3490:
second, the frame of reference, refers to an idealized system used to assign such numbers To avoid unnecessary restrictions, we can divorce this arrangement from metrical notions. Of special importance for our purposes is that each frame of reference has a definite state of motion at each event of spacetime. Within the context of special relativity and as long as we restrict ourselves to frames of reference in inertial motion, then little of importance depends on the difference between an inertial frame of reference and the inertial coordinate system it induces. This comfortable circumstance ceases immediately once we begin to consider frames of reference in nonuniform motion even within special relativity.âŠMore recently, to negotiate the obvious ambiguities of
Einsteinâs treatment, the notion of frame of reference has reappeared as a structure distinct from a coordinate system.
744:
1935:
1032:
3195:
757:
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points) or times (sets of simultaneous events). So the ideas of a space, a time, of rest and simultaneity, go inextricably together with that of frame. However, a mere shift of origin, or a purely spatial rotation of space coordinates results in a new coordinate system. So frames correspond at best
994:
A coordinate system is a mathematical concept, amounting to a choice of language used to describe observations. Consequently, an observer in an observational frame of reference can choose to employ any coordinate system (Cartesian, polar, curvilinear, generalized, ...) to describe observations made
3489:
In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas. The first is the notion of a coordinate system, understood simply as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods. The
3435:
As noted by
Brillouin, a distinction between mathematical sets of coordinates and physical frames of reference must be made. The ignorance of such distinction is the source of much confusion⊠the dependent functions such as velocity for example, are measured with respect to a physical reference
979:, which are only indirectly related to space and time. It seems useful to divorce the various aspects of a reference frame for the discussion below. We therefore take observational frames of reference, coordinate systems, and observational equipment as independent concepts, separated as below:
955:
rather than upon the coordinate choice or the character of the observations or observational apparatus. In this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. On the other hand, a
3920:
According to
Hawking and Ellis: "A manifold is a space locally similar to Euclidean space in that it can be covered by coordinate patches. This structure allows differentiation to be defined, but does not distinguish between different coordinate systems. Thus, the only concepts defined by the
2089:
or simply "lab frame." An example would be the frame in which the detectors for a particle accelerator are at rest. The lab frame in some experiments is an inertial frame, but it is not required to be (for example the laboratory on the surface of the Earth in many physics experiments is not
3256:: the reference frame is, in some sense, the "Euclidean space carried by the observer". Let us give a more mathematical definition:⊠the reference frame is... the set of all points in the Euclidean space with the rigid body motion of the observer. The frame, denoted
960:
may be employed for many purposes where the state of motion is not the primary concern. For example, a coordinate system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, the formulation of many problems in physics employs
1755:{\displaystyle \mathbf {e} _{i}(\mathbf {r} )=\lim _{\epsilon \rightarrow 0}{\frac {\mathbf {r} \left(x^{1},\ \dots ,\ x^{i}+\epsilon ,\ \dots ,\ x^{n}\right)-\mathbf {r} \left(x^{1},\ \dots ,\ x^{i},\ \dots ,\ x^{n}\right)}{\epsilon }},\quad i=1,\ \dots ,\ n,}
1978:
plus a coordinate lattice constructed to be an orthonormal right-handed set of spacelike vectors perpendicular to a timelike vector. See Doran. This restricted view is not used here, and is not universally adopted even in discussions of relativity. In
2049:
must be invoked to explain observations. An example is an observational frame of reference centered at a point on the Earth's surface. This frame of reference orbits around the center of the Earth, which introduces the fictitious forces known as the
1047:
Although the term "coordinate system" is often used (particularly by physicists) in a nontechnical sense, the term "coordinate system" does have a precise meaning in mathematics, and sometimes that is what the physicist means as well.
4220:
For example, MĂžller states: "Instead of
Cartesian coordinates we can obviously just as well employ general curvilinear coordinates for the fixation of points in physical space.âŠwe shall now introduce general "curvilinear" coordinates
1464:
1908:. There is no necessary connection between coordinate systems and physical motion (or any other aspect of reality). However, coordinate systems can include time as a coordinate, and can be used to describe motion. Thus,
1317:
1938:
Three frames of reference in special relativity. The black frame is at rest. The primed frame moves at 40% of light speed, and the double primed frame at 80%. Note the scissors-like change as speed increases.
98:
920:
The need to distinguish between the various meanings of "frame of reference" has led to a variety of terms. For example, sometimes the type of coordinate system is attached as a modifier, as in
2097:
In this connection it may be noted that the clocks and rods often used to describe observers' measurement equipment in thought, in practice are replaced by a much more complicated and indirect
1892:
1958:, is a physical concept related to an observer and the observer's state of motion. Here we adopt the view expressed by Kumar and Barve: an observational frame of reference is characterized
2132:
In fact, Einstein felt that clocks and rods were merely expedient measuring devices and they should be replaced by more fundamental entities based upon, for example, atoms and molecules.
999:
frame of reference. This viewpoint can be found elsewhere as well. Which is not to dispute that some coordinate systems may be a better choice for some observations than are others.
995:
from that frame of reference. A change in the choice of this coordinate system does not change an observer's state of motion, and so does not entail a change in the observer's
122:
3462:
3429:
3385:
3361:
3337:
3302:
3278:
3243:
1173:
2140:
The discussion is taken beyond simple space-time coordinate systems by
Brading and Castellani. Extension to coordinate systems using generalized coordinates underlies the
2094:"COM frame" in which calculations are sometimes simplified, since potentially all kinetic energy still present in the COM frame may be used for making new particles.
2898:
4167:
2180:
2090:
inertial). In particle physics experiments, it is often useful to transform energies and momenta of particles from the lab frame where they are measured, to the
4044:
2074:(for example, clocks and rods) attached to the frame (see Norton quote above). This question is not addressed in this article, and is of particular interest in
845:
is identified both mathematically (with numerical coordinate values) and physically (signaled by conventional markers). An important special case is that of
3924:
1004:
Choice of what to measure and with what observational apparatus is a matter separate from the observer's state of motion and choice of coordinate system.
4416:
1352:
4334:
4524:
4058:
3694:
As such, the coordinate system is a mathematical construct, a language, that may be related to motion, but has no necessary connection to motion.
2035:
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1212:
788:
4443:
4279:
3759:
4112:
3859:
4085:
2945:
2039:
377:
4014:
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The idea of a reference frame is really quite different from that of a coordinate system. Frames differ just when they define different
4306:
3180:
496:
2969:
2715:
2169:
2085:
In physics experiments, the frame of reference in which the laboratory measurement devices are at rest is usually referred to as the
869:, a reference frame may be defined with a reference point at the origin and a reference point at one unit distance along each of the
4362:
3732:
3636:
3809:
3677:
469:
2174:
2751:
2705:
54:
3705:
4507:
4204:
3869:
3537:
2889:
2373:
1962:. However, there is lack of unanimity on this point. In special relativity, the distinction is sometimes made between an
451:
31:
4389:
2626:
2192:
4040:
4016:
Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review
3591:
1810:
4534:
4480:
4453:
4426:
4399:
4372:
4344:
4316:
4289:
4259:
4177:
4149:
4139:
4122:
4095:
4068:
4024:
3990:
3934:
3902:
3819:
3769:
3742:
3715:
3687:
3646:
3601:
3581:
2997:
2631:
781:
2062:. (All of these forces including gravity disappear in a truly inertial reference frame, which is one of free-fall.)
2075:
2951:
743:
3653:. Less abstractly, a coordinate system in a space of n-dimensions is defined in terms of a basis set of vectors {
2664:
1766:
988:
117:
17:
1998:. An inertial frame of reference is defined as one in which all laws of physics take on their simplest form. In
2767:
372:
112:
3198:
2940:
2845:
2775:
2320:
1064:
885:
3560:
The distinction between macroscopic and microscopic frames shows up, for example, in electromagnetism where
2908:
2030:. These relativistic and Newtonian transformations are expressed in spaces of general dimension in terms of
4556:
2762:
1995:
774:
761:
522:
445:
367:
210:
3280:, is said to move with the observer.⊠The spatial positions of particles are labelled relative to a frame
3524:. Astrophysics and Space Science Library. Vol. 154. Dordrecht: Springer Netherlands. pp. 1â12.
3436:
frame, but one is free to choose any mathematical coordinate system in which the equations are specified.
2803:
2537:
2522:
2305:
2295:
2220:
2114:
1991:
1929:
1072:
932:
874:
866:
441:
242:
3573:
Electromagnetic and
Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media
220:
3090:
2532:
2485:
2265:
661:
550:
476:
336:
269:
3443:
3410:
3366:
3342:
3318:
3283:
3259:
3224:
3120:
2746:
2427:
1917:
1473:. At any selected point, tangents to the intersecting coordinate lines at that point define a set of
415:
1093:
896:), which implies that the observer is at rest in the frame, although not necessarily located at its
4194:
3145:
2695:
2455:
2366:
2290:
2225:
2091:
926:
847:
691:
535:
3564:
of various time and length scales are used to determine the current and charge densities entering
3395:
by carrying out, at each instant of time, the same coordinate transformation on the components of
2874:
2796:
2416:
2407:
2345:
2340:
2280:
2250:
2141:
2027:
1913:
1022:
963:
641:
405:
681:
3894:
3565:
3100:
2963:
2869:
2827:
2501:
2465:
2335:
2325:
2003:
1909:
1781:
1773:
826:
686:
503:
4497:
4421:(Reprint of 1977 edition by Prentice-Hall ed.). Courier Dover Publications. p. 313.
3791:
4251:
3982:
3561:
3105:
2710:
2649:
2031:
2011:
696:
671:
357:
175:
3010:
2936:
2780:
2382:
2210:
2149:
2145:
1203:
1199:
1035:
An observer O, situated at the origin of a local set of coordinates â a frame of reference
897:
881:
822:
716:
676:
584:
580:
572:
562:
352:
345:
101:
711:
8:
3921:
manifold structure are those that are independent of the choice of a coordinate system."
3085:
3080:
3070:
2902:
2851:
2669:
2654:
2480:
2411:
2359:
2330:
2270:
2240:
2153:
2079:
2059:
909:
842:
491:
432:
410:
155:
150:
145:
45:
3075:
3887:
3779:
3150:
3050:
2974:
2832:
2813:
2807:
2758:
2700:
2609:
2527:
2422:
2390:
2310:
2285:
2275:
2260:
1999:
1984:
1980:
1198:, they could be angles of relative rotations, linear displacements, or deformations of
1016:
621:
362:
205:
165:
3837:
General covariance and the foundations of general relativity: eight decades of dispute
930:. Sometimes the way it transforms to frames considered as related is emphasized as in
4530:
4503:
4476:
4449:
4422:
4395:
4368:
4340:
4312:
4285:
4255:
4244:
4200:
4173:
4145:
4118:
4091:
4064:
4020:
3986:
3975:
3930:
3898:
3865:
3815:
3765:
3738:
3711:
3683:
3642:
3597:
3577:
3543:
3533:
3160:
2992:
2984:
2585:
2542:
2300:
2235:
2230:
2215:
2078:, where the relation between observer and measurement is still under discussion (see
2055:
2045:
In contrast to the inertial frame, a non-inertial frame of reference is one in which
1182:, these numbers could be (for example) coefficients in a functional expansion like a
818:
631:
588:
545:
540:
481:
257:
247:
140:
3030:
4241:
3835:
3707:
Unification of
Classical, Quantum and Relativistic Mechanics and of the Four Forces
3525:
3065:
3055:
3002:
2979:
2557:
2086:
2046:
1905:
1026:
838:
830:
726:
706:
651:
646:
592:
567:
422:
280:
225:
200:
3125:
1780:. If the basis vectors are orthogonal at every point, the coordinate system is an
4010:
3944:
3517:
3513:
3140:
3115:
3040:
3035:
2918:
2879:
2841:
2785:
2659:
2595:
2255:
2157:
905:
901:
721:
666:
616:
611:
530:
4137:
3529:
2923:
1934:
3805:
3165:
2837:
2821:
2817:
2720:
2690:
2547:
2450:
2245:
2110:
2051:
1459:{\displaystyle x^{j}(x,y,z,\dots )=\mathrm {constant} ,\quad j=1,\ \dots ,\ n.}
1183:
1031:
984:
936:. Sometimes frames are distinguished by the scale of their observations, as in
748:
656:
557:
274:
888:
and the phenomenon under observation. In this context, the term often becomes
4550:
4475:(Reprint of the 1962 2nd ed.). Courier Dover Publications. p. 118.
3958:
is contained in an open set that is homeomorphic to an open set in
Euclidean
3922:
3547:
3155:
3135:
3130:
3045:
2913:
2741:
2685:
2517:
2470:
2106:
2019:
2015:
1788:
636:
463:
3757:
3315:. The corresponding set of axes, sharing the rigid body motion of the frame
3972:
3175:
3095:
3060:
2590:
2552:
2197:
1901:
1179:
975:
969:
701:
626:
315:
195:
3399:
objects (vectors and tensors) introduced to represent physical quantities
2010:. In Newtonian mechanics, a more restricted definition requires only that
2959:
2928:
2475:
1191:
1068:
884:, reference frames are used to specify the relationship between a moving
865:
reference points are sufficient to fully define a reference frame. Using
3170:
2736:
2580:
2575:
2315:
1983:
the use of general coordinate systems is common (see, for example, the
1798:
1312:{\displaystyle x^{j}=x^{j}(x,\ y,\ z,\ \dots ),\quad j=1,\ \dots ,\ n,}
486:
160:
3703:
3630:
2567:
2098:
2071:
1187:
912:, in which all possible coordinate times are essentially equivalent.
806:
508:
3734:
Handbook of
Continuum Mechanics: General Concepts, Thermoelasticity
3110:
2460:
2126:
2007:
1195:
1060:
1052:
427:
310:
285:
4468:
4242:
A. P. Lightman; W. H. Press; R. H. Price; S. A. Teukolsky (1975).
4008:
3245:
and various associated
Euclidean three-space coordinate systems :
2014:
holds true; that is, a Newtonian inertial frame is one in which a
1987:
solution for the gravitational field outside an isolated sphere).
1765:
which can be normalized to be of unit length. For more detail see
4360:
1056:
802:
400:
253:
170:
4332:
3884:
3506:
2790:
2118:
2102:
459:
305:
215:
4110:
3811:
What Spacetime Explains: Metaphysical essays on space and time
3221:
Here is a quotation applicable to moving observational frames
4441:
4138:
A. I. Borisenko; I. E. Tarapov; Richard A. Silverman (1979).
2351:
2122:
2023:
1202:. Here we will suppose these coordinates can be related to a
295:
290:
232:
4281:
Differential Geometry and Relativity Theory: an introduction
3617:
In very general terms, a coordinate system is a set of arcs
2070:
A further aspect of a frame of reference is the role of the
1900:
As is apparent from these remarks, a coordinate system is a
2771:
1342:
Cartesian coordinates of the point. Given these functions,
900:. A relativistic reference frame includes (or implies) the
300:
263:
4192:
3675:
3638:
L.S. Pontryagin: Selected Works Vol. 2: Topological Groups
3634:
904:, which does not equate across different reference frames
4391:
Analytical Mechanics for Relativity and Quantum Mechanics
3730:
93:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
4165:
3973:
Shigeyuki Morita; Teruko Nagase; Katsumi Nomizu (2001).
3923:
Stephen W. Hawking; George Francis Rayner Ellis (1973).
3857:
3588:. These distinctions also appear in thermodynamics. See
2181:
Lagrangian and Eulerian specification of the flow field
4199:. Cambridge University Press. p. §5.2.2, p. 133.
3761:
Essays on the Formal Aspects of Electromagnetic Theory
3339:, can be considered to give a physical realization of
1990:
There are two types of observational reference frame:
1804:
in the coordinate system in terms of its coordinates:
3758:
Patrick Cornille (Akhlesh Lakhtakia, editor) (1993).
3446:
3413:
3407:
and this on the utility of separating the notions of
3369:
3345:
3321:
3286:
3262:
3227:
1813:
1507:
1355:
1215:
1096:
924:. Sometimes the state of motion is emphasized, as in
57:
4083:
3569:
3981:. American Mathematical Society Bookstore. p.
1923:
1039:. The observer in this frame uses the coordinates (
991:) is a physical concept related to state of motion.
4414:
4387:
4243:
4231:. Oxford University Press. p. 222 and p. 233.
4060:Simulating and Generating Motions of Human Figures
3974:
3886:
3456:
3423:
3379:
3355:
3331:
3296:
3272:
3237:
1886:
1787:An important aspect of a coordinate system is its
1754:
1458:
1311:
1167:
92:
3512:
1051:A coordinate system in mathematics is a facet of
1043:) to describe a spacetime event, shown as a star.
4548:
4364:Symmetries in Physics: Philosophical Reflections
3864:. American Mathematical Society. p. 18 ff.
3704:J X Zheng-Johansson; Per-Ivar Johansson (2006).
3641:(3rd ed.). Gordon and Breach. p. 429.
1887:{\displaystyle (ds)^{2}=g_{ik}\ dx^{i}\ dx^{k},}
1535:
1083:-dimensional space are simply an ordered set of
4304:
4277:
908:to each other. The situation thus differs from
4529:. Cambridge University Press. p. 98 ff.
4522:
2367:
2026:, or is at rest. These frames are related by
782:
4361:Katherine Brading; Elena Castellani (2003).
4141:Vector and Tensor Analysis with Applications
4056:
4495:
4469:Alexander Solomonovich Kompaneyets (2003).
4462:
4367:. Cambridge University Press. p. 417.
4019:. Courier Dover Publications. p. 169.
3589:
1772:Coordinate surfaces, coordinate lines, and
4333:Guido Rizzi; Matteo Luca Ruggiero (2003).
4246:Problem Book in Relativity and Gravitation
4144:. Courier Dover Publications. p. 86.
4117:. Cambridge University Press. p. 84.
4104:
4077:
3929:. Cambridge University Press. p. 11.
3885:Arlan Ramsay; Robert D. Richtmyer (1995).
3814:. Cambridge University Press. p. 64.
3554:
3440:and this, also on the distinction between
2374:
2360:
1469:The intersection of these surfaces define
851:, a stationary or uniformly moving frame.
789:
775:
4502:. Cambridge University Press. Chapter 4.
4489:
4445:Classical Relativistic Many-Body Dynamics
4408:
4226:
4131:
4111:Wilford Zdunkowski; Andreas Bott (2003).
3861:Continuous symmetry: from Euclid to Klein
3611:
2170:International Terrestrial Reference Frame
4442:Matthew A. Trump; W. C. Schieve (1999).
4235:
3751:
3697:
2065:
1933:
1897:where repeated indices are summed over.
1030:
4516:
4394:. Oxford University Press. Chapter 16.
4354:
4298:
4033:
3954:-dimensional manifold if each point of
3926:The Large Scale Structure of Space-Time
3914:
3804:
2175:International Celestial Reference Frame
2101:that is connected to the nature of the
1186:. In a physical problem, they could be
14:
4549:
4250:. Princeton University Press. p.
4214:
4050:
3966:
3878:
3851:
3798:
4435:
4381:
4325:
4271:
4193:Chris Doran; Anthony Lasenby (2003).
4186:
4158:
3828:
3676:Edoardo Sernesi; J. Montaldi (1993).
2355:
1010:
471:Newton's law of universal gravitation
4311:. W W Norton & Co. p. 216.
4002:
3731:Jean Salençon; Stephen Lyle (2001).
3679:Linear Algebra: A Geometric Approach
4166:Arvind Kumar; Shrish Barve (2003).
3889:Introduction to Hyperbolic Geometry
3858:William Barker; Roger Howe (2008).
3724:
3635:Lev Semenovich PontriÍĄagin (1986).
3449:
3416:
3372:
3348:
3324:
3289:
3265:
3230:
2946:TolmanâOppenheimerâVolkoff equation
2899:FriedmannâLemaĂźtreâRobertsonâWalker
983:An observational frame (such as an
452:Mechanics of planar particle motion
60:
32:Frame of reference (disambiguation)
24:
4266:relativistic general coordinates.
3997:geometry axiom coordinate system.
3909:geometry axiom coordinate system.
2193:Frame fields in general relativity
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
951:is used when emphasis is upon the
25:
4568:
3764:. World Scientific. p. 149.
3248:We first introduce the notion of
2716:HamiltonâJacobiâEinstein equation
2135:
867:rectangular Cartesian coordinates
4196:Geometric Algebra for Physicists
3252:, itself related to the idea of
3194:
3193:
2154:classical relativistic mechanics
1944:observational frame of reference
1924:Observational frame of reference
1638:
1554:
1524:
1510:
1098:
949:observational frame of reference
890:observational frame of reference
756:
755:
742:
75:
4172:. Orient Longman. p. 115.
3710:. Nova Publishers. p. 13.
3457:{\displaystyle {\mathfrak {R}}}
3424:{\displaystyle {\mathfrak {R}}}
3387:, coordinates are changed from
3380:{\displaystyle {\mathfrak {R}}}
3356:{\displaystyle {\mathfrak {R}}}
3332:{\displaystyle {\mathfrak {R}}}
3297:{\displaystyle {\mathfrak {R}}}
3273:{\displaystyle {\mathfrak {R}}}
3238:{\displaystyle {\mathfrak {R}}}
2113:and that must be corrected for
1721:
1425:
1278:
1059:, in particular, a property of
989:non-inertial frame of reference
942:microscopic frames of reference
4169:How and Why in Basic Mechanics
4087:Lectures on General Relativity
3977:Geometry of Differential Forms
3941:A mathematical definition is:
3596:. MicroAnalytix. p. 205.
3215:
2523:Massâenergy equivalence (E=mc)
2381:
2109:that operate according to the
1824:
1814:
1542:
1528:
1520:
1390:
1366:
1346:are defined by the relations:
1272:
1239:
1206:system by a set of functions:
1168:{\displaystyle \mathbf {r} =.}
1159:
1105:
13:
1:
4336:Relativity in rotating frames
4084:Achilleus Papapetrou (1974).
3570:Kurt Edmund Oughstun (2006).
3499:
2321:Laboratory frame of reference
915:
894:observational reference frame
378:Koopmanâvon Neumann mechanics
4063:. Springer. pp. 12â13.
2163:
2006:, which are parametrized by
2002:these frames are related by
1970:. According to this view, a
1782:orthogonal coordinate system
922:Cartesian frame of reference
446:Non-inertial reference frame
7:
4415:Donald T Greenwood (1997).
4388:Oliver Davis Johns (2005).
3530:10.1007/978-94-009-0933-5_1
2538:Relativistic Doppler effect
2306:Material frame-indifference
2296:Inertial frame of reference
2221:Cartesian coordinate system
2203:
2198:Moving frame in mathematics
2115:gravitational time dilation
1960:only by its state of motion
1948:physical frame of reference
1930:Inertial frame of reference
933:Galilean frame of reference
927:rotating frame of reference
373:Appell's equation of motion
243:Inertial frame of reference
10:
4573:
4284:. CRC Press. p. 211.
4114:Dynamics of the Atmosphere
3009:In computational physics:
2533:Relativity of simultaneity
1927:
1918:coordinate transformations
1498:} at that point. That is:
1063:(for example, in physics,
1020:
1014:
947:In this article, the term
833:are specified by a set of
29:
27:Abstract coordinate system
3682:. CRC Press. p. 95.
3576:. Springer. p. 165.
3191:
3023:
2888:
2860:
2846:LenseâThirring precession
2729:
2678:
2640:
2619:
2608:
2566:
2510:
2494:
2436:
2428:Doubly special relativity
2400:
2389:
1946:, often referred to as a
848:inertial reference frames
4448:. Springer. p. 99.
4339:. Springer. p. 33.
4305:Richard Wolfson (2003).
4278:Richard L Faber (1983).
4229:The Theory of Relativity
3520:(1989). "Introduction".
3208:
2706:Post-Newtonian formalism
2696:Einstein field equations
2632:Mathematical formulation
2456:Hyperbolic orthogonality
2291:Geodetic reference frame
2226:Center-of-momentum frame
2092:center of momentum frame
2028:Galilean transformations
1914:Galilean transformations
536:Rotating reference frame
368:HamiltonâJacobi equation
4090:. Springer. p. 5.
3834:John D. Norton (1993).
3737:. Springer. p. 9.
3485:and from J. D. Norton:
2417:Galilean transformation
2408:Principle of relativity
2346:Quantum reference frame
2341:Principle of relativity
2281:Generalized coordinates
2251:Curvilinear coordinates
2004:Lorentz transformations
1910:Lorentz transformations
1797:, which determines the
1767:curvilinear coordinates
1023:Generalized coordinates
964:generalized coordinates
477:Newton's laws of motion
337:Newton's laws of motion
4523:Carlo Rovelli (2004).
3562:constitutive relations
3492:
3483:
3481:of coordinate systems.
3458:
3438:
3425:
3405:
3381:
3357:
3333:
3298:
3274:
3239:
2502:Lorentz transformation
2336:Orthogonal coordinates
2326:Lorentz transformation
2266:FrenetâSerret formulas
1939:
1902:mathematical construct
1888:
1756:
1460:
1313:
1169:
1044:
882:Einsteinian relativity
504:Simple harmonic motion
417:Euler's laws of motion
211:D'Alembert's principle
94:
4057:Katsu Yamane (2004).
4009:Granino Arthur Korn;
3487:
3466:
3459:
3433:
3426:
3382:
3358:
3334:
3299:
3275:
3246:
3240:
2970:WeylâLewisâPapapetrou
2711:Raychaudhuri equation
2650:Equivalence principle
2072:measurement apparatus
2066:Measurement apparatus
1937:
1928:Further information:
1889:
1757:
1461:
1314:
1170:
1034:
358:Hamiltonian mechanics
176:Statistical mechanics
95:
4499:Quantum Field Theory
4496:M Srednicki (2007).
3893:. Springer. p.
3590:Paul McEvoy (2002).
3568:. See, for example,
3444:
3411:
3367:
3343:
3319:
3284:
3260:
3225:
3011:Numerical relativity
2852:pulsar timing arrays
2211:Analytical mechanics
2179:In fluid mechanics,
2150:quantum field theory
1811:
1776:are components of a
1505:
1353:
1213:
1204:Cartesian coordinate
1094:
1065:configuration spaces
581:Angular acceleration
573:Rotational frequency
353:Lagrangian mechanics
346:Analytical mechanics
102:Second law of motion
55:
30:For other uses, see
4557:Frames of reference
4472:Theoretical Physics
3962:-dimensional space.
3566:Maxwell's equations
2903:Friedmann equations
2797:HulseâTaylor binary
2759:Gravitational waves
2655:Riemannian geometry
2481:Proper acceleration
2466:Maxwell's equations
2412:Galilean relativity
2271:Galilean invariance
2241:Classical mechanics
2080:measurement problem
2060:gravitational force
1344:coordinate surfaces
910:Galilean relativity
433:Harmonic oscillator
411:Equations of motion
46:Classical mechanics
40:Part of a series on
4418:Classical dynamics
4227:C. MĂžller (1952).
4041:Encarta definition
3790:has generic name (
3454:
3421:
3377:
3353:
3329:
3304:by establishing a
3294:
3270:
3235:
2952:ReissnerâNordström
2870:BransâDicke theory
2701:Linearized gravity
2528:Length contraction
2446:Frame of reference
2423:Special relativity
2311:Rod and frame test
2286:Generalized forces
2276:General relativity
2261:Dynamics (physics)
2012:Newton's first law
2000:special relativity
1981:general relativity
1952:frame of reference
1940:
1884:
1752:
1549:
1456:
1309:
1165:
1045:
1017:Coordinate systems
1011:Coordinate systems
811:frame of reference
749:Physics portal
363:Routhian mechanics
238:Frame of reference
90:
4509:978-0-521-86449-7
4225:in four-spaceâŠ."
4206:978-0-521-71595-9
3871:978-0-8218-3900-3
3539:978-94-010-6909-0
3306:coordinate system
3205:
3204:
3019:
3018:
2998:OzsvĂĄthâSchĂŒcking
2604:
2603:
2586:Minkowski diagram
2543:Thomas precession
2486:Relativistic mass
2301:Local coordinates
2236:Centripetal force
2231:Centrifugal force
2216:Applied mechanics
2076:quantum mechanics
2056:centrifugal force
2047:fictitious forces
1916:may be viewed as
1867:
1851:
1778:coordinate system
1745:
1736:
1716:
1696:
1687:
1671:
1662:
1618:
1609:
1587:
1578:
1534:
1449:
1440:
1302:
1293:
1268:
1259:
1250:
1194:amplitudes. In a
1148:
1139:
1123:
958:coordinate system
906:moving relatively
819:coordinate system
817:) is an abstract
799:
798:
546:Centrifugal force
541:Centripetal force
497:Euler's equations
482:Relative velocity
258:Moment of inertia
88:
62:
16:(Redirected from
4564:
4541:
4540:
4520:
4514:
4513:
4493:
4487:
4486:
4466:
4460:
4459:
4439:
4433:
4432:
4412:
4406:
4405:
4385:
4379:
4378:
4358:
4352:
4350:
4329:
4323:
4322:
4302:
4296:
4295:
4275:
4269:
4268:
4249:
4239:
4233:
4232:
4218:
4212:
4210:
4190:
4184:
4183:
4162:
4156:
4155:
4135:
4129:
4128:
4108:
4102:
4101:
4081:
4075:
4074:
4054:
4048:
4037:
4031:
4030:
4006:
4000:
3999:
3980:
3970:
3964:
3940:
3918:
3912:
3911:
3892:
3882:
3876:
3875:
3855:
3849:
3842:Rep. Prog. Phys.
3832:
3826:
3825:
3802:
3796:
3795:
3789:
3785:
3783:
3775:
3755:
3749:
3748:
3728:
3722:
3721:
3701:
3695:
3693:
3652:
3615:
3609:
3607:
3593:Classical Theory
3587:
3558:
3552:
3551:
3522:Reference Frames
3518:Mueller, Ivan I.
3510:
3493:
3463:
3461:
3460:
3455:
3453:
3452:
3430:
3428:
3427:
3422:
3420:
3419:
3386:
3384:
3383:
3378:
3376:
3375:
3362:
3360:
3359:
3354:
3352:
3351:
3338:
3336:
3335:
3330:
3328:
3327:
3303:
3301:
3300:
3295:
3293:
3292:
3279:
3277:
3276:
3271:
3269:
3268:
3244:
3242:
3241:
3236:
3234:
3233:
3219:
3197:
3196:
2980:van Stockum dust
2752:Two-body problem
2670:Mach's principle
2617:
2616:
2558:Terrell rotation
2398:
2397:
2376:
2369:
2362:
2353:
2352:
2331:Mach's principle
2148:formulations of
2087:laboratory frame
1906:axiomatic system
1893:
1891:
1890:
1885:
1880:
1879:
1865:
1864:
1863:
1849:
1848:
1847:
1832:
1831:
1761:
1759:
1758:
1753:
1743:
1734:
1717:
1712:
1711:
1707:
1706:
1705:
1694:
1685:
1681:
1680:
1669:
1660:
1656:
1655:
1641:
1633:
1629:
1628:
1627:
1616:
1607:
1597:
1596:
1585:
1576:
1572:
1571:
1557:
1551:
1548:
1527:
1519:
1518:
1513:
1471:coordinate lines
1465:
1463:
1462:
1457:
1447:
1438:
1421:
1365:
1364:
1318:
1316:
1315:
1310:
1300:
1291:
1266:
1257:
1248:
1238:
1237:
1225:
1224:
1174:
1172:
1171:
1166:
1158:
1157:
1146:
1137:
1133:
1132:
1121:
1117:
1116:
1101:
1027:Axes conventions
864:
839:geometric points
835:reference points
791:
784:
777:
764:
759:
758:
751:
747:
746:
652:Johann Bernoulli
647:Daniel Bernoulli
568:Tangential speed
472:
448:
423:Fictitious force
418:
270:Mechanical power
260:
201:Angular momentum
99:
97:
96:
91:
89:
87:
79:
78:
69:
64:
63:
37:
36:
21:
4572:
4571:
4567:
4566:
4565:
4563:
4562:
4561:
4547:
4546:
4545:
4544:
4537:
4526:Quantum Gravity
4521:
4517:
4510:
4494:
4490:
4483:
4467:
4463:
4456:
4440:
4436:
4429:
4413:
4409:
4402:
4386:
4382:
4375:
4359:
4355:
4347:
4330:
4326:
4319:
4308:Simply Einstein
4303:
4299:
4292:
4276:
4272:
4262:
4240:
4236:
4219:
4215:
4207:
4191:
4187:
4180:
4163:
4159:
4152:
4136:
4132:
4125:
4109:
4105:
4098:
4082:
4078:
4071:
4055:
4051:
4038:
4034:
4027:
4011:Theresa M. Korn
4007:
4003:
3993:
3971:
3967:
3945:Hausdorff space
3937:
3919:
3915:
3905:
3883:
3879:
3872:
3856:
3852:
3833:
3829:
3822:
3806:Nerlich, Graham
3803:
3799:
3787:
3786:
3777:
3776:
3772:
3756:
3752:
3745:
3729:
3725:
3718:
3702:
3698:
3690:
3673:
3666:
3659:
3649:
3629:) in a complex
3616:
3612:
3604:
3584:
3559:
3555:
3540:
3511:
3507:
3502:
3497:
3496:
3448:
3447:
3445:
3442:
3441:
3415:
3414:
3412:
3409:
3408:
3371:
3370:
3368:
3365:
3364:
3347:
3346:
3344:
3341:
3340:
3323:
3322:
3320:
3317:
3316:
3288:
3287:
3285:
3282:
3281:
3264:
3263:
3261:
3258:
3257:
3250:reference frame
3229:
3228:
3226:
3223:
3222:
3220:
3216:
3211:
3206:
3201:
3187:
3015:
2919:BKL singularity
2909:LemaĂźtreâTolman
2884:
2880:Quantum gravity
2862:
2856:
2842:geodetic effect
2816:(together with
2786:LISA Pathfinder
2725:
2674:
2660:Penrose diagram
2642:
2636:
2611:
2600:
2596:Minkowski space
2562:
2506:
2490:
2438:
2432:
2392:
2385:
2380:
2350:
2256:Datum reference
2206:
2166:
2158:quantum gravity
2138:
2068:
2032:representations
1932:
1926:
1875:
1871:
1859:
1855:
1840:
1836:
1827:
1823:
1812:
1809:
1808:
1795:
1701:
1697:
1676:
1672:
1651:
1647:
1646:
1642:
1637:
1623:
1619:
1592:
1588:
1567:
1563:
1562:
1558:
1553:
1552:
1550:
1538:
1523:
1514:
1509:
1508:
1506:
1503:
1502:
1497:
1490:
1483:
1396:
1360:
1356:
1354:
1351:
1350:
1233:
1229:
1220:
1216:
1214:
1211:
1210:
1190:coordinates or
1153:
1149:
1128:
1124:
1112:
1108:
1097:
1095:
1092:
1091:
1029:
1019:
1013:
953:state of motion
918:
902:coordinate time
859:
815:reference frame
795:
754:
741:
740:
733:
732:
731:
606:
598:
597:
577:
531:Circular motion
525:
515:
514:
513:
470:
440:
437:
416:
395:
387:
386:
383:
382:
340:
330:
322:
321:
320:
279:
275:Mechanical work
268:
252:
190:
182:
181:
180:
135:
127:
104:
80:
74:
70:
68:
59:
58:
56:
53:
52:
35:
28:
23:
22:
18:Reference frame
15:
12:
11:
5:
4570:
4560:
4559:
4543:
4542:
4535:
4515:
4508:
4488:
4481:
4461:
4454:
4434:
4427:
4407:
4400:
4380:
4373:
4353:
4345:
4324:
4317:
4297:
4290:
4270:
4260:
4234:
4213:
4205:
4185:
4178:
4157:
4150:
4130:
4123:
4103:
4096:
4076:
4069:
4049:
4032:
4025:
4001:
3991:
3965:
3935:
3913:
3903:
3877:
3870:
3850:
3827:
3820:
3797:
3770:
3750:
3743:
3723:
3716:
3696:
3688:
3671:
3664:
3657:
3647:
3610:
3602:
3582:
3553:
3538:
3514:Kovalevsky, J.
3504:
3503:
3501:
3498:
3495:
3494:
3451:
3418:
3374:
3350:
3326:
3291:
3267:
3232:
3213:
3212:
3210:
3207:
3203:
3202:
3192:
3189:
3188:
3186:
3185:
3178:
3173:
3168:
3163:
3158:
3153:
3148:
3143:
3138:
3133:
3128:
3123:
3118:
3113:
3108:
3106:Choquet-Bruhat
3103:
3098:
3093:
3088:
3083:
3078:
3073:
3068:
3063:
3058:
3053:
3048:
3043:
3038:
3033:
3027:
3025:
3021:
3020:
3017:
3016:
3014:
3013:
3006:
3005:
3000:
2995:
2988:
2987:
2982:
2977:
2972:
2967:
2958:Axisymmetric:
2955:
2954:
2949:
2943:
2932:
2931:
2926:
2921:
2916:
2911:
2906:
2897:Cosmological:
2894:
2892:
2886:
2885:
2883:
2882:
2877:
2872:
2866:
2864:
2858:
2857:
2855:
2854:
2849:
2838:frame-dragging
2835:
2830:
2825:
2822:Einstein rings
2818:Einstein cross
2811:
2800:
2799:
2794:
2788:
2783:
2778:
2765:
2755:
2754:
2749:
2744:
2739:
2733:
2731:
2727:
2726:
2724:
2723:
2721:Ernst equation
2718:
2713:
2708:
2703:
2698:
2693:
2691:BSSN formalism
2688:
2682:
2680:
2676:
2675:
2673:
2672:
2667:
2662:
2657:
2652:
2646:
2644:
2638:
2637:
2635:
2634:
2629:
2623:
2621:
2614:
2606:
2605:
2602:
2601:
2599:
2598:
2593:
2588:
2583:
2578:
2572:
2570:
2564:
2563:
2561:
2560:
2555:
2550:
2548:Ladder paradox
2545:
2540:
2535:
2530:
2525:
2520:
2514:
2512:
2508:
2507:
2505:
2504:
2498:
2496:
2492:
2491:
2489:
2488:
2483:
2478:
2473:
2468:
2463:
2458:
2453:
2451:Speed of light
2448:
2442:
2440:
2434:
2433:
2431:
2430:
2425:
2420:
2414:
2404:
2402:
2395:
2387:
2386:
2379:
2378:
2371:
2364:
2356:
2349:
2348:
2343:
2338:
2333:
2328:
2323:
2318:
2313:
2308:
2303:
2298:
2293:
2288:
2283:
2278:
2273:
2268:
2263:
2258:
2253:
2248:
2246:Coriolis force
2243:
2238:
2233:
2228:
2223:
2218:
2213:
2207:
2205:
2202:
2201:
2200:
2195:
2189:
2188:
2184:
2183:
2177:
2172:
2165:
2162:
2137:
2136:Generalization
2134:
2111:standard model
2067:
2064:
2052:Coriolis force
2040:Galilean group
2036:Poincaré group
1954:, or simply a
1925:
1922:
1895:
1894:
1883:
1878:
1874:
1870:
1862:
1858:
1854:
1846:
1843:
1839:
1835:
1830:
1826:
1822:
1819:
1816:
1793:
1763:
1762:
1751:
1748:
1742:
1739:
1733:
1730:
1727:
1724:
1720:
1715:
1710:
1704:
1700:
1693:
1690:
1684:
1679:
1675:
1668:
1665:
1659:
1654:
1650:
1645:
1640:
1636:
1632:
1626:
1622:
1615:
1612:
1606:
1603:
1600:
1595:
1591:
1584:
1581:
1575:
1570:
1566:
1561:
1556:
1547:
1544:
1541:
1537:
1533:
1530:
1526:
1522:
1517:
1512:
1495:
1488:
1481:
1467:
1466:
1455:
1452:
1446:
1443:
1437:
1434:
1431:
1428:
1424:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1363:
1359:
1320:
1319:
1308:
1305:
1299:
1296:
1290:
1287:
1284:
1281:
1277:
1274:
1271:
1265:
1262:
1256:
1253:
1247:
1244:
1241:
1236:
1232:
1228:
1223:
1219:
1184:Fourier series
1176:
1175:
1164:
1161:
1156:
1152:
1145:
1142:
1136:
1131:
1127:
1120:
1115:
1111:
1107:
1104:
1100:
1015:Main article:
1012:
1009:
1006:
1005:
1001:
1000:
992:
985:inertial frame
917:
914:
797:
796:
794:
793:
786:
779:
771:
768:
767:
766:
765:
752:
735:
734:
730:
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
664:
659:
654:
649:
644:
639:
634:
629:
624:
619:
614:
608:
607:
604:
603:
600:
599:
596:
595:
576:
575:
570:
565:
560:
558:Coriolis force
555:
554:
553:
543:
538:
533:
527:
526:
521:
520:
517:
516:
512:
511:
506:
501:
500:
499:
494:
484:
479:
474:
467:
456:
455:
454:
449:
436:
435:
430:
425:
420:
413:
408:
403:
397:
396:
393:
392:
389:
388:
385:
384:
381:
380:
375:
370:
365:
360:
355:
349:
343:
341:
334:
331:
328:
327:
324:
323:
319:
318:
313:
308:
303:
298:
293:
288:
283:
277:
272:
266:
261:
250:
245:
240:
235:
230:
229:
228:
223:
213:
208:
203:
198:
192:
191:
188:
187:
184:
183:
179:
178:
173:
168:
163:
158:
153:
148:
143:
137:
136:
133:
132:
129:
128:
126:
125:
120:
115:
109:
106:
105:
100:
86:
83:
77:
73:
67:
49:
48:
42:
41:
26:
9:
6:
4:
3:
2:
4569:
4558:
4555:
4554:
4552:
4538:
4536:0-521-83733-2
4532:
4528:
4527:
4519:
4511:
4505:
4501:
4500:
4492:
4484:
4482:0-486-49532-9
4478:
4474:
4473:
4465:
4457:
4455:0-7923-5737-X
4451:
4447:
4446:
4438:
4430:
4428:0-486-69690-1
4424:
4420:
4419:
4411:
4403:
4401:0-19-856726-X
4397:
4393:
4392:
4384:
4376:
4374:0-521-82137-1
4370:
4366:
4365:
4357:
4348:
4346:1-4020-1805-3
4342:
4338:
4337:
4328:
4320:
4318:0-393-05154-4
4314:
4310:
4309:
4301:
4293:
4291:0-8247-1749-X
4287:
4283:
4282:
4274:
4267:
4263:
4261:0-691-08162-X
4257:
4253:
4248:
4247:
4238:
4230:
4224:
4217:
4208:
4202:
4198:
4197:
4189:
4181:
4179:81-7371-420-7
4175:
4171:
4170:
4161:
4153:
4151:0-486-63833-2
4147:
4143:
4142:
4134:
4126:
4124:0-521-00666-X
4120:
4116:
4115:
4107:
4099:
4097:90-277-0540-2
4093:
4089:
4088:
4080:
4072:
4070:3-540-20317-6
4066:
4062:
4061:
4053:
4046:
4042:
4036:
4028:
4026:0-486-41147-8
4022:
4018:
4017:
4012:
4005:
3998:
3994:
3992:0-8218-1045-6
3988:
3984:
3979:
3978:
3969:
3963:
3959:
3955:
3951:
3950:is called an
3947:
3946:
3938:
3936:0-521-09906-4
3932:
3928:
3927:
3917:
3910:
3906:
3904:0-387-94339-0
3900:
3896:
3891:
3890:
3881:
3873:
3867:
3863:
3862:
3854:
3847:
3843:
3839:
3838:
3831:
3823:
3821:0-521-45261-9
3817:
3813:
3812:
3807:
3801:
3793:
3788:|author=
3781:
3773:
3771:981-02-0854-5
3767:
3763:
3762:
3754:
3746:
3744:3-540-41443-6
3740:
3736:
3735:
3727:
3719:
3717:1-59454-260-0
3713:
3709:
3708:
3700:
3691:
3689:0-412-40680-2
3685:
3681:
3680:
3670:
3663:
3656:
3650:
3648:2-88124-133-6
3644:
3640:
3639:
3632:
3628:
3624:
3620:
3614:
3605:
3603:1-930832-02-8
3599:
3595:
3594:
3585:
3583:0-387-34599-X
3579:
3575:
3574:
3567:
3563:
3557:
3549:
3545:
3541:
3535:
3531:
3527:
3523:
3519:
3515:
3509:
3505:
3491:
3486:
3482:
3480:
3475:
3471:
3465:
3437:
3432:
3404:
3402:
3401:in this frame
3398:
3394:
3390:
3363:. In a frame
3314:
3310:
3307:
3255:
3251:
3218:
3214:
3200:
3190:
3184:
3183:
3179:
3177:
3174:
3172:
3169:
3167:
3164:
3162:
3159:
3157:
3154:
3152:
3149:
3147:
3144:
3142:
3139:
3137:
3134:
3132:
3129:
3127:
3124:
3122:
3119:
3117:
3114:
3112:
3109:
3107:
3104:
3102:
3099:
3097:
3094:
3092:
3091:Chandrasekhar
3089:
3087:
3084:
3082:
3079:
3077:
3074:
3072:
3069:
3067:
3064:
3062:
3059:
3057:
3054:
3052:
3051:Schwarzschild
3049:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3028:
3026:
3022:
3012:
3008:
3007:
3004:
3001:
2999:
2996:
2994:
2990:
2989:
2986:
2983:
2981:
2978:
2976:
2973:
2971:
2968:
2965:
2961:
2957:
2956:
2953:
2950:
2947:
2944:
2942:
2938:
2937:Schwarzschild
2934:
2933:
2930:
2927:
2925:
2922:
2920:
2917:
2915:
2912:
2910:
2907:
2904:
2900:
2896:
2895:
2893:
2891:
2887:
2881:
2878:
2876:
2873:
2871:
2868:
2867:
2865:
2859:
2853:
2850:
2847:
2843:
2839:
2836:
2834:
2833:Shapiro delay
2831:
2829:
2826:
2823:
2819:
2815:
2812:
2809:
2805:
2802:
2801:
2798:
2795:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2776:collaboration
2773:
2769:
2766:
2764:
2760:
2757:
2756:
2753:
2750:
2748:
2745:
2743:
2742:Event horizon
2740:
2738:
2735:
2734:
2732:
2728:
2722:
2719:
2717:
2714:
2712:
2709:
2707:
2704:
2702:
2699:
2697:
2694:
2692:
2689:
2687:
2686:ADM formalism
2684:
2683:
2681:
2677:
2671:
2668:
2666:
2663:
2661:
2658:
2656:
2653:
2651:
2648:
2647:
2645:
2639:
2633:
2630:
2628:
2625:
2624:
2622:
2618:
2615:
2613:
2607:
2597:
2594:
2592:
2591:Biquaternions
2589:
2587:
2584:
2582:
2579:
2577:
2574:
2573:
2571:
2569:
2565:
2559:
2556:
2554:
2551:
2549:
2546:
2544:
2541:
2539:
2536:
2534:
2531:
2529:
2526:
2524:
2521:
2519:
2518:Time dilation
2516:
2515:
2513:
2509:
2503:
2500:
2499:
2497:
2493:
2487:
2484:
2482:
2479:
2477:
2474:
2472:
2471:Proper length
2469:
2467:
2464:
2462:
2459:
2457:
2454:
2452:
2449:
2447:
2444:
2443:
2441:
2435:
2429:
2426:
2424:
2421:
2418:
2415:
2413:
2409:
2406:
2405:
2403:
2399:
2396:
2394:
2388:
2384:
2377:
2372:
2370:
2365:
2363:
2358:
2357:
2354:
2347:
2344:
2342:
2339:
2337:
2334:
2332:
2329:
2327:
2324:
2322:
2319:
2317:
2314:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2284:
2282:
2279:
2277:
2274:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2254:
2252:
2249:
2247:
2244:
2242:
2239:
2237:
2234:
2232:
2229:
2227:
2224:
2222:
2219:
2217:
2214:
2212:
2209:
2208:
2199:
2196:
2194:
2191:
2190:
2186:
2185:
2182:
2178:
2176:
2173:
2171:
2168:
2167:
2161:
2159:
2155:
2151:
2147:
2143:
2133:
2130:
2128:
2124:
2120:
2116:
2112:
2108:
2107:atomic clocks
2104:
2100:
2095:
2093:
2088:
2083:
2081:
2077:
2073:
2063:
2061:
2057:
2053:
2048:
2043:
2041:
2037:
2033:
2029:
2025:
2021:
2020:straight line
2018:travels in a
2017:
2016:free particle
2013:
2009:
2005:
2001:
1997:
1993:
1988:
1986:
1985:Schwarzschild
1982:
1977:
1973:
1969:
1965:
1961:
1957:
1953:
1949:
1945:
1936:
1931:
1921:
1919:
1915:
1911:
1907:
1904:, part of an
1903:
1898:
1881:
1876:
1872:
1868:
1860:
1856:
1852:
1844:
1841:
1837:
1833:
1828:
1820:
1817:
1807:
1806:
1805:
1803:
1800:
1796:
1790:
1789:metric tensor
1785:
1783:
1779:
1775:
1774:basis vectors
1770:
1768:
1749:
1746:
1740:
1737:
1731:
1728:
1725:
1722:
1718:
1713:
1708:
1702:
1698:
1691:
1688:
1682:
1677:
1673:
1666:
1663:
1657:
1652:
1648:
1643:
1634:
1630:
1624:
1620:
1613:
1610:
1604:
1601:
1598:
1593:
1589:
1582:
1579:
1573:
1568:
1564:
1559:
1545:
1539:
1531:
1515:
1501:
1500:
1499:
1494:
1487:
1480:
1476:
1475:basis vectors
1472:
1453:
1450:
1444:
1441:
1435:
1432:
1429:
1426:
1422:
1393:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1361:
1357:
1349:
1348:
1347:
1345:
1341:
1337:
1333:
1329:
1325:
1306:
1303:
1297:
1294:
1288:
1285:
1282:
1279:
1275:
1269:
1263:
1260:
1254:
1251:
1245:
1242:
1234:
1230:
1226:
1221:
1217:
1209:
1208:
1207:
1205:
1201:
1197:
1193:
1189:
1185:
1181:
1178:In a general
1162:
1154:
1150:
1143:
1140:
1134:
1129:
1125:
1118:
1113:
1109:
1102:
1090:
1089:
1088:
1086:
1082:
1078:
1074:
1070:
1066:
1062:
1058:
1054:
1049:
1042:
1038:
1033:
1028:
1024:
1018:
1008:
1003:
1002:
998:
997:observational
993:
990:
986:
982:
981:
980:
978:
977:
972:
971:
966:
965:
959:
954:
950:
945:
943:
939:
935:
934:
929:
928:
923:
913:
911:
907:
903:
899:
895:
891:
887:
883:
878:
876:
872:
868:
862:
857:
852:
850:
849:
844:
840:
836:
832:
828:
824:
820:
816:
812:
808:
804:
792:
787:
785:
780:
778:
773:
772:
770:
769:
763:
753:
750:
745:
739:
738:
737:
736:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
613:
610:
609:
602:
601:
594:
590:
586:
582:
579:
578:
574:
571:
569:
566:
564:
561:
559:
556:
552:
549:
548:
547:
544:
542:
539:
537:
534:
532:
529:
528:
524:
519:
518:
510:
507:
505:
502:
498:
495:
493:
490:
489:
488:
485:
483:
480:
478:
475:
473:
468:
465:
461:
458:
457:
453:
450:
447:
443:
439:
438:
434:
431:
429:
426:
424:
421:
419:
414:
412:
409:
407:
404:
402:
399:
398:
391:
390:
379:
376:
374:
371:
369:
366:
364:
361:
359:
356:
354:
351:
350:
348:
347:
342:
339:
338:
333:
332:
326:
325:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
292:
289:
287:
284:
282:
278:
276:
273:
271:
267:
265:
262:
259:
255:
251:
249:
246:
244:
241:
239:
236:
234:
231:
227:
224:
222:
219:
218:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
193:
186:
185:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
147:
144:
142:
139:
138:
131:
130:
124:
121:
119:
116:
114:
111:
110:
108:
107:
103:
84:
81:
71:
65:
51:
50:
47:
44:
43:
39:
38:
33:
19:
4525:
4518:
4498:
4491:
4471:
4464:
4444:
4437:
4417:
4410:
4390:
4383:
4363:
4356:
4335:
4327:
4307:
4300:
4280:
4273:
4265:
4245:
4237:
4228:
4222:
4216:
4195:
4188:
4168:
4160:
4140:
4133:
4113:
4106:
4086:
4079:
4059:
4052:
4035:
4015:
4004:
3996:
3976:
3968:
3961:
3957:
3953:
3949:
3943:A connected
3942:
3925:
3916:
3908:
3888:
3880:
3860:
3853:
3848:, pp. 835-7.
3845:
3841:
3836:
3830:
3810:
3800:
3760:
3753:
3733:
3726:
3706:
3699:
3678:
3668:
3661:
3654:
3637:
3626:
3622:
3618:
3613:
3592:
3572:
3556:
3521:
3508:
3488:
3484:
3478:
3473:
3469:
3467:
3439:
3434:
3406:
3400:
3396:
3392:
3388:
3312:
3311:with origin
3308:
3305:
3253:
3249:
3247:
3217:
3181:
2875:KaluzaâKlein
2627:Introduction
2553:Twin paradox
2445:
2187:Other frames
2139:
2131:
2096:
2084:
2069:
2044:
2022:at constant
1996:non-inertial
1989:
1975:
1971:
1967:
1963:
1959:
1955:
1951:
1947:
1943:
1941:
1899:
1896:
1801:
1791:
1786:
1777:
1771:
1764:
1492:
1485:
1478:
1474:
1470:
1468:
1343:
1339:
1335:
1331:
1327:
1323:
1321:
1196:robot design
1180:Banach space
1177:
1084:
1080:
1076:
1069:phase spaces
1050:
1046:
1040:
1036:
1007:
996:
976:eigenvectors
974:
970:normal modes
968:
962:
957:
952:
948:
946:
941:
937:
931:
925:
921:
919:
893:
889:
879:
870:
860:
858:dimensions,
855:
853:
846:
834:
814:
810:
800:
591: /
587: /
585:displacement
583: /
444: /
406:Displacement
344:
335:
329:Formulations
316:Virtual work
256: /
237:
196:Acceleration
189:Fundamentals
4047:2009-10-31.
2964:KerrâNewman
2935:Spherical:
2804:Other tests
2747:Singularity
2679:Formulation
2641:Fundamental
2495:Formulation
2476:Proper time
2437:Fundamental
2142:Hamiltonian
2105:, and uses
2038:and of the
1192:normal mode
1075:of a point
1073:coordinates
938:macroscopic
873:coordinate
827:orientation
727:von Neumann
394:Core topics
3500:References
3116:Zel'dovich
3024:Scientists
3003:Alcubierre
2810:of Mercury
2808:precession
2737:Black hole
2620:Background
2612:relativity
2581:World line
2576:Light cone
2401:Background
2393:relativity
2383:Relativity
2316:Kinematics
2146:Lagrangian
1799:arc length
1087:numbers:
1041:x, y, z, t
1021:See also:
916:Definition
662:d'Alembert
642:Maupertuis
605:Scientists
487:Rigid body
161:Kinematics
3780:cite book
3631:Lie group
3548:0067-0057
3472:(sets of
3397:intrinsic
3086:Robertson
3071:Friedmann
3066:Eddington
3056:de Sitter
2890:Solutions
2768:detectors
2763:astronomy
2730:Phenomena
2665:Geodesics
2568:Spacetime
2511:Phenomena
2164:Instances
2099:metrology
1738:…
1714:ϵ
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807:astronomy
707:Liouville
589:frequency
509:Vibration
226:potential
151:Continuum
146:Celestial
123:Textbooks
4551:Category
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4013:(2000).
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3254:observer
3199:Category
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2863:theories
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2643:concepts
2461:Rapidity
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2204:See also
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2008:rapidity
1992:inertial
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563:Pendulum
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523:Rotation
492:dynamics
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428:Friction
311:Velocity
286:Momentum
166:Kinetics
156:Dynamics
134:Branches
118:Timeline
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248:Impulse
221:kinetic
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