Einstein field equations

1143: 746: 2309: 76: 1964: 7711:. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as 10762: 9902: 1156: 759: 7546: 971: 1276: 2304:{\displaystyle {\begin{aligned}g_{\mu \nu }&=\times \operatorname {diag} (-1,+1,+1,+1)\\{R^{\mu }}_{\alpha \beta \gamma }&=\times \left(\Gamma _{\alpha \gamma ,\beta }^{\mu }-\Gamma _{\alpha \beta ,\gamma }^{\mu }+\Gamma _{\sigma \beta }^{\mu }\Gamma _{\gamma \alpha }^{\sigma }-\Gamma _{\sigma \gamma }^{\mu }\Gamma _{\beta \alpha }^{\sigma }\right)\\G_{\mu \nu }&=\times \kappa T_{\mu \nu }\end{aligned}}} 7276: 7265: 8067: 6357: 7917: 6199: 9446: 7080: 3667: 3541: 7541:{\displaystyle {\begin{aligned}{F^{\alpha \beta }}_{;\beta }&=0\\F_{}&={\tfrac {1}{3}}\left(F_{\alpha \beta ;\gamma }+F_{\beta \gamma ;\alpha }+F_{\gamma \alpha ;\beta }\right)={\tfrac {1}{3}}\left(F_{\alpha \beta ,\gamma }+F_{\beta \gamma ,\alpha }+F_{\gamma \alpha ,\beta }\right)=0.\end{aligned}}} 7737:

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

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

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

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

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

4264: 4937: 4578: 4699: 5169: 3137: 6425: 3239: 5747: 3898: 4925: 8322:, the stress–energy tensor on the right side of the equation must be written with each component in units of energy density (i.e., energy per volume, equivalently pressure). In Einstein's original publication, the choice is 5603: 3342: 5345: 4392: 6846: 4397:

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

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

147: 1519: 5318: 1270: 6699: 4706: 5880: 5742: 1969: 7260:{\displaystyle G^{\alpha \beta }+\Lambda g^{\alpha \beta }={\frac {\kappa }{\mu _{0}}}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+{\tfrac {1}{4}}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right).} 4077: 1913:

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

8062:{\displaystyle g^{\alpha \kappa }={\frac {{\tfrac {1}{6}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }}{\det(g)}}\,.} 7756:

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

6636: 2770: 1943:, since both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic 888:

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

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

4316: 5659: 6688: 6352:{\displaystyle R_{00}=\Gamma _{00,\rho }^{\rho }-\Gamma _{\rho 0,0}^{\rho }+\Gamma _{\rho \lambda }^{\rho }\Gamma _{00}^{\lambda }-\Gamma _{0\lambda }^{\rho }\Gamma _{\rho 0}^{\lambda }.} 7912:{\displaystyle \det(g)={\tfrac {1}{24}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\alpha \kappa }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }} 2429: 4182: 6194:{\displaystyle K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)\approx K\left(\rho c^{4}-{\tfrac {1}{2}}\left(-\rho c^{2}\right)\left(-c^{2}\right)\right)={\tfrac {1}{2}}K\rho c^{4}\,.} 4482: 4591: 5079: 3405: 3060: 6769: 1717: 7075:{\displaystyle T^{\alpha \beta }=\,-{\frac {1}{\mu _{0}}}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+{\tfrac {1}{4}}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right)} 3169: 2908: 2426:

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

1941: 1595: 1552: 1383: 1349: 1315: 4840: 5507: 3273: 1751: 4323: 3662:{\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }-{R^{\gamma }}_{\beta \gamma \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0} 3536:{\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }+{R^{\gamma }}_{\beta \varepsilon \gamma ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0} 1407: 53: 6776: 1431: 1895:

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

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

1576: 5174: 4010: 3888:{\displaystyle g^{\beta \delta }\left(R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }\right)=0} 2931: 9817: 1044: 3041:

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

9842: 5244: 790: 4477: 8755: 5666: 9428: 5067:{\displaystyle {\frac {d^{2}x^{\alpha }}{d\tau ^{2}}}=-\Gamma _{\beta \gamma }^{\alpha }{\frac {dx^{\beta }}{d\tau }}{\frac {dx^{\gamma }}{d\tau }}\,.} 9226: 169: 3007:

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

8217: 6570:{\displaystyle \Phi _{,ii}\approx \Gamma _{00,i}^{i}\approx R_{00}=K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)\approx {\tfrac {1}{2}}K\rho c^{4}} 9453: 5868:{\displaystyle T_{\mu \nu }\approx \operatorname {diag} \left(T_{00},0,0,0\right)\approx \operatorname {diag} \left(\rho c^{4},0,0,0\right)\,.} 6580: 3998:{\displaystyle {R^{\delta }}_{\delta ;\varepsilon }-{R^{\delta }}_{\varepsilon ;\delta }+{R^{\gamma \delta }}_{\delta \varepsilon ;\gamma }=0} 82: 9770: 1446: 8786: 2558: 1207: 374: 5485:{\displaystyle \Phi _{,i}\approx \Gamma _{00}^{i}={\tfrac {1}{2}}g^{i\alpha }\left(g_{\alpha 0,0}+g_{0\alpha ,0}-g_{00,\alpha }\right)\,.} 10512: 6368: 10747: 9847: 9373: 6903: 1885: 1099: 727: 9203: 10536: 10282: 9403: 9263: 8123: 7698: 4273: 537: 379: 8963:

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

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

5608: 9887: 6924: 1074: 783: 492: 7768:. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the 6641: 10799: 9776: 3760:{\displaystyle R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0} 2751:{\displaystyle R_{\mu \nu }-{\frac {2}{D-2}}\Lambda g_{\mu \nu }=\kappa \left(T_{\mu \nu }-{\frac {1}{D-2}}Tg_{\mu \nu }\right).} 8877:

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

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

10318: 10272: 9631: 7270: 3046: 405: 239: 8679: 10456: 9213: 9105: 8765: 8293: 8108: 7716: 3351: 923: 444: 4828:{\displaystyle {\begin{aligned}\Phi _{,ii}&=4\pi G\rho \\{\frac {d^{2}x^{i}}{dt^{2}}}&=-\Phi _{,i}\,.\end{aligned}}} 9940: 6851: 5998:{\displaystyle T=g^{\alpha \beta }T_{\alpha \beta }\approx g^{00}T_{00}\approx -{\frac {1}{c^{2}}}\rho c^{4}=-\rho c^{2}\,} 939: 9191: 4406:

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

3356: 1582:. This is a symmetric second-degree tensor that depends on only the metric tensor and its first and second derivatives. 1188: 776: 10193: 9837: 9660: 9502: 9343: 9157: 7704: 6734: 1352: 1039: 885: 156: 10564: 10198: 9521: 9398: 9131: 8656: 8610: 8540: 8422: 8390: 8357: 8266: 8128: 8083:

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

1863: 1119: 182: 9636: 4166:{\displaystyle \left({R^{\gamma }}_{\varepsilon }-{\tfrac {1}{2}}{g^{\gamma }}_{\varepsilon }R\right)_{;\gamma }=0} 10518: 1142: 745: 462: 75: 10231: 9681: 2871:{\displaystyle R_{\mu \nu }-\Lambda g_{\mu \nu }=\kappa \left(T_{\mu \nu }-{\frac {1}{2}}T\,g_{\mu \nu }\right).} 897: 369: 5241:

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

10334: 9491: 9363: 7715:. Even so, there are several cases where the field equations have been solved completely, and those are called 6702:

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

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

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

908: 7553: 4422:, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is 10789: 10765: 10507: 10412: 10342: 9256: 9184: 8118: 7681:{\displaystyle F_{\alpha \beta }=A_{\alpha ;\beta }-A_{\beta ;\alpha }=A_{\alpha ,\beta }-A_{\beta ,\alpha }} 5237:

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

1944: 1859: 889: 456: 164: 10475: 502: 10329: 9469: 9358: 9353: 9071: 8556:

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

8113: 7712: 522: 4423: 10794: 10370: 10104: 10089: 9626: 9585: 9529: 9378: 9303: 9179: 8098: 934:. Further simplification is achieved in approximating the spacetime as having only small deviations from 763: 254: 174: 56: 9174: 7772:, ignoring higher-power terms. This linearization procedure can be used to investigate the phenomena of 10657: 10099: 10052: 9553: 9368: 8414: 4443: 3545:

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

1959:(MTW). The authors analyzed conventions that exist and classified these according to three signs ( ): 667: 215: 6714: 1695: 10687: 10313: 9994: 9461: 9408: 9388: 8093: 4451: 1685:{\displaystyle \kappa ={\frac {8\pi G}{c^{4}}}\approx 2.07665(5)\times 10^{-43}\,{\textrm {N}}^{-1},} 931: 467: 279: 2622:

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

10784: 10712: 10022: 9933: 9906: 9857: 9852: 9832: 9822: 9782: 9477: 9249: 4447: 2522:{\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }-\Lambda g_{\mu \nu }=-\kappa T_{\mu \nu }.} 1386: 858: 824: 702: 692: 542: 2883: 1916: 1841:{\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }.} 1527: 1358: 1324: 1290: 10441: 10363: 9983: 9974: 9862: 8080: 7773: 6731:, the vacuum field equations, also known as 'Einstein vacuum equations' (EVE), can be written as 3268:

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

1181: 687: 421: 6713:

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

10667: 10530: 10436: 10394: 10068: 10032: 9794: 9702: 9674: 9577: 9436: 9087: 6859: 6698: 4407: 4259:{\displaystyle \left(R^{\gamma \delta }-{\tfrac {1}{2}}g^{\gamma \delta }R\right)_{;\gamma }=0} 1727: 1014: 877: 657: 642: 487: 249: 8283: 4573:{\displaystyle \nabla ^{2}\Phi \left({\vec {x}},t\right)=4\pi G\rho \left({\vec {x}},t\right)} 2552: 2419:, Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) and Peacock (1999) are 1392: 22: 10672: 10277: 10216: 9867: 9827: 9667: 9616: 9611: 9537: 9348: 9328: 9318: 8103: 7087: 6916: 4694:{\displaystyle {\ddot {\vec {x}}}(t)={\vec {g}}=-\nabla \Phi \left({\vec {x}}(t),t\right)\,.} 3001: 2923: 1416: 1410: 1109: 1009: 892:

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

865: 632: 200: 5164:{\displaystyle {\frac {dx^{\beta }}{d\tau }}\approx \left({\frac {dt}{d\tau }},0,0,0\right)} 10577: 10503: 10347: 9949: 9746: 9695: 9688: 9641: 9313: 9286: 9200:— An explanation of Einstein's field equation, its derivation, and some of its consequences 9035: 8982: 8929: 8886: 8843: 8795: 8784:(1977). "Solutions of the Maxwell and Yang–Mills equations associated with Hopf fibrings". 8720: 8567: 8461: 8176: 8133: 7549: 3132:{\displaystyle T_{\mu \nu }^{\mathrm {(vac)} }=-{\frac {\Lambda }{\kappa }}g_{\mu \nu }\,.} 919: 452: 7742:. New solutions have been discovered using these methods by LeBlanc and Kohli and Haslam. 7575:

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

8: 10652: 10647: 10637: 10469: 10418: 10236: 10221: 10047: 9978: 9926: 7757: 7738:

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

7568: 6887: 4469:, which is the gravitational potential in joules per kilogram of the gravitational field 1862:, which dictates how freely falling matter moves through spacetime, form the core of the 943: 927: 647: 617: 612: 364: 10642: 9039: 8986: 8933: 8890: 8847: 8799: 8724: 8571: 8465: 8180: 3053:

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

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

3234:{\displaystyle \rho _{\mathrm {vac} }=-p_{\mathrm {vac} }={\frac {\Lambda }{\kappa }},} 3023: 2404: 1956: 1878: 1867: 1561: 1174: 1160: 1124: 986: 981: 881: 804: 697: 582: 507: 354: 259: 244: 205: 67: 8192: 4007:

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

3038:"that the introduction of the cosmological term was the biggest blunder of his life". 1881:

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

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

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

527: 302: 10597: 8898: 8740: 8587: 7572: 4920:{\displaystyle R_{\mu \nu }=K\left(T_{\mu \nu }-{\tfrac {1}{2}}Tg_{\mu \nu }\right)} 918:

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

577: 10632: 10622: 10569: 10546: 10124: 9758: 9740: 9418: 9055: 9043: 8990: 8937: 8894: 8851: 8803: 8781: 8728: 8687: 8575: 8558: 8479: 8469: 8184: 7769: 7739: 5238: 4411: 2911: 1874: 1579: 873: 869: 652: 607: 587: 532: 10692: 9227:

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

8642: 8235: 5598:{\displaystyle 2\Phi _{,i}\approx g^{ij}\left(-g_{00,j}\right)\approx -g_{00,i}\,} 3337:{\displaystyle \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0.} 2313:

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

672: 10707: 10682: 10607: 10602: 10485: 10446: 10408: 10352: 10226: 10162: 9788: 9720: 9485: 9393: 9338: 9308: 9272: 9141: 9115: 8831: 8254: 8213: 7765: 6855: 4387:{\displaystyle \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0} 4267: 3008: 2536: 1555: 1437: 1318: 1019: 935: 850: 831: 707: 682: 567: 562: 426: 307: 269: 10490: 9728: 6841:{\displaystyle R_{\mu \nu }={\frac {\Lambda }{{\frac {D}{2}}-1}}g_{\mu \nu }\,.} 477: 10732: 10404: 10388: 10384: 10287: 10257: 10114: 10017: 9545: 9323: 9079: 8994: 8336:, in which case the stress–energy tensor components have units of mass density. 8138: 5342:

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

4419: 4415: 1735: 1147: 912: 750: 717: 712: 400: 264: 9216:

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

9047: 8920:

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

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

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

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

926:

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

10778: 10722: 10702: 10697: 10612: 10480: 10308: 10252: 10084: 10037: 9764: 9734: 9621: 9231: 9097: 9083: 8188: 6863: 3146: 3015:

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

2531:

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

677: 592: 572: 497: 395: 274: 8288:(illustrated ed.). Springer Science & Business Media. p. 180. 637: 10742: 10662: 10627: 10157: 10119: 9872: 9752: 8648: 8638: 8493: 8367: 6873: 4431: 3768: 3411:

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

3142: 3035: 3019: 1882: 1114: 622: 602: 9016:

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

2394:{\displaystyle R_{\mu \nu }=\times \times {R^{\alpha }}_{\mu \alpha \nu }} 907:

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

10526: 10495: 10042: 9877: 9561: 9383: 8715: 5230:{\displaystyle {\frac {d}{dt}}\left({\frac {dt}{d\tau }}\right)\approx 0} 4067:{\displaystyle R_{;\varepsilon }-2{R^{\gamma }}_{\varepsilon ;\gamma }=0} 1094: 512: 482: 8474: 8347: 5663:

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

2993:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }\,,} 10737: 10303: 10147: 10142: 9882: 8855: 8807: 8383:

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

8222:

Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin

7727: 7557: 6920: 4458:

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

722: 284: 210: 142:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }} 9030: 8456: 8440:"Theorems on Existence and Global Dynamics for the Einstein Equations" 8308:

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

4465:

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

4266:

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

1514:{\displaystyle G_{\mu \nu }=R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu },} 10134: 8579: 7761: 7708: 5313:{\displaystyle {\frac {d^{2}x^{i}}{dt^{2}}}\approx -\Gamma _{00}^{i}} 4585: 3042: 1265:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }} 1004: 970: 962: 843: 820: 293: 8701:

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

8500: 10677: 10027: 9235: 7923:; and the inverse of the metric in 4 dimensions can be written as: 7731: 6869: 854: 662: 472: 312: 9241: 8977: 8408: 5737:{\displaystyle R_{00}=K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)} 1275: 1888:, which correspond to the freedom to choose a coordinate system. 1069: 893: 6890:

and manifolds with a Ricci tensor proportional to the metric as

6773:

In the case of nonzero cosmological constant, the equations are

1851:

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

10357: 7561: 5504:

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

1739: 1280: 1201:

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

835: 8259:

Spacetime and Geometry – An Introduction to General Relativity

9918: 9238:, Einstein field equation on the side of a train in Bolivia. 10338: 8555: 7082:

is used, then the Einstein field equations are called the

900:) in the resulting geometry are then calculated using the 9214:

Arch and scaffold: How Einstein found his field equations

9207: 8601:

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

8346:

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

8197: 1955:

The above form of the EFE is the standard established by

9236:"The impact of general relativity on the Atacama Desert" 6631:{\displaystyle {\tfrac {1}{2}}K\rho c^{4}=4\pi G\rho \,} 7093:, taken to be zero in conventional relativity theory): 6850:

The solutions to the vacuum field equations are called

6577:

which reduces to the Newtonian field equation provided

9194:— A simple introduction to Einstein's Field Equations. 7950: 7808: 7444: 7356: 7199: 7017: 6585: 6540: 6507: 6160: 6091: 6034: 5744:

the low speed and static field assumptions imply that

5705: 5384: 4885: 4209: 4110: 3263: 1877:. Each tensor has 10 independent components. The four 942:. These equations are used to study phenomena such as 9514:

Die Grundlagen der Einsteinschen Relativitäts-Theorie

8680:"Was Einstein's 'biggest blunder' a stellar success?" 7929: 7791: 7588: 7279: 7099: 6932: 6779: 6737: 6644: 6583: 6428: 6371: 6209: 6011: 5883: 5750: 5669: 5611: 5510: 5348: 5247: 5177: 5082: 4940: 4843: 4709: 4594: 4485: 4326: 4276: 4185: 4080: 4013: 3901: 3780: 3675: 3553: 3427: 3359: 3276: 3172: 3063: 2934: 2886: 2773: 2628: 2605:{\displaystyle R-{\frac {D}{2}}R+D\Lambda =\kappa T,} 2561: 2432: 2319: 1967: 1919: 1754: 1698: 1598: 1564: 1530: 1449: 1419: 1395: 1361: 1327: 1293: 1210: 85: 25: 19:"Einstein equation" redirects here. For the equation 8600: 8512: 8163:"The Foundation of the General Theory of Relativity" 9078: 8619: 8506: 8241: 6413:{\displaystyle R_{00}\approx \Gamma _{00,i}^{i}\,.} 857:and stress within that spacetime (expressed by the 9145: 9119: 8345: 8061: 7911: 7703:The solutions of the Einstein field equations are 7680: 7540: 7259: 7074: 6840: 6763: 6682: 6630: 6569: 6412: 6351: 6193: 5997: 5867: 5736: 5653: 5597: 5484: 5312: 5229: 5163: 5066: 4919: 4827: 4693: 4572: 4386: 4310: 4258: 4165: 4066: 3997: 3887: 3759: 3661: 3535: 3399: 3336: 3233: 3131: 2992: 2902: 2870: 2750: 2604: 2521: 2415:, Peebles (1980) and Efstathiou et al. (1990) are 2393: 2303: 1935: 1840: 1711: 1684: 1570: 1546: 1513: 1425: 1401: 1377: 1343: 1309: 1264: 141: 47: 9447:Investigations on the Theory of Brownian Movement 10776: 8404: 8402: 8040: 7792: 6361:Our simplifying assumptions make the squares of 3347:Derivation of local energy–momentum conservation 4437: 4311:{\displaystyle {G^{\alpha \beta }}_{;\beta }=0} 1873:The EFE is a tensor equation relating a set of 9454:Relativity: The Special and the General Theory 8829: 8644:My World Line : An Informal Autobiography 5654:{\displaystyle g_{00}\approx -c^{2}-2\Phi \,.} 9934: 9257: 8876: 8411:Exact Solutions of Einstein's Field Equations 8399: 6897: 6365:disappear together with the time derivatives 3009:universe that is not expanding or contracting 1182: 784: 8832:"A class of homogeneous cosmological models" 8787:International Journal of Theoretical Physics 7730:and to different models of evolution of the 7571:is zero. From the latter, it follows by the 6728: 6683:{\displaystyle K={\frac {8\pi G}{c^{4}}}\,.} 2917: 9204:Video Lecture on Einstein's Field Equations 8962: 9941: 9927: 9264: 9250: 8281: 3049:, and to explain this a positive value of 2546: 1189: 1175: 791: 777: 9029: 8976: 8714: 8533:The Large-scale Structure of the Universe 8483: 8473: 8455: 8055: 7699:Solutions of the Einstein field equations 6949: 6834: 6757: 6693: 6676: 6627: 6406: 6187: 5994: 5861: 5647: 5594: 5478: 5060: 4817: 4687: 3125: 2986: 2846: 1661: 896:trajectories of particles and radiation ( 9888:Emergency Committee of Atomic Scientists 9114: 9015: 8780: 8760:. Oxford University Press. p. 164. 8518: 8380: 8212: 8160: 7556:. The first equation asserts that the 4- 6697: 6203:From the definition of the Ricci tensor 4270:, gives, after relabelling the indices, 3011:. This effort was unsuccessful because: 2910:in the expression on the right with the 2616:is the spacetime dimension. Solving for 1433:is the Einstein gravitational constant. 1274: 9140: 8919: 8830:Ellis, G. F. R.; MacCallum, M. (1969). 8686:. University of Toronto. Archived from 8625: 8530: 8437: 8282:Grøn, Øyvind; Hervik, Sigbjorn (2007). 8253: 6904:Maxwell's equations in curved spacetime 6422:Combining the above equations together 4320:Using the EFE, this immediately gives, 2914:without significant loss of accuracy). 10777: 9632:Albert Einstein World Award of Science 9210:Physics Professor Edmund Bertschinger. 8700: 8694: 8352:(2d ed.). New York: McGraw-Hill. 8247: 8206: 3166:that are fixed constants and given by 3047:accelerating expansion of the universe 9922: 9245: 8753: 8637: 8631: 8531:Peebles, Phillip James Edwin (1980). 8374: 8218:"Die Feldgleichungen der Gravitation" 8156: 8154: 8109:Exact solutions in general relativity 7745: 4462:Derivation of Newton's law of gravity 3774:Next, contract again with the metric 2543:metric sign convention adopted here. 1075:Newton's law of universal gravitation 16:Field-equations in general relativity 8677: 8671: 8385:. Vintage Press. pp. 107, 233. 8261:. Addison Wesley. pp. 151–159. 6925:electromagnetic stress–energy tensor 4584:is the mass density. The orbit of a 1745:The EFE can thus also be written as 949: 10513:Tolman–Oppenheimer–Volkoff equation 10466:Friedmann–Lemaître–Robertson–Walker 9271: 8507:Misner, Thorne & Wheeler (1973) 8242:Misner, Thorne & Wheeler (1973) 7273:are also applicable in free space: 3400:{\displaystyle R_{\alpha \beta }=0} 3264:Conservation of energy and momentum 1858:These equations, together with the 868:are related to the distribution of 13: 9661:Albert Einstein: Creator and Rebel 9374:Einstein–Infeld–Hoffmann equations 9344:Einstein relation (kinetic theory) 9220: 9198:The Meaning of Einstein's Equation 8349:Introduction to general relativity 8151: 7779: 7116: 6798: 6449: 6430: 6386: 6329: 6311: 6293: 6275: 6248: 6224: 5644: 5515: 5366: 5350: 5296: 4989: 4805: 4715: 4647: 4644: 4496: 4487: 4328: 3278: 3220: 3209: 3206: 3203: 3185: 3182: 3179: 3104: 3087: 3084: 3081: 2951: 2790: 2663: 2587: 2555:of both sides of the EFE one gets 2478: 2220: 2202: 2181: 2163: 2136: 2109: 1950: 1800: 1396: 1227: 1040:Introduction to general relativity 102: 14: 10811: 10283:Hamilton–Jacobi–Einstein equation 9522:The Einstein Theory of Relativity 9167: 8129:Mathematics of general relativity 8124:Hamilton–Jacobi–Einstein equation 7567:is zero, and the second that its 7548:where the semicolon represents a 6764:{\displaystyle R_{\mu \nu }=0\,.} 4703:In tensor notation, these become 1875:symmetric 4 × 4 tensors 1728:Newtonian constant of gravitation 1120:Mathematics of general relativity 1045:Mathematics of general relativity 10761: 10760: 9901: 9900: 9637:Einstein Prize for Laser Science 7726:. It leads to the prediction of 2928:In the Einstein field equations 2553:trace with respect to the metric 1712:{\displaystyle {\textrm {m/J}},} 1155: 1154: 1141: 969: 830:The equations were published by 758: 757: 744: 74: 9682:Einstein: His Life and Universe 9009: 8956: 8913: 8870: 8823: 8774: 8747: 8594: 8549: 8524: 8431: 4401: 1587:Einstein gravitational constant 10800:Partial differential equations 10090:Mass–energy equivalence (E=mc) 9948: 9304:Mass–energy equivalence (E=mc) 9192:Caltech Tutorial on Relativity 9152:. Cambridge University Press. 8603:Particle Physics and Cosmology 8535:. Princeton University Press. 8339: 8302: 8275: 8049: 8043: 7801: 7795: 7343: 7328: 6908:If the energy–momentum tensor 6729:trace-reversed field equations 6706:If the energy–momentum tensor 5605:which is satisfied by letting 4673: 4667: 4661: 4632: 4620: 4614: 4603: 4553: 4510: 3386: 3371: 3090: 3078: 2360: 2351: 2345: 2336: 2275: 2266: 2097: 2088: 2049: 2013: 2001: 1992: 1945:partial differential equations 1642: 1636: 890:partial differential equations 1: 9065: 8119:History of general relativity 7713:post-Newtonian approximations 3352:differential Bianchi identity 2411:, whereas Weinberg (1972) is 9072:General relativity resources 8678:Wahl, Nicolle (2005-11-22). 8114:General relativity resources 7692: 4438:The correspondence principle 3767:using the definition of the 3022:showed that our universe is 2903:{\displaystyle g_{\mu \nu }} 2539:is used rather than the MTW 1936:{\displaystyle g_{\mu \nu }} 1547:{\displaystyle R_{\mu \nu }} 1378:{\displaystyle T_{\mu \nu }} 1344:{\displaystyle g_{\mu \nu }} 1310:{\displaystyle G_{\mu \nu }} 864:Analogously to the way that 805:general theory of relativity 7: 10105:Relativistic Doppler effect 9627:Albert Einstein Peace Prize 9404:Unsuccessful investigations 9180:Encyclopedia of Mathematics 8086: 7271:covariant Maxwell equations 3258: 3045:observations have shown an 2767:dimensions this reduces to 2405:Misner, Thorne, and Wheeler 1957:Misner, Thorne, and Wheeler 1906:is everywhere zero) define 1070:Introduction to gravitation 909:Newton's law of gravitation 255:Gravitational time dilation 10: 10816: 10576:In computational physics: 10100:Relativity of simultaneity 9554:Picasso at the Lapin Agile 9492:Russell–Einstein Manifesto 9364:Bose–Einstein correlations 8995:10.1103/physrevd.88.063518 8942:10.1088/0264-9381/14/8/025 8415:Cambridge University Press 7749: 7696: 7552:, and the brackets denote 7084:Einstein–Maxwell equations 6901: 6898:Einstein–Maxwell equations 4074:which can be rewritten as 2921: 375:Mathisson–Papapetrou–Dixon 216:Pseudo-Riemannian manifold 18: 10758: 10590: 10455: 10427: 10413:Lense–Thirring precession 10296: 10245: 10207: 10186: 10175: 10133: 10077: 10061: 10003: 9995:Doubly special relativity 9967: 9956: 9896: 9810: 9719: 9650: 9604: 9501: 9462:The Meaning of Relativity 9427: 9279: 9126:. John Wiley & Sons. 9122:Gravitation and Cosmology 9048:10.1007/s10714-006-0310-5 8899:10.1088/0264-9381/3/6/011 8733:10.1142/S0217751X02013113 8438:Rendall, Alan D. (2005). 8381:Weinberg, Steven (1993). 8161:Einstein, Albert (1916). 8094:Conformastatic spacetimes 4452:slow-motion approximation 4430:, which is linear in the 4175:A final contraction with 3241:where it is assumed that 2918:The cosmological constant 1100:Derivations of relativity 853:) with the local energy, 834:in 1915 in the form of a 819:) relate the geometry of 10273:Post-Newtonian formalism 10263:Einstein field equations 10199:Mathematical formulation 10023:Hyperbolic orthogonality 9853:Albert Einstein Archives 9783:Bernhard Caesar Einstein 9530:Relics: Einstein's Brain 9478:The Evolution of Physics 9359:Bose–Einstein statistics 9354:Bose–Einstein condensate 9334:Einstein field equations 8450:(1). Article number: 6. 8189:10.1002/andp.19163540702 8145: 4454:. In fact, the constant 4448:weak-field approximation 3141:This tensor describes a 3030:Einstein then abandoned 3000:the term containing the 1864:mathematical formulation 1402:{\displaystyle \Lambda } 1050:Einstein field equations 838:which related the local 809:Einstein field equations 380:Hamilton–Jacobi–Einstein 360:Einstein field equations 183:Mathematical formulation 48:{\displaystyle E=mc^{2}} 9984:Galilean transformation 9975:Principle of relativity 9863:Einstein Papers Project 9379:Einstein–de Haas effect 9088:Wheeler, John Archibald 8099:Einstein–Hilbert action 8081:Einstein-Hilbert action 7774:gravitational radiation 4478:Gauss's law for gravity 4444:Newton's law of gravity 3669:which is equivalent to 3157:and isotropic pressure 2547:Equivalent formulations 2407:classify themselves as 2403:With these definitions 1426:{\displaystyle \kappa } 1015:Lorentz transformations 823:to the distribution of 57:Mass–energy equivalence 10069:Lorentz transformation 9795:Thomas Martin Einstein 9703:Introducing Relativity 9675:Einstein for Beginners 9578:Einstein and Eddington 9369:Einstein–Cartan theory 8754:Brown, Harvey (2005). 8063: 7913: 7682: 7542: 7261: 7076: 6860:Schwarzschild solution 6842: 6765: 6715:vacuum field equations 6703: 6694:Vacuum field equations 6684: 6632: 6571: 6414: 6353: 6195: 5999: 5869: 5738: 5655: 5599: 5494:Our assumptions force 5486: 5314: 5231: 5165: 5068: 4921: 4829: 4695: 4574: 4424:Schrödinger's equation 4388: 4312: 4260: 4167: 4068: 3999: 3889: 3761: 3663: 3537: 3401: 3338: 3235: 3133: 2994: 2904: 2872: 2752: 2606: 2523: 2395: 2305: 1937: 1842: 1713: 1686: 1572: 1556:Ricci curvature tensor 1548: 1515: 1427: 1403: 1379: 1345: 1311: 1284: 1266: 866:electromagnetic fields 250:Gravitational redshift 143: 49: 10537:Weyl−Lewis−Papapetrou 10278:Raychaudhuri equation 10217:Equivalence principle 9868:Einstein refrigerator 9858:Einstein's Blackboard 9737:(second wife; cousin) 9668:Einstein and Religion 9617:Albert Einstein Medal 9612:Albert Einstein Award 9470:The World as I See It 9409:Wave–particle duality 9389:Bohr–Einstein debates 9349:Cosmological constant 9329:Equivalence principle 9319:Einstein coefficients 8216:(November 25, 1915). 8104:Equivalence principle 8064: 7914: 7764:approximates that of 7760:is very weak and the 7683: 7543: 7262: 7088:cosmological constant 7077: 6917:electromagnetic field 6886:, are referred to as 6843: 6766: 6701: 6685: 6633: 6572: 6415: 6354: 6196: 6000: 5870: 5739: 5656: 5600: 5487: 5320:where two factors of 5315: 5232: 5166: 5069: 4922: 4830: 4696: 4575: 4389: 4313: 4261: 4168: 4069: 4000: 3890: 3762: 3664: 3538: 3402: 3339: 3251:is defined as above. 3236: 3134: 3002:cosmological constant 2995: 2924:Cosmological constant 2905: 2873: 2753: 2607: 2524: 2396: 2306: 1938: 1843: 1714: 1687: 1573: 1549: 1516: 1428: 1411:cosmological constant 1404: 1380: 1346: 1312: 1278: 1267: 1110:Differential geometry 1010:Equivalence principle 922:. Special classes of 880:, the EFE relate the 538:Weyl−Lewis−Papapetrou 493:Kerr–Newman–de Sitter 313:Einstein–Rosen bridge 245:Gravitational lensing 201:Equivalence principle 144: 50: 10790:Equations of physics 10578:Numerical relativity 10419:pulsar timing arrays 9747:Hans Albert Einstein 9696:I Am Albert Einstein 9642:Einstein Prize (APS) 9314:Photoelectric effect 9287:Theory of relativity 9175:"Einstein equations" 9148:Cosmological Physics 8703:Int. J. Mod. Phys. A 8134:Numerical relativity 7927: 7789: 7586: 7550:covariant derivative 7277: 7097: 6930: 6888:Ricci-flat manifolds 6777: 6735: 6642: 6638:which will occur if 6581: 6426: 6369: 6207: 6009: 5881: 5748: 5667: 5609: 5508: 5346: 5245: 5175: 5080: 4938: 4841: 4707: 4592: 4483: 4324: 4274: 4183: 4078: 4011: 3899: 3778: 3673: 3551: 3425: 3357: 3274: 3170: 3061: 2932: 2884: 2771: 2626: 2559: 2430: 2317: 1965: 1917: 1752: 1696: 1596: 1562: 1528: 1447: 1417: 1393: 1387:stress–energy tensor 1359: 1325: 1291: 1208: 1088:Relevant mathematics 928:rotating black holes 859:stress–energy tensor 817:Einstein's equations 468:Einstein–Rosen waves 194:Fundamental concepts 83: 23: 10470:Friedmann equations 10364:Hulse–Taylor binary 10326:Gravitational waves 10222:Riemannian geometry 10048:Proper acceleration 10033:Maxwell's equations 9979:Galilean relativity 9399:Thought experiments 9040:2006GReGr..38.1233K 8987:2013PhRvD..88f3518K 8934:1997CQGra..14.2281L 8922:Class. Quantum Grav 8891:1986CQGra...3.1105H 8879:Class. Quantum Grav 8848:1969CMaPh..12..108E 8800:1977IJTP...16..561T 8757:Physical Relativity 8725:2002IJMPA..17S.180T 8605:. New York: Wiley. 8572:1990Natur.348..705E 8475:10.12942/lrr-2005-6 8466:2005LRR.....8....6R 8444:Living Rev. Relativ 8181:1916AnP...354..769E 7758:gravitational field 7569:exterior derivative 7554:anti-symmetrization 6468: 6405: 6345: 6327: 6306: 6291: 6270: 6243: 5379: 5309: 5005: 4927:for some constant, 4588:particle satisfies 4408:Maxwell's equations 3095: 2236: 2218: 2197: 2179: 2158: 2131: 957:Part of a series on 944:gravitational waves 878:Maxwell's equations 422:Kaluza–Klein theory 308:Minkowski spacetime 260:Gravitational waves 10795:General relativity 10519:Reissner–Nordström 10437:Brans–Dicke theory 10268:Linearized gravity 10095:Length contraction 10013:Frame of reference 9990:Special relativity 9848:Things named after 9710:Subtle is the Lord 9533:(1994 documentary) 9525:(1923 documentary) 9517:(1922 documentary) 9414:Gravitational wave 9297:General relativity 9292:Special relativity 9080:Misner, Charles W. 8856:10.1007/BF01645908 8808:10.1007/BF01811088 8641:(April 28, 1970). 8168:Annalen der Physik 8059: 7959: 7921:Levi-Civita symbol 7909: 7817: 7752:Linearized gravity 7746:The linearized EFE 7678: 7538: 7536: 7453: 7365: 7269:Additionally, the 7257: 7208: 7072: 7026: 6892:Einstein manifolds 6838: 6761: 6704: 6680: 6628: 6594: 6567: 6549: 6516: 6448: 6410: 6385: 6349: 6328: 6310: 6292: 6274: 6247: 6223: 6191: 6169: 6100: 6043: 5995: 5865: 5734: 5714: 5651: 5595: 5482: 5393: 5365: 5310: 5295: 5227: 5161: 5064: 4988: 4917: 4894: 4825: 4823: 4691: 4570: 4463: 4446:by using both the 4442:The EFE reduce to 4414:are linear in the 4384: 4308: 4256: 4218: 4163: 4119: 4064: 3995: 3885: 3757: 3659: 3533: 3397: 3348: 3334: 3245:has SI unit m and 3231: 3129: 3064: 2990: 2900: 2868: 2748: 2602: 2519: 2391: 2301: 2299: 2219: 2201: 2180: 2162: 2135: 2108: 1933: 1908:Einstein manifolds 1886:degrees of freedom 1879:Bianchi identities 1868:general relativity 1838: 1709: 1682: 1568: 1544: 1511: 1423: 1399: 1375: 1341: 1307: 1285: 1262: 1148:Physics portal 1125:Spacetime topology 1105:Spacetime diagrams 1033:General relativity 1005:Spacetime manifold 998:Spacetime concepts 987:General relativity 982:Special relativity 932:expanding universe 882:spacetime geometry 849:(expressed by the 751:Physics portal 523:Oppenheimer–Snyder 463:Reissner–Nordström 355:Linearized gravity 303:Spacetime diagrams 206:Special relativity 139: 68:General relativity 45: 10772: 10771: 10586: 10585: 10565:Ozsváth–Schücking 10171: 10170: 10153:Minkowski diagram 10110:Thomas precession 10053:Relativistic mass 9916: 9915: 9818:Awards and honors 9801:Siegbert Einstein 9689:Einstein's Cosmos 9107:978-0-7167-0344-0 9096:. San Francisco: 8782:Trautman, Andrzej 8767:978-0-19-927583-0 8295:978-0-387-69200-5 8053: 7958: 7816: 7452: 7364: 7207: 7150: 7025: 6968: 6872:with a vanishing 6819: 6810: 6674: 6593: 6548: 6515: 6168: 6099: 6042: 5960: 5713: 5392: 5287: 5215: 5191: 5136: 5108: 5058: 5031: 4980: 4933:geodesic equation 4893: 4792: 4664: 4635: 4611: 4606: 4556: 4513: 4461: 4428:quantum mechanics 4217: 4118: 3346: 3226: 3110: 2841: 2722: 2661: 2576: 2457: 1860:geodesic equation 1779: 1703: 1667: 1628: 1571:{\displaystyle R} 1490: 1279:EFE on a wall in 1199: 1198: 1063:Classical gravity 950:Mathematical form 938:, leading to the 902:geodesic equation 801: 800: 434: 433: 320: 319: 10807: 10764: 10763: 10547:van Stockum dust 10319:Two-body problem 10237:Mach's principle 10184: 10183: 10125:Terrell rotation 9965: 9964: 9943: 9936: 9929: 9920: 9919: 9904: 9903: 9803:(distant cousin) 9797:(great-grandson) 9759:Hermann Einstein 9741:Lieserl Einstein 9419:Tea leaf paradox 9266: 9259: 9252: 9243: 9242: 9188: 9163: 9151: 9142:Peacock, John A. 9137: 9125: 9116:Weinberg, Steven 9111: 9060: 9059: 9033: 9024:(8): 1233–1240. 9013: 9007: 9006: 8980: 8960: 8954: 8953: 8917: 8911: 8910: 8885:(6): 1105–1124. 8874: 8868: 8867: 8836:Comm. Math. Phys 8827: 8821: 8819: 8778: 8772: 8771: 8751: 8745: 8744: 8718: 8716:astro-ph/0202008 8698: 8692: 8691: 8675: 8669: 8668: 8666: 8665: 8635: 8629: 8623: 8617: 8616: 8598: 8592: 8591: 8580:10.1038/348705a0 8553: 8547: 8546: 8528: 8522: 8516: 8510: 8509:, p. 501ff. 8504: 8498: 8497: 8487: 8477: 8459: 8435: 8429: 8428: 8406: 8397: 8396: 8378: 8372: 8371: 8343: 8337: 8335: 8321: 8306: 8300: 8299: 8279: 8273: 8272: 8251: 8245: 8239: 8233: 8232: 8230: 8229: 8214:Einstein, Albert 8210: 8204: 8203: 8201: 8191:. Archived from 8158: 8078: 8068: 8066: 8065: 8060: 8054: 8052: 8038: 8037: 8036: 8024: 8023: 8011: 8010: 7998: 7997: 7979: 7978: 7960: 7951: 7947: 7942: 7941: 7918: 7916: 7915: 7910: 7908: 7907: 7895: 7894: 7882: 7881: 7869: 7868: 7856: 7855: 7837: 7836: 7818: 7809: 7770:Minkowski metric 7740:dynamical system 7687: 7685: 7684: 7679: 7677: 7676: 7658: 7657: 7639: 7638: 7620: 7619: 7601: 7600: 7581: 7566: 7547: 7545: 7544: 7539: 7537: 7527: 7523: 7522: 7521: 7500: 7499: 7478: 7477: 7454: 7445: 7439: 7435: 7434: 7433: 7412: 7411: 7390: 7389: 7366: 7357: 7347: 7346: 7308: 7307: 7299: 7298: 7297: 7266: 7264: 7263: 7258: 7253: 7249: 7248: 7247: 7235: 7234: 7222: 7221: 7209: 7200: 7194: 7193: 7188: 7187: 7186: 7175: 7174: 7169: 7168: 7167: 7151: 7149: 7148: 7136: 7131: 7130: 7112: 7111: 7092: 7081: 7079: 7078: 7073: 7071: 7067: 7066: 7065: 7053: 7052: 7040: 7039: 7027: 7018: 7012: 7011: 7006: 7005: 7004: 6993: 6992: 6987: 6986: 6985: 6969: 6967: 6966: 6954: 6945: 6944: 6914: 6885: 6852:vacuum solutions 6847: 6845: 6844: 6839: 6833: 6832: 6820: 6818: 6811: 6803: 6797: 6792: 6791: 6770: 6768: 6767: 6762: 6750: 6749: 6726: 6712: 6689: 6687: 6686: 6681: 6675: 6673: 6672: 6663: 6652: 6637: 6635: 6634: 6629: 6611: 6610: 6595: 6586: 6576: 6574: 6573: 6568: 6566: 6565: 6550: 6541: 6535: 6531: 6530: 6529: 6517: 6508: 6502: 6501: 6481: 6480: 6467: 6462: 6444: 6443: 6419: 6417: 6416: 6411: 6404: 6399: 6381: 6380: 6364: 6358: 6356: 6355: 6350: 6344: 6339: 6326: 6321: 6305: 6300: 6290: 6285: 6269: 6264: 6242: 6237: 6219: 6218: 6200: 6198: 6197: 6192: 6186: 6185: 6170: 6161: 6155: 6151: 6150: 6146: 6145: 6144: 6127: 6123: 6122: 6121: 6101: 6092: 6086: 6085: 6062: 6058: 6057: 6056: 6044: 6035: 6029: 6028: 6004: 6002: 6001: 5996: 5993: 5992: 5974: 5973: 5961: 5959: 5958: 5946: 5938: 5937: 5928: 5927: 5915: 5914: 5902: 5901: 5874: 5872: 5871: 5866: 5860: 5856: 5837: 5836: 5810: 5806: 5787: 5786: 5763: 5762: 5743: 5741: 5740: 5735: 5733: 5729: 5728: 5727: 5715: 5706: 5700: 5699: 5679: 5678: 5660: 5658: 5657: 5652: 5637: 5636: 5621: 5620: 5604: 5602: 5601: 5596: 5593: 5592: 5571: 5567: 5566: 5565: 5542: 5541: 5526: 5525: 5503: 5491: 5489: 5488: 5483: 5477: 5473: 5472: 5471: 5453: 5452: 5431: 5430: 5407: 5406: 5394: 5385: 5378: 5373: 5361: 5360: 5341: 5340: 5338: 5337: 5332: 5329: 5319: 5317: 5316: 5311: 5308: 5303: 5288: 5286: 5285: 5284: 5271: 5270: 5269: 5260: 5259: 5249: 5239:Minkowski metric 5236: 5234: 5233: 5228: 5220: 5216: 5214: 5206: 5198: 5192: 5190: 5179: 5170: 5168: 5167: 5162: 5160: 5156: 5137: 5135: 5127: 5119: 5109: 5107: 5099: 5098: 5097: 5084: 5073: 5071: 5070: 5065: 5059: 5057: 5049: 5048: 5047: 5034: 5032: 5030: 5022: 5021: 5020: 5007: 5004: 4999: 4981: 4979: 4978: 4977: 4964: 4963: 4962: 4953: 4952: 4942: 4930: 4926: 4924: 4923: 4918: 4916: 4912: 4911: 4910: 4895: 4886: 4880: 4879: 4856: 4855: 4834: 4832: 4831: 4826: 4824: 4816: 4815: 4793: 4791: 4790: 4789: 4776: 4775: 4774: 4765: 4764: 4754: 4729: 4728: 4700: 4698: 4697: 4692: 4686: 4682: 4666: 4665: 4657: 4637: 4636: 4628: 4613: 4612: 4607: 4599: 4597: 4583: 4579: 4577: 4576: 4571: 4569: 4565: 4558: 4557: 4549: 4526: 4522: 4515: 4514: 4506: 4495: 4494: 4475: 4468: 4457: 4412:electromagnetism 4393: 4391: 4390: 4385: 4377: 4376: 4368: 4367: 4366: 4349: 4348: 4336: 4335: 4317: 4315: 4314: 4309: 4301: 4300: 4292: 4291: 4290: 4265: 4263: 4262: 4257: 4249: 4248: 4240: 4236: 4232: 4231: 4219: 4210: 4204: 4203: 4178: 4172: 4170: 4169: 4164: 4156: 4155: 4147: 4143: 4139: 4138: 4133: 4132: 4131: 4120: 4111: 4105: 4104: 4099: 4098: 4097: 4073: 4071: 4070: 4065: 4057: 4056: 4045: 4044: 4043: 4026: 4025: 4004: 4002: 4001: 3996: 3988: 3987: 3973: 3972: 3971: 3954: 3953: 3942: 3941: 3940: 3926: 3925: 3914: 3913: 3912: 3894: 3892: 3891: 3886: 3878: 3874: 3873: 3872: 3855: 3854: 3853: 3839: 3838: 3817: 3816: 3793: 3792: 3766: 3764: 3763: 3758: 3750: 3749: 3732: 3731: 3730: 3716: 3715: 3694: 3693: 3668: 3666: 3665: 3660: 3652: 3651: 3634: 3633: 3632: 3618: 3617: 3600: 3599: 3598: 3584: 3583: 3566: 3565: 3564: 3542: 3540: 3539: 3534: 3526: 3525: 3508: 3507: 3506: 3492: 3491: 3474: 3473: 3472: 3458: 3457: 3440: 3439: 3438: 3420: 3410: 3406: 3404: 3403: 3398: 3390: 3389: 3350:Contracting the 3343: 3341: 3340: 3335: 3327: 3326: 3318: 3317: 3316: 3299: 3298: 3286: 3285: 3250: 3244: 3240: 3238: 3237: 3232: 3227: 3219: 3214: 3213: 3212: 3190: 3189: 3188: 3165: 3156: 3138: 3136: 3135: 3130: 3124: 3123: 3111: 3103: 3094: 3093: 3075: 3052: 3033: 3018:observations by 3006: 2999: 2997: 2996: 2991: 2985: 2984: 2966: 2965: 2947: 2946: 2912:Minkowski metric 2909: 2907: 2906: 2901: 2899: 2898: 2877: 2875: 2874: 2869: 2864: 2860: 2859: 2858: 2842: 2834: 2829: 2828: 2805: 2804: 2786: 2785: 2766: 2757: 2755: 2754: 2749: 2744: 2740: 2739: 2738: 2723: 2721: 2707: 2702: 2701: 2678: 2677: 2662: 2660: 2646: 2641: 2640: 2621: 2615: 2611: 2609: 2608: 2603: 2577: 2569: 2542: 2534: 2528: 2526: 2525: 2520: 2515: 2514: 2493: 2492: 2474: 2473: 2458: 2450: 2445: 2444: 2422: 2418: 2414: 2410: 2400: 2398: 2397: 2392: 2390: 2389: 2378: 2377: 2376: 2332: 2331: 2310: 2308: 2307: 2302: 2300: 2296: 2295: 2258: 2257: 2241: 2237: 2235: 2230: 2217: 2212: 2196: 2191: 2178: 2173: 2157: 2152: 2130: 2125: 2080: 2079: 2068: 2067: 2066: 1984: 1983: 1942: 1940: 1939: 1934: 1932: 1931: 1905: 1894: 1847: 1845: 1844: 1839: 1834: 1833: 1815: 1814: 1796: 1795: 1780: 1772: 1767: 1766: 1733: 1725: 1718: 1716: 1715: 1710: 1705: 1704: 1701: 1691: 1689: 1688: 1683: 1678: 1677: 1669: 1668: 1665: 1660: 1659: 1629: 1627: 1626: 1617: 1606: 1580:scalar curvature 1577: 1575: 1574: 1569: 1553: 1551: 1550: 1545: 1543: 1542: 1520: 1518: 1517: 1512: 1507: 1506: 1491: 1483: 1478: 1477: 1462: 1461: 1432: 1430: 1429: 1424: 1408: 1406: 1405: 1400: 1384: 1382: 1381: 1376: 1374: 1373: 1350: 1348: 1347: 1342: 1340: 1339: 1316: 1314: 1313: 1308: 1306: 1305: 1271: 1269: 1268: 1263: 1261: 1260: 1242: 1241: 1223: 1222: 1191: 1184: 1177: 1163: 1158: 1157: 1150: 1146: 1145: 1115:Curved spacetime 973: 954: 953: 847: 846: 815:; also known as 793: 786: 779: 766: 761: 760: 753: 749: 748: 533:van Stockum dust 518:Robertson–Walker 344: 343: 234: 233: 148: 146: 145: 140: 138: 137: 125: 117: 116: 98: 97: 78: 64: 63: 54: 52: 51: 46: 44: 43: 10815: 10814: 10810: 10809: 10808: 10806: 10805: 10804: 10785:Albert Einstein 10775: 10774: 10773: 10768: 10754: 10582: 10486:BKL singularity 10476:Lemaître–Tolman 10451: 10447:Quantum gravity 10429: 10423: 10409:geodetic effect 10383:(together with 10353:LISA Pathfinder 10292: 10241: 10227:Penrose diagram 10209: 10203: 10178: 10167: 10163:Minkowski space 10129: 10073: 10057: 10005: 9999: 9959: 9952: 9947: 9917: 9912: 9892: 9843:Religious views 9838:Political views 9806: 9791:(granddaughter) 9789:Evelyn Einstein 9777:Robert Einstein 9771:Eduard Einstein 9715: 9652: 9646: 9600: 9570:Einstein's Gift 9504: 9497: 9438:Annus mirabilis 9423: 9394:Teleparallelism 9339:Einstein radius 9309:Brownian motion 9275: 9273:Albert Einstein 9270: 9223: 9221:External images 9173: 9170: 9160: 9134: 9108: 9068: 9063: 9014: 9010: 8961: 8957: 8918: 8914: 8875: 8871: 8828: 8824: 8779: 8775: 8768: 8752: 8748: 8709:(S1): 180–196. 8699: 8695: 8676: 8672: 8663: 8661: 8659: 8636: 8632: 8624: 8620: 8613: 8599: 8595: 8554: 8550: 8543: 8529: 8525: 8519:Weinberg (1972) 8517: 8513: 8505: 8501: 8436: 8432: 8425: 8407: 8400: 8393: 8379: 8375: 8360: 8344: 8340: 8323: 8309: 8307: 8303: 8296: 8280: 8276: 8269: 8252: 8248: 8244:, p. 916 . 8240: 8236: 8227: 8225: 8211: 8207: 8195: 8159: 8152: 8148: 8143: 8089: 8072: 8039: 8029: 8025: 8016: 8012: 8003: 7999: 7984: 7980: 7965: 7961: 7949: 7948: 7946: 7934: 7930: 7928: 7925: 7924: 7900: 7896: 7887: 7883: 7874: 7870: 7861: 7857: 7842: 7838: 7823: 7819: 7807: 7790: 7787: 7786: 7782: 7780:Polynomial form 7766:Minkowski space 7754: 7748: 7717:exact solutions 7701: 7695: 7666: 7662: 7647: 7643: 7628: 7624: 7609: 7605: 7593: 7589: 7587: 7584: 7583: 7580: 7576: 7564: 7535: 7534: 7508: 7504: 7486: 7482: 7464: 7460: 7459: 7455: 7443: 7420: 7416: 7398: 7394: 7376: 7372: 7371: 7367: 7355: 7348: 7327: 7323: 7320: 7319: 7309: 7300: 7290: 7286: 7285: 7284: 7280: 7278: 7275: 7274: 7240: 7236: 7227: 7223: 7214: 7210: 7198: 7189: 7182: 7178: 7177: 7176: 7170: 7163: 7159: 7158: 7157: 7156: 7152: 7144: 7140: 7135: 7123: 7119: 7104: 7100: 7098: 7095: 7094: 7090: 7058: 7054: 7045: 7041: 7032: 7028: 7016: 7007: 7000: 6996: 6995: 6994: 6988: 6981: 6977: 6976: 6975: 6974: 6970: 6962: 6958: 6953: 6937: 6933: 6931: 6928: 6927: 6913: 6909: 6906: 6900: 6882: 6877: 6856:Minkowski space 6825: 6821: 6802: 6801: 6796: 6784: 6780: 6778: 6775: 6774: 6742: 6738: 6736: 6733: 6732: 6723: 6718: 6711: 6707: 6696: 6691: 6668: 6664: 6653: 6651: 6643: 6640: 6639: 6606: 6602: 6584: 6582: 6579: 6578: 6561: 6557: 6539: 6525: 6521: 6506: 6497: 6493: 6492: 6488: 6476: 6472: 6463: 6452: 6433: 6429: 6427: 6424: 6423: 6400: 6389: 6376: 6372: 6370: 6367: 6366: 6362: 6340: 6332: 6322: 6314: 6301: 6296: 6286: 6278: 6265: 6251: 6238: 6227: 6214: 6210: 6208: 6205: 6204: 6181: 6177: 6159: 6140: 6136: 6132: 6128: 6117: 6113: 6106: 6102: 6090: 6081: 6077: 6073: 6069: 6052: 6048: 6033: 6024: 6020: 6019: 6015: 6010: 6007: 6006: 5988: 5984: 5969: 5965: 5954: 5950: 5945: 5933: 5929: 5923: 5919: 5907: 5903: 5894: 5890: 5882: 5879: 5878: 5832: 5828: 5824: 5820: 5782: 5778: 5777: 5773: 5755: 5751: 5749: 5746: 5745: 5723: 5719: 5704: 5695: 5691: 5690: 5686: 5674: 5670: 5668: 5665: 5664: 5632: 5628: 5616: 5612: 5610: 5607: 5606: 5582: 5578: 5555: 5551: 5547: 5543: 5534: 5530: 5518: 5514: 5509: 5506: 5505: 5495: 5461: 5457: 5439: 5435: 5417: 5413: 5412: 5408: 5399: 5395: 5383: 5374: 5369: 5353: 5349: 5347: 5344: 5343: 5333: 5330: 5325: 5324: 5322: 5321: 5304: 5299: 5280: 5276: 5272: 5265: 5261: 5255: 5251: 5250: 5248: 5246: 5243: 5242: 5207: 5199: 5197: 5193: 5183: 5178: 5176: 5173: 5172: 5128: 5120: 5118: 5117: 5113: 5100: 5093: 5089: 5085: 5083: 5081: 5078: 5077: 5050: 5043: 5039: 5035: 5033: 5023: 5016: 5012: 5008: 5006: 5000: 4992: 4973: 4969: 4965: 4958: 4954: 4948: 4944: 4943: 4941: 4939: 4936: 4935: 4928: 4903: 4899: 4884: 4872: 4868: 4867: 4863: 4848: 4844: 4842: 4839: 4838: 4822: 4821: 4808: 4804: 4794: 4785: 4781: 4777: 4770: 4766: 4760: 4756: 4755: 4753: 4750: 4749: 4730: 4718: 4714: 4710: 4708: 4705: 4704: 4656: 4655: 4654: 4650: 4627: 4626: 4598: 4596: 4595: 4593: 4590: 4589: 4581: 4548: 4547: 4546: 4542: 4505: 4504: 4503: 4499: 4490: 4486: 4484: 4481: 4480: 4470: 4466: 4455: 4440: 4420:magnetic fields 4404: 4395: 4369: 4359: 4355: 4354: 4353: 4341: 4337: 4331: 4327: 4325: 4322: 4321: 4293: 4283: 4279: 4278: 4277: 4275: 4272: 4271: 4268:Einstein tensor 4241: 4224: 4220: 4208: 4196: 4192: 4191: 4187: 4186: 4184: 4181: 4180: 4176: 4148: 4134: 4127: 4123: 4122: 4121: 4109: 4100: 4093: 4089: 4088: 4087: 4086: 4082: 4081: 4079: 4076: 4075: 4046: 4039: 4035: 4034: 4033: 4018: 4014: 4012: 4009: 4008: 3974: 3964: 3960: 3959: 3958: 3943: 3936: 3932: 3931: 3930: 3915: 3908: 3904: 3903: 3902: 3900: 3897: 3896: 3856: 3849: 3845: 3844: 3843: 3825: 3821: 3803: 3799: 3798: 3794: 3785: 3781: 3779: 3776: 3775: 3733: 3726: 3722: 3721: 3720: 3702: 3698: 3680: 3676: 3674: 3671: 3670: 3635: 3628: 3624: 3623: 3622: 3601: 3594: 3590: 3589: 3588: 3567: 3560: 3556: 3555: 3554: 3552: 3549: 3548: 3509: 3502: 3498: 3497: 3496: 3475: 3468: 3464: 3463: 3462: 3441: 3434: 3430: 3429: 3428: 3426: 3423: 3422: 3417: 3412: 3408: 3364: 3360: 3358: 3355: 3354: 3319: 3309: 3305: 3304: 3303: 3291: 3287: 3281: 3277: 3275: 3272: 3271: 3266: 3261: 3246: 3242: 3218: 3202: 3201: 3197: 3178: 3177: 3173: 3171: 3168: 3167: 3164: 3158: 3155: 3149: 3116: 3112: 3102: 3077: 3076: 3068: 3062: 3059: 3058: 3050: 3034:, remarking to 3031: 3004: 2977: 2973: 2958: 2954: 2939: 2935: 2933: 2930: 2929: 2926: 2920: 2891: 2887: 2885: 2882: 2881: 2851: 2847: 2833: 2821: 2817: 2816: 2812: 2797: 2793: 2778: 2774: 2772: 2769: 2768: 2761: 2731: 2727: 2711: 2706: 2694: 2690: 2689: 2685: 2670: 2666: 2650: 2645: 2633: 2629: 2627: 2624: 2623: 2617: 2613: 2568: 2560: 2557: 2556: 2549: 2540: 2537:sign convention 2532: 2507: 2503: 2485: 2481: 2466: 2462: 2449: 2437: 2433: 2431: 2428: 2427: 2420: 2416: 2412: 2408: 2379: 2372: 2368: 2367: 2366: 2324: 2320: 2318: 2315: 2314: 2298: 2297: 2288: 2284: 2259: 2250: 2246: 2243: 2242: 2231: 2223: 2213: 2205: 2192: 2184: 2174: 2166: 2153: 2139: 2126: 2112: 2107: 2103: 2081: 2069: 2062: 2058: 2057: 2056: 2053: 2052: 1985: 1976: 1972: 1968: 1966: 1963: 1962: 1953: 1951:Sign convention 1924: 1920: 1918: 1915: 1914: 1904: 1896: 1892: 1826: 1822: 1807: 1803: 1788: 1784: 1771: 1759: 1755: 1753: 1750: 1749: 1731: 1723: 1700: 1699: 1697: 1694: 1693: 1670: 1664: 1663: 1662: 1652: 1648: 1622: 1618: 1607: 1605: 1597: 1594: 1593: 1563: 1560: 1559: 1535: 1531: 1529: 1526: 1525: 1499: 1495: 1482: 1470: 1466: 1454: 1450: 1448: 1445: 1444: 1438:Einstein tensor 1418: 1415: 1414: 1394: 1391: 1390: 1366: 1362: 1360: 1357: 1356: 1332: 1328: 1326: 1323: 1322: 1319:Einstein tensor 1298: 1294: 1292: 1289: 1288: 1253: 1249: 1234: 1230: 1215: 1211: 1209: 1206: 1205: 1195: 1166: 1153: 1140: 1139: 1131: 1130: 1129: 1089: 1081: 1080: 1079: 1064: 1056: 1055: 1054: 1034: 1026: 1025: 1024: 1020:Minkowski space 999: 991: 952: 924:exact solutions 851:Einstein tensor 841: 840: 836:tensor equation 832:Albert Einstein 797: 756: 743: 742: 735: 734: 558: 557: 548: 547: 503:Lemaître–Tolman 448: 447: 436: 435: 427:Quantum gravity 414:Advanced theory 341: 340: 339: 322: 321: 270:Geodetic effect 231: 230: 221: 220: 196: 195: 179: 149: 130: 126: 121: 109: 105: 90: 86: 84: 81: 80: 60: 39: 35: 24: 21: 20: 17: 12: 11: 5: 10813: 10803: 10802: 10797: 10792: 10787: 10770: 10769: 10759: 10756: 10755: 10753: 10752: 10745: 10740: 10735: 10730: 10725: 10720: 10715: 10710: 10705: 10700: 10695: 10690: 10685: 10680: 10675: 10673:Choquet-Bruhat 10670: 10665: 10660: 10655: 10650: 10645: 10640: 10635: 10630: 10625: 10620: 10615: 10610: 10605: 10600: 10594: 10592: 10588: 10587: 10584: 10583: 10581: 10580: 10573: 10572: 10567: 10562: 10555: 10554: 10549: 10544: 10539: 10534: 10525:Axisymmetric: 10522: 10521: 10516: 10510: 10499: 10498: 10493: 10488: 10483: 10478: 10473: 10464:Cosmological: 10461: 10459: 10453: 10452: 10450: 10449: 10444: 10439: 10433: 10431: 10425: 10424: 10422: 10421: 10416: 10405:frame-dragging 10402: 10397: 10392: 10389:Einstein rings 10385:Einstein cross 10378: 10367: 10366: 10361: 10355: 10350: 10345: 10332: 10322: 10321: 10316: 10311: 10306: 10300: 10298: 10294: 10293: 10291: 10290: 10288:Ernst equation 10285: 10280: 10275: 10270: 10265: 10260: 10258:BSSN formalism 10255: 10249: 10247: 10243: 10242: 10240: 10239: 10234: 10229: 10224: 10219: 10213: 10211: 10205: 10204: 10202: 10201: 10196: 10190: 10188: 10181: 10173: 10172: 10169: 10168: 10166: 10165: 10160: 10155: 10150: 10145: 10139: 10137: 10131: 10130: 10128: 10127: 10122: 10117: 10115:Ladder paradox 10112: 10107: 10102: 10097: 10092: 10087: 10081: 10079: 10075: 10074: 10072: 10071: 10065: 10063: 10059: 10058: 10056: 10055: 10050: 10045: 10040: 10035: 10030: 10025: 10020: 10018:Speed of light 10015: 10009: 10007: 10001: 10000: 9998: 9997: 9992: 9987: 9981: 9971: 9969: 9962: 9954: 9953: 9946: 9945: 9938: 9931: 9923: 9914: 9913: 9911: 9910: 9897: 9894: 9893: 9891: 9890: 9885: 9880: 9875: 9870: 9865: 9860: 9855: 9850: 9845: 9840: 9835: 9830: 9825: 9820: 9814: 9812: 9808: 9807: 9805: 9804: 9798: 9792: 9786: 9780: 9774: 9768: 9762: 9756: 9750: 9744: 9738: 9732: 9725: 9723: 9717: 9716: 9714: 9713: 9706: 9699: 9692: 9685: 9678: 9671: 9664: 9656: 9654: 9648: 9647: 9645: 9644: 9639: 9634: 9629: 9624: 9619: 9614: 9608: 9606: 9602: 9601: 9599: 9598: 9590: 9582: 9581:(2008 TV film) 9574: 9566: 9558: 9550: 9546:Young Einstein 9542: 9538:Insignificance 9534: 9526: 9518: 9509: 9507: 9499: 9498: 9496: 9495: 9489: 9486:Why Socialism? 9482: 9474: 9466: 9458: 9450: 9443: 9433: 9431: 9425: 9424: 9422: 9421: 9416: 9411: 9406: 9401: 9396: 9391: 9386: 9381: 9376: 9371: 9366: 9361: 9356: 9351: 9346: 9341: 9336: 9331: 9326: 9324:Einstein solid 9321: 9316: 9311: 9306: 9301: 9300: 9299: 9294: 9283: 9281: 9277: 9276: 9269: 9268: 9261: 9254: 9246: 9240: 9239: 9229: 9222: 9219: 9218: 9217: 9211: 9201: 9195: 9189: 9169: 9168:External links 9166: 9165: 9164: 9159:978-0521410724 9158: 9138: 9132: 9112: 9106: 9084:Thorne, Kip S. 9067: 9064: 9062: 9061: 9018:Gen. Rel. Grav 9008: 8955: 8912: 8869: 8842:(2): 108–141. 8822: 8794:(9): 561–565. 8773: 8766: 8746: 8693: 8690:on 2007-03-07. 8670: 8657: 8630: 8626:Peacock (1999) 8618: 8611: 8593: 8548: 8541: 8523: 8511: 8499: 8430: 8423: 8398: 8391: 8373: 8358: 8338: 8301: 8294: 8274: 8267: 8246: 8234: 8205: 8202:on 2012-02-06. 8149: 8147: 8144: 8142: 8141: 8139:Ricci calculus 8136: 8131: 8126: 8121: 8116: 8111: 8106: 8101: 8096: 8090: 8088: 8085: 8058: 8051: 8048: 8045: 8042: 8035: 8032: 8028: 8022: 8019: 8015: 8009: 8006: 8002: 7996: 7993: 7990: 7987: 7983: 7977: 7974: 7971: 7968: 7964: 7957: 7954: 7945: 7940: 7937: 7933: 7906: 7903: 7899: 7893: 7890: 7886: 7880: 7877: 7873: 7867: 7864: 7860: 7854: 7851: 7848: 7845: 7841: 7835: 7832: 7829: 7826: 7822: 7815: 7812: 7806: 7803: 7800: 7797: 7794: 7781: 7778: 7750:Main article: 7747: 7744: 7697:Main article: 7694: 7691: 7675: 7672: 7669: 7665: 7661: 7656: 7653: 7650: 7646: 7642: 7637: 7634: 7631: 7627: 7623: 7618: 7615: 7612: 7608: 7604: 7599: 7596: 7592: 7578: 7573:Poincaré lemma 7533: 7530: 7526: 7520: 7517: 7514: 7511: 7507: 7503: 7498: 7495: 7492: 7489: 7485: 7481: 7476: 7473: 7470: 7467: 7463: 7458: 7451: 7448: 7442: 7438: 7432: 7429: 7426: 7423: 7419: 7415: 7410: 7407: 7404: 7401: 7397: 7393: 7388: 7385: 7382: 7379: 7375: 7370: 7363: 7360: 7354: 7351: 7349: 7345: 7342: 7339: 7336: 7333: 7330: 7326: 7322: 7321: 7318: 7315: 7312: 7310: 7306: 7303: 7296: 7293: 7289: 7283: 7282: 7256: 7252: 7246: 7243: 7239: 7233: 7230: 7226: 7220: 7217: 7213: 7206: 7203: 7197: 7192: 7185: 7181: 7173: 7166: 7162: 7155: 7147: 7143: 7139: 7134: 7129: 7126: 7122: 7118: 7115: 7110: 7107: 7103: 7070: 7064: 7061: 7057: 7051: 7048: 7044: 7038: 7035: 7031: 7024: 7021: 7015: 7010: 7003: 6999: 6991: 6984: 6980: 6973: 6965: 6961: 6957: 6952: 6948: 6943: 6940: 6936: 6923:, i.e. if the 6915:is that of an 6911: 6899: 6896: 6880: 6837: 6831: 6828: 6824: 6817: 6814: 6809: 6806: 6800: 6795: 6790: 6787: 6783: 6760: 6756: 6753: 6748: 6745: 6741: 6721: 6709: 6695: 6692: 6679: 6671: 6667: 6662: 6659: 6656: 6650: 6647: 6626: 6623: 6620: 6617: 6614: 6609: 6605: 6601: 6598: 6592: 6589: 6564: 6560: 6556: 6553: 6547: 6544: 6538: 6534: 6528: 6524: 6520: 6514: 6511: 6505: 6500: 6496: 6491: 6487: 6484: 6479: 6475: 6471: 6466: 6461: 6458: 6455: 6451: 6447: 6442: 6439: 6436: 6432: 6409: 6403: 6398: 6395: 6392: 6388: 6384: 6379: 6375: 6348: 6343: 6338: 6335: 6331: 6325: 6320: 6317: 6313: 6309: 6304: 6299: 6295: 6289: 6284: 6281: 6277: 6273: 6268: 6263: 6260: 6257: 6254: 6250: 6246: 6241: 6236: 6233: 6230: 6226: 6222: 6217: 6213: 6190: 6184: 6180: 6176: 6173: 6167: 6164: 6158: 6154: 6149: 6143: 6139: 6135: 6131: 6126: 6120: 6116: 6112: 6109: 6105: 6098: 6095: 6089: 6084: 6080: 6076: 6072: 6068: 6065: 6061: 6055: 6051: 6047: 6041: 6038: 6032: 6027: 6023: 6018: 6014: 5991: 5987: 5983: 5980: 5977: 5972: 5968: 5964: 5957: 5953: 5949: 5944: 5941: 5936: 5932: 5926: 5922: 5918: 5913: 5910: 5906: 5900: 5897: 5893: 5889: 5886: 5864: 5859: 5855: 5852: 5849: 5846: 5843: 5840: 5835: 5831: 5827: 5823: 5819: 5816: 5813: 5809: 5805: 5802: 5799: 5796: 5793: 5790: 5785: 5781: 5776: 5772: 5769: 5766: 5761: 5758: 5754: 5732: 5726: 5722: 5718: 5712: 5709: 5703: 5698: 5694: 5689: 5685: 5682: 5677: 5673: 5650: 5646: 5643: 5640: 5635: 5631: 5627: 5624: 5619: 5615: 5591: 5588: 5585: 5581: 5577: 5574: 5570: 5564: 5561: 5558: 5554: 5550: 5546: 5540: 5537: 5533: 5529: 5524: 5521: 5517: 5513: 5481: 5476: 5470: 5467: 5464: 5460: 5456: 5451: 5448: 5445: 5442: 5438: 5434: 5429: 5426: 5423: 5420: 5416: 5411: 5405: 5402: 5398: 5391: 5388: 5382: 5377: 5372: 5368: 5364: 5359: 5356: 5352: 5307: 5302: 5298: 5294: 5291: 5283: 5279: 5275: 5268: 5264: 5258: 5254: 5226: 5223: 5219: 5213: 5210: 5205: 5202: 5196: 5189: 5186: 5182: 5159: 5155: 5152: 5149: 5146: 5143: 5140: 5134: 5131: 5126: 5123: 5116: 5112: 5106: 5103: 5096: 5092: 5088: 5063: 5056: 5053: 5046: 5042: 5038: 5029: 5026: 5019: 5015: 5011: 5003: 4998: 4995: 4991: 4987: 4984: 4976: 4972: 4968: 4961: 4957: 4951: 4947: 4915: 4909: 4906: 4902: 4898: 4892: 4889: 4883: 4878: 4875: 4871: 4866: 4862: 4859: 4854: 4851: 4847: 4820: 4814: 4811: 4807: 4803: 4800: 4797: 4795: 4788: 4784: 4780: 4773: 4769: 4763: 4759: 4752: 4751: 4748: 4745: 4742: 4739: 4736: 4733: 4731: 4727: 4724: 4721: 4717: 4713: 4712: 4690: 4685: 4681: 4678: 4675: 4672: 4669: 4663: 4660: 4653: 4649: 4646: 4643: 4640: 4634: 4631: 4625: 4622: 4619: 4616: 4610: 4605: 4602: 4568: 4564: 4561: 4555: 4552: 4545: 4541: 4538: 4535: 4532: 4529: 4525: 4521: 4518: 4512: 4509: 4502: 4498: 4493: 4489: 4460: 4439: 4436: 4403: 4400: 4383: 4380: 4375: 4372: 4365: 4362: 4358: 4352: 4347: 4344: 4340: 4334: 4330: 4307: 4304: 4299: 4296: 4289: 4286: 4282: 4255: 4252: 4247: 4244: 4239: 4235: 4230: 4227: 4223: 4216: 4213: 4207: 4202: 4199: 4195: 4190: 4162: 4159: 4154: 4151: 4146: 4142: 4137: 4130: 4126: 4117: 4114: 4108: 4103: 4096: 4092: 4085: 4063: 4060: 4055: 4052: 4049: 4042: 4038: 4032: 4029: 4024: 4021: 4017: 3994: 3991: 3986: 3983: 3980: 3977: 3970: 3967: 3963: 3957: 3952: 3949: 3946: 3939: 3935: 3929: 3924: 3921: 3918: 3911: 3907: 3884: 3881: 3877: 3871: 3868: 3865: 3862: 3859: 3852: 3848: 3842: 3837: 3834: 3831: 3828: 3824: 3820: 3815: 3812: 3809: 3806: 3802: 3797: 3791: 3788: 3784: 3756: 3753: 3748: 3745: 3742: 3739: 3736: 3729: 3725: 3719: 3714: 3711: 3708: 3705: 3701: 3697: 3692: 3689: 3686: 3683: 3679: 3658: 3655: 3650: 3647: 3644: 3641: 3638: 3631: 3627: 3621: 3616: 3613: 3610: 3607: 3604: 3597: 3593: 3587: 3582: 3579: 3576: 3573: 3570: 3563: 3559: 3532: 3529: 3524: 3521: 3518: 3515: 3512: 3505: 3501: 3495: 3490: 3487: 3484: 3481: 3478: 3471: 3467: 3461: 3456: 3453: 3450: 3447: 3444: 3437: 3433: 3415: 3396: 3393: 3388: 3385: 3382: 3379: 3376: 3373: 3370: 3367: 3363: 3345: 3333: 3330: 3325: 3322: 3315: 3312: 3308: 3302: 3297: 3294: 3290: 3284: 3280: 3265: 3262: 3260: 3257: 3230: 3225: 3222: 3217: 3211: 3208: 3205: 3200: 3196: 3193: 3187: 3184: 3181: 3176: 3162: 3153: 3147:energy density 3128: 3122: 3119: 3115: 3109: 3106: 3101: 3098: 3092: 3089: 3086: 3083: 3080: 3074: 3071: 3067: 3028: 3027: 3016: 2989: 2983: 2980: 2976: 2972: 2969: 2964: 2961: 2957: 2953: 2950: 2945: 2942: 2938: 2922:Main article: 2919: 2916: 2897: 2894: 2890: 2867: 2863: 2857: 2854: 2850: 2845: 2840: 2837: 2832: 2827: 2824: 2820: 2815: 2811: 2808: 2803: 2800: 2796: 2792: 2789: 2784: 2781: 2777: 2747: 2743: 2737: 2734: 2730: 2726: 2720: 2717: 2714: 2710: 2705: 2700: 2697: 2693: 2688: 2684: 2681: 2676: 2673: 2669: 2665: 2659: 2656: 2653: 2649: 2644: 2639: 2636: 2632: 2601: 2598: 2595: 2592: 2589: 2586: 2583: 2580: 2575: 2572: 2567: 2564: 2548: 2545: 2518: 2513: 2510: 2506: 2502: 2499: 2496: 2491: 2488: 2484: 2480: 2477: 2472: 2469: 2465: 2461: 2456: 2453: 2448: 2443: 2440: 2436: 2388: 2385: 2382: 2375: 2371: 2365: 2362: 2359: 2356: 2353: 2350: 2347: 2344: 2341: 2338: 2335: 2330: 2327: 2323: 2294: 2291: 2287: 2283: 2280: 2277: 2274: 2271: 2268: 2265: 2262: 2260: 2256: 2253: 2249: 2245: 2244: 2240: 2234: 2229: 2226: 2222: 2216: 2211: 2208: 2204: 2200: 2195: 2190: 2187: 2183: 2177: 2172: 2169: 2165: 2161: 2156: 2151: 2148: 2145: 2142: 2138: 2134: 2129: 2124: 2121: 2118: 2115: 2111: 2106: 2102: 2099: 2096: 2093: 2090: 2087: 2084: 2082: 2078: 2075: 2072: 2065: 2061: 2055: 2054: 2051: 2048: 2045: 2042: 2039: 2036: 2033: 2030: 2027: 2024: 2021: 2018: 2015: 2012: 2009: 2006: 2003: 2000: 1997: 1994: 1991: 1988: 1986: 1982: 1979: 1975: 1971: 1970: 1952: 1949: 1930: 1927: 1923: 1900: 1849: 1848: 1837: 1832: 1829: 1825: 1821: 1818: 1813: 1810: 1806: 1802: 1799: 1794: 1791: 1787: 1783: 1778: 1775: 1770: 1765: 1762: 1758: 1736:speed of light 1720: 1719: 1708: 1681: 1676: 1673: 1658: 1655: 1651: 1647: 1644: 1641: 1638: 1635: 1632: 1625: 1621: 1616: 1613: 1610: 1604: 1601: 1589:is defined as 1567: 1541: 1538: 1534: 1522: 1521: 1510: 1505: 1502: 1498: 1494: 1489: 1486: 1481: 1476: 1473: 1469: 1465: 1460: 1457: 1453: 1440:is defined as 1422: 1398: 1372: 1369: 1365: 1338: 1335: 1331: 1304: 1301: 1297: 1273: 1272: 1259: 1256: 1252: 1248: 1245: 1240: 1237: 1233: 1229: 1226: 1221: 1218: 1214: 1197: 1196: 1194: 1193: 1186: 1179: 1171: 1168: 1167: 1165: 1164: 1151: 1136: 1133: 1132: 1128: 1127: 1122: 1117: 1112: 1107: 1102: 1097: 1091: 1090: 1087: 1086: 1083: 1082: 1078: 1077: 1072: 1066: 1065: 1062: 1061: 1058: 1057: 1053: 1052: 1047: 1042: 1036: 1035: 1032: 1031: 1028: 1027: 1023: 1022: 1017: 1012: 1007: 1001: 1000: 997: 996: 993: 992: 990: 989: 984: 978: 975: 974: 966: 965: 959: 958: 951: 948: 940:linearized EFE 936:flat spacetime 913:speed of light 799: 798: 796: 795: 788: 781: 773: 770: 769: 768: 767: 754: 737: 736: 733: 732: 725: 720: 715: 710: 705: 700: 695: 690: 685: 680: 675: 670: 665: 660: 655: 650: 645: 640: 635: 630: 625: 620: 615: 610: 605: 600: 595: 590: 585: 580: 575: 570: 565: 559: 555: 554: 553: 550: 549: 546: 545: 540: 535: 530: 525: 520: 515: 510: 505: 500: 495: 490: 485: 480: 475: 470: 465: 460: 449: 443: 442: 441: 438: 437: 432: 431: 430: 429: 424: 416: 415: 411: 410: 409: 408: 406:Post-Newtonian 403: 398: 390: 389: 385: 384: 383: 382: 377: 372: 367: 362: 357: 349: 348: 342: 338: 337: 334: 330: 329: 328: 327: 324: 323: 318: 317: 316: 315: 310: 305: 297: 296: 290: 289: 288: 287: 282: 277: 272: 267: 265:Frame-dragging 262: 257: 252: 247: 242: 240:Kepler problem 232: 228: 227: 226: 223: 222: 219: 218: 213: 208: 203: 197: 193: 192: 191: 188: 187: 186: 185: 180: 178: 177: 172: 167: 161: 159: 151: 150: 136: 133: 129: 124: 120: 115: 112: 108: 104: 101: 96: 93: 89: 79: 71: 70: 42: 38: 34: 31: 28: 15: 9: 6: 4: 3: 2: 10812: 10801: 10798: 10796: 10793: 10791: 10788: 10786: 10783: 10782: 10780: 10767: 10757: 10751: 10750: 10746: 10744: 10741: 10739: 10736: 10734: 10731: 10729: 10726: 10724: 10721: 10719: 10716: 10714: 10711: 10709: 10706: 10704: 10701: 10699: 10696: 10694: 10691: 10689: 10686: 10684: 10681: 10679: 10676: 10674: 10671: 10669: 10666: 10664: 10661: 10659: 10658:Chandrasekhar 10656: 10654: 10651: 10649: 10646: 10644: 10641: 10639: 10636: 10634: 10631: 10629: 10626: 10624: 10621: 10619: 10618:Schwarzschild 10616: 10614: 10611: 10609: 10606: 10604: 10601: 10599: 10596: 10595: 10593: 10589: 10579: 10575: 10574: 10571: 10568: 10566: 10563: 10561: 10557: 10556: 10553: 10550: 10548: 10545: 10543: 10540: 10538: 10535: 10532: 10528: 10524: 10523: 10520: 10517: 10514: 10511: 10509: 10505: 10504:Schwarzschild 10501: 10500: 10497: 10494: 10492: 10489: 10487: 10484: 10482: 10479: 10477: 10474: 10471: 10467: 10463: 10462: 10460: 10458: 10454: 10448: 10445: 10443: 10440: 10438: 10435: 10434: 10432: 10426: 10420: 10417: 10414: 10410: 10406: 10403: 10401: 10400:Shapiro delay 10398: 10396: 10393: 10390: 10386: 10382: 10379: 10376: 10372: 10369: 10368: 10365: 10362: 10359: 10356: 10354: 10351: 10349: 10346: 10344: 10343:collaboration 10340: 10336: 10333: 10331: 10327: 10324: 10323: 10320: 10317: 10315: 10312: 10310: 10309:Event horizon 10307: 10305: 10302: 10301: 10299: 10295: 10289: 10286: 10284: 10281: 10279: 10276: 10274: 10271: 10269: 10266: 10264: 10261: 10259: 10256: 10254: 10253:ADM formalism 10251: 10250: 10248: 10244: 10238: 10235: 10233: 10230: 10228: 10225: 10223: 10220: 10218: 10215: 10214: 10212: 10206: 10200: 10197: 10195: 10192: 10191: 10189: 10185: 10182: 10180: 10174: 10164: 10161: 10159: 10158:Biquaternions 10156: 10154: 10151: 10149: 10146: 10144: 10141: 10140: 10138: 10136: 10132: 10126: 10123: 10121: 10118: 10116: 10113: 10111: 10108: 10106: 10103: 10101: 10098: 10096: 10093: 10091: 10088: 10086: 10085:Time dilation 10083: 10082: 10080: 10076: 10070: 10067: 10066: 10064: 10060: 10054: 10051: 10049: 10046: 10044: 10041: 10039: 10038:Proper length 10036: 10034: 10031: 10029: 10026: 10024: 10021: 10019: 10016: 10014: 10011: 10010: 10008: 10002: 9996: 9993: 9991: 9988: 9985: 9982: 9980: 9976: 9973: 9972: 9970: 9966: 9963: 9961: 9955: 9951: 9944: 9939: 9937: 9932: 9930: 9925: 9924: 9921: 9909: 9908: 9899: 9898: 9895: 9889: 9886: 9884: 9881: 9879: 9876: 9874: 9871: 9869: 9866: 9864: 9861: 9859: 9856: 9854: 9851: 9849: 9846: 9844: 9841: 9839: 9836: 9834: 9831: 9829: 9826: 9824: 9821: 9819: 9816: 9815: 9813: 9809: 9802: 9799: 9796: 9793: 9790: 9787: 9784: 9781: 9778: 9775: 9772: 9769: 9766: 9765:Maja Einstein 9763: 9760: 9757: 9754: 9751: 9748: 9745: 9742: 9739: 9736: 9735:Elsa Einstein 9733: 9730: 9727: 9726: 9724: 9722: 9718: 9712: 9711: 9707: 9705: 9704: 9700: 9698: 9697: 9693: 9691: 9690: 9686: 9684: 9683: 9679: 9677: 9676: 9672: 9670: 9669: 9665: 9663: 9662: 9658: 9657: 9655: 9649: 9643: 9640: 9638: 9635: 9633: 9630: 9628: 9625: 9623: 9622:Kalinga Prize 9620: 9618: 9615: 9613: 9610: 9609: 9607: 9603: 9596: 9595: 9591: 9589:(2017 series) 9588: 9587: 9583: 9580: 9579: 9575: 9572: 9571: 9567: 9564: 9563: 9559: 9556: 9555: 9551: 9548: 9547: 9543: 9540: 9539: 9535: 9532: 9531: 9527: 9524: 9523: 9519: 9516: 9515: 9511: 9510: 9508: 9506: 9500: 9493: 9490: 9487: 9483: 9480: 9479: 9475: 9472: 9471: 9467: 9464: 9463: 9459: 9456: 9455: 9451: 9448: 9444: 9441: 9439: 9435: 9434: 9432: 9430: 9426: 9420: 9417: 9415: 9412: 9410: 9407: 9405: 9402: 9400: 9397: 9395: 9392: 9390: 9387: 9385: 9382: 9380: 9377: 9375: 9372: 9370: 9367: 9365: 9362: 9360: 9357: 9355: 9352: 9350: 9347: 9345: 9342: 9340: 9337: 9335: 9332: 9330: 9327: 9325: 9322: 9320: 9317: 9315: 9312: 9310: 9307: 9305: 9302: 9298: 9295: 9293: 9290: 9289: 9288: 9285: 9284: 9282: 9278: 9274: 9267: 9262: 9260: 9255: 9253: 9248: 9247: 9244: 9237: 9233: 9232:Suzanne Imber 9230: 9228: 9225: 9224: 9215: 9212: 9209: 9205: 9202: 9199: 9196: 9193: 9190: 9186: 9182: 9181: 9176: 9172: 9171: 9161: 9155: 9150: 9149: 9143: 9139: 9135: 9133:0-471-92567-5 9129: 9124: 9123: 9117: 9113: 9109: 9103: 9099: 9098:W. H. Freeman 9095: 9094: 9089: 9085: 9081: 9077: 9076: 9075: 9073: 9057: 9053: 9049: 9045: 9041: 9037: 9032: 9031:gr-qc/0507026 9027: 9023: 9019: 9012: 9004: 9000: 8996: 8992: 8988: 8984: 8979: 8974: 8971:(6): 063518. 8970: 8966: 8959: 8951: 8947: 8943: 8939: 8935: 8931: 8927: 8923: 8916: 8908: 8904: 8900: 8896: 8892: 8888: 8884: 8880: 8873: 8865: 8861: 8857: 8853: 8849: 8845: 8841: 8837: 8833: 8826: 8817: 8813: 8809: 8805: 8801: 8797: 8793: 8789: 8788: 8783: 8777: 8769: 8763: 8759: 8758: 8750: 8742: 8738: 8734: 8730: 8726: 8722: 8717: 8712: 8708: 8704: 8697: 8689: 8685: 8681: 8674: 8660: 8658:0-670-50376-2 8654: 8650: 8646: 8645: 8640: 8639:Gamow, George 8634: 8627: 8622: 8614: 8612:0-471-60088-1 8608: 8604: 8597: 8589: 8585: 8581: 8577: 8573: 8569: 8566:(6303): 705. 8565: 8561: 8560: 8552: 8544: 8542:0-691-08239-1 8538: 8534: 8527: 8520: 8515: 8508: 8503: 8495: 8491: 8486: 8481: 8476: 8471: 8467: 8463: 8458: 8457:gr-qc/0505133 8453: 8449: 8445: 8441: 8434: 8426: 8424:0-521-46136-7 8420: 8416: 8412: 8405: 8403: 8394: 8392:0-09-922391-0 8388: 8384: 8377: 8369: 8365: 8361: 8359:0-07-000423-4 8355: 8351: 8350: 8342: 8334: 8330: 8326: 8320: 8316: 8312: 8305: 8297: 8291: 8287: 8286: 8278: 8270: 8268:0-8053-8732-3 8264: 8260: 8256: 8255:Carroll, Sean 8250: 8243: 8238: 8223: 8219: 8215: 8209: 8199: 8194: 8190: 8186: 8182: 8178: 8174: 8170: 8169: 8164: 8157: 8155: 8150: 8140: 8137: 8135: 8132: 8130: 8127: 8125: 8122: 8120: 8117: 8115: 8112: 8110: 8107: 8105: 8102: 8100: 8097: 8095: 8092: 8091: 8084: 8082: 8076: 8069: 8056: 8046: 8033: 8030: 8026: 8020: 8017: 8013: 8007: 8004: 8000: 7994: 7991: 7988: 7985: 7981: 7975: 7972: 7969: 7966: 7962: 7955: 7952: 7943: 7938: 7935: 7931: 7922: 7904: 7901: 7897: 7891: 7888: 7884: 7878: 7875: 7871: 7865: 7862: 7858: 7852: 7849: 7846: 7843: 7839: 7833: 7830: 7827: 7824: 7820: 7813: 7810: 7804: 7798: 7777: 7775: 7771: 7767: 7763: 7759: 7753: 7743: 7741: 7735: 7733: 7729: 7725: 7720: 7718: 7714: 7710: 7706: 7700: 7690: 7673: 7670: 7667: 7663: 7659: 7654: 7651: 7648: 7644: 7640: 7635: 7632: 7629: 7625: 7621: 7616: 7613: 7610: 7606: 7602: 7597: 7594: 7590: 7574: 7570: 7563: 7559: 7555: 7551: 7531: 7528: 7524: 7518: 7515: 7512: 7509: 7505: 7501: 7496: 7493: 7490: 7487: 7483: 7479: 7474: 7471: 7468: 7465: 7461: 7456: 7449: 7446: 7440: 7436: 7430: 7427: 7424: 7421: 7417: 7413: 7408: 7405: 7402: 7399: 7395: 7391: 7386: 7383: 7380: 7377: 7373: 7368: 7361: 7358: 7352: 7350: 7340: 7337: 7334: 7331: 7324: 7316: 7313: 7311: 7304: 7301: 7294: 7291: 7287: 7272: 7267: 7254: 7250: 7244: 7241: 7237: 7231: 7228: 7224: 7218: 7215: 7211: 7204: 7201: 7195: 7190: 7183: 7179: 7171: 7164: 7160: 7153: 7145: 7141: 7137: 7132: 7127: 7124: 7120: 7113: 7108: 7105: 7101: 7089: 7085: 7068: 7062: 7059: 7055: 7049: 7046: 7042: 7036: 7033: 7029: 7022: 7019: 7013: 7008: 7001: 6997: 6989: 6982: 6978: 6971: 6963: 6959: 6955: 6950: 6946: 6941: 6938: 6934: 6926: 6922: 6918: 6905: 6895: 6893: 6889: 6883: 6875: 6871: 6867: 6865: 6864:Kerr solution 6861: 6857: 6853: 6848: 6835: 6829: 6826: 6822: 6815: 6812: 6807: 6804: 6793: 6788: 6785: 6781: 6771: 6758: 6754: 6751: 6746: 6743: 6739: 6730: 6724: 6717:. By setting 6716: 6700: 6690: 6677: 6669: 6665: 6660: 6657: 6654: 6648: 6645: 6624: 6621: 6618: 6615: 6612: 6607: 6603: 6599: 6596: 6590: 6587: 6562: 6558: 6554: 6551: 6545: 6542: 6536: 6532: 6526: 6522: 6518: 6512: 6509: 6503: 6498: 6494: 6489: 6485: 6482: 6477: 6473: 6469: 6464: 6459: 6456: 6453: 6445: 6440: 6437: 6434: 6420: 6407: 6401: 6396: 6393: 6390: 6382: 6377: 6373: 6359: 6346: 6341: 6336: 6333: 6323: 6318: 6315: 6307: 6302: 6297: 6287: 6282: 6279: 6271: 6266: 6261: 6258: 6255: 6252: 6244: 6239: 6234: 6231: 6228: 6220: 6215: 6211: 6201: 6188: 6182: 6178: 6174: 6171: 6165: 6162: 6156: 6152: 6147: 6141: 6137: 6133: 6129: 6124: 6118: 6114: 6110: 6107: 6103: 6096: 6093: 6087: 6082: 6078: 6074: 6070: 6066: 6063: 6059: 6053: 6049: 6045: 6039: 6036: 6030: 6025: 6021: 6016: 6012: 5989: 5985: 5981: 5978: 5975: 5970: 5966: 5962: 5955: 5951: 5947: 5942: 5939: 5934: 5930: 5924: 5920: 5916: 5911: 5908: 5904: 5898: 5895: 5891: 5887: 5884: 5875: 5862: 5857: 5853: 5850: 5847: 5844: 5841: 5838: 5833: 5829: 5825: 5821: 5817: 5814: 5811: 5807: 5803: 5800: 5797: 5794: 5791: 5788: 5783: 5779: 5774: 5770: 5767: 5764: 5759: 5756: 5752: 5730: 5724: 5720: 5716: 5710: 5707: 5701: 5696: 5692: 5687: 5683: 5680: 5675: 5671: 5661: 5648: 5641: 5638: 5633: 5629: 5625: 5622: 5617: 5613: 5589: 5586: 5583: 5579: 5575: 5572: 5568: 5562: 5559: 5556: 5552: 5548: 5544: 5538: 5535: 5531: 5527: 5522: 5519: 5511: 5502: 5498: 5492: 5479: 5474: 5468: 5465: 5462: 5458: 5454: 5449: 5446: 5443: 5440: 5436: 5432: 5427: 5424: 5421: 5418: 5414: 5409: 5403: 5400: 5396: 5389: 5386: 5380: 5375: 5370: 5362: 5357: 5354: 5336: 5328: 5305: 5300: 5292: 5289: 5281: 5277: 5273: 5266: 5262: 5256: 5252: 5240: 5224: 5221: 5217: 5211: 5208: 5203: 5200: 5194: 5187: 5184: 5180: 5157: 5153: 5150: 5147: 5144: 5141: 5138: 5132: 5129: 5124: 5121: 5114: 5110: 5104: 5101: 5094: 5090: 5086: 5074: 5061: 5054: 5051: 5044: 5040: 5036: 5027: 5024: 5017: 5013: 5009: 5001: 4996: 4993: 4985: 4982: 4974: 4970: 4966: 4959: 4955: 4949: 4945: 4934: 4913: 4907: 4904: 4900: 4896: 4890: 4887: 4881: 4876: 4873: 4869: 4864: 4860: 4857: 4852: 4849: 4845: 4835: 4818: 4812: 4809: 4801: 4798: 4796: 4786: 4782: 4778: 4771: 4767: 4761: 4757: 4746: 4743: 4740: 4737: 4734: 4732: 4725: 4722: 4719: 4701: 4688: 4683: 4679: 4676: 4670: 4658: 4651: 4641: 4638: 4629: 4623: 4617: 4608: 4600: 4587: 4566: 4562: 4559: 4550: 4543: 4539: 4536: 4533: 4530: 4527: 4523: 4519: 4516: 4507: 4500: 4491: 4479: 4473: 4459: 4453: 4449: 4445: 4435: 4433: 4429: 4425: 4421: 4417: 4413: 4409: 4399: 4394: 4381: 4378: 4373: 4370: 4363: 4360: 4356: 4350: 4345: 4342: 4338: 4332: 4318: 4305: 4302: 4297: 4294: 4287: 4284: 4280: 4269: 4253: 4250: 4245: 4242: 4237: 4233: 4228: 4225: 4221: 4214: 4211: 4205: 4200: 4197: 4193: 4188: 4173: 4160: 4157: 4152: 4149: 4144: 4140: 4135: 4128: 4124: 4115: 4112: 4106: 4101: 4094: 4090: 4083: 4061: 4058: 4053: 4050: 4047: 4040: 4036: 4030: 4027: 4022: 4019: 4015: 4005: 3992: 3989: 3984: 3981: 3978: 3975: 3968: 3965: 3961: 3955: 3950: 3947: 3944: 3937: 3933: 3927: 3922: 3919: 3916: 3909: 3905: 3882: 3879: 3875: 3869: 3866: 3863: 3860: 3857: 3850: 3846: 3840: 3835: 3832: 3829: 3826: 3822: 3818: 3813: 3810: 3807: 3804: 3800: 3795: 3789: 3786: 3782: 3772: 3770: 3754: 3751: 3746: 3743: 3740: 3737: 3734: 3727: 3723: 3717: 3712: 3709: 3706: 3703: 3699: 3695: 3690: 3687: 3684: 3681: 3677: 3656: 3653: 3648: 3645: 3642: 3639: 3636: 3629: 3625: 3619: 3614: 3611: 3608: 3605: 3602: 3595: 3591: 3585: 3580: 3577: 3574: 3571: 3568: 3561: 3557: 3546: 3543: 3530: 3527: 3522: 3519: 3516: 3513: 3510: 3503: 3499: 3493: 3488: 3485: 3482: 3479: 3476: 3469: 3465: 3459: 3454: 3451: 3448: 3445: 3442: 3435: 3431: 3418: 3394: 3391: 3383: 3380: 3377: 3374: 3368: 3365: 3361: 3353: 3344: 3331: 3328: 3323: 3320: 3313: 3310: 3306: 3300: 3295: 3292: 3288: 3282: 3269: 3256: 3252: 3249: 3228: 3223: 3215: 3198: 3194: 3191: 3174: 3161: 3152: 3148: 3144: 3139: 3126: 3120: 3117: 3113: 3107: 3099: 3096: 3072: 3069: 3065: 3054: 3048: 3044: 3039: 3037: 3025: 3021: 3017: 3014: 3013: 3012: 3010: 3003: 2987: 2981: 2978: 2974: 2970: 2967: 2962: 2959: 2955: 2948: 2943: 2940: 2936: 2925: 2915: 2913: 2895: 2892: 2888: 2878: 2865: 2861: 2855: 2852: 2848: 2843: 2838: 2835: 2830: 2825: 2822: 2818: 2813: 2809: 2806: 2801: 2798: 2794: 2787: 2782: 2779: 2775: 2764: 2758: 2745: 2741: 2735: 2732: 2728: 2724: 2718: 2715: 2712: 2708: 2703: 2698: 2695: 2691: 2686: 2682: 2679: 2674: 2671: 2667: 2657: 2654: 2651: 2647: 2642: 2637: 2634: 2630: 2620: 2599: 2596: 2593: 2590: 2584: 2581: 2578: 2573: 2570: 2565: 2562: 2554: 2544: 2538: 2529: 2516: 2511: 2508: 2504: 2500: 2497: 2494: 2489: 2486: 2482: 2475: 2470: 2467: 2463: 2459: 2454: 2451: 2446: 2441: 2438: 2434: 2424: 2406: 2401: 2386: 2383: 2380: 2373: 2369: 2363: 2357: 2354: 2348: 2342: 2339: 2333: 2328: 2325: 2321: 2311: 2292: 2289: 2285: 2281: 2278: 2272: 2269: 2263: 2261: 2254: 2251: 2247: 2238: 2232: 2227: 2224: 2214: 2209: 2206: 2198: 2193: 2188: 2185: 2175: 2170: 2167: 2159: 2154: 2149: 2146: 2143: 2140: 2132: 2127: 2122: 2119: 2116: 2113: 2104: 2100: 2094: 2091: 2085: 2083: 2076: 2073: 2070: 2063: 2059: 2046: 2043: 2040: 2037: 2034: 2031: 2028: 2025: 2022: 2019: 2016: 2010: 2007: 2004: 1998: 1995: 1989: 1987: 1980: 1977: 1973: 1960: 1958: 1948: 1946: 1928: 1925: 1921: 1911: 1909: 1903: 1899: 1889: 1887: 1884: 1880: 1876: 1871: 1869: 1865: 1861: 1856: 1852: 1835: 1830: 1827: 1823: 1819: 1816: 1811: 1808: 1804: 1797: 1792: 1789: 1785: 1781: 1776: 1773: 1768: 1763: 1760: 1756: 1748: 1747: 1746: 1743: 1741: 1737: 1729: 1706: 1679: 1674: 1671: 1656: 1653: 1649: 1645: 1639: 1633: 1630: 1623: 1619: 1614: 1611: 1608: 1602: 1599: 1592: 1591: 1590: 1588: 1583: 1581: 1565: 1557: 1539: 1536: 1532: 1508: 1503: 1500: 1496: 1492: 1487: 1484: 1479: 1474: 1471: 1467: 1463: 1458: 1455: 1451: 1443: 1442: 1441: 1439: 1434: 1420: 1412: 1388: 1370: 1367: 1363: 1354: 1353:metric tensor 1336: 1333: 1329: 1320: 1302: 1299: 1295: 1283:, Netherlands 1282: 1277: 1257: 1254: 1250: 1246: 1243: 1238: 1235: 1231: 1224: 1219: 1216: 1212: 1204: 1203: 1202: 1192: 1187: 1185: 1180: 1178: 1173: 1172: 1170: 1169: 1162: 1152: 1149: 1144: 1138: 1137: 1135: 1134: 1126: 1123: 1121: 1118: 1116: 1113: 1111: 1108: 1106: 1103: 1101: 1098: 1096: 1093: 1092: 1085: 1084: 1076: 1073: 1071: 1068: 1067: 1060: 1059: 1051: 1048: 1046: 1043: 1041: 1038: 1037: 1030: 1029: 1021: 1018: 1016: 1013: 1011: 1008: 1006: 1003: 1002: 995: 994: 988: 985: 983: 980: 979: 977: 976: 972: 968: 967: 964: 961: 960: 956: 955: 947: 945: 941: 937: 933: 929: 925: 921: 916: 914: 910: 905: 903: 899: 895: 891: 887: 886:metric tensor 883: 879: 875: 871: 867: 862: 860: 856: 852: 848: 845: 837: 833: 828: 826: 822: 818: 814: 810: 806: 794: 789: 787: 782: 780: 775: 774: 772: 771: 765: 755: 752: 747: 741: 740: 739: 738: 731: 730: 726: 724: 721: 719: 716: 714: 711: 709: 706: 704: 701: 699: 696: 694: 691: 689: 686: 684: 681: 679: 676: 674: 671: 669: 668:Chandrasekhar 666: 664: 661: 659: 656: 654: 651: 649: 646: 644: 641: 639: 636: 634: 631: 629: 626: 624: 621: 619: 616: 614: 611: 609: 606: 604: 601: 599: 596: 594: 591: 589: 586: 584: 583:Schwarzschild 581: 579: 576: 574: 571: 569: 566: 564: 561: 560: 552: 551: 544: 543:Hartle–Thorne 541: 539: 536: 534: 531: 529: 526: 524: 521: 519: 516: 514: 511: 509: 506: 504: 501: 499: 496: 494: 491: 489: 486: 484: 481: 479: 476: 474: 471: 469: 466: 464: 461: 458: 454: 453:Schwarzschild 451: 450: 446: 440: 439: 428: 425: 423: 420: 419: 418: 417: 413: 412: 407: 404: 402: 399: 397: 394: 393: 392: 391: 387: 386: 381: 378: 376: 373: 371: 368: 366: 363: 361: 358: 356: 353: 352: 351: 350: 346: 345: 335: 332: 331: 326: 325: 314: 311: 309: 306: 304: 301: 300: 299: 298: 295: 292: 291: 286: 283: 281: 278: 276: 275:Event horizon 273: 271: 268: 266: 263: 261: 258: 256: 253: 251: 248: 246: 243: 241: 238: 237: 236: 235: 225: 224: 217: 214: 212: 209: 207: 204: 202: 199: 198: 190: 189: 184: 181: 176: 173: 171: 168: 166: 163: 162: 160: 158: 155: 154: 153: 152: 134: 131: 127: 122: 118: 113: 110: 106: 99: 94: 91: 87: 77: 73: 72: 69: 66: 65: 62: 58: 40: 36: 32: 29: 26: 10748: 10442:Kaluza–Klein 10262: 10194:Introduction 10120:Twin paradox 9905: 9873:Einsteinhaus 9753:Pauline Koch 9731:(first wife) 9729:Mileva Marić 9708: 9701: 9694: 9687: 9680: 9673: 9666: 9659: 9592: 9584: 9576: 9568: 9560: 9552: 9544: 9536: 9528: 9520: 9512: 9476: 9468: 9460: 9452: 9437: 9333: 9178: 9147: 9121: 9091: 9069: 9021: 9017: 9011: 8968: 8965:Phys. Rev. D 8964: 8958: 8925: 8921: 8915: 8882: 8878: 8872: 8839: 8835: 8825: 8791: 8785: 8776: 8756: 8749: 8706: 8702: 8696: 8688:the original 8683: 8673: 8662:. Retrieved 8649:Viking Adult 8643: 8633: 8621: 8602: 8596: 8563: 8557: 8551: 8532: 8526: 8514: 8502: 8447: 8443: 8433: 8410: 8382: 8376: 8348: 8341: 8332: 8328: 8324: 8318: 8314: 8310: 8304: 8284: 8277: 8258: 8249: 8237: 8226:. Retrieved 8221: 8208: 8193:the original 8172: 8166: 8074: 8070: 7783: 7755: 7736: 7721: 7702: 7268: 7083: 6907: 6878: 6874:Ricci tensor 6868: 6849: 6772: 6719: 6705: 6421: 6360: 6202: 5876: 5662: 5500: 5496: 5493: 5334: 5326: 5075: 4836: 4702: 4586:free-falling 4471: 4464: 4441: 4432:wavefunction 4405: 4402:Nonlinearity 4396: 4319: 4174: 4006: 3773: 3769:Ricci tensor 3547: 3544: 3413: 3349: 3270: 3267: 3253: 3247: 3159: 3150: 3143:vacuum state 3140: 3055: 3043:astronomical 3040: 3036:George Gamow 3029: 3020:Edwin Hubble 2927: 2879: 2762: 2759: 2618: 2550: 2530: 2425: 2402: 2312: 1961: 1954: 1912: 1901: 1897: 1890: 1883:gauge-fixing 1872: 1857: 1853: 1850: 1744: 1721: 1586: 1584: 1523: 1435: 1286: 1200: 1049: 917: 906: 863: 839: 829: 816: 812: 808: 802: 728: 688:Raychaudhuri 359: 157:Introduction 61: 10531:Kerr–Newman 10502:Spherical: 10371:Other tests 10314:Singularity 10246:Formulation 10208:Fundamental 10062:Formulation 10043:Proper time 10004:Fundamental 9878:Einsteinium 9651:Books about 9597:(2023 film) 9594:Oppenheimer 9573:(2003 play) 9565:(1994 film) 9557:(1993 play) 9549:(1988 film) 9541:(1985 film) 9384:EPR paradox 9093:Gravitation 8928:(8): 2281. 7728:black holes 2551:Taking the 1855:spacetime. 1095:Four-vector 827:within it. 703:van Stockum 633:Oppenheimer 488:Kerr–Newman 280:Singularity 10779:Categories 10683:Zel'dovich 10591:Scientists 10570:Alcubierre 10377:of Mercury 10375:precession 10304:Black hole 10187:Background 10179:relativity 10148:World line 10143:Light cone 9968:Background 9960:relativity 9950:Relativity 9883:Max Talmey 9785:(grandson) 9743:(daughter) 9503:In popular 9066:References 8664:2007-03-14 8228:2017-08-21 8175:(7): 769. 7919:using the 7582:such that 7558:divergence 6921:free space 6902:See also: 4931:, and the 842:spacetime 556:Scientists 388:Formalisms 336:Formalisms 285:Black hole 211:World line 10653:Robertson 10638:Friedmann 10633:Eddington 10623:de Sitter 10457:Solutions 10335:detectors 10330:astronomy 10297:Phenomena 10232:Geodesics 10135:Spacetime 10078:Phenomena 9185:EMS Press 9003:119178273 8978:1304.8042 8950:250876974 8907:250907312 8864:122577276 8816:123364248 8684:News@UofT 8224:: 844–847 8034:ν 8031:δ 8021:μ 8018:γ 8008:λ 8005:β 7995:ν 7992:μ 7989:λ 7986:κ 7982:ε 7976:δ 7973:γ 7970:β 7967:α 7963:ε 7939:κ 7936:α 7905:ν 7902:δ 7892:μ 7889:γ 7879:λ 7876:β 7866:κ 7863:α 7853:ν 7850:μ 7847:λ 7844:κ 7840:ε 7834:δ 7831:γ 7828:β 7825:α 7821:ε 7762:spacetime 7724:cosmology 7709:spacetime 7693:Solutions 7674:α 7668:β 7660:− 7655:β 7649:α 7636:α 7630:β 7622:− 7617:β 7611:α 7598:β 7595:α 7519:β 7513:α 7510:γ 7497:α 7491:γ 7488:β 7475:γ 7469:β 7466:α 7431:β 7425:α 7422:γ 7409:α 7403:γ 7400:β 7387:γ 7381:β 7378:α 7341:γ 7335:β 7332:α 7305:β 7295:β 7292:α 7245:τ 7242:ψ 7232:τ 7229:ψ 7219:β 7216:α 7191:β 7184:ψ 7172:ψ 7165:α 7142:μ 7138:κ 7128:β 7125:α 7117:Λ 7109:β 7106:α 7063:τ 7060:ψ 7050:τ 7047:ψ 7037:β 7034:α 7009:β 7002:ψ 6990:ψ 6983:α 6960:μ 6951:− 6942:β 6939:α 6870:Manifolds 6830:ν 6827:μ 6813:− 6799:Λ 6789:ν 6786:μ 6747:ν 6744:μ 6658:π 6625:ρ 6619:π 6600:ρ 6555:ρ 6537:≈ 6504:− 6470:≈ 6450:Γ 6446:≈ 6431:Φ 6387:Γ 6383:≈ 6342:λ 6334:ρ 6330:Γ 6324:ρ 6319:λ 6312:Γ 6308:− 6303:λ 6294:Γ 6288:ρ 6283:λ 6280:ρ 6276:Γ 6267:ρ 6253:ρ 6249:Γ 6245:− 6240:ρ 6235:ρ 6225:Γ 6175:ρ 6134:− 6111:ρ 6108:− 6088:− 6075:ρ 6064:≈ 6031:− 6005:and thus 5982:ρ 5979:− 5963:ρ 5943:− 5940:≈ 5917:≈ 5912:β 5909:α 5899:β 5896:α 5826:ρ 5818:⁡ 5812:≈ 5771:⁡ 5765:≈ 5760:ν 5757:μ 5702:− 5645:Φ 5639:− 5626:− 5623:≈ 5576:− 5573:≈ 5549:− 5528:≈ 5516:Φ 5469:α 5455:− 5444:α 5419:α 5404:α 5367:Γ 5363:≈ 5351:Φ 5297:Γ 5293:− 5290:≈ 5222:≈ 5212:τ 5171:and thus 5133:τ 5111:≈ 5105:τ 5095:β 5055:τ 5045:γ 5028:τ 5018:β 5002:α 4997:γ 4994:β 4990:Γ 4986:− 4971:τ 4960:α 4908:ν 4905:μ 4882:− 4877:ν 4874:μ 4853:ν 4850:μ 4806:Φ 4802:− 4747:ρ 4741:π 4716:Φ 4662:→ 4648:Φ 4645:∇ 4642:− 4633:→ 4609:¨ 4604:→ 4554:→ 4540:ρ 4534:π 4511:→ 4497:Φ 4488:∇ 4374:β 4364:β 4361:α 4346:β 4343:α 4333:β 4329:∇ 4298:β 4288:β 4285:α 4246:γ 4229:δ 4226:γ 4206:− 4201:δ 4198:γ 4153:γ 4136:ε 4129:γ 4107:− 4102:ε 4095:γ 4054:γ 4048:ε 4041:γ 4028:− 4023:ε 3985:γ 3979:ε 3976:δ 3969:δ 3966:γ 3951:δ 3945:ε 3938:δ 3928:− 3923:ε 3917:δ 3910:δ 3870:γ 3864:ε 3861:δ 3858:β 3851:γ 3836:δ 3830:ε 3827:β 3819:− 3814:ε 3808:δ 3805:β 3790:δ 3787:β 3747:γ 3741:ε 3738:δ 3735:β 3728:γ 3713:δ 3707:ε 3704:β 3696:− 3691:ε 3685:δ 3682:β 3649:γ 3643:ε 3640:δ 3637:β 3630:γ 3615:δ 3609:ε 3606:γ 3603:β 3596:γ 3586:− 3581:ε 3575:δ 3572:γ 3569:β 3562:γ 3523:γ 3517:ε 3514:δ 3511:β 3504:γ 3489:δ 3483:γ 3480:ε 3477:β 3470:γ 3455:ε 3449:δ 3446:γ 3443:β 3436:γ 3384:ε 3378:δ 3375:γ 3369:β 3366:α 3324:β 3314:β 3311:α 3296:β 3293:α 3283:β 3279:∇ 3224:κ 3221:Λ 3195:− 3175:ρ 3121:ν 3118:μ 3108:κ 3105:Λ 3100:− 3073:ν 3070:μ 3024:expanding 2982:ν 2979:μ 2971:κ 2963:ν 2960:μ 2952:Λ 2944:ν 2941:μ 2896:ν 2893:μ 2856:ν 2853:μ 2831:− 2826:ν 2823:μ 2810:κ 2802:ν 2799:μ 2791:Λ 2788:− 2783:ν 2780:μ 2736:ν 2733:μ 2716:− 2704:− 2699:ν 2696:μ 2683:κ 2675:ν 2672:μ 2664:Λ 2655:− 2643:− 2638:ν 2635:μ 2594:κ 2588:Λ 2566:− 2541:(− + + +) 2533:(+ − − −) 2512:ν 2509:μ 2501:κ 2498:− 2490:ν 2487:μ 2479:Λ 2476:− 2471:ν 2468:μ 2447:− 2442:ν 2439:μ 2387:ν 2384:α 2381:μ 2374:α 2364:× 2349:× 2329:ν 2326:μ 2293:ν 2290:μ 2282:κ 2279:× 2255:ν 2252:μ 2233:σ 2228:α 2225:β 2221:Γ 2215:μ 2210:γ 2207:σ 2203:Γ 2199:− 2194:σ 2189:α 2186:γ 2182:Γ 2176:μ 2171:β 2168:σ 2164:Γ 2155:μ 2150:γ 2144:β 2141:α 2137:Γ 2133:− 2128:μ 2123:β 2117:γ 2114:α 2110:Γ 2101:× 2077:γ 2074:β 2071:α 2064:μ 2017:− 2011:⁡ 2005:× 1981:ν 1978:μ 1929:ν 1926:μ 1831:ν 1828:μ 1820:κ 1812:ν 1809:μ 1801:Λ 1793:ν 1790:μ 1769:− 1764:ν 1761:μ 1672:− 1654:− 1646:× 1631:≈ 1612:π 1600:κ 1540:ν 1537:μ 1504:ν 1501:μ 1480:− 1475:ν 1472:μ 1459:ν 1456:μ 1421:κ 1397:Λ 1371:ν 1368:μ 1337:ν 1334:μ 1303:ν 1300:μ 1258:ν 1255:μ 1247:κ 1239:ν 1236:μ 1228:Λ 1220:ν 1217:μ 963:Spacetime 898:geodesics 844:curvature 821:spacetime 648:Robertson 613:Friedmann 608:Eddington 598:Nordström 588:de Sitter 445:Solutions 370:Geodesics 365:Friedmann 347:Equations 333:Equations 294:Spacetime 229:Phenomena 135:ν 132:μ 123:κ 114:ν 111:μ 103:Λ 95:ν 92:μ 10766:Category 10643:Lemaître 10608:Einstein 10598:Poincaré 10558:Others: 10542:Taub–NUT 10508:interior 10430:theories 10428:Advanced 10395:redshift 10210:concepts 10028:Rapidity 10006:concepts 9907:Category 9833:Memorial 9779:(cousin) 9767:(sister) 9761:(father) 9755:(mother) 9653:Einstein 9488:" (1949) 9449:" (1905) 9144:(1999). 9118:(1972). 9090:(1973). 8741:16669258 8588:12988317 8494:28179868 8257:(2004). 8087:See also 7732:universe 6862:and the 4450:and the 4416:electric 3259:Features 3145:with an 1161:Category 930:and the 920:symmetry 894:inertial 874:currents 855:momentum 764:Category 628:Lemaître 593:Reissner 578:Poincaré 563:Einstein 508:Taub–NUT 473:Wormhole 457:interior 170:Timeline 10708:Hawking 10703:Penrose 10688:Novikov 10668:Wheeler 10613:Hilbert 10603:Lorentz 10560:pp-wave 10381:lensing 10177:General 9958:Special 9811:Related 9505:culture 9280:Physics 9187:, 2001 9056:6263993 9036:Bibcode 8983:Bibcode 8930:Bibcode 8887:Bibcode 8844:Bibcode 8796:Bibcode 8721:Bibcode 8568:Bibcode 8485:5256071 8462:Bibcode 8368:1046135 8177:Bibcode 7705:metrics 7560:of the 6854:. Flat 6727:in the 5339:⁠ 5323:⁠ 3895:to get 2535:metric 2421:(− + −) 2417:(− + +) 2413:(+ − −) 2409:(+ + +) 1734:is the 1726:is the 1634:2.07665 1578:is the 1554:is the 1409:is the 1385:is the 1351:is the 1317:is the 870:charges 803:In the 683:Hawking 678:Penrose 653:Bardeen 643:Wheeler 573:Hilbert 568:Lorentz 528:pp-wave 165:History 10749:others 10738:Thorne 10728:Misner 10713:Taylor 10698:Geroch 10693:Ehlers 10663:Zwicky 10481:Kasner 9721:Family 9605:Prizes 9586:Genius 9494:(1955) 9481:(1938) 9473:(1934) 9465:(1922) 9457:(1916) 9442:(1905) 9440:papers 9156: 9130: 9104: 9054: 9001: 8948: 8905: 8862: 8814: 8764: 8739: 8655: 8609: 8586: 8559:Nature 8539: 8492: 8482: 8421: 8389: 8366: 8356: 8292: 8265: 7562:2-form 7086:(with 4580:where 4476:, see 4179:gives 2612:where 1740:vacuum 1722:where 1558:, and 1524:where 1287:where 1281:Leiden 1159: 825:matter 807:, the 762: 729:others 723:Thorne 713:Newman 693:Taylor 673:Ehlers 658:Walker 623:Zwicky 498:Kasner 55:, see 10743:Weiss 10723:Bondi 10718:Hulse 10648:Milne 10552:discs 10496:Milne 10491:Gödel 10348:Virgo 9828:House 9823:Brain 9773:(son) 9749:(son) 9429:Works 9052:S2CID 9026:arXiv 8999:S2CID 8973:arXiv 8946:S2CID 8903:S2CID 8860:S2CID 8812:S2CID 8737:S2CID 8711:arXiv 8584:S2CID 8452:arXiv 8146:Notes 4474:= −∇Φ 3407:with 698:Hulse 638:Gödel 618:Milne 513:Milne 478:Gödel 175:Tests 10678:Kerr 10628:Weyl 10527:Kerr 10387:and 10341:and 10339:LIGO 9562:I.Q. 9154:ISBN 9128:ISBN 9102:ISBN 9070:See 8762:ISBN 8653:ISBN 8607:ISBN 8537:ISBN 8490:PMID 8419:ISBN 8387:ISBN 8364:OCLC 8354:ISBN 8290:ISBN 8263:ISBN 8073:det( 5815:diag 5768:diag 4418:and 2008:diag 1730:and 1585:The 1436:The 1413:and 876:via 872:and 708:Taub 663:Kerr 603:Weyl 483:Kerr 401:BSSN 10733:Yau 10358:GEO 9208:MIT 9206:by 9044:doi 8991:doi 8938:doi 8895:doi 8852:doi 8804:doi 8729:doi 8576:doi 8564:348 8480:PMC 8470:doi 8327:= 8 8313:= 8 8198:PDF 8185:doi 8173:354 8041:det 7793:det 7707:of 6919:in 6884:= 0 6725:= 0 5877:So 4426:of 4410:of 3419:= 0 3163:vac 3154:vac 2765:= 4 2760:In 1866:of 1738:in 1702:m/J 1692:or 861:). 813:EFE 718:Yau 396:ADM 10781:: 10407:/ 10373:: 10328:: 9234:, 9183:, 9177:, 9100:. 9086:; 9082:; 9074:. 9050:. 9042:. 9034:. 9022:38 9020:. 8997:. 8989:. 8981:. 8969:88 8967:. 8944:. 8936:. 8926:14 8924:. 8901:. 8893:. 8881:. 8858:. 8850:. 8840:12 8838:. 8834:. 8810:. 8802:. 8792:16 8790:. 8735:. 8727:. 8719:. 8707:17 8705:. 8682:. 8651:. 8647:. 8582:. 8574:. 8562:. 8488:. 8478:. 8468:. 8460:. 8446:. 8442:. 8417:. 8413:. 8401:^ 8362:. 8329:πG 8315:πG 8220:. 8183:. 8171:. 8165:. 8153:^ 7814:24 7776:. 7734:. 7719:. 7532:0. 6912:μν 6894:. 6881:μν 6876:, 6866:. 6722:μν 6710:μν 6527:00 6499:00 6478:00 6454:00 6391:00 6378:00 6298:00 6229:00 6216:00 6054:00 6026:00 5935:00 5925:00 5784:00 5725:00 5697:00 5676:00 5618:00 5584:00 5557:00 5499:= 5463:00 5371:00 5335:dτ 5327:dt 5301:00 4434:. 3771:. 3421:, 3416:;γ 3332:0. 2423:. 1947:. 1910:. 1902:μν 1870:. 1742:. 1657:43 1650:10 1389:, 1355:, 1321:, 946:. 915:. 904:. 10533:) 10529:( 10515:) 10506:( 10472:) 10468:( 10415:) 10411:( 10391:) 10360:) 10337:( 9986:) 9977:( 9942:e 9935:t 9928:v 9484:" 9445:" 9265:e 9258:t 9251:v 9162:. 9136:. 9110:. 9058:. 9046:: 9038:: 9028:: 9005:. 8993:: 8985:: 8975:: 8952:. 8940:: 8932:: 8909:. 8897:: 8889:: 8883:3 8866:. 8854:: 8846:: 8820:. 8818:. 8806:: 8798:: 8770:. 8743:. 8731:: 8723:: 8713:: 8667:. 8628:. 8615:. 8590:. 8578:: 8570:: 8545:. 8521:. 8496:. 8472:: 8464:: 8454:: 8448:8 8427:. 8395:. 8370:. 8333:c 8331:/ 8325:κ 8319:c 8317:/ 8311:κ 8298:. 8271:. 8231:. 8200:) 8196:( 8187:: 8179:: 8077:) 8075:g 8057:. 8050:) 8047:g 8044:( 8027:g 8014:g 8001:g 7956:6 7953:1 7944:= 7932:g 7898:g 7885:g 7872:g 7859:g 7811:1 7805:= 7802:) 7799:g 7796:( 7671:, 7664:A 7652:, 7645:A 7641:= 7633:; 7626:A 7614:; 7607:A 7603:= 7591:F 7579:α 7577:A 7565:F 7529:= 7525:) 7516:, 7506:F 7502:+ 7494:, 7484:F 7480:+ 7472:, 7462:F 7457:( 7450:3 7447:1 7441:= 7437:) 7428:; 7418:F 7414:+ 7406:; 7396:F 7392:+ 7384:; 7374:F 7369:( 7362:3 7359:1 7353:= 7344:] 7338:; 7329:[ 7325:F 7317:0 7314:= 7302:; 7288:F 7255:. 7251:) 7238:F 7225:F 7212:g 7205:4 7202:1 7196:+ 7180:F 7161:F 7154:( 7146:0 7133:= 7121:g 7114:+ 7102:G 7091:Λ 7069:) 7056:F 7043:F 7030:g 7023:4 7020:1 7014:+ 6998:F 6979:F 6972:( 6964:0 6956:1 6947:= 6935:T 6910:T 6879:R 6836:. 6823:g 6816:1 6808:2 6805:D 6794:= 6782:R 6759:. 6755:0 6752:= 6740:R 6720:T 6708:T 6678:. 6670:4 6666:c 6661:G 6655:8 6649:= 6646:K 6622:G 6616:4 6613:= 6608:4 6604:c 6597:K 6591:2 6588:1 6563:4 6559:c 6552:K 6546:2 6543:1 6533:) 6523:g 6519:T 6513:2 6510:1 6495:T 6490:( 6486:K 6483:= 6474:R 6465:i 6460:i 6457:, 6441:i 6438:i 6435:, 6408:. 6402:i 6397:i 6394:, 6374:R 6363:Γ 6347:. 6337:0 6316:0 6272:+ 6262:0 6259:, 6256:0 6232:, 6221:= 6212:R 6189:. 6183:4 6179:c 6172:K 6166:2 6163:1 6157:= 6153:) 6148:) 6142:2 6138:c 6130:( 6125:) 6119:2 6115:c 6104:( 6097:2 6094:1 6083:4 6079:c 6071:( 6067:K 6060:) 6050:g 6046:T 6040:2 6037:1 6022:T 6017:( 6013:K 5990:2 5986:c 5976:= 5971:4 5967:c 5956:2 5952:c 5948:1 5931:T 5921:g 5905:T 5892:g 5888:= 5885:T 5863:. 5858:) 5854:0 5851:, 5848:0 5845:, 5842:0 5839:, 5834:4 5830:c 5822:( 5808:) 5804:0 5801:, 5798:0 5795:, 5792:0 5789:, 5780:T 5775:( 5753:T 5731:) 5721:g 5717:T 5711:2 5708:1 5693:T 5688:( 5684:K 5681:= 5672:R 5649:. 5642:2 5634:2 5630:c 5614:g 5590:i 5587:, 5580:g 5569:) 5563:j 5560:, 5553:g 5545:( 5539:j 5536:i 5532:g 5523:i 5520:, 5512:2 5501:i 5497:α 5480:. 5475:) 5466:, 5459:g 5450:0 5447:, 5441:0 5437:g 5433:+ 5428:0 5425:, 5422:0 5415:g 5410:( 5401:i 5397:g 5390:2 5387:1 5381:= 5376:i 5358:i 5355:, 5331:/ 5306:i 5282:2 5278:t 5274:d 5267:i 5263:x 5257:2 5253:d 5225:0 5218:) 5209:d 5204:t 5201:d 5195:( 5188:t 5185:d 5181:d 5158:) 5154:0 5151:, 5148:0 5145:, 5142:0 5139:, 5130:d 5125:t 5122:d 5115:( 5102:d 5091:x 5087:d 5062:. 5052:d 5041:x 5037:d 5025:d 5014:x 5010:d 4983:= 4975:2 4967:d 4956:x 4950:2 4946:d 4929:K 4914:) 4901:g 4897:T 4891:2 4888:1 4870:T 4865:( 4861:K 4858:= 4846:R 4819:. 4813:i 4810:, 4799:= 4787:2 4783:t 4779:d 4772:i 4768:x 4762:2 4758:d 4744:G 4738:4 4735:= 4726:i 4723:i 4720:, 4689:. 4684:) 4680:t 4677:, 4674:) 4671:t 4668:( 4659:x 4652:( 4639:= 4630:g 4624:= 4621:) 4618:t 4615:( 4601:x 4582:ρ 4567:) 4563:t 4560:, 4551:x 4544:( 4537:G 4531:4 4528:= 4524:) 4520:t 4517:, 4508:x 4501:( 4492:2 4472:g 4467:Φ 4456:G 4382:0 4379:= 4371:; 4357:T 4351:= 4339:T 4306:0 4303:= 4295:; 4281:G 4254:0 4251:= 4243:; 4238:) 4234:R 4222:g 4215:2 4212:1 4194:R 4189:( 4177:g 4161:0 4158:= 4150:; 4145:) 4141:R 4125:g 4116:2 4113:1 4091:R 4084:( 4062:0 4059:= 4051:; 4037:R 4031:2 4020:; 4016:R 3993:0 3990:= 3982:; 3962:R 3956:+ 3948:; 3934:R 3920:; 3906:R 3883:0 3880:= 3876:) 3867:; 3847:R 3841:+ 3833:; 3823:R 3811:; 3801:R 3796:( 3783:g 3755:0 3752:= 3744:; 3724:R 3718:+ 3710:; 3700:R 3688:; 3678:R 3657:0 3654:= 3646:; 3626:R 3620:+ 3612:; 3592:R 3578:; 3558:R 3531:0 3528:= 3520:; 3500:R 3494:+ 3486:; 3466:R 3460:+ 3452:; 3432:R 3414:g 3409:g 3395:0 3392:= 3387:] 3381:; 3372:[ 3362:R 3329:= 3321:; 3307:T 3301:= 3289:T 3248:κ 3243:Λ 3229:, 3216:= 3210:c 3207:a 3204:v 3199:p 3192:= 3186:c 3183:a 3180:v 3160:p 3151:ρ 3127:. 3114:g 3097:= 3091:) 3088:c 3085:a 3082:v 3079:( 3066:T 3051:Λ 3032:Λ 3026:. 3005:Λ 2988:, 2975:T 2968:= 2956:g 2949:+ 2937:G 2889:g 2866:. 2862:) 2849:g 2844:T 2839:2 2836:1 2819:T 2814:( 2807:= 2795:g 2776:R 2763:D 2746:. 2742:) 2729:g 2725:T 2719:2 2713:D 2709:1 2692:T 2687:( 2680:= 2668:g 2658:2 2652:D 2648:2 2631:R 2619:R 2614:D 2600:, 2597:T 2591:= 2585:D 2582:+ 2579:R 2574:2 2571:D 2563:R 2517:. 2505:T 2495:= 2483:g 2464:g 2460:R 2455:2 2452:1 2435:R 2370:R 2361:] 2358:3 2355:S 2352:[ 2346:] 2343:2 2340:S 2337:[ 2334:= 2322:R 2286:T 2276:] 2273:3 2270:S 2267:[ 2264:= 2248:G 2239:) 2160:+ 2147:, 2120:, 2105:( 2098:] 2095:2 2092:S 2089:[ 2086:= 2060:R 2050:) 2047:1 2044:+ 2041:, 2038:1 2035:+ 2032:, 2029:1 2026:+ 2023:, 2020:1 2014:( 2002:] 1999:1 1996:S 1993:[ 1990:= 1974:g 1922:g 1898:T 1893:n 1836:. 1824:T 1817:= 1805:g 1798:+ 1786:g 1782:R 1777:2 1774:1 1757:R 1732:c 1724:G 1707:, 1680:, 1675:1 1666:N 1643:) 1640:5 1637:( 1624:4 1620:c 1615:G 1609:8 1603:= 1566:R 1533:R 1509:, 1497:g 1493:R 1488:2 1485:1 1468:R 1464:= 1452:G 1364:T 1330:g 1296:G 1251:T 1244:= 1232:g 1225:+ 1213:G 1190:e 1183:t 1176:v 811:( 792:e 785:t 778:v 459:) 455:( 128:T 119:= 107:g 100:+ 88:G 59:. 41:2 37:c 33:m 30:= 27:E

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Einstein field equations

Index