 9543: 9542: 9538: 9535: 9534: 9530: 9527: 9526: 9522: 9521: 9519: 9517: 9511: 9504: 9501: 9498: 9494: 9491: 9490: 9486: 9483: 9482: 9478: 9475: 9474: 9470: 9467: 9466: 9462: 9459: 9455: 9452: 9450: 9446: 9445: 9443: 9441: 9437: 9431: 9428: 9426: 9423: 9421: 9418: 9416: 9413: 9411: 9408: 9406: 9403: 9401: 9398: 9396: 9393: 9391: 9388: 9386: 9383: 9381: 9378: 9376: 9373: 9371: 9368: 9366: 9363: 9361: 9358: 9356: 9353: 9351: 9348: 9346: 9343: 9341: 9338: 9336: 9333: 9331: 9328: 9326: 9323: 9321: 9318: 9316: 9313: 9309: 9306: 9304: 9301: 9300: 9299: 9296: 9295: 9293: 9289: 9285: 9278: 9273: 9271: 9266: 9264: 9259: 9258: 9255: 9248: 9244: 9243:Suzanne Imber 9241: 9239: 9236: 9235: 9226: 9223: 9220: 9216: 9213: 9210: 9207: 9204: 9201: 9197: 9193: 9192: 9187: 9183: 9182: 9172: 9166: 9161: 9160: 9154: 9150: 9146: 9144:0-471-92567-5 9140: 9135: 9134: 9128: 9124: 9120: 9114: 9110: 9109:W. H. Freeman 9106: 9105: 9100: 9096: 9092: 9088: 9087: 9086: 9084: 9068: 9064: 9060: 9056: 9052: 9048: 9043: 9042:gr-qc/0507026 9038: 9034: 9030: 9023: 9015: 9011: 9007: 9003: 8999: 8995: 8990: 8985: 8982:(6): 063518. 8981: 8977: 8970: 8962: 8958: 8954: 8950: 8946: 8942: 8938: 8934: 8927: 8919: 8915: 8911: 8907: 8903: 8899: 8895: 8891: 8884: 8876: 8872: 8868: 8864: 8860: 8856: 8852: 8848: 8844: 8837: 8828: 8824: 8820: 8816: 8812: 8808: 8804: 8800: 8799: 8794: 8788: 8780: 8774: 8770: 8769: 8761: 8753: 8749: 8745: 8741: 8737: 8733: 8728: 8723: 8719: 8715: 8708: 8700: 8696: 8692: 8685: 8671: 8669:0-670-50376-2 8665: 8661: 8657: 8656: 8651: 8650:Gamow, George 8645: 8638: 8633: 8625: 8623:0-471-60088-1 8619: 8615: 8608: 8600: 8596: 8592: 8588: 8584: 8580: 8577:(6303): 705. 8576: 8572: 8571: 8563: 8555: 8553:0-691-08239-1 8549: 8545: 8538: 8531: 8526: 8519: 8514: 8506: 8502: 8497: 8492: 8487: 8482: 8478: 8474: 8469: 8468:gr-qc/0505133 8464: 8460: 8456: 8452: 8445: 8437: 8435:0-521-46136-7 8431: 8427: 8423: 8416: 8414: 8405: 8403:0-09-922391-0 8399: 8395: 8388: 8380: 8376: 8372: 8370:0-07-000423-4 8366: 8362: 8361: 8353: 8345: 8341: 8337: 8331: 8327: 8323: 8316: 8308: 8302: 8298: 8297: 8289: 8281: 8279:0-8053-8732-3 8275: 8271: 8267: 8266:Carroll, Sean 8261: 8254: 8249: 8234: 8230: 8226: 8220: 8210: 8205: 8201: 8197: 8193: 8189: 8185: 8181: 8180: 8175: 8168: 8166: 8161: 8151: 8148: 8146: 8143: 8141: 8138: 8136: 8133: 8131: 8128: 8126: 8123: 8121: 8118: 8116: 8113: 8111: 8108: 8106: 8103: 8102: 8095: 8093: 8087: 8080: 8067: 8057: 8044: 8041: 8037: 8031: 8028: 8024: 8018: 8015: 8011: 8005: 8002: 7999: 7996: 7992: 7986: 7983: 7980: 7977: 7973: 7966: 7963: 7954: 7949: 7946: 7942: 7933: 7915: 7912: 7908: 7902: 7899: 7895: 7889: 7886: 7882: 7876: 7873: 7869: 7863: 7860: 7857: 7854: 7850: 7844: 7841: 7838: 7835: 7831: 7824: 7821: 7815: 7809: 7788: 7786: 7782: 7778: 7774: 7770: 7764: 7754: 7752: 7746: 7744: 7740: 7736: 7731: 7729: 7725: 7721: 7717: 7711: 7701: 7684: 7681: 7678: 7674: 7670: 7665: 7662: 7659: 7655: 7651: 7646: 7643: 7640: 7636: 7632: 7627: 7624: 7621: 7617: 7613: 7608: 7605: 7601: 7585: 7581: 7574: 7570: 7566: 7562: 7542: 7539: 7535: 7529: 7526: 7523: 7520: 7516: 7512: 7507: 7504: 7501: 7498: 7494: 7490: 7485: 7482: 7479: 7476: 7472: 7467: 7460: 7457: 7451: 7447: 7441: 7438: 7435: 7432: 7428: 7424: 7419: 7416: 7413: 7410: 7406: 7402: 7397: 7394: 7391: 7388: 7384: 7379: 7372: 7369: 7363: 7361: 7351: 7348: 7345: 7342: 7335: 7327: 7324: 7322: 7315: 7312: 7305: 7302: 7298: 7283: 7278: 7265: 7261: 7255: 7252: 7248: 7242: 7239: 7235: 7229: 7226: 7222: 7215: 7212: 7206: 7201: 7194: 7190: 7182: 7175: 7171: 7164: 7156: 7152: 7148: 7143: 7138: 7135: 7131: 7124: 7119: 7116: 7112: 7100: 7096: 7079: 7073: 7070: 7066: 7060: 7057: 7053: 7047: 7044: 7040: 7033: 7030: 7024: 7019: 7012: 7008: 7000: 6993: 6989: 6982: 6974: 6970: 6966: 6961: 6957: 6952: 6949: 6945: 6937: 6933: 6929: 6916: 6906: 6904: 6900: 6894: 6886: 6882: 6878: 6876: 6875:Kerr solution 6872: 6868: 6864: 6859: 6846: 6840: 6837: 6833: 6826: 6823: 6818: 6815: 6804: 6799: 6796: 6792: 6782: 6769: 6765: 6762: 6757: 6754: 6750: 6741: 6735: 6728:. By setting 6727: 6711: 6701: 6688: 6680: 6676: 6671: 6668: 6665: 6659: 6656: 6635: 6632: 6629: 6626: 6623: 6618: 6614: 6610: 6607: 6601: 6598: 6573: 6569: 6565: 6562: 6556: 6553: 6547: 6543: 6537: 6533: 6529: 6523: 6520: 6514: 6509: 6505: 6500: 6496: 6493: 6488: 6484: 6480: 6475: 6470: 6467: 6464: 6456: 6451: 6448: 6445: 6431: 6418: 6412: 6407: 6404: 6401: 6393: 6388: 6384: 6370: 6357: 6352: 6347: 6344: 6334: 6329: 6326: 6318: 6313: 6308: 6298: 6293: 6290: 6282: 6277: 6272: 6269: 6266: 6263: 6255: 6250: 6245: 6242: 6239: 6231: 6226: 6222: 6212: 6199: 6193: 6189: 6185: 6182: 6176: 6173: 6167: 6163: 6158: 6152: 6148: 6144: 6140: 6135: 6129: 6125: 6121: 6118: 6114: 6107: 6104: 6098: 6093: 6089: 6085: 6081: 6077: 6074: 6070: 6064: 6060: 6056: 6050: 6047: 6041: 6036: 6032: 6027: 6023: 6000: 5996: 5992: 5989: 5986: 5981: 5977: 5973: 5966: 5962: 5958: 5953: 5950: 5945: 5941: 5935: 5931: 5927: 5922: 5919: 5915: 5909: 5906: 5902: 5898: 5895: 5886: 5873: 5868: 5864: 5861: 5858: 5855: 5852: 5849: 5844: 5840: 5836: 5832: 5828: 5825: 5822: 5818: 5814: 5811: 5808: 5805: 5802: 5799: 5794: 5790: 5785: 5781: 5778: 5775: 5770: 5767: 5763: 5741: 5735: 5731: 5727: 5721: 5718: 5712: 5707: 5703: 5698: 5694: 5691: 5686: 5682: 5672: 5659: 5652: 5649: 5644: 5640: 5636: 5633: 5628: 5624: 5600: 5597: 5594: 5590: 5586: 5583: 5579: 5573: 5570: 5567: 5563: 5559: 5555: 5549: 5546: 5542: 5538: 5533: 5530: 5522: 5513: 5509: 5503: 5490: 5485: 5479: 5476: 5473: 5469: 5465: 5460: 5457: 5454: 5451: 5447: 5443: 5438: 5435: 5432: 5429: 5425: 5420: 5414: 5411: 5407: 5400: 5397: 5391: 5386: 5381: 5373: 5368: 5365: 5347: 5339: 5316: 5311: 5303: 5300: 5292: 5288: 5284: 5277: 5273: 5267: 5263: 5251: 5235: 5232: 5228: 5222: 5219: 5214: 5211: 5205: 5198: 5195: 5191: 5168: 5164: 5161: 5158: 5155: 5152: 5149: 5143: 5140: 5135: 5132: 5125: 5121: 5115: 5112: 5105: 5101: 5097: 5085: 5072: 5065: 5062: 5055: 5051: 5047: 5038: 5035: 5028: 5024: 5020: 5012: 5007: 5004: 4996: 4993: 4985: 4981: 4977: 4970: 4966: 4960: 4956: 4945: 4924: 4918: 4915: 4911: 4907: 4901: 4898: 4892: 4887: 4884: 4880: 4875: 4871: 4868: 4863: 4860: 4856: 4846: 4829: 4823: 4820: 4812: 4809: 4807: 4797: 4793: 4789: 4782: 4778: 4772: 4768: 4757: 4754: 4751: 4748: 4745: 4743: 4736: 4733: 4730: 4712: 4699: 4694: 4690: 4687: 4681: 4669: 4662: 4652: 4649: 4640: 4634: 4628: 4619: 4611: 4598: 4577: 4573: 4570: 4561: 4554: 4550: 4547: 4544: 4541: 4538: 4534: 4530: 4527: 4518: 4511: 4502: 4490: 4484: 4470: 4464: 4460: 4456: 4446: 4444: 4440: 4436: 4432: 4428: 4424: 4420: 4410: 4405: 4392: 4389: 4384: 4381: 4374: 4371: 4367: 4361: 4356: 4353: 4349: 4343: 4329: 4316: 4313: 4308: 4305: 4298: 4295: 4291: 4280: 4264: 4261: 4256: 4253: 4248: 4244: 4239: 4236: 4232: 4225: 4222: 4216: 4211: 4208: 4204: 4199: 4184: 4171: 4168: 4163: 4160: 4155: 4151: 4146: 4139: 4135: 4126: 4123: 4117: 4112: 4105: 4101: 4094: 4072: 4069: 4064: 4061: 4058: 4051: 4047: 4041: 4038: 4033: 4030: 4026: 4016: 4003: 4000: 3995: 3992: 3989: 3986: 3979: 3976: 3972: 3966: 3961: 3958: 3955: 3948: 3944: 3938: 3933: 3930: 3927: 3920: 3916: 3893: 3890: 3886: 3880: 3877: 3874: 3871: 3868: 3861: 3857: 3851: 3846: 3843: 3840: 3837: 3833: 3829: 3824: 3821: 3818: 3815: 3811: 3806: 3800: 3797: 3793: 3783: 3781: 3765: 3762: 3757: 3754: 3751: 3748: 3745: 3738: 3734: 3728: 3723: 3720: 3717: 3714: 3710: 3706: 3701: 3698: 3695: 3692: 3688: 3667: 3664: 3659: 3656: 3653: 3650: 3647: 3640: 3636: 3630: 3625: 3622: 3619: 3616: 3613: 3606: 3602: 3596: 3591: 3588: 3585: 3582: 3579: 3572: 3568: 3557: 3554: 3541: 3538: 3533: 3530: 3527: 3524: 3521: 3514: 3510: 3504: 3499: 3496: 3493: 3490: 3487: 3480: 3476: 3470: 3465: 3462: 3459: 3456: 3453: 3446: 3442: 3429: 3405: 3402: 3394: 3391: 3388: 3385: 3379: 3376: 3372: 3364: 3355: 3342: 3339: 3334: 3331: 3324: 3321: 3317: 3311: 3306: 3303: 3299: 3293: 3280: 3267: 3263: 3260: 3239: 3234: 3226: 3209: 3205: 3202: 3185: 3172: 3163: 3159: 3155: 3150: 3137: 3131: 3128: 3124: 3118: 3110: 3107: 3083: 3080: 3076: 3065: 3059: 3055: 3050: 3048: 3036: 3032: 3028: 3025: 3024: 3023: 3021: 3014: 2998: 2992: 2989: 2985: 2981: 2978: 2973: 2970: 2966: 2959: 2954: 2951: 2947: 2936: 2926: 2924: 2906: 2903: 2899: 2889: 2876: 2872: 2866: 2863: 2859: 2854: 2849: 2846: 2841: 2836: 2833: 2829: 2824: 2820: 2817: 2812: 2809: 2805: 2798: 2793: 2790: 2786: 2775: 2769: 2756: 2752: 2746: 2743: 2739: 2735: 2729: 2726: 2723: 2719: 2714: 2709: 2706: 2702: 2697: 2693: 2690: 2685: 2682: 2678: 2668: 2665: 2662: 2658: 2653: 2648: 2645: 2641: 2631: 2610: 2607: 2604: 2601: 2595: 2592: 2589: 2584: 2581: 2576: 2573: 2565: 2555: 2549: 2540: 2527: 2522: 2519: 2515: 2511: 2508: 2505: 2500: 2497: 2493: 2486: 2481: 2478: 2474: 2470: 2465: 2462: 2457: 2452: 2449: 2445: 2435: 2417: 2412: 2397: 2394: 2391: 2384: 2380: 2374: 2368: 2365: 2359: 2353: 2350: 2344: 2339: 2336: 2332: 2322: 2303: 2300: 2296: 2292: 2289: 2283: 2280: 2274: 2272: 2265: 2262: 2258: 2249: 2243: 2238: 2235: 2225: 2220: 2217: 2209: 2204: 2199: 2196: 2186: 2181: 2178: 2170: 2165: 2160: 2157: 2154: 2151: 2143: 2138: 2133: 2130: 2127: 2124: 2115: 2111: 2105: 2102: 2096: 2094: 2087: 2084: 2081: 2074: 2070: 2057: 2054: 2051: 2048: 2045: 2042: 2039: 2036: 2033: 2030: 2027: 2021: 2018: 2015: 2009: 2006: 2000: 1998: 1991: 1988: 1984: 1971: 1969: 1959: 1957: 1939: 1936: 1932: 1922: 1920: 1914: 1910: 1900: 1898: 1895: 1891: 1887: 1882: 1880: 1876: 1872: 1867: 1863: 1846: 1841: 1838: 1834: 1830: 1827: 1822: 1819: 1815: 1808: 1803: 1800: 1796: 1792: 1787: 1784: 1779: 1774: 1771: 1767: 1759: 1758: 1757: 1754: 1752: 1748: 1740: 1717: 1690: 1685: 1682: 1667: 1664: 1660: 1656: 1650: 1644: 1641: 1634: 1630: 1625: 1622: 1619: 1613: 1610: 1603: 1602: 1601: 1599: 1594: 1592: 1576: 1568: 1550: 1547: 1543: 1519: 1514: 1511: 1507: 1503: 1498: 1495: 1490: 1485: 1482: 1478: 1474: 1469: 1466: 1462: 1454: 1453: 1452: 1450: 1445: 1431: 1423: 1399: 1381: 1378: 1374: 1365: 1364:metric tensor 1347: 1344: 1340: 1331: 1313: 1310: 1306: 1294:, Netherlands 1293: 1288: 1268: 1265: 1261: 1257: 1254: 1249: 1246: 1242: 1235: 1230: 1227: 1223: 1215: 1214: 1213: 1203: 1198: 1196: 1191: 1189: 1184: 1183: 1181: 1180: 1173: 1163: 1160: 1155: 1149: 1148: 1146: 1145: 1137: 1134: 1132: 1129: 1127: 1124: 1122: 1119: 1117: 1114: 1112: 1109: 1107: 1104: 1103: 1096: 1095: 1087: 1084: 1082: 1079: 1078: 1071: 1070: 1062: 1059: 1057: 1054: 1052: 1049: 1048: 1041: 1040: 1032: 1029: 1027: 1024: 1022: 1019: 1017: 1014: 1013: 1006: 1005: 999: 996: 994: 991: 990: 988: 987: 983: 979: 978: 975: 972: 971: 967: 966: 958: 956: 952: 948: 944: 940: 936: 932: 927: 925: 921: 916: 914: 910: 906: 902: 898: 897:metric tensor 894: 890: 886: 882: 878: 873: 871: 867: 863: 859: 856: 848: 844: 839: 837: 833: 829: 825: 821: 817: 805: 800: 798: 793: 791: 786: 785: 783: 782: 776: 766: 763: 758: 752: 751: 750: 749: 742: 741: 737: 735: 732: 730: 727: 725: 722: 720: 717: 715: 712: 710: 707: 705: 702: 700: 697: 695: 692: 690: 687: 685: 682: 680: 679:Chandrasekhar 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: 607: 605: 602: 600: 597: 595: 594:Schwarzschild 592: 590: 587: 585: 582: 580: 577: 575: 572: 571: 563: 562: 555: 554:Hartle–Thorne 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: 485: 482: 480: 477: 475: 472: 469: 465: 464:Schwarzschild 462: 461: 457: 451: 450: 439: 436: 434: 431: 430: 429: 428: 424: 423: 418: 415: 413: 410: 408: 405: 404: 403: 402: 398: 397: 392: 389: 387: 384: 382: 379: 377: 374: 372: 369: 367: 364: 363: 362: 361: 357: 356: 346: 343: 342: 337: 336: 325: 322: 320: 317: 315: 312: 311: 310: 309: 306: 303: 302: 297: 294: 292: 289: 287: 286:Event horizon 284: 282: 279: 277: 274: 272: 269: 267: 264: 262: 259: 257: 254: 252: 249: 248: 247: 246: 236: 235: 228: 225: 223: 220: 218: 215: 213: 210: 209: 201: 200: 195: 192: 187: 184: 182: 179: 177: 174: 173: 171: 169: 166: 165: 164: 163: 145: 142: 138: 133: 129: 124: 121: 117: 110: 105: 102: 98: 88: 84: 83: 80: 77: 76: 73: 69: 51: 47: 43: 40: 37: 19: 10759: 10453:Kaluza–Klein 10273: 10205:Introduction 10131:Twin paradox 9916: 9884:Einsteinhaus 9764:Pauline Koch 9742:(first wife) 9740:Mileva Marić 9719: 9712: 9705: 9698: 9691: 9684: 9677: 9670: 9603: 9595: 9587: 9579: 9571: 9563: 9555: 9547: 9539: 9531: 9523: 9487: 9479: 9471: 9463: 9448: 9344: 9189: 9158: 9132: 9102: 9080: 9032: 9028: 9022: 8979: 8976:Phys. Rev. D 8975: 8969: 8936: 8932: 8926: 8893: 8889: 8883: 8850: 8846: 8836: 8802: 8796: 8787: 8767: 8760: 8717: 8713: 8707: 8699:the original 8694: 8684: 8673:. Retrieved 8660:Viking Adult 8654: 8644: 8632: 8613: 8607: 8574: 8568: 8562: 8543: 8537: 8525: 8513: 8458: 8454: 8444: 8421: 8393: 8387: 8359: 8352: 8343: 8339: 8335: 8329: 8325: 8321: 8315: 8295: 8288: 8269: 8260: 8248: 8237:. Retrieved 8232: 8219: 8204:the original 8183: 8177: 8085: 8081: 7794: 7766: 7747: 7732: 7713: 7279: 7094: 6918: 6889: 6885:Ricci tensor 6879: 6860: 6783: 6730: 6716: 6432: 6371: 6213: 5887: 5673: 5511: 5507: 5504: 5345: 5337: 5086: 4847: 4713: 4597:free-falling 4482: 4475: 4452: 4443:wavefunction 4416: 4413:Nonlinearity 4407: 4330: 4185: 4017: 3784: 3780:Ricci tensor 3558: 3555: 3424: 3360: 3281: 3278: 3264: 3258: 3170: 3161: 3154:vacuum state 3151: 3066: 3054:astronomical 3051: 3047:George Gamow 3040: 3031:Edwin Hubble 2938: 2890: 2773: 2770: 2629: 2561: 2541: 2436: 2413: 2323: 1972: 1965: 1923: 1912: 1908: 1901: 1894:gauge-fixing 1883: 1868: 1864: 1861: 1755: 1732: 1597: 1595: 1534: 1446: 1297: 1211: 1060: 928: 917: 874: 850: 840: 827: 823: 819: 813: 739: 699:Raychaudhuri 370: 168:Introduction 72: 10542:Kerr–Newman 10513:Spherical: 10382:Other tests 10325:Singularity 10257:Formulation 10219:Fundamental 10073:Formulation 10054:Proper time 10015:Fundamental 9889:Einsteinium 9662:Books about 9608:(2023 film) 9605:Oppenheimer 9584:(2003 play) 9576:(1994 film) 9568:(1993 play) 9560:(1988 film) 9552:(1985 film) 9395:EPR paradox 9104:Gravitation 8939:(8): 2281. 7739:black holes 2562:Taking the 1866:spacetime. 1106:Four-vector 838:within it. 714:van Stockum 644:Oppenheimer 499:Kerr–Newman 291:Singularity 10790:Categories 10694:Zel'dovich 10602:Scientists 10581:Alcubierre 10388:of Mercury 10386:precession 10315:Black hole 10198:Background 10190:relativity 10159:World line 10154:Light cone 9979:Background 9971:relativity 9961:Relativity 9894:Max Talmey 9796:(grandson) 9754:(daughter) 9514:In popular 9077:References 8675:2007-03-14 8239:2017-08-21 8186:(7): 769. 7930:using the 7593:such that 7569:divergence 6932:free space 6913:See also: 4942:, and the 853:spacetime 567:Scientists 399:Formalisms 347:Formalisms 296:Black hole 222:World line 10664:Robertson 10649:Friedmann 10644:Eddington 10634:de Sitter 10468:Solutions 10346:detectors 10341:astronomy 10308:Phenomena 10243:Geodesics 10146:Spacetime 10089:Phenomena 9196:EMS Press 9014:119178273 8989:1304.8042 8961:250876974 8918:250907312 8875:122577276 8827:123364248 8695:News@UofT 8235:: 844–847 8045:ν 8042:δ 8032:μ 8029:γ 8019:λ 8016:β 8006:ν 8003:μ 8000:λ 7997:κ 7993:ε 7987:δ 7984:γ 7981:β 7978:α 7974:ε 7950:κ 7947:α 7916:ν 7913:δ 7903:μ 7900:γ 7890:λ 7887:β 7877:κ 7874:α 7864:ν 7861:μ 7858:λ 7855:κ 7851:ε 7845:δ 7842:γ 7839:β 7836:α 7832:ε 7773:spacetime 7735:cosmology 7720:spacetime 7704:Solutions 7685:α 7679:β 7671:− 7666:β 7660:α 7647:α 7641:β 7633:− 7628:β 7622:α 7609:β 7606:α 7530:β 7524:α 7521:γ 7508:α 7502:γ 7499:β 7486:γ 7480:β 7477:α 7442:β 7436:α 7433:γ 7420:α 7414:γ 7411:β 7398:γ 7392:β 7389:α 7352:γ 7346:β 7343:α 7316:β 7306:β 7303:α 7256:τ 7253:ψ 7243:τ 7240:ψ 7230:β 7227:α 7202:β 7195:ψ 7183:ψ 7176:α 7153:μ 7149:κ 7139:β 7136:α 7128:Λ 7120:β 7117:α 7074:τ 7071:ψ 7061:τ 7058:ψ 7048:β 7045:α 7020:β 7013:ψ 7001:ψ 6994:α 6971:μ 6962:− 6953:β 6950:α 6881:Manifolds 6841:ν 6838:μ 6824:− 6810:Λ 6800:ν 6797:μ 6758:ν 6755:μ 6669:π 6636:ρ 6630:π 6611:ρ 6566:ρ 6548:≈ 6515:− 6481:≈ 6461:Γ 6457:≈ 6442:Φ 6398:Γ 6394:≈ 6353:λ 6345:ρ 6341:Γ 6335:ρ 6330:λ 6323:Γ 6319:− 6314:λ 6305:Γ 6299:ρ 6294:λ 6291:ρ 6287:Γ 6278:ρ 6264:ρ 6260:Γ 6256:− 6251:ρ 6246:ρ 6236:Γ 6186:ρ 6145:− 6122:ρ 6119:− 6099:− 6086:ρ 6075:≈ 6042:− 6016:and thus 5993:ρ 5990:− 5974:ρ 5954:− 5951:≈ 5928:≈ 5923:β 5920:α 5910:β 5907:α 5837:ρ 5829:⁡ 5823:≈ 5782:⁡ 5776:≈ 5771:ν 5768:μ 5713:− 5656:Φ 5650:− 5637:− 5634:≈ 5587:− 5584:≈ 5560:− 5539:≈ 5527:Φ 5480:α 5466:− 5455:α 5430:α 5415:α 5378:Γ 5374:≈ 5362:Φ 5308:Γ 5304:− 5301:≈ 5233:≈ 5223:τ 5182:and thus 5144:τ 5122:≈ 5116:τ 5106:β 5066:τ 5056:γ 5039:τ 5029:β 5013:α 5008:γ 5005:β 5001:Γ 4997:− 4982:τ 4971:α 4919:ν 4916:μ 4893:− 4888:ν 4885:μ 4864:ν 4861:μ 4817:Φ 4813:− 4758:ρ 4752:π 4727:Φ 4673:→ 4659:Φ 4656:∇ 4653:− 4644:→ 4620:¨ 4615:→ 4565:→ 4551:ρ 4545:π 4522:→ 4508:Φ 4499:∇ 4385:β 4375:β 4372:α 4357:β 4354:α 4344:β 4340:∇ 4309:β 4299:β 4296:α 4257:γ 4240:δ 4237:γ 4217:− 4212:δ 4209:γ 4164:γ 4147:ε 4140:γ 4118:− 4113:ε 4106:γ 4065:γ 4059:ε 4052:γ 4039:− 4034:ε 3996:γ 3990:ε 3987:δ 3980:δ 3977:γ 3962:δ 3956:ε 3949:δ 3939:− 3934:ε 3928:δ 3921:δ 3881:γ 3875:ε 3872:δ 3869:β 3862:γ 3847:δ 3841:ε 3838:β 3830:− 3825:ε 3819:δ 3816:β 3801:δ 3798:β 3758:γ 3752:ε 3749:δ 3746:β 3739:γ 3724:δ 3718:ε 3715:β 3707:− 3702:ε 3696:δ 3693:β 3660:γ 3654:ε 3651:δ 3648:β 3641:γ 3626:δ 3620:ε 3617:γ 3614:β 3607:γ 3597:− 3592:ε 3586:δ 3583:γ 3580:β 3573:γ 3534:γ 3528:ε 3525:δ 3522:β 3515:γ 3500:δ 3494:γ 3491:ε 3488:β 3481:γ 3466:ε 3460:δ 3457:γ 3454:β 3447:γ 3395:ε 3389:δ 3386:γ 3380:β 3377:α 3335:β 3325:β 3322:α 3307:β 3304:α 3294:β 3290:∇ 3235:κ 3232:Λ 3206:− 3186:ρ 3132:ν 3129:μ 3119:κ 3116:Λ 3111:− 3084:ν 3081:μ 3035:expanding 2993:ν 2990:μ 2982:κ 2974:ν 2971:μ 2963:Λ 2955:ν 2952:μ 2907:ν 2904:μ 2867:ν 2864:μ 2842:− 2837:ν 2834:μ 2821:κ 2813:ν 2810:μ 2802:Λ 2799:− 2794:ν 2791:μ 2747:ν 2744:μ 2727:− 2715:− 2710:ν 2707:μ 2694:κ 2686:ν 2683:μ 2675:Λ 2666:− 2654:− 2649:ν 2646:μ 2605:κ 2599:Λ 2577:− 2552:(− + + +) 2544:(+ − − −) 2523:ν 2520:μ 2512:κ 2509:− 2501:ν 2498:μ 2490:Λ 2487:− 2482:ν 2479:μ 2458:− 2453:ν 2450:μ 2398:ν 2395:α 2392:μ 2385:α 2375:× 2360:× 2340:ν 2337:μ 2304:ν 2301:μ 2293:κ 2290:× 2266:ν 2263:μ 2244:σ 2239:α 2236:β 2232:Γ 2226:μ 2221:γ 2218:σ 2214:Γ 2210:− 2205:σ 2200:α 2197:γ 2193:Γ 2187:μ 2182:β 2179:σ 2175:Γ 2166:μ 2161:γ 2155:β 2152:α 2148:Γ 2144:− 2139:μ 2134:β 2128:γ 2125:α 2121:Γ 2112:× 2088:γ 2085:β 2082:α 2075:μ 2028:− 2022:⁡ 2016:× 1992:ν 1989:μ 1940:ν 1937:μ 1842:ν 1839:μ 1831:κ 1823:ν 1820:μ 1812:Λ 1804:ν 1801:μ 1780:− 1775:ν 1772:μ 1683:− 1665:− 1657:× 1642:≈ 1623:π 1611:κ 1551:ν 1548:μ 1515:ν 1512:μ 1491:− 1486:ν 1483:μ 1470:ν 1467:μ 1432:κ 1408:Λ 1382:ν 1379:μ 1348:ν 1345:μ 1314:ν 1311:μ 1269:ν 1266:μ 1258:κ 1250:ν 1247:μ 1239:Λ 1231:ν 1228:μ 974:Spacetime 909:geodesics 855:curvature 832:spacetime 659:Robertson 624:Friedmann 619:Eddington 609:Nordström 599:de Sitter 456:Solutions 381:Geodesics 376:Friedmann 358:Equations 344:Equations 305:Spacetime 240:Phenomena 146:ν 143:μ 134:κ 125:ν 122:μ 114:Λ 106:ν 103:μ 10777:Category 10654:Lemaître 10619:Einstein 10609:Poincaré 10569:Others: 10553:Taub–NUT 10519:interior 10441:theories 10439:Advanced 10406:redshift 10221:concepts 10039:Rapidity 10017:concepts 9918:Category 9844:Memorial 9790:(cousin) 9778:(sister) 9772:(father) 9766:(mother) 9664:Einstein 9499:" (1949) 9460:" (1905) 9155:(1999). 9129:(1972). 9101:(1973). 8752:16669258 8599:12988317 8505:28179868 8268:(2004). 8098:See also 7743:universe 6873:and the 4461:and the 4427:electric 3270:Features 3156:with an 1172:Category 941:and the 931:symmetry 905:inertial 885:currents 866:momentum 775:Category 639:Lemaître 604:Reissner 589:Poincaré 574:Einstein 519:Taub–NUT 484:Wormhole 468:interior 181:Timeline 10719:Hawking 10714:Penrose 10699:Novikov 10679:Wheeler 10624:Hilbert 10614:Lorentz 10571:pp-wave 10392:lensing 10188:General 9969:Special 9822:Related 9516:culture 9291:Physics 9198:, 2001 9067:6263993 9047:Bibcode 8994:Bibcode 8941:Bibcode 8898:Bibcode 8855:Bibcode 8807:Bibcode 8732:Bibcode 8579:Bibcode 8496:5256071 8473:Bibcode 8379:1046135 8188:Bibcode 7716:metrics 7571:of the 6865:. Flat 6738:in the 5350:⁠ 5334:⁠ 3906:to get 2546:metric 2432:(− + −) 2428:(− + +) 2424:(+ − −) 2420:(+ + +) 1745:is the 1737:is the 1645:2.07665 1589:is the 1565:is the 1420:is the 1396:is the 1362:is the 1328:is the 881:charges 814:In the 694:Hawking 689:Penrose 664:Bardeen 654:Wheeler 584:Hilbert 579:Lorentz 539:pp-wave 176:History 10760:others 10749:Thorne 10739:Misner 10724:Taylor 10709:Geroch 10704:Ehlers 10674:Zwicky 10492:Kasner 9732:Family 9616:Prizes 9597:Genius 9505:(1955) 9492:(1938) 9484:(1934) 9476:(1922) 9468:(1916) 9453:(1905) 9451:papers 9167: 9141: 9115: 9065: 9012: 8959: 8916: 8873: 8825: 8775: 8750: 8666: 8620: 8597: 8570:Nature 8550: 8503: 8493: 8432: 8400: 8377: 8367: 8303: 8276: 7573:2-form 7097:(with 4591:where 4487:, see 4190:gives 2623:where 1751:vacuum 1733:where 1569:, and 1535:where 1298:where 1292:Leiden 1170: 836:matter 818:, the 773: 740:others 734:Thorne 724:Newman 704:Taylor 684:Ehlers 669:Walker 634:Zwicky 509:Kasner 66:, see 10754:Weiss 10734:Bondi 10729:Hulse 10659:Milne 10563:discs 10507:Milne 10502:Gödel 10359:Virgo 9839:House 9834:Brain 9784:(son) 9760:(son) 9440:Works 9063:S2CID 9037:arXiv 9010:S2CID 8984:arXiv 8957:S2CID 8914:S2CID 8871:S2CID 8823:S2CID 8748:S2CID 8722:arXiv 8595:S2CID 8463:arXiv 8157:Notes 4485:= −∇Φ 3418:with 709:Hulse 649:Gödel 629:Milne 524:Milne 489:Gödel 186:Tests 10689:Kerr 10639:Weyl 10538:Kerr 10398:and 10352:and 10350:LIGO 9573:I.Q. 9165:ISBN 9139:ISBN 9113:ISBN 9081:See 8773:ISBN 8664:ISBN 8618:ISBN 8548:ISBN 8501:PMID 8430:ISBN 8398:ISBN 8375:OCLC 8365:ISBN 8301:ISBN 8274:ISBN 8084:det( 5826:diag 5779:diag 4429:and 2019:diag 1741:and 1596:The 1447:The 1424:and 887:via 883:and 719:Taub 674:Kerr 614:Weyl 494:Kerr 412:BSSN 10744:Yau 10369:GEO 9219:MIT 9217:by 9055:doi 9002:doi 8949:doi 8906:doi 8863:doi 8815:doi 8740:doi 8587:doi 8575:348 8491:PMC 8481:doi 8338:= 8 8324:= 8 8209:PDF 8196:doi 8184:354 8052:det 7804:det 7718:of 6930:in 6895:= 0 6736:= 0 5888:So 4437:of 4421:of 3430:= 0 3174:vac 3165:vac 2776:= 4 2771:In 1877:of 1749:in 1713:m/J 1703:or 872:). 824:EFE 729:Yau 407:ADM 10792:: 10418:/ 10384:: 10339:: 9245:, 9194:, 9188:, 9111:. 9097:; 9093:; 9085:. 9061:. 9053:. 9045:. 9033:38 9031:. 9008:. 9000:. 8992:. 8980:88 8978:. 8955:. 8947:. 8937:14 8935:. 8912:. 8904:. 8892:. 8869:. 8861:. 8851:12 8849:. 8845:. 8821:. 8813:. 8803:16 8801:. 8746:. 8738:. 8730:. 8718:17 8716:. 8693:. 8662:. 8658:. 8593:. 8585:. 8573:. 8499:. 8489:. 8479:. 8471:. 8457:. 8453:. 8428:. 8424:. 8412:^ 8373:. 8340:πG 8326:πG 8231:. 8194:. 8182:. 8176:. 8164:^ 7825:24 7787:. 7745:. 7730:. 7543:0. 6923:μν 6905:. 6892:μν 6887:, 6877:. 6733:μν 6721:μν 6538:00 6510:00 6489:00 6465:00 6402:00 6389:00 6309:00 6240:00 6227:00 6065:00 6037:00 5946:00 5936:00 5795:00 5736:00 5708:00 5687:00 5629:00 5595:00 5568:00 5510:= 5474:00 5382:00 5346:dτ 5338:dt 5312:00 4445:. 3782:. 3432:, 3427:;γ 3343:0. 2434:. 1958:. 1921:. 1913:μν 1881:. 1753:. 1668:43 1661:10 1400:, 1366:, 1332:, 957:. 926:. 915:. 10544:) 10540:( 10526:) 10517:( 10483:) 10479:( 10426:) 10422:( 10402:) 10371:) 10348:( 9997:) 9988:( 9953:e 9946:t 9939:v 9495:" 9456:" 9276:e 9269:t 9262:v 9173:. 9147:. 9121:. 9069:. 9057:: 9049:: 9039:: 9016:. 9004:: 8996:: 8986:: 8963:. 8951:: 8943:: 8920:. 8908:: 8900:: 8894:3 8877:. 8865:: 8857:: 8831:. 8829:. 8817:: 8809:: 8781:. 8754:. 8742:: 8734:: 8724:: 8678:. 8639:. 8626:. 8601:. 8589:: 8581:: 8556:. 8532:. 8507:. 8483:: 8475:: 8465:: 8459:8 8438:. 8406:. 8381:. 8344:c 8342:/ 8336:κ 8330:c 8328:/ 8322:κ 8309:. 8282:. 8242:. 8211:) 8207:( 8198:: 8190:: 8088:) 8086:g 8068:. 8061:) 8058:g 8055:( 8038:g 8025:g 8012:g 7967:6 7964:1 7955:= 7943:g 7909:g 7896:g 7883:g 7870:g 7822:1 7816:= 7813:) 7810:g 7807:( 7682:, 7675:A 7663:, 7656:A 7652:= 7644:; 7637:A 7625:; 7618:A 7614:= 7602:F 7590:α 7588:A 7576:F 7540:= 7536:) 7527:, 7517:F 7513:+ 7505:, 7495:F 7491:+ 7483:, 7473:F 7468:( 7461:3 7458:1 7452:= 7448:) 7439:; 7429:F 7425:+ 7417:; 7407:F 7403:+ 7395:; 7385:F 7380:( 7373:3 7370:1 7364:= 7355:] 7349:; 7340:[ 7336:F 7328:0 7325:= 7313:; 7299:F 7266:. 7262:) 7249:F 7236:F 7223:g 7216:4 7213:1 7207:+ 7191:F 7172:F 7165:( 7157:0 7144:= 7132:g 7125:+ 7113:G 7102:Λ 7080:) 7067:F 7054:F 7041:g 7034:4 7031:1 7025:+ 7009:F 6990:F 6983:( 6975:0 6967:1 6958:= 6946:T 6921:T 6890:R 6847:. 6834:g 6827:1 6819:2 6816:D 6805:= 6793:R 6770:. 6766:0 6763:= 6751:R 6731:T 6719:T 6689:. 6681:4 6677:c 6672:G 6666:8 6660:= 6657:K 6633:G 6627:4 6624:= 6619:4 6615:c 6608:K 6602:2 6599:1 6574:4 6570:c 6563:K 6557:2 6554:1 6544:) 6534:g 6530:T 6524:2 6521:1 6506:T 6501:( 6497:K 6494:= 6485:R 6476:i 6471:i 6468:, 6452:i 6449:i 6446:, 6419:. 6413:i 6408:i 6405:, 6385:R 6374:Γ 6358:. 6348:0 6327:0 6283:+ 6273:0 6270:, 6267:0 6243:, 6232:= 6223:R 6200:. 6194:4 6190:c 6183:K 6177:2 6174:1 6168:= 6164:) 6159:) 6153:2 6149:c 6141:( 6136:) 6130:2 6126:c 6115:( 6108:2 6105:1 6094:4 6090:c 6082:( 6078:K 6071:) 6061:g 6057:T 6051:2 6048:1 6033:T 6028:( 6024:K 6001:2 5997:c 5987:= 5982:4 5978:c 5967:2 5963:c 5959:1 5942:T 5932:g 5916:T 5903:g 5899:= 5896:T 5874:. 5869:) 5865:0 5862:, 5859:0 5856:, 5853:0 5850:, 5845:4 5841:c 5833:( 5819:) 5815:0 5812:, 5809:0 5806:, 5803:0 5800:, 5791:T 5786:( 5764:T 5742:) 5732:g 5728:T 5722:2 5719:1 5704:T 5699:( 5695:K 5692:= 5683:R 5660:. 5653:2 5645:2 5641:c 5625:g 5601:i 5598:, 5591:g 5580:) 5574:j 5571:, 5564:g 5556:( 5550:j 5547:i 5543:g 5534:i 5531:, 5523:2 5512:i 5508:α 5491:. 5486:) 5477:, 5470:g 5461:0 5458:, 5452:0 5448:g 5444:+ 5439:0 5436:, 5433:0 5426:g 5421:( 5412:i 5408:g 5401:2 5398:1 5392:= 5387:i 5369:i 5366:, 5342:/ 5317:i 5293:2 5289:t 5285:d 5278:i 5274:x 5268:2 5264:d 5236:0 5229:) 5220:d 5215:t 5212:d 5206:( 5199:t 5196:d 5192:d 5169:) 5165:0 5162:, 5159:0 5156:, 5153:0 5150:, 5141:d 5136:t 5133:d 5126:( 5113:d 5102:x 5098:d 5073:. 5063:d 5052:x 5048:d 5036:d 5025:x 5021:d 4994:= 4986:2 4978:d 4967:x 4961:2 4957:d 4940:K 4925:) 4912:g 4908:T 4902:2 4899:1 4881:T 4876:( 4872:K 4869:= 4857:R 4830:. 4824:i 4821:, 4810:= 4798:2 4794:t 4790:d 4783:i 4779:x 4773:2 4769:d 4755:G 4749:4 4746:= 4737:i 4734:i 4731:, 4700:. 4695:) 4691:t 4688:, 4685:) 4682:t 4679:( 4670:x 4663:( 4650:= 4641:g 4635:= 4632:) 4629:t 4626:( 4612:x 4593:ρ 4578:) 4574:t 4571:, 4562:x 4555:( 4548:G 4542:4 4539:= 4535:) 4531:t 4528:, 4519:x 4512:( 4503:2 4483:g 4478:Φ 4467:G 4393:0 4390:= 4382:; 4368:T 4362:= 4350:T 4317:0 4314:= 4306:; 4292:G 4265:0 4262:= 4254:; 4249:) 4245:R 4233:g 4226:2 4223:1 4205:R 4200:( 4188:g 4172:0 4169:= 4161:; 4156:) 4152:R 4136:g 4127:2 4124:1 4102:R 4095:( 4073:0 4070:= 4062:; 4048:R 4042:2 4031:; 4027:R 4004:0 4001:= 3993:; 3973:R 3967:+ 3959:; 3945:R 3931:; 3917:R 3894:0 3891:= 3887:) 3878:; 3858:R 3852:+ 3844:; 3834:R 3822:; 3812:R 3807:( 3794:g 3766:0 3763:= 3755:; 3735:R 3729:+ 3721:; 3711:R 3699:; 3689:R 3668:0 3665:= 3657:; 3637:R 3631:+ 3623:; 3603:R 3589:; 3569:R 3542:0 3539:= 3531:; 3511:R 3505:+ 3497:; 3477:R 3471:+ 3463:; 3443:R 3425:g 3420:g 3406:0 3403:= 3398:] 3392:; 3383:[ 3373:R 3340:= 3332:; 3318:T 3312:= 3300:T 3259:κ 3254:Λ 3240:, 3227:= 3221:c 3218:a 3215:v 3210:p 3203:= 3197:c 3194:a 3191:v 3171:p 3162:ρ 3138:. 3125:g 3108:= 3102:) 3099:c 3096:a 3093:v 3090:( 3077:T 3062:Λ 3043:Λ 3037:. 3016:Λ 2999:, 2986:T 2979:= 2967:g 2960:+ 2948:G 2900:g 2877:. 2873:) 2860:g 2855:T 2850:2 2847:1 2830:T 2825:( 2818:= 2806:g 2787:R 2774:D 2757:. 2753:) 2740:g 2736:T 2730:2 2724:D 2720:1 2703:T 2698:( 2691:= 2679:g 2669:2 2663:D 2659:2 2642:R 2630:R 2625:D 2611:, 2608:T 2602:= 2596:D 2593:+ 2590:R 2585:2 2582:D 2574:R 2528:. 2516:T 2506:= 2494:g 2475:g 2471:R 2466:2 2463:1 2446:R 2381:R 2372:] 2369:3 2366:S 2363:[ 2357:] 2354:2 2351:S 2348:[ 2345:= 2333:R 2297:T 2287:] 2284:3 2281:S 2278:[ 2275:= 2259:G 2250:) 2171:+ 2158:, 2131:, 2116:( 2109:] 2106:2 2103:S 2100:[ 2097:= 2071:R 2061:) 2058:1 2055:+ 2052:, 2049:1 2046:+ 2043:, 2040:1 2037:+ 2034:, 2031:1 2025:( 2013:] 2010:1 2007:S 2004:[ 2001:= 1985:g 1933:g 1909:T 1904:n 1847:. 1835:T 1828:= 1816:g 1809:+ 1797:g 1793:R 1788:2 1785:1 1768:R 1743:c 1735:G 1718:, 1691:, 1686:1 1677:N 1654:) 1651:5 1648:( 1635:4 1631:c 1626:G 1620:8 1614:= 1577:R 1544:R 1520:, 1508:g 1504:R 1499:2 1496:1 1479:R 1475:= 1463:G 1375:T 1341:g 1307:G 1262:T 1255:= 1243:g 1236:+ 1224:G 1201:e 1194:t 1187:v 822:( 803:e 796:t 789:v 470:) 466:( 139:T 130:= 118:g 111:+ 99:G 70:. 52:2 48:c 44:m 41:= 38:E 20:)

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Einstein field equations

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