737:
4051:
67:
3715:
750:
1669:. One can obtain intuitive geodesic and field equations in those situations as well . Unlike circular orbits, however, the speed of the particles in elliptic or hyperbolic trajectories is not constant. We therefore do not have a constant speed with which to scale the curvature. Therefore, in anticipation of the transition to relativistic mechanics, the trajectories and curvatures are scaled with the
6231:
4046:{\displaystyle {\begin{Vmatrix}{\mathcal {M}}_{1}^{1}&{\mathcal {M}}_{1}^{2}&{\mathcal {M}}_{1}^{3}\\{\mathcal {M}}_{2}^{1}&{\mathcal {M}}_{2}^{2}&{\mathcal {M}}_{2}^{3}\\{\mathcal {M}}_{3}^{1}&{\mathcal {M}}_{3}^{2}&{\mathcal {M}}_{3}^{3}\end{Vmatrix}}={\begin{Vmatrix}1&0&0\\0&\cos(\theta )&\sin(\theta )\\0&-\sin(\theta )&\cos(\theta )\end{Vmatrix}}}
2795:
6427:
4345:
world line (deviation from being vertical) is the velocity of the particle on that section of the world line. So at a bend in the world line the particle is being accelerated. Note how the view of spacetime changes when the observer accelerates, changing the instantaneous inertial frame of reference. These changes are governed by the
Lorentz transformations. Also note that:
2951:
5497:
5961:
2525:
5503:
In an arbitrarily rotating spacecraft, the curvature of space is due to two terms, one due to the mass density and one due to the arbitrary rotation of the spacecraft. The arbitrary rotation is non-physical and must be eliminated in any real physical theory of gravitation. In
General Relativity this
4344:
for the observer's current event, and intersect at that event. The small dots are other arbitrary events in the spacetime. For the observer's current instantaneous inertial frame of reference, the vertical direction indicates the time and the horizontal direction indicates distance. The slope of the
2218:
In the case that the separation of the particle is radial, the curvature is negative. This will cause the particles to separate rather than to be drawn toward each other as in the case in which they have the same radius. This is easy to understand. Outer orbits travel slower than inner orbits. This
1495:
of the spacecraft coincident with one of the particles. That particle would be at rest with respect to the spacecraft. An observer in the spacecraft would have no indication that the craft was orbiting Earth. The observer is only allowed to measure the behavior of the particles in the frame of the
6399:
The geodesic and field equations simply are a restatement of Newton's Law of
Gravitation as seen from a local frame of reference co-moving with the mass within the local frame. This picture contains many of the elements of General Relativity, including the concept that particles travel along
2806:
5220:
4683:
4521:
4813:
6430:
World line of a circular orbit about the Earth depicted in two spatial dimensions X and Y (the plane of the orbit) and a time dimension, usually put as the vertical axis. Note that the orbit about the Earth is (almost) a circle in space, but its worldline is a helix in
6400:
geodesics in a curved space (spacetime in the relativistic case) and that the curvature is due to the presence of mass density (mass/energy density in the relativistic case). This picture also contains some of the mathematical machinery of
General Relativity such as
800:
A unique feature of the gravitational force is that all massive objects accelerate in the same manner in a gravitational field. This is often expressed as "The gravitational mass is equal to the inertial mass." This allows us to think of gravity as a curvature of
6226:{\displaystyle {\begin{Vmatrix}R_{1}^{1}&R_{1}^{2}&R_{1}^{3}\\R_{2}^{1}&R_{2}^{2}&R_{2}^{3}\\R_{3}^{1}&R_{3}^{2}&R_{3}^{3}\end{Vmatrix}}={\begin{Vmatrix}R_{\perp }&0&0\\0&R_{\perp }&0\\0&0&R_{\|}\end{Vmatrix}}}
2790:{\displaystyle {\begin{Vmatrix}R_{1}^{1}&R_{1}^{2}&R_{1}^{3}\\R_{2}^{1}&R_{2}^{2}&R_{2}^{3}\\R_{3}^{1}&R_{3}^{2}&R_{3}^{3}\end{Vmatrix}}={\begin{Vmatrix}R_{\perp }&0&0\\0&R_{\perp }&0\\0&0&R_{\|}\end{Vmatrix}}}
1280:
In this example, the field equation is simply a statement of the
Newtonian concept that centripetal force is equal to gravitational force for circular orbits. We refer to this expression as a field equation in order to highlight the similarities with the
5211:
5914:
4310:
5062:
5686:
4945:
4200:
1979:
5775:
3542:
2131:
2946:{\displaystyle {\begin{Vmatrix}h^{1}&h^{2}&h^{3}\end{Vmatrix}}={\begin{Vmatrix}\mathbf {h} \cdot \mathbf {\hat {x}} &\mathbf {h} \cdot \mathbf {\hat {y}} &\mathbf {h} \cdot \mathbf {\hat {z}} \end{Vmatrix}}}
1782:
5492:{\displaystyle {d^{2}{\bar {h}}^{i} \over ds^{2}}+2\Gamma _{j}^{i}{d{\bar {h}}^{i} \over ds}+{d\Gamma _{j}^{i} \over ds}{\bar {h}}^{j}+\Gamma _{j}^{i}\Gamma _{k}^{j}{\bar {h}}^{k}+{\bar {R}}_{j}^{i}{\bar {h}}^{j}=0}
138:
1858:
1098:
3624:
3216:
direction, nor can he/she know which direction is the velocity with respect to earth. The orientation of the spacecraft may be quite different from the simple coordinate system in which the ceiling is in the
4532:
4379:
4316:
The curvature in the new coordinate system is non-diagonal. The inverse problem of transforming an arbitrary coordinate system into a diagonal system can be performed mathematically with the process of
6310:
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2478:
51:
as our prototype. This has the advantage that we know the kinetics of circular orbits. This allows us to calculate curvature of orbits in space directly and compare the results with dynamical forces.
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6388:
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direction and the front of the craft is in a direction coplanar with the radius and the velocity. We can transform our simple coordinates to an arbitrarily oriented coordinate system through
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A requirement of
General Relativity is that all measurements must be made locally. Imagine that the particles are inside a windowless spacecraft co-orbiting the Earth with the
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2004:
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The space craft may tumble about its center of mass. In that case the rotation matrix is time dependent. If the rotation matrix is time dependent, then it does not
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2012:
718:
528:
370:
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813:
If initially parallel paths of two particles on nearby geodesics remain parallel within some accuracy, then spacetime is flat to within that accuracy.
483:
4355:• events pass through the light cone lines due to the progression of proper time, but not due to the change of views caused by the accelerations, and
2319:
but still in the plane of the trajectory. Unlike in a circular orbit, this craft no longer necessarily points in the direction of the velocity. The
1701:
774:
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827:
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396:
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Co-planar elliptic orbits. The particle in the outer orbit travels slower than the particle in the inner orbit. They will separate with time.
435:
73:
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1033:
767:
3554:
6507:
147:
4678:{\displaystyle {d{\bar {h}}^{k} \over ds}+\Gamma _{j}^{k}{\bar {h}}^{j}\ {\stackrel {\mathrm {def} }{=}}\ {D{\bar {h}}^{k} \over Ds}}
4516:{\displaystyle {\mathcal {M}}_{i}^{k}{dh^{i} \over ds}={\mathcal {M}}_{i}^{k}{d{\bar {\mathcal {M}}}_{j}^{i}{\bar {h}}^{j} \over ds}}
6628:
6604:
6579:
6558:
6533:
6492:
994:
about the earth and with respect to each other. They are at their maximum distance from each other as they cross the equator. Their
173:
453:
4808:{\displaystyle \Gamma _{j}^{k}\ {\stackrel {\mathrm {def} }{=}}\ {\mathcal {M}}_{i}^{k}{d{\bar {\mathcal {M}}}_{j}^{i} \over ds}}
66:
6243:
3337:
2391:
360:
1628:
is the position vector for the second particle. An observer in the craft would think the second particle was oscillating in a
6781:
6318:
3113:
3047:
2959:
493:
2484:
1106:
447:
155:
513:
17:
6621:
6975:
6918:
896:
754:
245:
165:
6409:
5505:
1360:
6749:
6722:
6596:
4350:
1286:
658:
206:
1427:
458:
270:
533:
6855:
693:
683:
350:
4347:• the balls on the world line before/after future/past accelerations are more spaced out due to time dilation.
412:
6440:
5922:
1282:
678:
5206:{\displaystyle {d{\bar {u}}^{i} \over ds}+\Gamma _{j}^{i}{\bar {u}}^{j}+{\bar {R}}_{j}^{i}{\bar {h}}^{j}=0}
3409:
478:
6939:
6801:
2235:
A local coordinate system for a space craft co-moving with one of the particles can again be defined. The
1296:
The position of the moving particle with respect to the particle at rest in the co-moving reference frame.
1233:
991:
960:
648:
633:
240:
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6697:
4367:
623:
191:
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4349:• events which were simultaneous before an acceleration are at different times afterwards (due to the
2483:
where the Latin indices are over the spatial directions in the co-moving system, and we have used the
6903:
6727:
6687:
4332:
of a rapidly accelerating observer. In this animation, the dashed line is the spacetime trajectory ("
3633:
443:
5556:
for all i and j. The only time-dependent rotations allowed are those generated by the mass density.
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3259:
6878:
6737:
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6717:
6692:
6669:
6645:
4324:
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In this co-moving coordinate system the curvature tensor is diagonal. This is not true in general.
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1008:
498:
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sense, Fermi–Walker transport is simply a statement that the spacecraft is not allowed to tumble
5509:
4819:
4080:
If the rotation matrix does not depend on time then the geodesic equation becomes, upon rotation
1633:
688:
573:
345:
250:
235:
196:
58:
32:
28:
3187:
The co-moving spacecraft has no windows. An observer is not able to tell which direction is the
1787:
one can obtain the geodesic equation for the separation of two particles in nearby trajectories
5909:{\displaystyle {\bar {R}}_{j}^{i}={\mathcal {M}}_{k}^{i}R_{l}^{k}{\bar {\mathcal {M}}}_{j}^{l}}
4305:{\displaystyle {\bar {R}}_{j}^{i}={\mathcal {M}}_{k}^{i}R_{l}^{k}{\bar {\mathcal {M}}}_{j}^{l}}
2490:
6949:
6898:
6843:
6823:
6709:
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887:
568:
518:
293:
5783:
4060:
3672:
6888:
6828:
6754:
6475:
4357:• the world line always remains within the future and past light cones of the current event.
2186:
2161:
1662:
663:
643:
598:
578:
523:
5057:{\displaystyle {\bar {u}}^{i}\ {\stackrel {\mathrm {def} }{=}}\ {D{\bar {h}}^{i} \over Ds}}
4951:
which is the same as before with the exception that the derivatives have been generalized.
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6764:
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673:
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553:
417:
298:
260:
48:
886:. Since the orbits are circular, the gravitational force on the particles must equal the
468:
1636:
due to the choice of frames as opposed to a physical acceleration due to actual forces.
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966:
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708:
703:
391:
255:
4057:
This is a rotation in the y-z plane. The inverse is obtained by switching the sign of
6969:
6550:
5681:{\displaystyle {D^{2}{\bar {h}}^{i} \over Ds^{2}}+{\bar {R}}_{j}^{i}{\bar {h}}^{j}=0}
4940:{\displaystyle {D^{2}{\bar {h}}^{i} \over Ds^{2}}+{\bar {R}}_{j}^{i}{\bar {h}}^{j}=0}
4195:{\displaystyle {d^{2}{\bar {h}}^{i} \over ds^{2}}+{\bar {R}}_{j}^{i}{\bar {h}}^{j}=0}
2352:
The geodesic equation in a radial gravitational field can be described succinctly in
668:
628:
583:
563:
488:
386:
265:
1974:{\displaystyle R=R_{\perp }={GM \over {c^{2}r^{3}}}={4\pi G \over {3c^{2}}}\rho (r)}
1289:, which is the usual characterization of the field equation in Newtonian mechanics.
6873:
6502:
6480:
5770:{\displaystyle {D \over Ds}\ {\stackrel {\mathrm {def} }{=}}\ {d \over ds}+\Gamma }
3537:{\displaystyle {\bar {\mathcal {M}}}_{j}^{i}{\mathcal {M}}_{k}^{j}=\delta _{k}^{i}}
2356:
notation in the co-moving frame in which the ceiling of the space craft is in the
2126:{\displaystyle R=R_{\|}=-{2GM \over {c^{2}r^{3}}}=-{8\pi G \over {3c^{2}}}\rho (r)}
613:
593:
4337:
843:
503:
473:
1665:
trajectories in a plane that contains the earth center. The orbits need not be
6893:
6883:
6444:
4341:
4333:
4329:
1777:{\displaystyle {d^{2}\mathbf {r} \over dt^{2}}=-{GM \over r^{3}}\mathbf {r} }
995:
713:
275:
201:
6744:
6613:
6448:
1483:
The curvature of the particle trajectories is proportional to mass density.
1168:
802:
284:
40:
36:
6443:
to the relativistic realm in which trajectories in space are replaced with
3669:
A simple rotation matrix that rotates the coordinate axis through an angle
6426:
1519:-direction is toward the ceiling of the craft and this is directed along
653:
463:
303:
43:
and that curvature affects the motion of masses can be illustrated in a
3250:. This, however, destroys the diagonal nature of the curvature matrix.
1561:-direction is toward the front of the craft and is in the direction of
1499:
In this example, we can define a local coordinate system such that the
1002:
133:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
6545:
Misner, Charles; Thorne, Kip S. & Wheeler, John
Archibald (1973).
1207:
The curvature of the trajectory is generated by the mass of the earth
6848:
6451:. The equations are also generalized to more complicated curvatures.
6401:
4373:
In that case, the rotation of the separation velocity can be written
2353:
1853:{\displaystyle {d^{2}\mathbf {h} \over d\tau ^{2}}+R\mathbf {h} =0}
1093:{\displaystyle {d^{2}\mathbf {h} \over d\tau ^{2}}+R\mathbf {h} =0}
6425:
4323:
2226:
1648:
1291:
839:
Consider the situation in which there are two particles in nearby
825:
3619:{\displaystyle h^{i}={\bar {\mathcal {M}}}_{j}^{i}{\bar {h}}^{j}}
984:
6617:
4362:
Time dependent rotation of the local frame: Christoffel symbols
4336:") of a particle. The balls are placed at regular intervals of
2297:-direction, toward the front of the craft, is perpendicular to
5929:
5885:
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4773:
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1632:
generated by a gravitational field. This is an example of a
5560:
General geodesic and field equations in a
Newtonian setting
4959:
The velocity in the frame of the spacecraft can be written
2487:
in which repeated indices are summed. The curvature tensor
35:
domain. In particular, the idea that mass–energy generates
6305:{\displaystyle R_{\perp }={4\pi G \over {3c^{2}}}\rho (r)}
3395:{\displaystyle {\bar {h}}^{i}={\mathcal {M}}_{j}^{i}h^{j}}
2473:{\displaystyle {d^{2}h^{i} \over ds^{2}}+R_{j}^{i}h^{j}=0}
6498:
Basic introduction to the mathematics of curved spacetime
2231:
Local "diagonal" coordinate system for an elliptic orbit.
1305:
Mass can be written in terms of the average mass density
18:
Transition from
Newtonian mechanics to general relativity
6383:{\displaystyle R_{\|}=-{8\pi G \over {3c^{2}}}\rho (r)}
4340:
along the world line. The solid diagonal lines are the
2255:-direction, toward the ceiling, is in the direction of
6144:
5970:
3934:
3724:
2868:
2815:
2708:
2534:
6574:(Fourth Revised English ed.). Oxford: Pergamon.
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3557:
3470:
3443:
3412:
3340:
3315:
3286:
3262:
3223:
3193:
3153:
3140:{\displaystyle \mathbf {h} \cdot \mathbf {\hat {z}} }
3116:
3087:
3074:{\displaystyle \mathbf {h} \cdot \mathbf {\hat {y}} }
3050:
3021:
2999:
2986:{\displaystyle \mathbf {h} \cdot \mathbf {\hat {x}} }
2962:
2809:
2528:
2493:
2394:
2362:
2325:
2303:
2283:
2261:
2241:
2195:
2164:
2142:
2015:
1990:
1872:
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1589:
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1505:
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969:
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872:
852:
76:
3309:
is related to the separation vector before rotation
1160:{\displaystyle R={1 \over r^{2}}{v^{2} \over c^{2}}}
1027:
can be shown to be given by the "geodesic equation"
835:
The geodesic and field equations for circular orbits
817:
Two nearby particles in a radial gravitational field
6927:
6864:
6792:
6708:
6680:
6652:
6521:
6382:
6304:
6237:The curvature is proportional to the mass density
6225:
5947:
5908:
5792:
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5680:
5545:
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5205:
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1998:
1973:
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1019:
975:
944:
878:
858:
796:The equivalence of gravitational and inertial mass
132:
5955:is a rotation matrix and the curvature tensor is
4328:Diagram 1. Changing views of spacetime along the
3293:
3230:
3200:
3160:
3131:
3094:
3065:
3028:
2977:
2929:
2907:
2885:
2369:
2339:-direction is toward the left side of the craft.
1640:General motion in the earth's gravitational field
1603:-direction is toward the left side of the craft.
1285:. This equation is in a much different form than
945:{\displaystyle {v^{2} \over r}={GM \over r^{2}}}
1984:if the particle separation is perpendicular to
1301:Relationship between curvature and mass density
1410:{\displaystyle M={4\pi \rho (r)r^{3} \over 3}}
1227:. This is represented by the "field equation"
6629:
6422:Theoretical motivation for general relativity
775:
8:
6570:Landau, L. D. & Lifshitz, E. M. (1975).
6327:
6210:
2774:
2170:
2027:
1472:{\displaystyle R={4\pi G \over {3}}\rho (r)}
6636:
6622:
6614:
6524:Relativity: The Special and General Theory
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4228:
4217:
4216:
4213:
4180:
4169:
4168:
4161:
4156:
4145:
4144:
4131:
4116:
4105:
4104:
4097:
4090:
4088:
4062:
3929:
3912:
3907:
3901:
3900:
3891:
3886:
3880:
3879:
3870:
3865:
3859:
3858:
3847:
3842:
3836:
3835:
3826:
3821:
3815:
3814:
3805:
3800:
3794:
3793:
3782:
3777:
3771:
3770:
3761:
3756:
3750:
3749:
3740:
3735:
3729:
3728:
3719:
3717:
3694:
3674:
3646:
3641:
3635:
3610:
3599:
3598:
3591:
3586:
3575:
3573:
3572:
3562:
3556:
3528:
3523:
3510:
3505:
3499:
3498:
3491:
3486:
3475:
3473:
3472:
3469:
3445:
3444:
3442:
3416:
3414:
3413:
3411:
3386:
3376:
3371:
3365:
3364:
3354:
3343:
3342:
3339:
3316:
3314:
3288:
3287:
3285:
3264:
3263:
3261:
3225:
3224:
3222:
3195:
3194:
3192:
3155:
3154:
3152:
3126:
3125:
3117:
3115:
3089:
3088:
3086:
3060:
3059:
3051:
3049:
3023:
3022:
3020:
3000:
2998:
2972:
2971:
2963:
2961:
2924:
2923:
2915:
2902:
2901:
2893:
2880:
2879:
2871:
2863:
2846:
2834:
2822:
2810:
2808:
2773:
2744:
2715:
2703:
2686:
2681:
2669:
2664:
2652:
2647:
2633:
2628:
2616:
2611:
2599:
2594:
2580:
2575:
2563:
2558:
2546:
2541:
2529:
2527:
2503:
2498:
2492:
2458:
2448:
2443:
2427:
2412:
2402:
2395:
2393:
2364:
2363:
2361:
2324:
2304:
2302:
2282:
2262:
2260:
2240:
2196:
2194:
2169:
2163:
2143:
2141:
2102:
2094:
2081:
2066:
2056:
2051:
2038:
2026:
2014:
1991:
1989:
1950:
1942:
1929:
1917:
1907:
1902:
1892:
1883:
1871:
1839:
1824:
1810:
1804:
1797:
1795:
1769:
1761:
1747:
1732:
1718:
1712:
1705:
1703:
1677:
1613:
1611:
1588:
1568:
1566:
1546:
1526:
1524:
1504:
1450:
1437:
1429:
1395:
1370:
1362:
1339:
1310:
1258:
1253:
1243:
1235:
1212:
1176:
1149:
1139:
1133:
1125:
1116:
1108:
1079:
1064:
1050:
1044:
1037:
1035:
1012:
1010:
968:
934:
920:
906:
900:
898:
871:
851:
121:
112:
100:
81:
75:
3183:Arbitrary orientation of the local frame
822:Newtonian mechanics for circular orbits
404:
378:
337:
283:
57:
6834:Atomic, molecular, and optical physics
5948:{\displaystyle {\mathcal {M}}_{j}^{i}}
2800:and the separation vector is given by
3430:{\displaystyle {\bar {\mathcal {M}}}}
7:
1645:Elliptic and hyberbolic trajectories
1270:{\displaystyle R={GM \over {r^{3}}}}
6435:General relativity generalizes the
3302:{\displaystyle \mathbf {\bar {h}} }
3239:{\displaystyle \mathbf {\hat {r}} }
3209:{\displaystyle \mathbf {\hat {r}} }
3169:{\displaystyle \mathbf {\hat {z}} }
3103:{\displaystyle \mathbf {\hat {y}} }
3037:{\displaystyle \mathbf {\hat {x}} }
2378:{\displaystyle \mathbf {\hat {r}} }
1639:
830:Circular orbits at the same radius.
6508:Frame fields in general relativity
5787:
5764:
5735:
5732:
5729:
5523:
5404:
5389:
5341:
5281:
5118:
5007:
5004:
5001:
4732:
4729:
4726:
4699:
4628:
4625:
4622:
4576:
1657:More generally, particles move in
93:
25:
6493:Mathematics of general relativity
6395:Overview of the Newtonian picture
5546:{\displaystyle \Gamma _{j}^{i}=0}
2136:if the separation is parallel to
1695:From Newton's Law of Gravitation
3317:
3290:
3280:such that the separation vector
3227:
3197:
3157:
3128:
3118:
3091:
3062:
3052:
3025:
3001:
2974:
2964:
2926:
2916:
2904:
2894:
2882:
2872:
2366:
2305:
2263:
2197:
2144:
1992:
1840:
1811:
1770:
1719:
1614:
1569:
1527:
1080:
1051:
1013:
749:
748:
735:
65:
6955:Timeline of physics discoveries
3655:{\displaystyle \delta _{k}^{i}}
3253:Rotations are performed with a
6377:
6371:
6299:
6293:
6218:
6141:
6131:
5967:
5889:
5826:
5660:
5636:
5596:
5504:is done with a process called
5471:
5447:
5425:
5373:
5308:
5244:
5185:
5161:
5139:
5091:
5068:The geodesic equation becomes
5031:
4976:
4955:Arbitrariness in the curvature
4919:
4895:
4855:
4825:The geodesic equation becomes
4777:
4652:
4597:
4549:
4490:
4466:
4285:
4222:
4174:
4150:
4110:
4038:
4032:
4026:
4015:
4009:
3988:
3982:
3971:
3965:
3931:
3921:
3721:
3604:
3580:
3480:
3454:{\displaystyle {\mathcal {M}}}
3421:
3348:
3273:{\displaystyle {\mathcal {C}}}
2938:
2865:
2855:
2812:
2782:
2705:
2695:
2531:
2219:leads to particle separation.
2120:
2114:
1968:
1962:
1466:
1460:
1388:
1382:
1321:
1315:
846:orbits of the Earth at radius
809:Test for flatness in spacetime
27:Some of the basic concepts of
1:
2485:Einstein summation convention
6593:General Theory of Relativity
3324:{\displaystyle \mathbf {h} }
3008:{\displaystyle \mathbf {h} }
2312:{\displaystyle \mathbf {r} }
2270:{\displaystyle \mathbf {r} }
2204:{\displaystyle \mathbf {h} }
2151:{\displaystyle \mathbf {r} }
1999:{\displaystyle \mathbf {r} }
1621:{\displaystyle \mathbf {h} }
1576:{\displaystyle \mathbf {v} }
1534:{\displaystyle \mathbf {r} }
1020:{\displaystyle \mathbf {h} }
31:can be outlined outside the
6919:Quantum information science
6416:Relativistic generalization
1421:The field equation becomes
1003:Newton's Law of Gravitation
246:Gravitational time dilation
6992:
6750:Classical electromagnetism
6597:Princeton University Press
6572:Classical Theory of Fields
6419:
4370:with the time derivative.
4351:relativity of simultaneity
1606:In this frame, the vector
1334:inside a sphere of radius
366:Mathisson–Papapetrou–Dixon
207:Pseudo-Riemannian manifold
2512:{\displaystyle R_{j}^{i}}
6856:Condensed matter physics
3147:is the component in the
3081:is the component in the
2158:. In the calculation of
1327:{\displaystyle \rho (r)}
1193:{\displaystyle \tau =ct}
998:intersect at the poles.
371:Hamilton–Jacobi–Einstein
351:Einstein field equations
174:Mathematical formulation
6591:P. A. M. Dirac (1996).
5793:{\displaystyle \Gamma }
4070:{\displaystyle \theta }
3682:{\displaystyle \theta }
2223:Local coordinate system
1863:and the field equation
1634:coordinate acceleration
1283:Einstein field equation
6940:Nobel Prize in Physics
6802:Relativistic mechanics
6432:
6410:Fermi–Walker transport
6384:
6306:
6227:
5949:
5910:
5794:
5771:
5682:
5547:
5506:Fermi–Walker transport
5493:
5207:
5058:
4941:
4809:
4679:
4517:
4358:
4306:
4196:
4071:
4047:
3703:
3683:
3656:
3620:
3538:
3455:
3431:
3396:
3325:
3303:
3274:
3240:
3210:
3170:
3141:
3104:
3075:
3038:
3009:
2987:
2947:
2791:
2513:
2474:
2379:
2333:
2313:
2291:
2271:
2249:
2232:
2205:
2179:
2178:{\displaystyle R_{\|}}
2152:
2127:
2000:
1975:
1854:
1778:
1686:
1654:
1622:
1597:
1577:
1555:
1535:
1513:
1473:
1411:
1348:
1328:
1297:
1271:
1221:
1194:
1171:of the trajectory and
1161:
1094:
1021:
1005:the separation vector
992:simple harmonic motion
990:The particles execute
977:
961:gravitational constant
946:
880:
860:
831:
241:Gravitational redshift
134:
6945:Philosophy of physics
6520:Einstein, A. (1961).
6429:
6385:
6307:
6228:
5950:
5911:
5795:
5772:
5683:
5548:
5494:
5208:
5059:
4942:
4810:
4680:
4518:
4327:
4307:
4197:
4072:
4048:
3704:
3684:
3657:
3621:
3539:
3456:
3432:
3397:
3326:
3304:
3275:
3241:
3211:
3171:
3142:
3105:
3076:
3039:
3010:
2988:
2948:
2792:
2514:
2475:
2380:
2348:Simple diagonal frame
2334:
2314:
2292:
2272:
2250:
2230:
2206:
2180:
2153:
2128:
2001:
1976:
1855:
1779:
1687:
1652:
1623:
1598:
1578:
1556:
1536:
1514:
1474:
1412:
1349:
1329:
1295:
1272:
1222:
1195:
1162:
1095:
1022:
978:
947:
881:
861:
829:
529:Weyl−Lewis−Papapetrou
484:Kerr–Newman–de Sitter
304:Einstein–Rosen bridge
236:Gravitational lensing
192:Equivalence principle
135:
6904:Mathematical physics
6319:
6244:
5962:
5923:
5816:
5784:
5698:
5573:
5519:
5221:
5075:
4966:
4832:
4695:
4533:
4380:
4212:
4087:
4061:
3716:
3693:
3673:
3634:
3555:
3468:
3441:
3410:
3338:
3313:
3284:
3260:
3221:
3191:
3151:
3114:
3085:
3048:
3019:
2997:
2993:is the component of
2960:
2807:
2526:
2491:
2392:
2360:
2323:
2301:
2281:
2259:
2239:
2193:
2162:
2140:
2013:
1988:
1870:
1794:
1702:
1676:
1610:
1587:
1565:
1545:
1523:
1503:
1428:
1361:
1338:
1309:
1234:
1211:
1175:
1107:
1034:
1009:
967:
897:
870:
850:
459:Einstein–Rosen waves
185:Fundamental concepts
74:
6879:Atmospheric physics
6718:Classical mechanics
6646:branches of physics
6528:. New York: Crown.
6487:Related mathematics
6406:Christoffel symbols
6127:
6110:
6093:
6074:
6057:
6040:
6021:
6004:
5987:
5944:
5905:
5879:
5864:
5842:
5652:
5536:
5463:
5417:
5402:
5354:
5294:
5177:
5131:
4911:
4793:
4761:
4712:
4589:
4482:
4450:
4401:
4301:
4275:
4260:
4238:
4166:
3917:
3896:
3875:
3852:
3831:
3810:
3787:
3766:
3745:
3651:
3596:
3533:
3515:
3496:
3381:
2691:
2674:
2657:
2638:
2621:
2604:
2585:
2568:
2551:
2508:
2453:
2215:term was retained.
413:Kaluza–Klein theory
299:Minkowski spacetime
251:Gravitational waves
6976:General relativity
6935:History of physics
6433:
6380:
6302:
6223:
6217:
6130:
6113:
6096:
6079:
6060:
6043:
6026:
6007:
5990:
5973:
5945:
5926:
5906:
5880:
5865:
5846:
5819:
5802:Christoffel symbol
5790:
5767:
5678:
5629:
5543:
5522:
5489:
5440:
5403:
5388:
5340:
5280:
5203:
5154:
5117:
5054:
4937:
4888:
4820:Christoffel symbol
4805:
4768:
4743:
4698:
4675:
4575:
4513:
4457:
4432:
4383:
4359:
4302:
4276:
4261:
4242:
4215:
4192:
4143:
4067:
4043:
4037:
3920:
3899:
3878:
3857:
3834:
3813:
3792:
3769:
3748:
3727:
3699:
3679:
3652:
3637:
3616:
3571:
3534:
3519:
3497:
3471:
3451:
3427:
3392:
3363:
3321:
3299:
3270:
3236:
3206:
3166:
3137:
3100:
3071:
3034:
3005:
2983:
2943:
2937:
2854:
2787:
2781:
2694:
2677:
2660:
2643:
2624:
2607:
2590:
2571:
2554:
2537:
2509:
2494:
2470:
2439:
2375:
2343:Tensor description
2329:
2309:
2287:
2267:
2245:
2233:
2201:
2175:
2148:
2123:
1996:
1971:
1850:
1774:
1682:
1655:
1618:
1593:
1573:
1551:
1531:
1509:
1487:Local measurements
1469:
1407:
1354:by the expression
1344:
1324:
1298:
1267:
1217:
1204:c times the time.
1190:
1157:
1090:
1017:
973:
942:
876:
856:
832:
742:Physics portal
514:Oppenheimer–Snyder
454:Reissner–Nordström
346:Linearized gravity
294:Spacetime diagrams
197:Special relativity
130:
59:General relativity
29:general relativity
6963:
6962:
6950:Physics education
6899:Materials science
6866:Interdisciplinary
6824:Quantum mechanics
6549:. San Francisco:
6437:geodesic equation
6366:
6288:
5892:
5829:
5759:
5745:
5740:
5718:
5714:
5663:
5639:
5624:
5599:
5565:Geodesic equation
5474:
5450:
5428:
5376:
5364:
5329:
5311:
5272:
5247:
5188:
5164:
5142:
5112:
5094:
5052:
5034:
5017:
5012:
4990:
4979:
4922:
4898:
4883:
4858:
4803:
4780:
4742:
4737:
4715:
4673:
4655:
4638:
4633:
4611:
4600:
4570:
4552:
4511:
4493:
4469:
4427:
4288:
4225:
4177:
4153:
4138:
4113:
3702:{\displaystyle x}
3607:
3583:
3483:
3424:
3351:
3296:
3233:
3203:
3163:
3134:
3097:
3068:
3031:
2980:
2932:
2910:
2888:
2434:
2372:
2332:{\displaystyle y}
2290:{\displaystyle x}
2248:{\displaystyle z}
2109:
2073:
1957:
1924:
1831:
1767:
1739:
1685:{\displaystyle c}
1596:{\displaystyle y}
1554:{\displaystyle x}
1512:{\displaystyle z}
1455:
1405:
1347:{\displaystyle r}
1265:
1220:{\displaystyle M}
1155:
1131:
1071:
976:{\displaystyle M}
940:
915:
888:centripetal force
879:{\displaystyle v}
859:{\displaystyle r}
792:
791:
425:
424:
311:
310:
16:(Redirected from
6983:
6889:Chemical physics
6829:Particle physics
6755:Classical optics
6638:
6631:
6624:
6615:
6610:
6585:
6564:
6539:
6527:
6476:Bernhard Riemann
6389:
6387:
6386:
6381:
6367:
6365:
6364:
6363:
6350:
6339:
6331:
6330:
6311:
6309:
6308:
6303:
6289:
6287:
6286:
6285:
6272:
6261:
6256:
6255:
6232:
6230:
6229:
6224:
6222:
6221:
6214:
6213:
6185:
6184:
6156:
6155:
6135:
6134:
6126:
6121:
6109:
6104:
6092:
6087:
6073:
6068:
6056:
6051:
6039:
6034:
6020:
6015:
6003:
5998:
5986:
5981:
5954:
5952:
5951:
5946:
5943:
5938:
5933:
5932:
5915:
5913:
5912:
5907:
5904:
5899:
5894:
5893:
5888:
5883:
5878:
5873:
5863:
5858:
5853:
5852:
5841:
5836:
5831:
5830:
5822:
5799:
5797:
5796:
5791:
5776:
5774:
5773:
5768:
5760:
5758:
5747:
5743:
5742:
5741:
5739:
5738:
5726:
5721:
5716:
5715:
5713:
5702:
5687:
5685:
5684:
5679:
5671:
5670:
5665:
5664:
5656:
5651:
5646:
5641:
5640:
5632:
5625:
5623:
5622:
5621:
5608:
5607:
5606:
5601:
5600:
5592:
5588:
5587:
5577:
5552:
5550:
5549:
5544:
5535:
5530:
5498:
5496:
5495:
5490:
5482:
5481:
5476:
5475:
5467:
5462:
5457:
5452:
5451:
5443:
5436:
5435:
5430:
5429:
5421:
5416:
5411:
5401:
5396:
5384:
5383:
5378:
5377:
5369:
5365:
5363:
5355:
5353:
5348:
5335:
5330:
5328:
5320:
5319:
5318:
5313:
5312:
5304:
5296:
5293:
5288:
5273:
5271:
5270:
5269:
5256:
5255:
5254:
5249:
5248:
5240:
5236:
5235:
5225:
5212:
5210:
5209:
5204:
5196:
5195:
5190:
5189:
5181:
5176:
5171:
5166:
5165:
5157:
5150:
5149:
5144:
5143:
5135:
5130:
5125:
5113:
5111:
5103:
5102:
5101:
5096:
5095:
5087:
5079:
5063:
5061:
5060:
5055:
5053:
5051:
5043:
5042:
5041:
5036:
5035:
5027:
5019:
5015:
5014:
5013:
5011:
5010:
4998:
4993:
4988:
4987:
4986:
4981:
4980:
4972:
4946:
4944:
4943:
4938:
4930:
4929:
4924:
4923:
4915:
4910:
4905:
4900:
4899:
4891:
4884:
4882:
4881:
4880:
4867:
4866:
4865:
4860:
4859:
4851:
4847:
4846:
4836:
4814:
4812:
4811:
4806:
4804:
4802:
4794:
4792:
4787:
4782:
4781:
4776:
4771:
4763:
4760:
4755:
4750:
4749:
4740:
4739:
4738:
4736:
4735:
4723:
4718:
4713:
4711:
4706:
4684:
4682:
4681:
4676:
4674:
4672:
4664:
4663:
4662:
4657:
4656:
4648:
4640:
4636:
4635:
4634:
4632:
4631:
4619:
4614:
4609:
4608:
4607:
4602:
4601:
4593:
4588:
4583:
4571:
4569:
4561:
4560:
4559:
4554:
4553:
4545:
4537:
4522:
4520:
4519:
4514:
4512:
4510:
4502:
4501:
4500:
4495:
4494:
4486:
4481:
4476:
4471:
4470:
4465:
4460:
4452:
4449:
4444:
4439:
4438:
4428:
4426:
4418:
4417:
4416:
4403:
4400:
4395:
4390:
4389:
4311:
4309:
4308:
4303:
4300:
4295:
4290:
4289:
4284:
4279:
4274:
4269:
4259:
4254:
4249:
4248:
4237:
4232:
4227:
4226:
4218:
4201:
4199:
4198:
4193:
4185:
4184:
4179:
4178:
4170:
4165:
4160:
4155:
4154:
4146:
4139:
4137:
4136:
4135:
4122:
4121:
4120:
4115:
4114:
4106:
4102:
4101:
4091:
4076:
4074:
4073:
4068:
4052:
4050:
4049:
4044:
4042:
4041:
3925:
3924:
3916:
3911:
3906:
3905:
3895:
3890:
3885:
3884:
3874:
3869:
3864:
3863:
3851:
3846:
3841:
3840:
3830:
3825:
3820:
3819:
3809:
3804:
3799:
3798:
3786:
3781:
3776:
3775:
3765:
3760:
3755:
3754:
3744:
3739:
3734:
3733:
3708:
3706:
3705:
3700:
3688:
3686:
3685:
3680:
3661:
3659:
3658:
3653:
3650:
3645:
3625:
3623:
3622:
3617:
3615:
3614:
3609:
3608:
3600:
3595:
3590:
3585:
3584:
3579:
3574:
3567:
3566:
3543:
3541:
3540:
3535:
3532:
3527:
3514:
3509:
3504:
3503:
3495:
3490:
3485:
3484:
3479:
3474:
3460:
3458:
3457:
3452:
3450:
3449:
3436:
3434:
3433:
3428:
3426:
3425:
3420:
3415:
3401:
3399:
3398:
3393:
3391:
3390:
3380:
3375:
3370:
3369:
3359:
3358:
3353:
3352:
3344:
3331:by the relation
3330:
3328:
3327:
3322:
3320:
3308:
3306:
3305:
3300:
3298:
3297:
3289:
3279:
3277:
3276:
3271:
3269:
3268:
3245:
3243:
3242:
3237:
3235:
3234:
3226:
3215:
3213:
3212:
3207:
3205:
3204:
3196:
3175:
3173:
3172:
3167:
3165:
3164:
3156:
3146:
3144:
3143:
3138:
3136:
3135:
3127:
3121:
3109:
3107:
3106:
3101:
3099:
3098:
3090:
3080:
3078:
3077:
3072:
3070:
3069:
3061:
3055:
3043:
3041:
3040:
3035:
3033:
3032:
3024:
3014:
3012:
3011:
3006:
3004:
2992:
2990:
2989:
2984:
2982:
2981:
2973:
2967:
2952:
2950:
2949:
2944:
2942:
2941:
2934:
2933:
2925:
2919:
2912:
2911:
2903:
2897:
2890:
2889:
2881:
2875:
2859:
2858:
2851:
2850:
2839:
2838:
2827:
2826:
2796:
2794:
2793:
2788:
2786:
2785:
2778:
2777:
2749:
2748:
2720:
2719:
2699:
2698:
2690:
2685:
2673:
2668:
2656:
2651:
2637:
2632:
2620:
2615:
2603:
2598:
2584:
2579:
2567:
2562:
2550:
2545:
2518:
2516:
2515:
2510:
2507:
2502:
2479:
2477:
2476:
2471:
2463:
2462:
2452:
2447:
2435:
2433:
2432:
2431:
2418:
2417:
2416:
2407:
2406:
2396:
2384:
2382:
2381:
2376:
2374:
2373:
2365:
2338:
2336:
2335:
2330:
2318:
2316:
2315:
2310:
2308:
2296:
2294:
2293:
2288:
2276:
2274:
2273:
2268:
2266:
2254:
2252:
2251:
2246:
2210:
2208:
2207:
2202:
2200:
2184:
2182:
2181:
2176:
2174:
2173:
2157:
2155:
2154:
2149:
2147:
2132:
2130:
2129:
2124:
2110:
2108:
2107:
2106:
2093:
2082:
2074:
2072:
2071:
2070:
2061:
2060:
2050:
2039:
2031:
2030:
2005:
2003:
2002:
1997:
1995:
1980:
1978:
1977:
1972:
1958:
1956:
1955:
1954:
1941:
1930:
1925:
1923:
1922:
1921:
1912:
1911:
1901:
1893:
1888:
1887:
1859:
1857:
1856:
1851:
1843:
1832:
1830:
1829:
1828:
1815:
1814:
1809:
1808:
1798:
1783:
1781:
1780:
1775:
1773:
1768:
1766:
1765:
1756:
1748:
1740:
1738:
1737:
1736:
1723:
1722:
1717:
1716:
1706:
1691:
1689:
1688:
1683:
1627:
1625:
1624:
1619:
1617:
1602:
1600:
1599:
1594:
1582:
1580:
1579:
1574:
1572:
1560:
1558:
1557:
1552:
1540:
1538:
1537:
1532:
1530:
1518:
1516:
1515:
1510:
1478:
1476:
1475:
1470:
1456:
1454:
1449:
1438:
1416:
1414:
1413:
1408:
1406:
1401:
1400:
1399:
1371:
1353:
1351:
1350:
1345:
1333:
1331:
1330:
1325:
1276:
1274:
1273:
1268:
1266:
1264:
1263:
1262:
1252:
1244:
1226:
1224:
1223:
1218:
1199:
1197:
1196:
1191:
1166:
1164:
1163:
1158:
1156:
1154:
1153:
1144:
1143:
1134:
1132:
1130:
1129:
1117:
1099:
1097:
1096:
1091:
1083:
1072:
1070:
1069:
1068:
1055:
1054:
1049:
1048:
1038:
1026:
1024:
1023:
1018:
1016:
982:
980:
979:
974:
951:
949:
948:
943:
941:
939:
938:
929:
921:
916:
911:
910:
901:
885:
883:
882:
877:
865:
863:
862:
857:
784:
777:
770:
757:
752:
751:
744:
740:
739:
524:van Stockum dust
509:Robertson–Walker
335:
225:
139:
137:
136:
131:
129:
128:
116:
108:
107:
89:
88:
69:
55:
47:setting. We use
21:
6991:
6990:
6986:
6985:
6984:
6982:
6981:
6980:
6966:
6965:
6964:
6959:
6923:
6909:Medical physics
6860:
6819:Nuclear physics
6788:
6782:Non-equilibrium
6704:
6676:
6648:
6642:
6607:
6590:
6582:
6569:
6561:
6544:
6536:
6519:
6516:
6489:
6466:Albert Einstein
6462:
6457:
6424:
6418:
6397:
6355:
6340:
6322:
6317:
6316:
6277:
6262:
6247:
6242:
6241:
6216:
6215:
6205:
6203:
6198:
6192:
6191:
6186:
6176:
6174:
6168:
6167:
6162:
6157:
6147:
6140:
6129:
6128:
6111:
6094:
6076:
6075:
6058:
6041:
6023:
6022:
6005:
5988:
5966:
5960:
5959:
5921:
5920:
5814:
5813:
5810:
5782:
5781:
5751:
5706:
5696:
5695:
5653:
5613:
5609:
5589:
5579:
5578:
5571:
5570:
5567:
5562:
5517:
5516:
5464:
5418:
5366:
5356:
5336:
5321:
5301:
5297:
5261:
5257:
5237:
5227:
5226:
5219:
5218:
5178:
5132:
5104:
5084:
5080:
5073:
5072:
5044:
5024:
5020:
4969:
4964:
4963:
4957:
4912:
4872:
4868:
4848:
4838:
4837:
4830:
4829:
4795:
4764:
4693:
4692:
4665:
4645:
4641:
4590:
4562:
4542:
4538:
4531:
4530:
4503:
4483:
4453:
4419:
4408:
4404:
4378:
4377:
4364:
4356:
4354:
4348:
4346:
4319:diagonalization
4210:
4209:
4167:
4127:
4123:
4103:
4093:
4092:
4085:
4084:
4059:
4058:
4036:
4035:
4018:
3998:
3992:
3991:
3974:
3957:
3951:
3950:
3945:
3940:
3930:
3919:
3918:
3897:
3876:
3854:
3853:
3832:
3811:
3789:
3788:
3767:
3746:
3720:
3714:
3713:
3691:
3690:
3671:
3670:
3664:Kronecker delta
3632:
3631:
3597:
3558:
3553:
3552:
3466:
3465:
3439:
3438:
3408:
3407:
3382:
3341:
3336:
3335:
3311:
3310:
3282:
3281:
3258:
3257:
3255:rotation matrix
3219:
3218:
3189:
3188:
3185:
3149:
3148:
3112:
3111:
3110:direction, and
3083:
3082:
3046:
3045:
3017:
3016:
2995:
2994:
2958:
2957:
2936:
2935:
2913:
2891:
2864:
2853:
2852:
2842:
2840:
2830:
2828:
2818:
2811:
2805:
2804:
2780:
2779:
2769:
2767:
2762:
2756:
2755:
2750:
2740:
2738:
2732:
2731:
2726:
2721:
2711:
2704:
2693:
2692:
2675:
2658:
2640:
2639:
2622:
2605:
2587:
2586:
2569:
2552:
2530:
2524:
2523:
2489:
2488:
2454:
2423:
2419:
2408:
2398:
2397:
2390:
2389:
2358:
2357:
2350:
2345:
2321:
2320:
2299:
2298:
2279:
2278:
2257:
2256:
2237:
2236:
2225:
2191:
2190:
2185:the radius was
2165:
2160:
2159:
2138:
2137:
2098:
2083:
2062:
2052:
2040:
2022:
2011:
2010:
1986:
1985:
1946:
1931:
1913:
1903:
1894:
1879:
1868:
1867:
1820:
1816:
1800:
1799:
1792:
1791:
1757:
1749:
1728:
1724:
1708:
1707:
1700:
1699:
1674:
1673:
1647:
1642:
1608:
1607:
1585:
1584:
1563:
1562:
1543:
1542:
1521:
1520:
1501:
1500:
1489:
1439:
1426:
1425:
1391:
1372:
1359:
1358:
1336:
1335:
1307:
1306:
1303:
1254:
1245:
1232:
1231:
1209:
1208:
1173:
1172:
1145:
1135:
1121:
1105:
1104:
1060:
1056:
1040:
1039:
1032:
1031:
1007:
1006:
965:
964:
930:
922:
902:
895:
894:
868:
867:
848:
847:
837:
824:
819:
811:
798:
793:
788:
747:
734:
733:
726:
725:
549:
548:
539:
538:
494:Lemaître–Tolman
439:
438:
427:
426:
418:Quantum gravity
405:Advanced theory
332:
331:
330:
313:
312:
261:Geodetic effect
222:
221:
212:
211:
187:
186:
170:
140:
117:
96:
77:
72:
71:
49:circular orbits
23:
22:
15:
12:
11:
5:
6989:
6987:
6979:
6978:
6968:
6967:
6961:
6960:
6958:
6957:
6952:
6947:
6942:
6937:
6931:
6929:
6925:
6924:
6922:
6921:
6916:
6911:
6906:
6901:
6896:
6891:
6886:
6881:
6876:
6870:
6868:
6862:
6861:
6859:
6858:
6853:
6852:
6851:
6846:
6841:
6831:
6826:
6821:
6816:
6815:
6814:
6809:
6798:
6796:
6790:
6789:
6787:
6786:
6785:
6784:
6779:
6772:Thermodynamics
6769:
6768:
6767:
6762:
6752:
6747:
6742:
6741:
6740:
6735:
6730:
6725:
6714:
6712:
6706:
6705:
6703:
6702:
6701:
6700:
6690:
6684:
6682:
6678:
6677:
6675:
6674:
6673:
6672:
6662:
6656:
6654:
6650:
6649:
6643:
6641:
6640:
6633:
6626:
6618:
6612:
6611:
6605:
6587:
6586:
6580:
6566:
6565:
6559:
6541:
6540:
6534:
6515:
6512:
6511:
6510:
6505:
6500:
6495:
6488:
6485:
6484:
6483:
6478:
6473:
6468:
6461:
6458:
6456:
6453:
6441:field equation
6420:Main article:
6417:
6414:
6396:
6393:
6392:
6391:
6379:
6376:
6373:
6370:
6362:
6358:
6354:
6349:
6346:
6343:
6337:
6334:
6329:
6325:
6313:
6312:
6301:
6298:
6295:
6292:
6284:
6280:
6276:
6271:
6268:
6265:
6259:
6254:
6250:
6235:
6234:
6220:
6212:
6208:
6204:
6202:
6199:
6197:
6194:
6193:
6190:
6187:
6183:
6179:
6175:
6173:
6170:
6169:
6166:
6163:
6161:
6158:
6154:
6150:
6146:
6145:
6143:
6138:
6133:
6125:
6120:
6116:
6112:
6108:
6103:
6099:
6095:
6091:
6086:
6082:
6078:
6077:
6072:
6067:
6063:
6059:
6055:
6050:
6046:
6042:
6038:
6033:
6029:
6025:
6024:
6019:
6014:
6010:
6006:
6002:
5997:
5993:
5989:
5985:
5980:
5976:
5972:
5971:
5969:
5942:
5937:
5931:
5917:
5916:
5903:
5898:
5891:
5887:
5877:
5872:
5868:
5862:
5857:
5851:
5845:
5840:
5835:
5828:
5825:
5809:
5808:Field equation
5806:
5789:
5778:
5777:
5766:
5763:
5757:
5754:
5750:
5737:
5734:
5731:
5725:
5712:
5709:
5705:
5689:
5688:
5677:
5674:
5669:
5662:
5659:
5650:
5645:
5638:
5635:
5628:
5620:
5616:
5612:
5605:
5598:
5595:
5586:
5582:
5566:
5563:
5561:
5558:
5554:
5553:
5542:
5539:
5534:
5529:
5525:
5501:
5500:
5488:
5485:
5480:
5473:
5470:
5461:
5456:
5449:
5446:
5439:
5434:
5427:
5424:
5415:
5410:
5406:
5400:
5395:
5391:
5387:
5382:
5375:
5372:
5362:
5359:
5352:
5347:
5343:
5339:
5333:
5327:
5324:
5317:
5310:
5307:
5300:
5292:
5287:
5283:
5279:
5276:
5268:
5264:
5260:
5253:
5246:
5243:
5234:
5230:
5215:
5214:
5202:
5199:
5194:
5187:
5184:
5175:
5170:
5163:
5160:
5153:
5148:
5141:
5138:
5129:
5124:
5120:
5116:
5110:
5107:
5100:
5093:
5090:
5083:
5066:
5065:
5050:
5047:
5040:
5033:
5030:
5023:
5009:
5006:
5003:
4997:
4985:
4978:
4975:
4956:
4953:
4949:
4948:
4936:
4933:
4928:
4921:
4918:
4909:
4904:
4897:
4894:
4887:
4879:
4875:
4871:
4864:
4857:
4854:
4845:
4841:
4818:is known as a
4816:
4815:
4801:
4798:
4791:
4786:
4779:
4775:
4767:
4759:
4754:
4748:
4734:
4731:
4728:
4722:
4710:
4705:
4701:
4686:
4685:
4671:
4668:
4661:
4654:
4651:
4644:
4630:
4627:
4624:
4618:
4606:
4599:
4596:
4587:
4582:
4578:
4574:
4568:
4565:
4558:
4551:
4548:
4541:
4526:which becomes
4524:
4523:
4509:
4506:
4499:
4492:
4489:
4480:
4475:
4468:
4464:
4456:
4448:
4443:
4437:
4431:
4425:
4422:
4415:
4411:
4407:
4399:
4394:
4388:
4363:
4360:
4314:
4313:
4299:
4294:
4287:
4283:
4273:
4268:
4264:
4258:
4253:
4247:
4241:
4236:
4231:
4224:
4221:
4203:
4202:
4191:
4188:
4183:
4176:
4173:
4164:
4159:
4152:
4149:
4142:
4134:
4130:
4126:
4119:
4112:
4109:
4100:
4096:
4066:
4055:
4054:
4040:
4034:
4031:
4028:
4025:
4022:
4019:
4017:
4014:
4011:
4008:
4005:
4002:
3999:
3997:
3994:
3993:
3990:
3987:
3984:
3981:
3978:
3975:
3973:
3970:
3967:
3964:
3961:
3958:
3956:
3953:
3952:
3949:
3946:
3944:
3941:
3939:
3936:
3935:
3933:
3928:
3923:
3915:
3910:
3904:
3898:
3894:
3889:
3883:
3877:
3873:
3868:
3862:
3856:
3855:
3850:
3845:
3839:
3833:
3829:
3824:
3818:
3812:
3808:
3803:
3797:
3791:
3790:
3785:
3780:
3774:
3768:
3764:
3759:
3753:
3747:
3743:
3738:
3732:
3726:
3725:
3723:
3698:
3678:
3649:
3644:
3640:
3628:
3627:
3613:
3606:
3603:
3594:
3589:
3582:
3578:
3570:
3565:
3561:
3546:
3545:
3531:
3526:
3522:
3518:
3513:
3508:
3502:
3494:
3489:
3482:
3478:
3461:is defined by
3448:
3423:
3419:
3404:
3403:
3389:
3385:
3379:
3374:
3368:
3362:
3357:
3350:
3347:
3319:
3295:
3292:
3267:
3232:
3229:
3202:
3199:
3184:
3181:
3162:
3159:
3133:
3130:
3124:
3120:
3096:
3093:
3067:
3064:
3058:
3054:
3030:
3027:
3003:
2979:
2976:
2970:
2966:
2954:
2953:
2940:
2931:
2928:
2922:
2918:
2914:
2909:
2906:
2900:
2896:
2892:
2887:
2884:
2878:
2874:
2870:
2869:
2867:
2862:
2857:
2849:
2845:
2841:
2837:
2833:
2829:
2825:
2821:
2817:
2816:
2814:
2798:
2797:
2784:
2776:
2772:
2768:
2766:
2763:
2761:
2758:
2757:
2754:
2751:
2747:
2743:
2739:
2737:
2734:
2733:
2730:
2727:
2725:
2722:
2718:
2714:
2710:
2709:
2707:
2702:
2697:
2689:
2684:
2680:
2676:
2672:
2667:
2663:
2659:
2655:
2650:
2646:
2642:
2641:
2636:
2631:
2627:
2623:
2619:
2614:
2610:
2606:
2602:
2597:
2593:
2589:
2588:
2583:
2578:
2574:
2570:
2566:
2561:
2557:
2553:
2549:
2544:
2540:
2536:
2535:
2533:
2506:
2501:
2497:
2481:
2480:
2469:
2466:
2461:
2457:
2451:
2446:
2442:
2438:
2430:
2426:
2422:
2415:
2411:
2405:
2401:
2371:
2368:
2349:
2346:
2344:
2341:
2328:
2307:
2286:
2265:
2244:
2224:
2221:
2199:
2172:
2168:
2146:
2134:
2133:
2122:
2119:
2116:
2113:
2105:
2101:
2097:
2092:
2089:
2086:
2080:
2077:
2069:
2065:
2059:
2055:
2049:
2046:
2043:
2037:
2034:
2029:
2025:
2021:
2018:
1994:
1982:
1981:
1970:
1967:
1964:
1961:
1953:
1949:
1945:
1940:
1937:
1934:
1928:
1920:
1916:
1910:
1906:
1900:
1897:
1891:
1886:
1882:
1878:
1875:
1861:
1860:
1849:
1846:
1842:
1838:
1835:
1827:
1823:
1819:
1813:
1807:
1803:
1785:
1784:
1772:
1764:
1760:
1755:
1752:
1746:
1743:
1735:
1731:
1727:
1721:
1715:
1711:
1681:
1671:speed of light
1646:
1643:
1641:
1638:
1630:potential well
1616:
1592:
1571:
1550:
1529:
1508:
1493:center of mass
1488:
1485:
1481:
1480:
1468:
1465:
1462:
1459:
1453:
1448:
1445:
1442:
1436:
1433:
1419:
1418:
1404:
1398:
1394:
1390:
1387:
1384:
1381:
1378:
1375:
1369:
1366:
1343:
1323:
1320:
1317:
1314:
1302:
1299:
1278:
1277:
1261:
1257:
1251:
1248:
1242:
1239:
1216:
1202:speed of light
1189:
1186:
1183:
1180:
1152:
1148:
1142:
1138:
1128:
1124:
1120:
1115:
1112:
1101:
1100:
1089:
1086:
1082:
1078:
1075:
1067:
1063:
1059:
1053:
1047:
1043:
1015:
987:of the earth.
972:
953:
952:
937:
933:
928:
925:
919:
914:
909:
905:
875:
855:
836:
833:
823:
820:
818:
815:
810:
807:
797:
794:
790:
789:
787:
786:
779:
772:
764:
761:
760:
759:
758:
745:
728:
727:
724:
723:
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:
571:
566:
561:
556:
550:
546:
545:
544:
541:
540:
537:
536:
531:
526:
521:
516:
511:
506:
501:
496:
491:
486:
481:
476:
471:
466:
461:
456:
451:
440:
434:
433:
432:
429:
428:
423:
422:
421:
420:
415:
407:
406:
402:
401:
400:
399:
397:Post-Newtonian
394:
389:
381:
380:
376:
375:
374:
373:
368:
363:
358:
353:
348:
340:
339:
333:
329:
328:
325:
321:
320:
319:
318:
315:
314:
309:
308:
307:
306:
301:
296:
288:
287:
281:
280:
279:
278:
273:
268:
263:
258:
256:Frame-dragging
253:
248:
243:
238:
233:
231:Kepler problem
223:
219:
218:
217:
214:
213:
210:
209:
204:
199:
194:
188:
184:
183:
182:
179:
178:
177:
176:
171:
169:
168:
163:
158:
152:
150:
142:
141:
127:
124:
120:
115:
111:
106:
103:
99:
95:
92:
87:
84:
80:
70:
62:
61:
53:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6988:
6977:
6974:
6973:
6971:
6956:
6953:
6951:
6948:
6946:
6943:
6941:
6938:
6936:
6933:
6932:
6930:
6926:
6920:
6917:
6915:
6914:Ocean physics
6912:
6910:
6907:
6905:
6902:
6900:
6897:
6895:
6892:
6890:
6887:
6885:
6882:
6880:
6877:
6875:
6872:
6871:
6869:
6867:
6863:
6857:
6854:
6850:
6849:Modern optics
6847:
6845:
6842:
6840:
6837:
6836:
6835:
6832:
6830:
6827:
6825:
6822:
6820:
6817:
6813:
6810:
6808:
6805:
6804:
6803:
6800:
6799:
6797:
6795:
6791:
6783:
6780:
6778:
6775:
6774:
6773:
6770:
6766:
6763:
6761:
6758:
6757:
6756:
6753:
6751:
6748:
6746:
6743:
6739:
6736:
6734:
6731:
6729:
6726:
6724:
6721:
6720:
6719:
6716:
6715:
6713:
6711:
6707:
6699:
6698:Computational
6696:
6695:
6694:
6691:
6689:
6686:
6685:
6683:
6679:
6671:
6668:
6667:
6666:
6663:
6661:
6658:
6657:
6655:
6651:
6647:
6639:
6634:
6632:
6627:
6625:
6620:
6619:
6616:
6608:
6606:0-691-01146-X
6602:
6598:
6594:
6589:
6588:
6583:
6581:0-08-018176-7
6577:
6573:
6568:
6567:
6562:
6560:0-7167-0344-0
6556:
6552:
6551:W. H. Freeman
6548:
6543:
6542:
6537:
6535:0-517-02961-8
6531:
6526:
6525:
6518:
6517:
6513:
6509:
6506:
6504:
6501:
6499:
6496:
6494:
6491:
6490:
6486:
6482:
6479:
6477:
6474:
6472:
6469:
6467:
6464:
6463:
6459:
6454:
6452:
6450:
6446:
6442:
6438:
6428:
6423:
6415:
6413:
6411:
6407:
6403:
6394:
6374:
6368:
6360:
6356:
6352:
6347:
6344:
6341:
6335:
6332:
6323:
6315:
6314:
6296:
6290:
6282:
6278:
6274:
6269:
6266:
6263:
6257:
6252:
6248:
6240:
6239:
6238:
6206:
6200:
6195:
6188:
6181:
6177:
6171:
6164:
6159:
6152:
6148:
6136:
6123:
6118:
6114:
6106:
6101:
6097:
6089:
6084:
6080:
6070:
6065:
6061:
6053:
6048:
6044:
6036:
6031:
6027:
6017:
6012:
6008:
6000:
5995:
5991:
5983:
5978:
5974:
5958:
5957:
5956:
5940:
5935:
5901:
5896:
5875:
5870:
5866:
5860:
5855:
5843:
5838:
5833:
5823:
5812:
5811:
5807:
5805:
5803:
5761:
5755:
5752:
5748:
5723:
5710:
5707:
5703:
5694:
5693:
5692:
5675:
5672:
5667:
5657:
5648:
5643:
5633:
5626:
5618:
5614:
5610:
5603:
5593:
5584:
5580:
5569:
5568:
5564:
5559:
5557:
5540:
5537:
5532:
5527:
5515:
5514:
5513:
5511:
5507:
5486:
5483:
5478:
5468:
5459:
5454:
5444:
5437:
5432:
5422:
5413:
5408:
5398:
5393:
5385:
5380:
5370:
5360:
5357:
5350:
5345:
5337:
5331:
5325:
5322:
5315:
5305:
5298:
5290:
5285:
5277:
5274:
5266:
5262:
5258:
5251:
5241:
5232:
5228:
5217:
5216:
5200:
5197:
5192:
5182:
5173:
5168:
5158:
5151:
5146:
5136:
5127:
5122:
5114:
5108:
5105:
5098:
5088:
5081:
5071:
5070:
5069:
5048:
5045:
5038:
5028:
5021:
4995:
4983:
4973:
4962:
4961:
4960:
4954:
4952:
4934:
4931:
4926:
4916:
4907:
4902:
4892:
4885:
4877:
4873:
4869:
4862:
4852:
4843:
4839:
4828:
4827:
4826:
4823:
4821:
4799:
4796:
4789:
4784:
4765:
4757:
4752:
4720:
4708:
4703:
4691:
4690:
4689:
4669:
4666:
4659:
4649:
4642:
4616:
4604:
4594:
4585:
4580:
4572:
4566:
4563:
4556:
4546:
4539:
4529:
4528:
4527:
4507:
4504:
4497:
4487:
4478:
4473:
4454:
4446:
4441:
4429:
4423:
4420:
4413:
4409:
4405:
4397:
4392:
4376:
4375:
4374:
4371:
4369:
4361:
4352:
4343:
4339:
4335:
4331:
4326:
4322:
4320:
4297:
4292:
4271:
4266:
4262:
4256:
4251:
4239:
4234:
4229:
4219:
4208:
4207:
4206:
4189:
4186:
4181:
4171:
4162:
4157:
4147:
4140:
4132:
4128:
4124:
4117:
4107:
4098:
4094:
4083:
4082:
4081:
4078:
4064:
4029:
4023:
4020:
4012:
4006:
4003:
4000:
3995:
3985:
3979:
3976:
3968:
3962:
3959:
3954:
3947:
3942:
3937:
3926:
3913:
3908:
3892:
3887:
3871:
3866:
3848:
3843:
3827:
3822:
3806:
3801:
3783:
3778:
3762:
3757:
3741:
3736:
3712:
3711:
3710:
3696:
3676:
3667:
3665:
3647:
3642:
3638:
3611:
3601:
3592:
3587:
3568:
3563:
3559:
3551:
3550:
3549:
3548:which yields
3529:
3524:
3520:
3516:
3511:
3506:
3492:
3487:
3464:
3463:
3462:
3387:
3383:
3377:
3372:
3360:
3355:
3345:
3334:
3333:
3332:
3256:
3251:
3249:
3182:
3180:
3177:
3122:
3056:
2968:
2920:
2898:
2876:
2860:
2847:
2843:
2835:
2831:
2823:
2819:
2803:
2802:
2801:
2770:
2764:
2759:
2752:
2745:
2741:
2735:
2728:
2723:
2716:
2712:
2700:
2687:
2682:
2678:
2670:
2665:
2661:
2653:
2648:
2644:
2634:
2629:
2625:
2617:
2612:
2608:
2600:
2595:
2591:
2581:
2576:
2572:
2564:
2559:
2555:
2547:
2542:
2538:
2522:
2521:
2520:
2504:
2499:
2495:
2486:
2467:
2464:
2459:
2455:
2449:
2444:
2440:
2436:
2428:
2424:
2420:
2413:
2409:
2403:
2399:
2388:
2387:
2386:
2355:
2347:
2342:
2340:
2326:
2284:
2242:
2229:
2222:
2220:
2216:
2214:
2188:
2166:
2117:
2111:
2103:
2099:
2095:
2090:
2087:
2084:
2078:
2075:
2067:
2063:
2057:
2053:
2047:
2044:
2041:
2035:
2032:
2023:
2019:
2016:
2009:
2008:
2007:
1965:
1959:
1951:
1947:
1943:
1938:
1935:
1932:
1926:
1918:
1914:
1908:
1904:
1898:
1895:
1889:
1884:
1880:
1876:
1873:
1866:
1865:
1864:
1847:
1844:
1836:
1833:
1825:
1821:
1817:
1805:
1801:
1790:
1789:
1788:
1762:
1758:
1753:
1750:
1744:
1741:
1733:
1729:
1725:
1713:
1709:
1698:
1697:
1696:
1693:
1679:
1672:
1668:
1664:
1660:
1651:
1644:
1637:
1635:
1631:
1604:
1590:
1548:
1506:
1497:
1494:
1486:
1484:
1463:
1457:
1451:
1446:
1443:
1440:
1434:
1431:
1424:
1423:
1422:
1402:
1396:
1392:
1385:
1379:
1376:
1373:
1367:
1364:
1357:
1356:
1355:
1341:
1318:
1312:
1300:
1294:
1290:
1288:
1284:
1259:
1255:
1249:
1246:
1240:
1237:
1230:
1229:
1228:
1214:
1205:
1203:
1187:
1184:
1181:
1178:
1170:
1150:
1146:
1140:
1136:
1126:
1122:
1118:
1113:
1110:
1087:
1084:
1076:
1073:
1065:
1061:
1057:
1045:
1041:
1030:
1029:
1028:
1004:
999:
997:
993:
988:
986:
970:
962:
958:
935:
931:
926:
923:
917:
912:
907:
903:
893:
892:
891:
889:
873:
853:
845:
842:
834:
828:
821:
816:
814:
808:
806:
804:
795:
785:
780:
778:
773:
771:
766:
765:
763:
762:
756:
746:
743:
738:
732:
731:
730:
729:
722:
721:
717:
715:
712:
710:
707:
705:
702:
700:
697:
695:
692:
690:
687:
685:
682:
680:
677:
675:
672:
670:
667:
665:
662:
660:
659:Chandrasekhar
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:
592:
590:
587:
585:
582:
580:
577:
575:
574:Schwarzschild
572:
570:
567:
565:
562:
560:
557:
555:
552:
551:
543:
542:
535:
534:Hartle–Thorne
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:
470:
467:
465:
462:
460:
457:
455:
452:
449:
445:
444:Schwarzschild
442:
441:
437:
431:
430:
419:
416:
414:
411:
410:
409:
408:
403:
398:
395:
393:
390:
388:
385:
384:
383:
382:
377:
372:
369:
367:
364:
362:
359:
357:
354:
352:
349:
347:
344:
343:
342:
341:
336:
326:
323:
322:
317:
316:
305:
302:
300:
297:
295:
292:
291:
290:
289:
286:
282:
277:
274:
272:
269:
267:
266:Event horizon
264:
262:
259:
257:
254:
252:
249:
247:
244:
242:
239:
237:
234:
232:
229:
228:
227:
226:
216:
215:
208:
205:
203:
200:
198:
195:
193:
190:
189:
181:
180:
175:
172:
167:
164:
162:
159:
157:
154:
153:
151:
149:
146:
145:
144:
143:
125:
122:
118:
113:
109:
104:
101:
97:
90:
85:
82:
78:
68:
64:
63:
60:
56:
52:
50:
46:
42:
38:
34:
30:
19:
6874:Astrophysics
6688:Experimental
6592:
6571:
6546:
6523:
6503:Tidal tensor
6481:Enrico Fermi
6434:
6398:
6236:
5918:
5779:
5690:
5555:
5502:
5067:
4958:
4950:
4824:
4817:
4687:
4525:
4372:
4365:
4315:
4204:
4079:
4056:
3668:
3629:
3547:
3406:The inverse
3405:
3252:
3186:
3178:
2955:
2799:
2519:is given by
2482:
2351:
2234:
2217:
2189:in terms of
2135:
1983:
1862:
1786:
1694:
1656:
1605:
1498:
1490:
1482:
1420:
1304:
1279:
1206:
1102:
1000:
996:trajectories
989:
956:
954:
838:
812:
799:
719:
679:Raychaudhuri
148:Introduction
33:relativistic
26:
6777:Statistical
6693:Theoretical
6670:Engineering
6547:Gravitation
6471:Élie Cartan
6460:Biographies
6445:world lines
4342:light cones
4338:proper time
3176:direction.
3044:direction,
2211:. Only the
1287:Gauss's law
694:van Stockum
624:Oppenheimer
479:Kerr–Newman
271:Singularity
6894:Geophysics
6884:Biophysics
6728:Analytical
6681:Approaches
6514:References
6431:spacetime.
4334:world line
4330:world line
3689:about the
2385:direction
1663:hyberbolic
866:and speed
547:Scientists
379:Formalisms
327:Formalisms
276:Black hole
202:World line
6844:Molecular
6745:Acoustics
6738:Continuum
6733:Celestial
6723:Newtonian
6710:Classical
6653:Divisions
6449:spacetime
6369:ρ
6345:π
6336:−
6328:‖
6291:ρ
6267:π
6253:⊥
6211:‖
6182:⊥
6153:⊥
5890:¯
5827:¯
5788:Γ
5765:Γ
5661:¯
5637:¯
5597:¯
5524:Γ
5510:Euclidean
5472:¯
5448:¯
5426:¯
5405:Γ
5390:Γ
5374:¯
5342:Γ
5309:¯
5282:Γ
5245:¯
5186:¯
5162:¯
5140:¯
5119:Γ
5092:¯
5032:¯
4977:¯
4920:¯
4896:¯
4856:¯
4778:¯
4700:Γ
4653:¯
4598:¯
4577:Γ
4550:¯
4491:¯
4467:¯
4286:¯
4223:¯
4175:¯
4151:¯
4111:¯
4065:θ
4030:θ
4024:
4013:θ
4007:
4001:−
3986:θ
3980:
3969:θ
3963:
3709:-axis is
3677:θ
3639:δ
3605:¯
3581:¯
3521:δ
3481:¯
3422:¯
3349:¯
3294:¯
3248:rotations
3231:^
3201:^
3161:^
3132:^
3123:⋅
3095:^
3066:^
3057:⋅
3029:^
2978:^
2969:⋅
2930:^
2921:⋅
2908:^
2899:⋅
2886:^
2877:⋅
2775:‖
2746:⊥
2717:⊥
2370:^
2171:‖
2112:ρ
2088:π
2079:−
2036:−
2028:‖
1960:ρ
1936:π
1885:⊥
1822:τ
1745:−
1458:ρ
1444:π
1380:ρ
1377:π
1313:ρ
1179:τ
1169:curvature
1062:τ
803:spacetime
639:Robertson
604:Friedmann
599:Eddington
589:Nordström
579:de Sitter
436:Solutions
361:Geodesics
356:Friedmann
338:Equations
324:Equations
285:Spacetime
220:Phenomena
126:ν
123:μ
114:κ
105:ν
102:μ
94:Λ
86:ν
83:μ
45:Newtonian
37:curvature
6970:Category
6455:See also
6439:and the
6219:‖
6142:‖
6132:‖
5968:‖
4039:‖
3932:‖
3922:‖
3722:‖
2939:‖
2866:‖
2856:‖
2813:‖
2783:‖
2706:‖
2696:‖
2532:‖
2187:expanded
1667:circular
1659:elliptic
841:circular
755:Category
619:Lemaître
584:Reissner
569:Poincaré
554:Einstein
499:Taub–NUT
464:Wormhole
448:interior
161:Timeline
6928:Related
6812:General
6807:Special
6665:Applied
6402:tensors
5508:. In a
4368:commute
3662:is the
3015:in the
1496:craft.
1200:is the
1167:is the
983:is the
959:is the
674:Hawking
669:Penrose
644:Bardeen
634:Wheeler
564:Hilbert
559:Lorentz
519:pp-wave
156:History
6839:Atomic
6794:Modern
6644:Major
6603:
6578:
6557:
6532:
6408:, and
5919:where
5744:
5717:
5691:where
5016:
4989:
4741:
4714:
4688:where
4637:
4610:
4205:where
2956:where
2354:tensor
2277:. The
2213:linear
1583:. The
1541:. The
1103:where
955:where
753:
720:others
714:Thorne
704:Newman
684:Taylor
664:Ehlers
649:Walker
614:Zwicky
489:Kasner
5800:is a
3630:Here
1001:From
844:polar
689:Hulse
629:Gödel
609:Milne
504:Milne
469:Gödel
166:Tests
41:space
6765:Wave
6660:Pure
6601:ISBN
6576:ISBN
6555:ISBN
6530:ISBN
5780:and
2006:and
985:mass
963:and
699:Taub
654:Kerr
594:Weyl
474:Kerr
392:BSSN
6760:Ray
6447:in
4021:cos
4004:sin
3977:sin
3960:cos
3437:of
1661:or
805:.
709:Yau
387:ADM
39:in
6972::
6599:.
6595:.
6553:.
6412:.
6404:,
5804:.
4822:.
4353:),
4321:.
4077:.
3666:.
1692:.
890:,
6637:e
6630:t
6623:v
6609:.
6584:.
6563:.
6538:.
6390:.
6378:)
6375:r
6372:(
6361:2
6357:c
6353:3
6348:G
6342:8
6333:=
6324:R
6300:)
6297:r
6294:(
6283:2
6279:c
6275:3
6270:G
6264:4
6258:=
6249:R
6233:.
6207:R
6201:0
6196:0
6189:0
6178:R
6172:0
6165:0
6160:0
6149:R
6137:=
6124:3
6119:3
6115:R
6107:2
6102:3
6098:R
6090:1
6085:3
6081:R
6071:3
6066:2
6062:R
6054:2
6049:2
6045:R
6037:1
6032:2
6028:R
6018:3
6013:1
6009:R
6001:2
5996:1
5992:R
5984:1
5979:1
5975:R
5941:i
5936:j
5930:M
5902:l
5897:j
5886:M
5876:k
5871:l
5867:R
5861:i
5856:k
5850:M
5844:=
5839:i
5834:j
5824:R
5762:+
5756:s
5753:d
5749:d
5736:f
5733:e
5730:d
5724:=
5711:s
5708:D
5704:D
5676:0
5673:=
5668:j
5658:h
5649:i
5644:j
5634:R
5627:+
5619:2
5615:s
5611:D
5604:i
5594:h
5585:2
5581:D
5541:0
5538:=
5533:i
5528:j
5499:.
5487:0
5484:=
5479:j
5469:h
5460:i
5455:j
5445:R
5438:+
5433:k
5423:h
5414:j
5409:k
5399:i
5394:j
5386:+
5381:j
5371:h
5361:s
5358:d
5351:i
5346:j
5338:d
5332:+
5326:s
5323:d
5316:i
5306:h
5299:d
5291:i
5286:j
5278:2
5275:+
5267:2
5263:s
5259:d
5252:i
5242:h
5233:2
5229:d
5213:.
5201:0
5198:=
5193:j
5183:h
5174:i
5169:j
5159:R
5152:+
5147:j
5137:u
5128:i
5123:j
5115:+
5109:s
5106:d
5099:i
5089:u
5082:d
5064:.
5049:s
5046:D
5039:i
5029:h
5022:D
5008:f
5005:e
5002:d
4996:=
4984:i
4974:u
4947:,
4935:0
4932:=
4927:j
4917:h
4908:i
4903:j
4893:R
4886:+
4878:2
4874:s
4870:D
4863:i
4853:h
4844:2
4840:D
4800:s
4797:d
4790:i
4785:j
4774:M
4766:d
4758:k
4753:i
4747:M
4733:f
4730:e
4727:d
4721:=
4709:k
4704:j
4670:s
4667:D
4660:k
4650:h
4643:D
4629:f
4626:e
4623:d
4617:=
4605:j
4595:h
4586:k
4581:j
4573:+
4567:s
4564:d
4557:k
4547:h
4540:d
4508:s
4505:d
4498:j
4488:h
4479:i
4474:j
4463:M
4455:d
4447:k
4442:i
4436:M
4430:=
4424:s
4421:d
4414:i
4410:h
4406:d
4398:k
4393:i
4387:M
4312:.
4298:l
4293:j
4282:M
4272:k
4267:l
4263:R
4257:i
4252:k
4246:M
4240:=
4235:i
4230:j
4220:R
4190:0
4187:=
4182:j
4172:h
4163:i
4158:j
4148:R
4141:+
4133:2
4129:s
4125:d
4118:i
4108:h
4099:2
4095:d
4053:.
4033:)
4027:(
4016:)
4010:(
3996:0
3989:)
3983:(
3972:)
3966:(
3955:0
3948:0
3943:0
3938:1
3927:=
3914:3
3909:3
3903:M
3893:2
3888:3
3882:M
3872:1
3867:3
3861:M
3849:3
3844:2
3838:M
3828:2
3823:2
3817:M
3807:1
3802:2
3796:M
3784:3
3779:1
3773:M
3763:2
3758:1
3752:M
3742:1
3737:1
3731:M
3697:x
3648:i
3643:k
3626:.
3612:j
3602:h
3593:i
3588:j
3577:M
3569:=
3564:i
3560:h
3544:,
3530:i
3525:k
3517:=
3512:j
3507:k
3501:M
3493:i
3488:j
3477:M
3447:M
3418:M
3402:.
3388:j
3384:h
3378:i
3373:j
3367:M
3361:=
3356:i
3346:h
3318:h
3291:h
3266:C
3228:r
3198:r
3158:z
3129:z
3119:h
3092:y
3063:y
3053:h
3026:x
3002:h
2975:x
2965:h
2927:z
2917:h
2905:y
2895:h
2883:x
2873:h
2861:=
2848:3
2844:h
2836:2
2832:h
2824:1
2820:h
2771:R
2765:0
2760:0
2753:0
2742:R
2736:0
2729:0
2724:0
2713:R
2701:=
2688:3
2683:3
2679:R
2671:2
2666:3
2662:R
2654:1
2649:3
2645:R
2635:3
2630:2
2626:R
2618:2
2613:2
2609:R
2601:1
2596:2
2592:R
2582:3
2577:1
2573:R
2565:2
2560:1
2556:R
2548:1
2543:1
2539:R
2505:i
2500:j
2496:R
2468:0
2465:=
2460:j
2456:h
2450:i
2445:j
2441:R
2437:+
2429:2
2425:s
2421:d
2414:i
2410:h
2404:2
2400:d
2367:r
2327:y
2306:r
2285:x
2264:r
2243:z
2198:h
2167:R
2145:r
2121:)
2118:r
2115:(
2104:2
2100:c
2096:3
2091:G
2085:8
2076:=
2068:3
2064:r
2058:2
2054:c
2048:M
2045:G
2042:2
2033:=
2024:R
2020:=
2017:R
1993:r
1969:)
1966:r
1963:(
1952:2
1948:c
1944:3
1939:G
1933:4
1927:=
1919:3
1915:r
1909:2
1905:c
1899:M
1896:G
1890:=
1881:R
1877:=
1874:R
1848:0
1845:=
1841:h
1837:R
1834:+
1826:2
1818:d
1812:h
1806:2
1802:d
1771:r
1763:3
1759:r
1754:M
1751:G
1742:=
1734:2
1730:t
1726:d
1720:r
1714:2
1710:d
1680:c
1615:h
1591:y
1570:v
1549:x
1528:r
1507:z
1479:.
1467:)
1464:r
1461:(
1452:3
1447:G
1441:4
1435:=
1432:R
1417:.
1403:3
1397:3
1393:r
1389:)
1386:r
1383:(
1374:4
1368:=
1365:M
1342:r
1322:)
1319:r
1316:(
1260:3
1256:r
1250:M
1247:G
1241:=
1238:R
1215:M
1188:t
1185:c
1182:=
1151:2
1147:c
1141:2
1137:v
1127:2
1123:r
1119:1
1114:=
1111:R
1088:0
1085:=
1081:h
1077:R
1074:+
1066:2
1058:d
1052:h
1046:2
1042:d
1014:h
971:M
957:G
936:2
932:r
927:M
924:G
918:=
913:r
908:2
904:v
874:v
854:r
783:e
776:t
769:v
450:)
446:(
119:T
110:=
98:g
91:+
79:G
20:)
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