8268:
4297:
3627:
5691:(relative to the mass contained in the sphere of matter closer to the origin) is equal to a constant related to the curvature of the universe. In other words, the energy (relative to the origin) of a co-moving particle in free-fall is conserved. General relativity merely adds a connection between the spatial curvature of the universe and the energy of such a particle: positive total energy implies negative curvature and negative total energy implies positive curvature.
8244:
4310:
559:
36:
8304:
8328:
8280:
571:
8316:
1206:
8292:
6060:
dipole through studies of radio galaxies and quasars show disagreement in the magnitude. Taken at face value, these observations are at odds with the
Universe being described by the FLRW metric. Moreover, one can argue that there is a maximum value to the Hubble constant within an FLRW cosmology tolerated by current observations,
1020:
1734:
4801:
model. Because the FLRW model assumes homogeneity, some popular accounts mistakenly assert that the Big Bang model cannot account for the observed lumpiness of the universe. In a strictly FLRW model, there are no clusters of galaxies or stars, since these are objects much denser than a typical part
6059:
and its generalization, astrophysicists now agree that the early universe is almost homogeneous and isotropic (when averaged over a very large scale) and thus nearly a FLRW spacetime. That being said, attempts to confirm the purely kinematic interpretation of the Cosmic
Microwave Background (CMB)
5774:
from the UK explored the problem further during the 1930s. In 1935 Robertson and Walker rigorously proved that the FLRW metric is the only one on a spacetime that is spatially homogeneous and isotropic (as noted above, this is a geometric result and is not tied specifically to the equations of
4802:
of the universe. Nonetheless, the FLRW model is used as a first approximation for the evolution of the real, lumpy universe because it is simple to calculate, and models that calculate the lumpiness in the universe are added onto the FLRW models as extensions. Most cosmologists agree that the
2119:
3605:
4642:
4788:
5045:
1494:
924:
2721:
2878:
1201:{\displaystyle \mathrm {d} \mathbf {\Sigma } ^{2}={\frac {\mathrm {d} r^{2}}{1-kr^{2}}}+r^{2}\mathrm {d} \mathbf {\Omega } ^{2},\quad {\text{where }}\mathrm {d} \mathbf {\Omega } ^{2}=\mathrm {d} \theta ^{2}+\sin ^{2}\theta \,\mathrm {d} \phi ^{2}.}
2347:
5671:
3416:
1369:
A disadvantage of reduced circumference coordinates is that they cover only half of the 3-sphere in the case of positive curvature—circumferences beyond that point begin to decrease, leading to degeneracy. (This is not a problem if space is
5552:
1532:
4919:
3104:
3249:
6033:
or anisotropic? Is the CMB dipole purely kinematic, or does it signal a possible breakdown of the FLRW metric? Even if the cosmological principle is correct, is the
Friedmann–Lemaître–Robertson–Walker metric valid in the late
5680:) is the amount that leaves through the sides due to the expansion of the universe plus the mass equivalent of the work done by pressure against the material being expelled. This is the conservation of mass–energy (
5197:
3698:
5735:, acted as the scientific referee of Friedmann's work. Eventually Einstein acknowledged the correctness of Friedmann's calculations, but failed to appreciate the physical significance of Friedmann's predictions.
4497:
1918:
3427:
4513:
2202:
5240:
4648:
4925:
4360:
Einstein's field equations are not used in deriving the general form for the metric: it follows from the geometric properties of homogeneity and isotropy. However, determining the time evolution of
5931:
2948:
5413:
1810:
1524:
1309:
988:
2495:
1400:
2445:
6861:
Abdalla, Elcio; et al. (June 2022). "Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies".
5868:
809:
4814:. As of 2003, the theoretical implications of the various extensions to the FLRW model appear to be well understood, and the goal is to make these consistent with observations from
2559:
6095:, and depending on how local determinations converge, this may point to a breakdown of the FLRW metric in the late universe, necessitating an explanation beyond the FLRW metric.
2732:
2399:
949:
5293:
2261:
5558:
3255:
5345:
7097:. The original work is Ehlers, J., Geren, P., Sachs, R.K.: Isotropic solutions of Einstein-Liouville equations. J. Math. Phys. 9, 1344 (1968). For the generalization, see
2233:
5115:
4419:
1729:{\displaystyle S_{k}(r)={\begin{cases}{\sqrt {k}}^{\,-1}\sin(r{\sqrt {k}}),&k>0\\r,&k=0\\{\sqrt {|k|}}^{\,-1}\sinh(r{\sqrt {|k|}}),&k<0.\end{cases}}}
7014:
Krishnan, Chethan; Mohayaee, Roya; Colgáin, Eoin Ó; Sheikh-Jabbari, M. M.; Yin, Lu (16 September 2021). "Does Hubble
Tension Signal a Breakdown in FLRW Cosmology?".
5436:
6085:
5998:
2547:
4387:
2888:
In more general FLRW space using spherical coordinates (called "reduced-circumference polar coordinates" above), the surviving components of the Ricci tensor are
4836:
2954:
6603:(1931), "Expansion of the universe, A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulæ",
5974:
5954:
5068:
3110:
7253:
Krishnan, Chethan; Mohayaee, Roya; Ó Colgáin, Eoin; Sheikh-Jabbari, M. M.; Yin, Lu (2021-05-25). "Does Hubble tension signal a breakdown in FLRW cosmology?".
727:. The general form of the metric follows from the geometric properties of homogeneity and isotropy; Einstein's field equations are only needed to derive the
4341:
5430:
This is due to McCrea and Milne, although sometimes incorrectly ascribed to
Friedmann. The Friedmann equations are equivalent to this pair of equations:
531:
6605:
5760:
8348:
3720:
5687:
The second equation says that the kinetic energy (seen from the origin) of a particle of unit mass moving with the expansion plus its (negative)
8353:
6820:
6639:(1927), "Un univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques",
2114:{\displaystyle S_{k}(r)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}k^{n}r^{2n+1}}{(2n+1)!}}=r-{\frac {kr^{3}}{6}}+{\frac {k^{2}r^{5}}{120}}-\cdots }
6029:
and assumed by all models that use the
Friedmann–Lemaître–Robertson–Walker metric, including the current version of ΛCDM, or is the universe
5743:
5151:
3633:
3600:{\displaystyle R={\frac {6}{c^{2}}}\left({\frac {{\ddot {a}}(t)}{a(t)}}+{\frac {{\dot {a}}^{2}(t)}{a^{2}(t)}}+{\frac {k}{a^{2}(t)}}\right).}
803:
of space. It also assumes that the spatial component of the metric can be time-dependent. The generic metric that meets these conditions is
4434:
3925:
601:
7994:
5750:(Annals of the Scientific Society of Brussels). In the face of the observational evidence for the expansion of the universe obtained by
4637:{\displaystyle \left({\frac {\dot {a}}{a}}\right)^{2}+{\frac {kc^{2}}{a^{2}}}-{\frac {\Lambda c^{2}}{3}}={\frac {\kappa c^{4}}{3}}\rho }
8229:
4278:
8018:
7764:
7141:
Siewert, Thilo M.; Schmidt-Rubart, Matthias; Schwarz, Dominik J. (2021). "Cosmic radio dipole: Estimators and frequency dependence".
5249:
can be interpreted as arising from a form of energy that has negative pressure, equal in magnitude to its (positive) energy density:
4088:
3930:
963:. It is normally written as a function of three spatial coordinates, but there are several conventions for doing so, detailed below.
4783:{\displaystyle 2{\frac {\ddot {a}}{a}}+\left({\frac {\dot {a}}{a}}\right)^{2}+{\frac {kc^{2}}{a^{2}}}-\Lambda c^{2}=-\kappa c^{2}p.}
4334:
4043:
6229:. Springer Praxis books in astronomy and planetary science (2. ed., reprinted ed.). Chichester, UK: Praxis Publ. p. 61.
5040:{\displaystyle {\frac {\ddot {a}}{a}}=-{\frac {\kappa c^{4}}{6}}\left(\rho +{\frac {3p}{c^{2}}}\right)+{\frac {\Lambda c^{2}}{3}}}
2130:
7081:
5203:
7800:
7754:
6411:
Rosu, Haret C.; Mancas, S. C.; Chen, Pisin (2015-05-05). "Barotropic FRW cosmologies with
Chiellini damping in comoving time".
3956:
3790:
7938:
7387:
7321:
7091:
6309:
6284:
6259:
6234:
5794:) that assume that the only contributions to stress–energy are cold matter ("dust"), radiation, and a cosmological constant.
3995:
731:
of the universe as a function of time. Depending on geographical or historical preferences, the set of the four scientists –
692:
7422:
7194:
Secrest, Nathan J.; Hausegger, Sebastian von; Rameez, Mohamed; Mohayaee, Roya; Sarkar, Subir; Colin, Jacques (2021-02-25).
6048:
6037:
5879:
335:
6325:
4327:
6115:
multiple connectedness in the positive curvature case and says that "we are still free to restore" simple connectedness.
7675:
6207:
6056:
5727:
published his work, it remained relatively unnoticed by his contemporaries. Friedmann was in direct communication with
3707:
2894:
8046:
7680:
5806:
3733:
1489:{\displaystyle \mathrm {d} \mathbf {\Sigma } ^{2}=\mathrm {d} r^{2}+S_{k}(r)^{2}\,\mathrm {d} \mathbf {\Omega } ^{2}}
1354:
544:
5367:
1788:
1502:
1287:
966:
796:
704:
594:
8267:
8000:
5088:
The second equation states that both the energy density and the pressure cause the expansion rate of the universe
4296:
4013:
3626:
2452:
7713:
6915:
6180:
Ellis, G. F. R.; van Elst, H. (1999). "Cosmological models (Cargèse lectures 1998)". In Marc Lachièze-Rey (ed.).
3920:
2405:
919:{\displaystyle -c^{2}\mathrm {d} \tau ^{2}=-c^{2}\mathrm {d} t^{2}+{a(t)}^{2}\mathrm {d} \mathbf {\Sigma } ^{2},}
177:
7816:
2716:{\displaystyle R_{tt}=-3{\frac {\ddot {a}}{a}},\quad R_{xx}=R_{yy}=R_{zz}=c^{-2}(a{\ddot {a}}+2{\dot {a}}^{2})}
539:
263:
253:
5823:
5357:
In fact, in order to get a term that causes an acceleration of the universe expansion, it is enough to have a
1823:
may be taken to belong to the set {−1 ,0, +1} (for negative, zero, and positive curvature respectively). Then
8247:
7989:
7894:
7824:
4422:
4007:
3715:
182:
105:
7957:
5721:
first derived the main results of the FLRW model in 1922 and 1924. Although the prestigious physics journal
4053:
7811:
6942:
Migkas, K.; Schellenberger, G.; Reiprich, T. H.; Pacaud, F.; Ramos-Ceja, M. E.; Lovisari, L. (April 2020).
5676:
The first equation says that the decrease in the mass contained in a fixed cube (whose side is momentarily
5121:, with pressure playing a similar role to that of energy (or mass) density, according to the principles of
4073:
2873:{\displaystyle R=6c^{-2}\left({\frac {{\ddot {a}}(t)}{a(t)}}+{\frac {{\dot {a}}^{2}(t)}{a^{2}(t)}}\right).}
8358:
7852:
7586:
7571:
7100:
7077:
6000:
is the density of space of this universe. The numerical value of
Einstein's radius is of the order of 10
5681:
5078:
4314:
3805:
3725:
2342:{\displaystyle \mathrm {d} \mathbf {\Sigma } ^{2}=\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}.}
587:
563:
110:
5666:{\displaystyle {\frac {{\dot {a}}^{2}}{2}}-{\frac {\kappa c^{4}a^{3}\rho }{6a}}=-{\frac {kc^{2}}{2}}\,.}
5117:
to decrease, i.e., both cause a deceleration in the expansion of the universe. This is a consequence of
3411:{\displaystyle R_{\phi \phi }=r^{2}(c^{-2}(a(t){\ddot {a}}(t)+2{\dot {a}}^{2}(t))+2k)\sin ^{2}(\theta )}
2365:
932:
8258:
8139:
7581:
7534:
4815:
4504:
4218:
3766:
523:
329:
309:
117:
62:
5255:
8169:
7795:
7476:
6944:"Probing cosmic isotropy with a new X-ray galaxy cluster sample through the L X – T scaling relation"
4018:
3830:
1273:
712:
155:
5873:
in the
Friedmann equation, the radius of curvature of space of this universe (Einstein's radius) is
5723:
5077:
The first equation can be derived also from thermodynamical considerations and is equivalent to the
1563:
8194:
7744:
7504:
7415:
5311:
5085:(which is implicitly assumed in the derivation of the Friedmann–Lemaître–Robertson–Walker metric).
4429:
4253:
4243:
4093:
3910:
1003:
728:
696:
324:
89:
5698:
term is assumed to be treated as dark energy and thus merged into the density and pressure terms.
8363:
7923:
7845:
7465:
7456:
6030:
5071:
4811:
4238:
3972:
2214:
292:
172:
5091:
8149:
8012:
7918:
7876:
7550:
7514:
6815:
6026:
5977:
5810:
5782:
since they proved its generic properties, is different from the dynamical "Friedmann–Lemaître"
5771:
5547:{\displaystyle -a^{3}{\dot {\rho }}=3a^{2}{\dot {a}}\rho +{\frac {3a^{2}p{\dot {a}}}{c^{2}}}\,}
5419:
4392:
4208:
4193:
4038:
3800:
744:
724:
1214:
is a constant representing the curvature of the space. There are two common unit conventions:
8154:
7759:
7698:
6779:
6738:
6699:
6224:
6013:
5695:
5246:
5142:
5126:
4183:
3751:
497:
299:
241:
7164:
7103:(2007), "Proving Almost-Homogeneity of the Universe: An Almost Ehlers-Geren-Sachs Theorem",
6969:
8059:
7985:
7829:
7431:
7344:
7272:
7217:
7160:
7112:
7033:
6965:
6880:
6829:
6788:
6747:
6708:
6677:
6648:
6614:
6565:
6519:
6481:
6430:
6368:
6195:
6146:
6063:
5983:
5746:, arrived independently at results similar to those of Friedmann and published them in the
5688:
4914:{\displaystyle {\dot {\rho }}=-3{\frac {\dot {a}}{a}}\left(\rho +{\frac {p}{c^{2}}}\right)}
4003:
3099:{\displaystyle R_{rr}={\frac {c^{-2}(a(t){\ddot {a}}(t)+2{\dot {a}}^{2}(t))+2k}{1-kr^{2}}}}
2520:
510:
482:
304:
7382:(Repr ed.). Oxford : New York: Clarendon Press ; Oxford University Press.
4363:
3244:{\displaystyle R_{\theta \theta }=r^{2}(c^{-2}(a(t){\ddot {a}}(t)+2{\dot {a}}^{2}(t))+2k)}
787:. The FLRW model was developed independently by the named authors in the 1920s and 1930s.
8:
8332:
8134:
8129:
8119:
7951:
7900:
7718:
7703:
7529:
7460:
7408:
6920:
6774:
6733:
6694:
6553:
6507:
6052:
5767:
5718:
5358:
4803:
4500:
4355:
4198:
4168:
4163:
3915:
1851:
1371:
740:
732:
402:
372:
319:
275:
167:
57:
8124:
7348:
7276:
7221:
7116:
7037:
6884:
6833:
6792:
6751:
6712:
6681:
6665:
6652:
6636:
6618:
6600:
6569:
6556:(1924). "Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes".
6523:
6485:
6434:
6372:
6199:
6150:
6047:, uses the FLRW metric. By combining the observation data from some experiments such as
5739:
4389:
does require
Einstein's field equations together with a way of calculating the density,
4178:
4148:
736:
442:
8320:
8308:
8199:
8099:
8023:
7881:
7862:
7856:
7807:
7749:
7658:
7576:
7494:
7471:
7439:
7376:
7288:
7262:
7235:
7207:
7176:
7150:
7057:
7023:
6989:
6955:
6896:
6870:
6581:
6535:
6454:
6420:
6392:
6358:
6349:
Rosu, H. C.; Ojeda-May, P. (June 2006). "Supersymmetry of FRW Barotropic Cosmologies".
6185:
6162:
6136:
5959:
5939:
5351:
5122:
5053:
4248:
4133:
4058:
3905:
3810:
3795:
3756:
3618:
1763:
1238:
780:
700:
462:
432:
397:
367:
314:
258:
27:
7396:(See Chapter 23 for a particularly clear and concise introduction to the FLRW models.)
6472:
McCrea, W. H.; Milne, E. A. (1934). "Newtonian universes and the curvature of space".
16:
Metric based on the exact solution of Einstein's field equations of general relativity
8209:
8041:
8033:
7634:
7591:
7383:
7362:
7317:
7292:
7239:
7180:
7087:
7061:
7049:
6993:
6981:
6900:
6585:
6539:
6446:
6396:
6384:
6305:
6280:
6255:
6230:
6203:
6166:
6158:
5706:
5082:
4078:
3853:
1898:
1813:
1312:
688:
492:
8079:
6943:
6458:
6025:
Is the universe homogeneous and isotropic at large enough scales, as claimed by the
4128:
8284:
8114:
8104:
8051:
8028:
7606:
7352:
7280:
7225:
7168:
7120:
7041:
6973:
6888:
6837:
6796:
6755:
6716:
6622:
6573:
6527:
6489:
6438:
6376:
6154:
6044:
5814:
5758:, and in 1930–31 Lemaître's paper was translated into English and published in the
5755:
4830:
The pair of equations given above is equivalent to the following pair of equations
4798:
4507:
is similarly assumed to be isotropic and homogeneous. The resulting equations are:
4203:
4158:
4138:
4083:
960:
952:
784:
625:
575:
377:
213:
82:
8174:
7172:
6977:
4223:
387:
362:
8189:
8164:
8089:
8084:
7967:
7928:
7890:
7834:
7708:
7644:
7311:
6127:
5728:
4258:
4233:
4118:
4113:
3977:
3858:
3820:
956:
502:
437:
422:
407:
392:
382:
246:
143:
7972:
6892:
4028:
1260:
because it is equal to the measured circumference of a circle (at that value of
8272:
8214:
7886:
7870:
7866:
7769:
7739:
7596:
7499:
7284:
7230:
7195:
7045:
4301:
4268:
4263:
3951:
3815:
720:
487:
447:
6841:
6493:
6442:
6380:
6332:
8342:
8204:
8184:
8179:
8094:
7962:
7790:
7734:
7566:
7519:
7366:
7357:
7332:
7053:
6985:
6627:
6450:
6388:
4228:
4143:
4123:
4048:
3946:
3825:
2208:
1257:
472:
457:
357:
4188:
783:, although such a description is also associated with the further developed
8296:
8224:
8144:
8109:
7639:
7601:
5751:
5702:
4173:
4153:
2550:
1910:
1014:
In reduced-circumference polar coordinates the spatial metric has the form
477:
452:
427:
412:
268:
5775:
general relativity, which were always assumed by Friedmann and Lemaître).
8008:
7977:
7524:
5299:
5130:
5118:
4063:
4033:
225:
218:
5742:, a Belgian priest, astronomer and periodic professor of physics at the
2549:
FLRW space using Cartesian coordinates, the surviving components of the
8219:
7785:
7629:
7624:
7252:
6590:
English trans. in 'General Relativity and Gravitation' 1999 vol.31, 31–
6577:
6531:
6001:
5817:
that was supposed to represent our universe in idealized form. Putting
5192:{\displaystyle \rho \rightarrow \rho -{\frac {\Lambda }{\kappa c^{2}}}}
4273:
3835:
3761:
3693:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
467:
6363:
6190:
6141:
1785:) is often chosen to equal 1 in the present cosmological era, so that
1284:) is often chosen to equal 1 in the present cosmological era, so that
7616:
7013:
6875:
5709:
effects. So they may cause a deviation from the Friedmann equations.
4492:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }}
3844:
708:
417:
5754:
in the late 1920s, Lemaître's results were noticed in particular by
8159:
7509:
7267:
7212:
7155:
7125:
7028:
6960:
6801:
6760:
6720:
6425:
6017:
4794:
4213:
4023:
3863:
800:
716:
150:
52:
45:
6941:
2504:
is one of the radial coordinates defined above, but this is rare.
951:
ranges over a 3-dimensional space of uniform curvature, that is,
1326:(for negative, zero, and positive curvature respectively). Then
7839:
7193:
637:
7400:
6277:
Spacetime and geometry: an introduction to general relativity
7140:
6125:
Lachieze-Rey, M.; Luminet, J.-P. (1995). "Cosmic Topology".
2197:{\displaystyle S_{k}(r)=r\;\mathrm {sinc} \,(r{\sqrt {k}}),}
35:
7820:
5235:{\displaystyle p\rightarrow p+{\frac {\Lambda }{\kappa }}.}
4819:
1722:
675:
666:
657:
646:
8291:
7313:
The measure of the universe: a history of modern cosmology
6279:. New York: Cambridge University Press. pp. 329–333.
5145:
term can be omitted if we make the following replacements
1009:
6184:. NATO Science Series C. Vol. 541. pp. 1–116.
6111:
For an early reference, see Robertson (1935); Robertson
4810:, i.e., a model that follows the FLRW metric apart from
2235:
is one of the imaginary, zero or real square roots of
8256:
6254:. Chicago: University of Chicago Press. p. 116.
6066:
5986:
5962:
5942:
5926:{\displaystyle R_{\text{E}}=c/{\sqrt {4\pi G\rho }},}
5882:
5826:
5561:
5439:
5370:
5314:
5258:
5206:
5154:
5094:
5056:
4928:
4839:
4651:
4516:
4437:
4395:
4366:
3636:
3430:
3258:
3113:
2957:
2897:
2735:
2562:
2523:
2455:
2408:
2368:
2264:
2217:
2133:
1921:
1791:
1535:
1505:
1403:
1290:
1023:
969:
935:
812:
649:
631:
7098:
6304:. Chicago: University of Chicago Press. p. 97.
5797:
1746:
may be taken to have units of length, in which case
1374:, i.e. a 3-sphere with opposite points identified.)
1221:
may be taken to have units of length, in which case
672:
669:
654:
640:
634:
7196:"A Test of the Cosmological Principle with Quasars"
7139:See Siewert et al. for a recent summary of results
6124:
663:
660:
643:
628:
7375:
6079:
5992:
5968:
5948:
5925:
5862:
5665:
5546:
5407:
5339:
5287:
5234:
5191:
5109:
5062:
5039:
4913:
4782:
4636:
4491:
4413:
4381:
3692:
3599:
3410:
3243:
3098:
2942:
2872:
2715:
2541:
2489:
2439:
2393:
2341:
2227:
2196:
2113:
1804:
1739:As before, there are two common unit conventions:
1728:
1518:
1488:
1303:
1200:
982:
943:
918:
7337:Monthly Notices of the Royal Astronomical Society
6606:Monthly Notices of the Royal Astronomical Society
5778:This solution, often called the Robertson–Walker
5761:Monthly Notices of the Royal Astronomical Society
1893:Though it is usually defined piecewise as above,
8340:
6818:(1937), "On Milne's theory of world-structure",
2943:{\displaystyle R_{tt}=-3{\frac {\ddot {a}}{a}},}
994:– all of the time dependence is in the function
7333:"Classification of Uniform Cosmological Models"
6670:Annales de la Société Scientifique de Bruxelles
6641:Annales de la Société Scientifique de Bruxelles
6222:
5748:Annales de la Société Scientifique de Bruxelles
5081:, assuming the expansion of the universe is an
1394:is proportional to radial distance; this gives
7007:
6821:Proceedings of the London Mathematical Society
6777:(1936), "Kinematics and world structure III",
6410:
5408:{\displaystyle p<-{\frac {\rho c^{2}}{3}}.}
4793:These equations are the basis of the standard
1805:{\displaystyle \mathrm {d} \mathbf {\Sigma } }
1519:{\displaystyle \mathrm {d} \mathbf {\Omega } }
1304:{\displaystyle \mathrm {d} \mathbf {\Sigma } }
983:{\displaystyle \mathrm {d} \mathbf {\Sigma } }
795:The FLRW metric starts with the assumption of
7416:
7373:
6856:
6854:
6852:
6850:
6736:(1936), "Kinematics and world structure II",
6348:
6179:
6043:The current standard model of cosmology, the
5298:which is an equation of state of vacuum with
4335:
1377:
595:
7075:
6351:International Journal of Theoretical Physics
5070:, the spatial curvature index, serving as a
2490:{\displaystyle z=r\sin \theta \sin \phi \,,}
6471:
6019:
5684:) contained within a part of the universe.
5425:
2440:{\displaystyle y=r\sin \theta \cos \phi \,}
2239:. These definitions are valid for all
7423:
7409:
6847:
6697:(1935), "Kinematics and world structure",
4342:
4328:
2512:
2159:
616:Friedmann–Lemaître–Robertson–Walker metric
602:
588:
34:
7356:
7266:
7229:
7211:
7154:
7124:
7027:
6959:
6874:
6800:
6773:
6759:
6732:
6693:
6626:
6552:
6506:
6424:
6362:
6189:
6140:
5659:
5543:
5281:
4797:cosmological model including the current
2483:
2436:
2387:
2174:
1664:
1575:
1468:
1179:
7330:
6913:
6664:
6635:
6599:
6510:(1922). "Über die Krümmung des Raumes".
5863:{\displaystyle {\dot {a}}={\ddot {a}}=0}
5136:
4428:This metric has an analytic solution to
2883:
2246:
1873:is essentially a third angle along with
8349:Coordinate charts in general relativity
7083:The large scale structure of space-time
6860:
6274:
6223:Bergström, Lars; Goobar, Ariel (2008).
6182:Theoretical and Observational Cosmology
5131:causes an acceleration in the expansion
1264:), centered at the origin, divided by 2
1010:Reduced-circumference polar coordinates
8341:
6814:
6173:
1854:of the space, and may also be written
1357:of the space, and may also be written
775:). This model is sometimes called the
8354:Exact solutions in general relativity
7404:
7309:
6216:
6118:
6916:"Do We Live in a Lopsided Universe?"
6299:
6249:
7995:Tolman–Oppenheimer–Volkoff equation
7948:Friedmann–Lemaître–Robertson–Walker
6863:Journal of High Energy Astrophysics
6226:Cosmology and particle astrophysics
6038:(more unsolved problems in physics)
5786:, which are specific solutions for
5712:
13:
7303:
6668:(1933), "l'Univers en expansion",
5221:
5169:
5018:
4742:
4587:
4454:
3653:
2394:{\displaystyle x=r\cos \theta \,,}
2322:
2304:
2286:
2266:
2170:
2167:
2164:
2161:
1960:
1793:
1507:
1470:
1425:
1405:
1322:may be taken to belong to the set
1292:
1181:
1146:
1126:
1100:
1048:
1025:
971:
944:{\displaystyle \mathbf {\Sigma } }
897:
858:
827:
330:2dF Galaxy Redshift Survey ("2dF")
14:
8375:
7765:Hamilton–Jacobi–Einstein equation
7378:Introducing Einstein's relativity
6007:
5978:Newtonian constant of gravitation
5803:Einstein's radius of the universe
5798:Einstein's radius of the universe
5738:Friedmann died in 1925. In 1927,
5418:Such a field is sometimes called
5305:An attempt to generalize this to
4825:
790:
545:Timeline of cosmological theories
310:Cosmic Background Explorer (COBE)
8326:
8314:
8302:
8290:
8278:
8266:
8243:
8242:
7316:. New York: Dover Publications.
6914:Billings, Lee (April 15, 2020).
6474:Quarterly Journal of Mathematics
6409:Their solutions can be found in
5288:{\displaystyle p=-\rho c^{2}\,,}
4309:
4308:
4295:
3625:
2272:
1798:
1512:
1476:
1411:
1297:
1256:is sometimes called the reduced
1132:
1106:
1031:
976:
937:
903:
624:
569:
558:
557:
7246:
7187:
7133:
7068:
6935:
6907:
6808:
6767:
6726:
6687:
6658:
6593:
6546:
6500:
6465:
6403:
4812:primordial density fluctuations
2605:
2207:where sinc is the unnormalized
1119:
325:Sloan Digital Sky Survey (SDSS)
178:Future of an expanding universe
7572:Mass–energy equivalence (E=mc)
7430:
7086:, Cambridge University Press,
6342:
6318:
6293:
6268:
6243:
6105:
5689:gravitational potential energy
5354:without further modification.
5210:
5158:
4423:cosmological equation of state
4405:
4399:
4376:
4370:
3583:
3577:
3552:
3546:
3531:
3525:
3494:
3488:
3480:
3474:
3405:
3399:
3383:
3371:
3368:
3362:
3334:
3328:
3313:
3307:
3301:
3285:
3238:
3226:
3223:
3217:
3189:
3183:
3168:
3162:
3156:
3140:
3060:
3057:
3051:
3023:
3017:
3002:
2996:
2990:
2856:
2850:
2835:
2829:
2798:
2792:
2784:
2778:
2710:
2667:
2536:
2524:
2188:
2175:
2150:
2144:
2033:
2018:
1978:
1968:
1938:
1932:
1909:. It can also be written as a
1766:of the space at the time when
1702:
1696:
1688:
1679:
1657:
1649:
1603:
1590:
1552:
1546:
1459:
1452:
1241:of the space at the time when
886:
880:
540:History of the Big Bang theory
336:Wilkinson Microwave Anisotropy
1:
7255:Classical and Quantum Gravity
7099:Stoeger, W. R.; Maartens, R;
7016:Classical and Quantum Gravity
6098:
6004:, or 10 billion light years.
5744:Catholic University of Leuven
5340:{\displaystyle p=w\rho c^{2}}
532:Discovery of cosmic microwave
183:Ultimate fate of the universe
7143:Astronomy & Astrophysics
6948:Astronomy & Astrophysics
6331:. p. 23. Archived from
6159:10.1016/0370-1573(94)00085-H
6055:with theoretical results of
3610:
2507:
1885:may be used instead of
1835:) has units of length. When
1338:) has units of length. When
715:(or otherwise, contracting)
7:
7587:Relativistic Doppler effect
7173:10.1051/0004-6361/202039840
6978:10.1051/0004-6361/201936602
6893:10.1016/j.jheap.2022.04.002
6020:Unsolved problem in physics
5682:first law of thermodynamics
5079:first law of thermodynamics
4806:is well approximated by an
3806:Gravitational time dilation
2228:{\displaystyle {\sqrt {k}}}
1390:coordinates the coordinate
747:– are variously grouped as
300:Black Hole Initiative (BHI)
10:
8380:
8058:In computational physics:
7582:Relativity of simultaneity
7310:North, John David (1990).
6057:Ehlers–Geren–Sachs theorem
6011:
5110:{\displaystyle {\dot {a}}}
4430:Einstein's field equations
4353:
3926:Mathisson–Papapetrou–Dixon
3767:Pseudo-Riemannian manifold
3421:and the Ricci scalar is
2726:and the Ricci scalar is
1819:Alternatively, as before,
1378:Hyperspherical coordinates
753:Friedmann–Robertson–Walker
63:Chronology of the universe
8240:
8072:
7937:
7909:
7895:Lense–Thirring precession
7778:
7727:
7689:
7668:
7657:
7615:
7559:
7543:
7485:
7477:Doubly special relativity
7449:
7438:
7200:The Astrophysical Journal
6443:10.1142/S021773231550100x
6381:10.1007/s10773-006-9123-2
6275:Carroll, Sean M. (2019).
5717:The Soviet mathematician
4414:{\displaystyle \rho (t),}
2255:= 0 one may write simply
1274:Schwarzschild coordinates
703:. The metric describes a
156:Expansion of the universe
7755:Post-Newtonian formalism
7745:Einstein field equations
7681:Mathematical formulation
7505:Hyperbolic orthogonality
7331:Harrison, E. R. (1967).
7285:10.1088/1361-6382/ac1a81
7231:10.3847/2041-8213/abdd40
7046:10.1088/1361-6382/ac1a81
6558:Zeitschrift für Physik A
6512:Zeitschrift für Physik A
6413:Modern Physics Letters A
6300:Wald, Robert M. (1984).
6250:Wald, Robert M. (1984).
5426:Newtonian interpretation
5074:for the first equation.
3931:Hamilton–Jacobi–Einstein
3911:Einstein field equations
3734:Mathematical formulation
2352:This can be extended to
1750:has units of length and
1225:has units of length and
697:Einstein field equations
320:Planck space observatory
106:Gravitational wave (GWB)
7466:Galilean transformation
7457:Principle of relativity
7374:D'Inverno, Ray (1992).
7165:2021A&A...653A...9S
6970:2020A&A...636A..15M
6842:10.1112/plms/s2-42.1.90
6494:10.1093/qmath/os-5.1.73
5956:is the speed of light,
5072:constant of integration
173:Inhomogeneous cosmology
7551:Lorentz transformation
7358:10.1093/mnras/137.1.69
6628:10.1093/mnras/91.5.483
6081:
6027:cosmological principle
5994:
5970:
5950:
5927:
5864:
5772:Arthur Geoffrey Walker
5733:Zeitschrift für Physik
5724:Zeitschrift für Physik
5667:
5548:
5409:
5341:
5289:
5236:
5193:
5111:
5064:
5041:
4915:
4784:
4638:
4505:energy–momentum tensor
4493:
4415:
4383:
3801:Gravitational redshift
3694:
3601:
3412:
3245:
3100:
2944:
2874:
2717:
2543:
2491:
2441:
2395:
2343:
2229:
2198:
2115:
1964:
1806:
1777:. Where appropriate,
1730:
1520:
1490:
1305:
1276:). Where appropriate,
1202:
984:
945:
920:
745:Arthur Geoffrey Walker
723:, but not necessarily
8019:Weyl−Lewis−Papapetrou
7760:Raychaudhuri equation
7699:Equivalence principle
7076:Hawking, Stephen W.;
6780:Astrophysical Journal
6739:Astrophysical Journal
6700:Astrophysical Journal
6082:
6080:{\displaystyle H_{0}}
6014:Shape of the universe
5995:
5993:{\displaystyle \rho }
5971:
5951:
5928:
5865:
5705:, one cannot neglect
5696:cosmological constant
5668:
5549:
5410:
5342:
5290:
5247:cosmological constant
5237:
5194:
5143:cosmological constant
5137:Cosmological constant
5129:, on the other hand,
5127:cosmological constant
5112:
5065:
5042:
4916:
4785:
4639:
4494:
4416:
4384:
4089:Weyl−Lewis−Papapetrou
4044:Kerr–Newman–de Sitter
3864:Einstein–Rosen bridge
3796:Gravitational lensing
3752:Equivalence principle
3695:
3602:
3413:
3246:
3101:
2945:
2884:Spherical coordinates
2875:
2718:
2544:
2542:{\displaystyle (k=0)}
2513:Cartesian coordinates
2492:
2442:
2396:
2344:
2247:Cartesian coordinates
2230:
2199:
2116:
1944:
1807:
1731:
1521:
1491:
1306:
1203:
985:
946:
921:
264:Large-scale structure
242:Shape of the universe
8060:Numerical relativity
7901:pulsar timing arrays
6554:Friedmann, Alexander
6508:Friedmann, Alexander
6064:
5984:
5960:
5940:
5880:
5824:
5731:, who, on behalf of
5559:
5437:
5368:
5312:
5256:
5204:
5152:
5092:
5054:
4926:
4837:
4649:
4514:
4435:
4393:
4382:{\displaystyle a(t)}
4364:
4019:Einstein–Rosen waves
3745:Fundamental concepts
3634:
3428:
3256:
3111:
2955:
2895:
2733:
2560:
2521:
2453:
2406:
2366:
2262:
2215:
2131:
1919:
1789:
1533:
1503:
1401:
1388:curvature-normalized
1288:
1021:
967:
933:
810:
576:Astronomy portal
534:background radiation
511:List of cosmologists
7952:Friedmann equations
7846:Hulse–Taylor binary
7808:Gravitational waves
7704:Riemannian geometry
7530:Proper acceleration
7515:Maxwell's equations
7461:Galilean relativity
7349:1967MNRAS.137...69H
7277:2021CQGra..38r4001K
7222:2021ApJ...908L..51S
7117:1995ApJ...443....1S
7078:Ellis, George F. R.
7038:2021CQGra..38r4001K
6954:(April 2020): A15.
6921:Scientific American
6885:2022JHEAp..34...49A
6834:1937PLMS...42...90W
6793:1936ApJ....83..257R
6752:1936ApJ....83..187R
6713:1935ApJ....82..284R
6682:1933ASSB...53...51L
6653:1927ASSB...47...49L
6619:1931MNRAS..91..483L
6570:1924ZPhy...21..326F
6524:1922ZPhy...10..377F
6486:1934QJMat...5...73M
6435:2015MPLA...3050100R
6373:2006IJTP...45.1152R
6200:1999ASIC..541....1E
6151:1995PhR...254..135L
5813:, a long-abandoned
5811:Einstein's universe
5807:radius of curvature
5768:Howard P. Robertson
5719:Alexander Friedmann
4804:observable universe
4501:Friedmann equations
4356:Friedmann equations
3973:Kaluza–Klein theory
3859:Minkowski spacetime
3811:Gravitational waves
1862:). Note that when
1852:radius of curvature
1355:radius of curvature
990:does not depend on
741:Howard P. Robertson
733:Alexander Friedmann
276:Structure formation
168:Friedmann equations
58:Age of the universe
22:Part of a series on
8359:Physical cosmology
8001:Reissner–Nordström
7919:Brans–Dicke theory
7750:Linearized gravity
7577:Length contraction
7495:Frame of reference
7472:Special relativity
7074:See pp. 351ff. in
6578:10.1007/BF01328280
6532:10.1007/BF01332580
6302:General relativity
6252:General relativity
6077:
5990:
5966:
5946:
5923:
5860:
5663:
5544:
5405:
5352:general invariance
5337:
5285:
5232:
5189:
5123:general relativity
5107:
5060:
5037:
4911:
4780:
4634:
4489:
4411:
4379:
4302:Physics portal
4074:Oppenheimer–Snyder
4014:Reissner–Nordström
3906:Linearized gravity
3854:Spacetime diagrams
3757:Special relativity
3690:
3619:General relativity
3597:
3408:
3241:
3096:
2940:
2870:
2713:
2539:
2487:
2437:
2391:
2339:
2225:
2194:
2111:
1802:
1764:Gaussian curvature
1726:
1721:
1516:
1486:
1301:
1239:Gaussian curvature
1198:
980:
941:
916:
769:Friedmann–Lemaître
701:general relativity
315:Dark Energy Survey
259:Large quasar group
28:Physical cosmology
8254:
8253:
8068:
8067:
8047:Ozsváth–Schücking
7653:
7652:
7635:Minkowski diagram
7592:Thomas precession
7535:Relativistic mass
7389:978-0-19-859686-8
7323:978-0-486-66517-7
7093:978-0-521-09906-6
6666:Lemaître, Georges
6637:Lemaître, Georges
6601:Lemaître, Georges
6311:978-0-226-87032-8
6286:978-1-108-48839-6
6261:978-0-226-87032-8
6236:978-3-540-32924-4
5969:{\displaystyle G}
5949:{\displaystyle c}
5918:
5890:
5851:
5836:
5657:
5629:
5586:
5574:
5541:
5527:
5490:
5462:
5400:
5227:
5187:
5133:of the universe.
5104:
5083:adiabatic process
5063:{\displaystyle k}
5035:
5005:
4972:
4944:
4939:
4904:
4876:
4871:
4849:
4808:almost FLRW model
4737:
4695:
4690:
4670:
4665:
4629:
4604:
4579:
4537:
4532:
4352:
4351:
3985:
3984:
3871:
3870:
3587:
3556:
3516:
3498:
3471:
3452:
3353:
3325:
3208:
3180:
3094:
3042:
3014:
2935:
2930:
2860:
2820:
2802:
2775:
2701:
2682:
2600:
2595:
2223:
2186:
2103:
2071:
2040:
1899:analytic function
1814:comoving distance
1700:
1661:
1601:
1572:
1526:is as before and
1313:comoving distance
1123:
1084:
1002:), known as the "
612:
611:
283:
282:
125:
124:
8371:
8331:
8330:
8329:
8319:
8318:
8317:
8307:
8306:
8305:
8295:
8294:
8283:
8282:
8281:
8271:
8270:
8262:
8246:
8245:
8029:van Stockum dust
7801:Two-body problem
7719:Mach's principle
7666:
7665:
7607:Terrell rotation
7447:
7446:
7425:
7418:
7411:
7402:
7401:
7393:
7381:
7370:
7360:
7327:
7297:
7296:
7270:
7250:
7244:
7243:
7233:
7215:
7191:
7185:
7184:
7158:
7137:
7131:
7129:
7128:
7096:
7072:
7066:
7065:
7031:
7011:
7005:
7004:
7002:
7000:
6963:
6939:
6933:
6932:
6930:
6928:
6911:
6905:
6904:
6878:
6858:
6845:
6844:
6812:
6806:
6805:
6804:
6775:Robertson, H. P.
6771:
6765:
6764:
6763:
6734:Robertson, H. P.
6730:
6724:
6723:
6695:Robertson, H. P.
6691:
6685:
6684:
6662:
6656:
6655:
6631:
6630:
6597:
6591:
6589:
6550:
6544:
6543:
6504:
6498:
6497:
6469:
6463:
6462:
6428:
6407:
6401:
6400:
6366:
6357:(6): 1152–1157.
6346:
6340:
6339:
6338:on Jan 11, 2020.
6337:
6330:
6322:
6316:
6315:
6297:
6291:
6290:
6272:
6266:
6265:
6247:
6241:
6240:
6220:
6214:
6213:
6193:
6177:
6171:
6170:
6144:
6122:
6116:
6109:
6094:
6092:
6086:
6084:
6083:
6078:
6076:
6075:
6045:Lambda-CDM model
6021:
5999:
5997:
5996:
5991:
5975:
5973:
5972:
5967:
5955:
5953:
5952:
5947:
5932:
5930:
5929:
5924:
5919:
5905:
5903:
5892:
5891:
5888:
5869:
5867:
5866:
5861:
5853:
5852:
5844:
5838:
5837:
5829:
5770:from the US and
5756:Arthur Eddington
5740:Georges Lemaître
5713:Name and history
5672:
5670:
5669:
5664:
5658:
5653:
5652:
5651:
5638:
5630:
5628:
5620:
5616:
5615:
5606:
5605:
5592:
5587:
5582:
5581:
5576:
5575:
5567:
5563:
5553:
5551:
5550:
5545:
5542:
5540:
5539:
5530:
5529:
5528:
5520:
5514:
5513:
5500:
5492:
5491:
5483:
5480:
5479:
5464:
5463:
5455:
5452:
5451:
5414:
5412:
5411:
5406:
5401:
5396:
5395:
5394:
5381:
5346:
5344:
5343:
5338:
5336:
5335:
5294:
5292:
5291:
5286:
5280:
5279:
5241:
5239:
5238:
5233:
5228:
5220:
5198:
5196:
5195:
5190:
5188:
5186:
5185:
5184:
5168:
5116:
5114:
5113:
5108:
5106:
5105:
5097:
5069:
5067:
5066:
5061:
5046:
5044:
5043:
5038:
5036:
5031:
5030:
5029:
5016:
5011:
5007:
5006:
5004:
5003:
4994:
4986:
4973:
4968:
4967:
4966:
4953:
4945:
4940:
4932:
4930:
4920:
4918:
4917:
4912:
4910:
4906:
4905:
4903:
4902:
4890:
4877:
4872:
4864:
4862:
4851:
4850:
4842:
4789:
4787:
4786:
4781:
4773:
4772:
4754:
4753:
4738:
4736:
4735:
4726:
4725:
4724:
4711:
4706:
4705:
4700:
4696:
4691:
4683:
4681:
4671:
4666:
4658:
4656:
4643:
4641:
4640:
4635:
4630:
4625:
4624:
4623:
4610:
4605:
4600:
4599:
4598:
4585:
4580:
4578:
4577:
4568:
4567:
4566:
4553:
4548:
4547:
4542:
4538:
4533:
4525:
4523:
4498:
4496:
4495:
4490:
4488:
4487:
4469:
4468:
4450:
4449:
4420:
4418:
4417:
4412:
4388:
4386:
4385:
4380:
4344:
4337:
4330:
4317:
4312:
4311:
4304:
4300:
4299:
4084:van Stockum dust
4069:Robertson–Walker
3895:
3894:
3785:
3784:
3699:
3697:
3696:
3691:
3689:
3688:
3676:
3668:
3667:
3649:
3648:
3629:
3615:
3614:
3606:
3604:
3603:
3598:
3593:
3589:
3588:
3586:
3576:
3575:
3562:
3557:
3555:
3545:
3544:
3534:
3524:
3523:
3518:
3517:
3509:
3504:
3499:
3497:
3483:
3473:
3472:
3464:
3460:
3453:
3451:
3450:
3438:
3417:
3415:
3414:
3409:
3395:
3394:
3361:
3360:
3355:
3354:
3346:
3327:
3326:
3318:
3300:
3299:
3284:
3283:
3271:
3270:
3250:
3248:
3247:
3242:
3216:
3215:
3210:
3209:
3201:
3182:
3181:
3173:
3155:
3154:
3139:
3138:
3126:
3125:
3105:
3103:
3102:
3097:
3095:
3093:
3092:
3091:
3072:
3050:
3049:
3044:
3043:
3035:
3016:
3015:
3007:
2989:
2988:
2975:
2970:
2969:
2949:
2947:
2946:
2941:
2936:
2931:
2923:
2921:
2910:
2909:
2879:
2877:
2876:
2871:
2866:
2862:
2861:
2859:
2849:
2848:
2838:
2828:
2827:
2822:
2821:
2813:
2808:
2803:
2801:
2787:
2777:
2776:
2768:
2764:
2757:
2756:
2722:
2720:
2719:
2714:
2709:
2708:
2703:
2702:
2694:
2684:
2683:
2675:
2666:
2665:
2650:
2649:
2634:
2633:
2618:
2617:
2601:
2596:
2588:
2586:
2575:
2574:
2548:
2546:
2545:
2540:
2496:
2494:
2493:
2488:
2446:
2444:
2443:
2438:
2400:
2398:
2397:
2392:
2358:
2348:
2346:
2345:
2340:
2335:
2334:
2325:
2317:
2316:
2307:
2299:
2298:
2289:
2281:
2280:
2275:
2269:
2234:
2232:
2231:
2226:
2224:
2219:
2203:
2201:
2200:
2195:
2187:
2182:
2173:
2143:
2142:
2120:
2118:
2117:
2112:
2104:
2099:
2098:
2097:
2088:
2087:
2077:
2072:
2067:
2066:
2065:
2052:
2041:
2039:
2016:
2015:
2014:
1996:
1995:
1986:
1985:
1966:
1963:
1958:
1931:
1930:
1868:
1841:
1827:is unitless and
1811:
1809:
1808:
1803:
1801:
1796:
1776:
1758:) is unitless.
1735:
1733:
1732:
1727:
1725:
1724:
1701:
1699:
1691:
1686:
1672:
1671:
1662:
1660:
1652:
1647:
1602:
1597:
1583:
1582:
1573:
1568:
1545:
1544:
1525:
1523:
1522:
1517:
1515:
1510:
1495:
1493:
1492:
1487:
1485:
1484:
1479:
1473:
1467:
1466:
1451:
1450:
1438:
1437:
1428:
1420:
1419:
1414:
1408:
1344:
1330:is unitless and
1325:
1310:
1308:
1307:
1302:
1300:
1295:
1267:
1251:
1207:
1205:
1204:
1199:
1194:
1193:
1184:
1172:
1171:
1159:
1158:
1149:
1141:
1140:
1135:
1129:
1124:
1121:
1115:
1114:
1109:
1103:
1098:
1097:
1085:
1083:
1082:
1081:
1062:
1061:
1060:
1051:
1045:
1040:
1039:
1034:
1028:
989:
987:
986:
981:
979:
974:
961:hyperbolic space
953:elliptical space
950:
948:
947:
942:
940:
925:
923:
922:
917:
912:
911:
906:
900:
895:
894:
889:
871:
870:
861:
856:
855:
840:
839:
830:
825:
824:
785:Lambda-CDM model
761:Robertson–Walker
737:Georges Lemaître
725:simply connected
686:
685:
682:
678:
677:
674:
671:
668:
665:
662:
659:
656:
652:
651:
648:
645:
642:
639:
636:
633:
630:
604:
597:
590:
574:
573:
572:
561:
560:
254:Galaxy formation
214:Lambda-CDM model
203:
202:
195:Components
77:
76:
38:
19:
18:
8379:
8378:
8374:
8373:
8372:
8370:
8369:
8368:
8339:
8338:
8337:
8327:
8325:
8315:
8313:
8303:
8301:
8289:
8279:
8277:
8265:
8257:
8255:
8250:
8236:
8064:
7968:BKL singularity
7958:Lemaître–Tolman
7933:
7929:Quantum gravity
7911:
7905:
7891:geodetic effect
7865:(together with
7835:LISA Pathfinder
7774:
7723:
7709:Penrose diagram
7691:
7685:
7660:
7649:
7645:Minkowski space
7611:
7555:
7539:
7487:
7481:
7441:
7434:
7429:
7390:
7324:
7306:
7304:Further reading
7301:
7300:
7251:
7247:
7192:
7188:
7138:
7134:
7094:
7073:
7069:
7012:
7008:
6998:
6996:
6940:
6936:
6926:
6924:
6912:
6908:
6859:
6848:
6813:
6809:
6772:
6768:
6731:
6727:
6692:
6688:
6663:
6659:
6633:translated from
6598:
6594:
6551:
6547:
6505:
6501:
6470:
6466:
6419:(20): 1550100.
6408:
6404:
6347:
6343:
6335:
6328:
6324:
6323:
6319:
6312:
6298:
6294:
6287:
6273:
6269:
6262:
6248:
6244:
6237:
6221:
6217:
6210:
6178:
6174:
6128:Physics Reports
6123:
6119:
6110:
6106:
6101:
6093:1 km/s/Mpc
6090:
6088:
6071:
6067:
6065:
6062:
6061:
6041:
6040:
6035:
6023:
6016:
6010:
5985:
5982:
5981:
5961:
5958:
5957:
5941:
5938:
5937:
5904:
5899:
5887:
5883:
5881:
5878:
5877:
5843:
5842:
5828:
5827:
5825:
5822:
5821:
5800:
5729:Albert Einstein
5715:
5647:
5643:
5639:
5637:
5621:
5611:
5607:
5601:
5597:
5593:
5591:
5577:
5566:
5565:
5564:
5562:
5560:
5557:
5556:
5535:
5531:
5519:
5518:
5509:
5505:
5501:
5499:
5482:
5481:
5475:
5471:
5454:
5453:
5447:
5443:
5438:
5435:
5434:
5428:
5390:
5386:
5382:
5380:
5369:
5366:
5365:
5361:that satisfies
5350:would not have
5331:
5327:
5313:
5310:
5309:
5275:
5271:
5257:
5254:
5253:
5245:Therefore, the
5219:
5205:
5202:
5201:
5180:
5176:
5172:
5167:
5153:
5150:
5149:
5139:
5096:
5095:
5093:
5090:
5089:
5055:
5052:
5051:
5025:
5021:
5017:
5015:
4999:
4995:
4987:
4985:
4978:
4974:
4962:
4958:
4954:
4952:
4931:
4929:
4927:
4924:
4923:
4898:
4894:
4889:
4882:
4878:
4863:
4861:
4841:
4840:
4838:
4835:
4834:
4828:
4768:
4764:
4749:
4745:
4731:
4727:
4720:
4716:
4712:
4710:
4701:
4682:
4680:
4676:
4675:
4657:
4655:
4650:
4647:
4646:
4619:
4615:
4611:
4609:
4594:
4590:
4586:
4584:
4573:
4569:
4562:
4558:
4554:
4552:
4543:
4524:
4522:
4518:
4517:
4515:
4512:
4511:
4480:
4476:
4461:
4457:
4442:
4438:
4436:
4433:
4432:
4394:
4391:
4390:
4365:
4362:
4361:
4358:
4348:
4307:
4294:
4293:
4286:
4285:
4109:
4108:
4099:
4098:
4054:Lemaître–Tolman
3999:
3998:
3987:
3986:
3978:Quantum gravity
3965:Advanced theory
3892:
3891:
3890:
3873:
3872:
3821:Geodetic effect
3782:
3781:
3772:
3771:
3747:
3746:
3730:
3700:
3681:
3677:
3672:
3660:
3656:
3641:
3637:
3635:
3632:
3631:
3613:
3571:
3567:
3566:
3561:
3540:
3536:
3535:
3519:
3508:
3507:
3506:
3505:
3503:
3484:
3463:
3462:
3461:
3459:
3458:
3454:
3446:
3442:
3437:
3429:
3426:
3425:
3390:
3386:
3356:
3345:
3344:
3343:
3317:
3316:
3292:
3288:
3279:
3275:
3263:
3259:
3257:
3254:
3253:
3211:
3200:
3199:
3198:
3172:
3171:
3147:
3143:
3134:
3130:
3118:
3114:
3112:
3109:
3108:
3087:
3083:
3073:
3045:
3034:
3033:
3032:
3006:
3005:
2981:
2977:
2976:
2974:
2962:
2958:
2956:
2953:
2952:
2922:
2920:
2902:
2898:
2896:
2893:
2892:
2886:
2844:
2840:
2839:
2823:
2812:
2811:
2810:
2809:
2807:
2788:
2767:
2766:
2765:
2763:
2762:
2758:
2749:
2745:
2734:
2731:
2730:
2704:
2693:
2692:
2691:
2674:
2673:
2658:
2654:
2642:
2638:
2626:
2622:
2610:
2606:
2587:
2585:
2567:
2563:
2561:
2558:
2557:
2522:
2519:
2518:
2515:
2510:
2454:
2451:
2450:
2407:
2404:
2403:
2367:
2364:
2363:
2353:
2330:
2326:
2321:
2312:
2308:
2303:
2294:
2290:
2285:
2276:
2271:
2270:
2265:
2263:
2260:
2259:
2249:
2218:
2216:
2213:
2212:
2181:
2160:
2138:
2134:
2132:
2129:
2128:
2093:
2089:
2083:
2079:
2078:
2076:
2061:
2057:
2053:
2051:
2017:
2001:
1997:
1991:
1987:
1981:
1977:
1967:
1965:
1959:
1948:
1926:
1922:
1920:
1917:
1916:
1863:
1836:
1797:
1792:
1790:
1787:
1786:
1767:
1720:
1719:
1708:
1695:
1687:
1685:
1663:
1656:
1648:
1646:
1645:
1642:
1641:
1630:
1621:
1620:
1609:
1596:
1574:
1567:
1566:
1559:
1558:
1540:
1536:
1534:
1531:
1530:
1511:
1506:
1504:
1501:
1500:
1480:
1475:
1474:
1469:
1462:
1458:
1446:
1442:
1433:
1429:
1424:
1415:
1410:
1409:
1404:
1402:
1399:
1398:
1380:
1339:
1323:
1318:Alternatively,
1296:
1291:
1289:
1286:
1285:
1265:
1242:
1233:) is unitless.
1189:
1185:
1180:
1167:
1163:
1154:
1150:
1145:
1136:
1131:
1130:
1125:
1120:
1110:
1105:
1104:
1099:
1093:
1089:
1077:
1073:
1063:
1056:
1052:
1047:
1046:
1044:
1035:
1030:
1029:
1024:
1022:
1019:
1018:
1012:
975:
970:
968:
965:
964:
957:Euclidean space
936:
934:
931:
930:
907:
902:
901:
896:
890:
876:
875:
866:
862:
857:
851:
847:
835:
831:
826:
820:
816:
811:
808:
807:
793:
680:
653:
627:
623:
608:
570:
568:
550:
549:
536:
533:
526:
524:Subject history
516:
515:
507:
352:
344:
343:
340:
337:
295:
285:
284:
247:Galaxy filament
200:
188:
187:
139:
134:Expansion
127:
126:
111:Microwave (CMB)
90:Nucleosynthesis
74:
17:
12:
11:
5:
8377:
8367:
8366:
8364:Metric tensors
8361:
8356:
8351:
8336:
8335:
8323:
8311:
8299:
8287:
8275:
8252:
8251:
8241:
8238:
8237:
8235:
8234:
8227:
8222:
8217:
8212:
8207:
8202:
8197:
8192:
8187:
8182:
8177:
8172:
8167:
8162:
8157:
8155:Choquet-Bruhat
8152:
8147:
8142:
8137:
8132:
8127:
8122:
8117:
8112:
8107:
8102:
8097:
8092:
8087:
8082:
8076:
8074:
8070:
8069:
8066:
8065:
8063:
8062:
8055:
8054:
8049:
8044:
8037:
8036:
8031:
8026:
8021:
8016:
8007:Axisymmetric:
8004:
8003:
7998:
7992:
7981:
7980:
7975:
7970:
7965:
7960:
7955:
7946:Cosmological:
7943:
7941:
7935:
7934:
7932:
7931:
7926:
7921:
7915:
7913:
7907:
7906:
7904:
7903:
7898:
7887:frame-dragging
7884:
7879:
7874:
7871:Einstein rings
7867:Einstein cross
7860:
7849:
7848:
7843:
7837:
7832:
7827:
7814:
7804:
7803:
7798:
7793:
7788:
7782:
7780:
7776:
7775:
7773:
7772:
7770:Ernst equation
7767:
7762:
7757:
7752:
7747:
7742:
7740:BSSN formalism
7737:
7731:
7729:
7725:
7724:
7722:
7721:
7716:
7711:
7706:
7701:
7695:
7693:
7687:
7686:
7684:
7683:
7678:
7672:
7670:
7663:
7655:
7654:
7651:
7650:
7648:
7647:
7642:
7637:
7632:
7627:
7621:
7619:
7613:
7612:
7610:
7609:
7604:
7599:
7597:Ladder paradox
7594:
7589:
7584:
7579:
7574:
7569:
7563:
7561:
7557:
7556:
7554:
7553:
7547:
7545:
7541:
7540:
7538:
7537:
7532:
7527:
7522:
7517:
7512:
7507:
7502:
7500:Speed of light
7497:
7491:
7489:
7483:
7482:
7480:
7479:
7474:
7469:
7463:
7453:
7451:
7444:
7436:
7435:
7428:
7427:
7420:
7413:
7405:
7399:
7398:
7388:
7371:
7328:
7322:
7305:
7302:
7299:
7298:
7261:(18): 184001.
7245:
7186:
7132:
7126:10.1086/175496
7092:
7067:
7022:(18): 184001.
7006:
6934:
6906:
6846:
6807:
6802:10.1086/143726
6766:
6761:10.1086/143716
6725:
6721:10.1086/143681
6686:
6657:
6613:(5): 483–490,
6592:
6564:(1): 326–332.
6545:
6518:(1): 377–386.
6499:
6464:
6402:
6341:
6317:
6310:
6292:
6285:
6267:
6260:
6242:
6235:
6215:
6209:978-0792359463
6208:
6172:
6135:(3): 135–214.
6117:
6103:
6102:
6100:
6097:
6074:
6070:
6036:
6024:
6018:
6009:
6008:Current status
6006:
5989:
5965:
5945:
5934:
5933:
5922:
5917:
5914:
5911:
5908:
5902:
5898:
5895:
5886:
5871:
5870:
5859:
5856:
5850:
5847:
5841:
5835:
5832:
5799:
5796:
5714:
5711:
5674:
5673:
5662:
5656:
5650:
5646:
5642:
5636:
5633:
5627:
5624:
5619:
5614:
5610:
5604:
5600:
5596:
5590:
5585:
5580:
5573:
5570:
5554:
5538:
5534:
5526:
5523:
5517:
5512:
5508:
5504:
5498:
5495:
5489:
5486:
5478:
5474:
5470:
5467:
5461:
5458:
5450:
5446:
5442:
5427:
5424:
5416:
5415:
5404:
5399:
5393:
5389:
5385:
5379:
5376:
5373:
5348:
5347:
5334:
5330:
5326:
5323:
5320:
5317:
5296:
5295:
5284:
5278:
5274:
5270:
5267:
5264:
5261:
5243:
5242:
5231:
5226:
5223:
5218:
5215:
5212:
5209:
5199:
5183:
5179:
5175:
5171:
5166:
5163:
5160:
5157:
5138:
5135:
5103:
5100:
5059:
5048:
5047:
5034:
5028:
5024:
5020:
5014:
5010:
5002:
4998:
4993:
4990:
4984:
4981:
4977:
4971:
4965:
4961:
4957:
4951:
4948:
4943:
4938:
4935:
4921:
4909:
4901:
4897:
4893:
4888:
4885:
4881:
4875:
4870:
4867:
4860:
4857:
4854:
4848:
4845:
4827:
4826:Interpretation
4824:
4791:
4790:
4779:
4776:
4771:
4767:
4763:
4760:
4757:
4752:
4748:
4744:
4741:
4734:
4730:
4723:
4719:
4715:
4709:
4704:
4699:
4694:
4689:
4686:
4679:
4674:
4669:
4664:
4661:
4654:
4644:
4633:
4628:
4622:
4618:
4614:
4608:
4603:
4597:
4593:
4589:
4583:
4576:
4572:
4565:
4561:
4557:
4551:
4546:
4541:
4536:
4531:
4528:
4521:
4486:
4483:
4479:
4475:
4472:
4467:
4464:
4460:
4456:
4453:
4448:
4445:
4441:
4410:
4407:
4404:
4401:
4398:
4378:
4375:
4372:
4369:
4354:Main article:
4350:
4349:
4347:
4346:
4339:
4332:
4324:
4321:
4320:
4319:
4318:
4305:
4288:
4287:
4284:
4283:
4276:
4271:
4266:
4261:
4256:
4251:
4246:
4241:
4236:
4231:
4226:
4221:
4216:
4211:
4206:
4201:
4196:
4191:
4186:
4181:
4176:
4171:
4166:
4161:
4156:
4151:
4146:
4141:
4136:
4131:
4126:
4121:
4116:
4110:
4106:
4105:
4104:
4101:
4100:
4097:
4096:
4091:
4086:
4081:
4076:
4071:
4066:
4061:
4056:
4051:
4046:
4041:
4036:
4031:
4026:
4021:
4016:
4011:
4000:
3994:
3993:
3992:
3989:
3988:
3983:
3982:
3981:
3980:
3975:
3967:
3966:
3962:
3961:
3960:
3959:
3957:Post-Newtonian
3954:
3949:
3941:
3940:
3936:
3935:
3934:
3933:
3928:
3923:
3918:
3913:
3908:
3900:
3899:
3893:
3889:
3888:
3885:
3881:
3880:
3879:
3878:
3875:
3874:
3869:
3868:
3867:
3866:
3861:
3856:
3848:
3847:
3841:
3840:
3839:
3838:
3833:
3828:
3823:
3818:
3816:Frame-dragging
3813:
3808:
3803:
3798:
3793:
3791:Kepler problem
3783:
3779:
3778:
3777:
3774:
3773:
3770:
3769:
3764:
3759:
3754:
3748:
3744:
3743:
3742:
3739:
3738:
3737:
3736:
3731:
3729:
3728:
3723:
3718:
3712:
3710:
3702:
3701:
3687:
3684:
3680:
3675:
3671:
3666:
3663:
3659:
3655:
3652:
3647:
3644:
3640:
3630:
3622:
3621:
3612:
3609:
3608:
3607:
3596:
3592:
3585:
3582:
3579:
3574:
3570:
3565:
3560:
3554:
3551:
3548:
3543:
3539:
3533:
3530:
3527:
3522:
3515:
3512:
3502:
3496:
3493:
3490:
3487:
3482:
3479:
3476:
3470:
3467:
3457:
3449:
3445:
3441:
3436:
3433:
3419:
3418:
3407:
3404:
3401:
3398:
3393:
3389:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3359:
3352:
3349:
3342:
3339:
3336:
3333:
3330:
3324:
3321:
3315:
3312:
3309:
3306:
3303:
3298:
3295:
3291:
3287:
3282:
3278:
3274:
3269:
3266:
3262:
3251:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3214:
3207:
3204:
3197:
3194:
3191:
3188:
3185:
3179:
3176:
3170:
3167:
3164:
3161:
3158:
3153:
3150:
3146:
3142:
3137:
3133:
3129:
3124:
3121:
3117:
3106:
3090:
3086:
3082:
3079:
3076:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3048:
3041:
3038:
3031:
3028:
3025:
3022:
3019:
3013:
3010:
3004:
3001:
2998:
2995:
2992:
2987:
2984:
2980:
2973:
2968:
2965:
2961:
2950:
2939:
2934:
2929:
2926:
2919:
2916:
2913:
2908:
2905:
2901:
2885:
2882:
2881:
2880:
2869:
2865:
2858:
2855:
2852:
2847:
2843:
2837:
2834:
2831:
2826:
2819:
2816:
2806:
2800:
2797:
2794:
2791:
2786:
2783:
2780:
2774:
2771:
2761:
2755:
2752:
2748:
2744:
2741:
2738:
2724:
2723:
2712:
2707:
2700:
2697:
2690:
2687:
2681:
2678:
2672:
2669:
2664:
2661:
2657:
2653:
2648:
2645:
2641:
2637:
2632:
2629:
2625:
2621:
2616:
2613:
2609:
2604:
2599:
2594:
2591:
2584:
2581:
2578:
2573:
2570:
2566:
2538:
2535:
2532:
2529:
2526:
2514:
2511:
2509:
2506:
2498:
2497:
2486:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2448:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2401:
2390:
2386:
2383:
2380:
2377:
2374:
2371:
2350:
2349:
2338:
2333:
2329:
2324:
2320:
2315:
2311:
2306:
2302:
2297:
2293:
2288:
2284:
2279:
2274:
2268:
2248:
2245:
2222:
2205:
2204:
2193:
2190:
2185:
2180:
2177:
2172:
2169:
2166:
2163:
2158:
2155:
2152:
2149:
2146:
2141:
2137:
2122:
2121:
2110:
2107:
2102:
2096:
2092:
2086:
2082:
2075:
2070:
2064:
2060:
2056:
2050:
2047:
2044:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2013:
2010:
2007:
2004:
2000:
1994:
1990:
1984:
1980:
1976:
1973:
1970:
1962:
1957:
1954:
1951:
1947:
1943:
1940:
1937:
1934:
1929:
1925:
1891:
1890:
1817:
1800:
1795:
1737:
1736:
1723:
1718:
1715:
1712:
1709:
1707:
1704:
1698:
1694:
1690:
1684:
1681:
1678:
1675:
1670:
1667:
1659:
1655:
1651:
1644:
1643:
1640:
1637:
1634:
1631:
1629:
1626:
1623:
1622:
1619:
1616:
1613:
1610:
1608:
1605:
1600:
1595:
1592:
1589:
1586:
1581:
1578:
1571:
1565:
1564:
1562:
1557:
1554:
1551:
1548:
1543:
1539:
1514:
1509:
1497:
1496:
1483:
1478:
1472:
1465:
1461:
1457:
1454:
1449:
1445:
1441:
1436:
1432:
1427:
1423:
1418:
1413:
1407:
1384:hyperspherical
1379:
1376:
1367:
1366:
1316:
1299:
1294:
1209:
1208:
1197:
1192:
1188:
1183:
1178:
1175:
1170:
1166:
1162:
1157:
1153:
1148:
1144:
1139:
1134:
1128:
1118:
1113:
1108:
1102:
1096:
1092:
1088:
1080:
1076:
1072:
1069:
1066:
1059:
1055:
1050:
1043:
1038:
1033:
1027:
1011:
1008:
978:
973:
939:
927:
926:
915:
910:
905:
899:
893:
888:
885:
882:
879:
874:
869:
865:
860:
854:
850:
846:
843:
838:
834:
829:
823:
819:
815:
792:
791:General metric
789:
777:Standard Model
721:path-connected
693:exact solution
610:
609:
607:
606:
599:
592:
584:
581:
580:
579:
578:
566:
552:
551:
548:
547:
542:
537:
530:
527:
522:
521:
518:
517:
514:
513:
506:
505:
500:
495:
490:
485:
480:
475:
470:
465:
460:
455:
450:
445:
440:
435:
430:
425:
420:
415:
410:
405:
400:
395:
390:
385:
380:
375:
370:
365:
360:
354:
353:
350:
349:
346:
345:
342:
341:
334:
332:
327:
322:
317:
312:
307:
302:
296:
291:
290:
287:
286:
281:
280:
279:
278:
266:
261:
256:
244:
236:
235:
231:
230:
229:
228:
216:
208:
207:
201:
194:
193:
190:
189:
186:
185:
180:
175:
170:
158:
153:
140:
133:
132:
129:
128:
123:
122:
121:
120:
118:Neutrino (CNB)
108:
100:
99:
95:
94:
93:
92:
75:
73:Early universe
72:
71:
68:
67:
66:
65:
60:
55:
40:
39:
31:
30:
24:
23:
15:
9:
6:
4:
3:
2:
8376:
8365:
8362:
8360:
8357:
8355:
8352:
8350:
8347:
8346:
8344:
8334:
8324:
8322:
8312:
8310:
8300:
8298:
8293:
8288:
8286:
8276:
8274:
8269:
8264:
8263:
8260:
8249:
8239:
8233:
8232:
8228:
8226:
8223:
8221:
8218:
8216:
8213:
8211:
8208:
8206:
8203:
8201:
8198:
8196:
8193:
8191:
8188:
8186:
8183:
8181:
8178:
8176:
8173:
8171:
8168:
8166:
8163:
8161:
8158:
8156:
8153:
8151:
8148:
8146:
8143:
8141:
8140:Chandrasekhar
8138:
8136:
8133:
8131:
8128:
8126:
8123:
8121:
8118:
8116:
8113:
8111:
8108:
8106:
8103:
8101:
8100:Schwarzschild
8098:
8096:
8093:
8091:
8088:
8086:
8083:
8081:
8078:
8077:
8075:
8071:
8061:
8057:
8056:
8053:
8050:
8048:
8045:
8043:
8039:
8038:
8035:
8032:
8030:
8027:
8025:
8022:
8020:
8017:
8014:
8010:
8006:
8005:
8002:
7999:
7996:
7993:
7991:
7987:
7986:Schwarzschild
7983:
7982:
7979:
7976:
7974:
7971:
7969:
7966:
7964:
7961:
7959:
7956:
7953:
7949:
7945:
7944:
7942:
7940:
7936:
7930:
7927:
7925:
7922:
7920:
7917:
7916:
7914:
7908:
7902:
7899:
7896:
7892:
7888:
7885:
7883:
7882:Shapiro delay
7880:
7878:
7875:
7872:
7868:
7864:
7861:
7858:
7854:
7851:
7850:
7847:
7844:
7841:
7838:
7836:
7833:
7831:
7828:
7826:
7825:collaboration
7822:
7818:
7815:
7813:
7809:
7806:
7805:
7802:
7799:
7797:
7794:
7792:
7791:Event horizon
7789:
7787:
7784:
7783:
7781:
7777:
7771:
7768:
7766:
7763:
7761:
7758:
7756:
7753:
7751:
7748:
7746:
7743:
7741:
7738:
7736:
7735:ADM formalism
7733:
7732:
7730:
7726:
7720:
7717:
7715:
7712:
7710:
7707:
7705:
7702:
7700:
7697:
7696:
7694:
7688:
7682:
7679:
7677:
7674:
7673:
7671:
7667:
7664:
7662:
7656:
7646:
7643:
7641:
7640:Biquaternions
7638:
7636:
7633:
7631:
7628:
7626:
7623:
7622:
7620:
7618:
7614:
7608:
7605:
7603:
7600:
7598:
7595:
7593:
7590:
7588:
7585:
7583:
7580:
7578:
7575:
7573:
7570:
7568:
7567:Time dilation
7565:
7564:
7562:
7558:
7552:
7549:
7548:
7546:
7542:
7536:
7533:
7531:
7528:
7526:
7523:
7521:
7520:Proper length
7518:
7516:
7513:
7511:
7508:
7506:
7503:
7501:
7498:
7496:
7493:
7492:
7490:
7484:
7478:
7475:
7473:
7470:
7467:
7464:
7462:
7458:
7455:
7454:
7452:
7448:
7445:
7443:
7437:
7433:
7426:
7421:
7419:
7414:
7412:
7407:
7406:
7403:
7397:
7391:
7385:
7380:
7379:
7372:
7368:
7364:
7359:
7354:
7350:
7346:
7342:
7338:
7334:
7329:
7325:
7319:
7315:
7314:
7308:
7307:
7294:
7290:
7286:
7282:
7278:
7274:
7269:
7264:
7260:
7256:
7249:
7241:
7237:
7232:
7227:
7223:
7219:
7214:
7209:
7205:
7201:
7197:
7190:
7182:
7178:
7174:
7170:
7166:
7162:
7157:
7152:
7148:
7144:
7136:
7127:
7122:
7118:
7114:
7110:
7106:
7105:Astrophys. J.
7102:
7101:Ellis, George
7095:
7089:
7085:
7084:
7079:
7071:
7063:
7059:
7055:
7051:
7047:
7043:
7039:
7035:
7030:
7025:
7021:
7017:
7010:
6995:
6991:
6987:
6983:
6979:
6975:
6971:
6967:
6962:
6957:
6953:
6949:
6945:
6938:
6923:
6922:
6917:
6910:
6902:
6898:
6894:
6890:
6886:
6882:
6877:
6872:
6868:
6864:
6857:
6855:
6853:
6851:
6843:
6839:
6835:
6831:
6828:(1): 90–127,
6827:
6823:
6822:
6817:
6816:Walker, A. G.
6811:
6803:
6798:
6794:
6790:
6786:
6782:
6781:
6776:
6770:
6762:
6757:
6753:
6749:
6745:
6741:
6740:
6735:
6729:
6722:
6718:
6714:
6710:
6706:
6702:
6701:
6696:
6690:
6683:
6679:
6675:
6671:
6667:
6661:
6654:
6650:
6646:
6642:
6638:
6634:
6629:
6624:
6620:
6616:
6612:
6608:
6607:
6602:
6596:
6587:
6583:
6579:
6575:
6571:
6567:
6563:
6560:(in German).
6559:
6555:
6549:
6541:
6537:
6533:
6529:
6525:
6521:
6517:
6513:
6509:
6503:
6495:
6491:
6487:
6483:
6479:
6475:
6468:
6460:
6456:
6452:
6448:
6444:
6440:
6436:
6432:
6427:
6422:
6418:
6414:
6406:
6398:
6394:
6390:
6386:
6382:
6378:
6374:
6370:
6365:
6364:gr-qc/0510004
6360:
6356:
6352:
6345:
6334:
6327:
6321:
6313:
6307:
6303:
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6288:
6282:
6278:
6271:
6263:
6257:
6253:
6246:
6238:
6232:
6228:
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6211:
6205:
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6197:
6192:
6191:gr-qc/9812046
6187:
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6176:
6168:
6164:
6160:
6156:
6152:
6148:
6143:
6142:gr-qc/9605010
6138:
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6121:
6114:
6108:
6104:
6096:
6072:
6068:
6058:
6054:
6050:
6046:
6039:
6032:
6031:inhomogeneous
6028:
6015:
6005:
6003:
5987:
5979:
5963:
5943:
5920:
5915:
5912:
5909:
5906:
5900:
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5555:
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5521:
5515:
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5502:
5496:
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5468:
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5355:
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5318:
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5308:
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5272:
5268:
5265:
5262:
5259:
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5248:
5229:
5224:
5216:
5213:
5207:
5200:
5181:
5177:
5173:
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5155:
5148:
5147:
5146:
5144:
5134:
5132:
5128:
5124:
5120:
5101:
5098:
5086:
5084:
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5075:
5073:
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5032:
5026:
5022:
5012:
5008:
5000:
4996:
4991:
4988:
4982:
4979:
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4963:
4959:
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4949:
4946:
4941:
4936:
4933:
4922:
4907:
4899:
4895:
4891:
4886:
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4879:
4873:
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4858:
4855:
4852:
4846:
4843:
4833:
4832:
4831:
4823:
4821:
4817:
4813:
4809:
4805:
4800:
4796:
4777:
4774:
4769:
4765:
4761:
4758:
4755:
4750:
4746:
4739:
4732:
4728:
4721:
4717:
4713:
4707:
4702:
4697:
4692:
4687:
4684:
4677:
4672:
4667:
4662:
4659:
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4606:
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4502:
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4470:
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4462:
4458:
4451:
4446:
4443:
4439:
4431:
4426:
4424:
4408:
4402:
4396:
4373:
4367:
4357:
4345:
4340:
4338:
4333:
4331:
4326:
4325:
4323:
4322:
4316:
4306:
4303:
4298:
4292:
4291:
4290:
4289:
4282:
4281:
4277:
4275:
4272:
4270:
4267:
4265:
4262:
4260:
4257:
4255:
4252:
4250:
4247:
4245:
4242:
4240:
4237:
4235:
4232:
4230:
4227:
4225:
4222:
4220:
4219:Chandrasekhar
4217:
4215:
4212:
4210:
4207:
4205:
4202:
4200:
4197:
4195:
4192:
4190:
4187:
4185:
4182:
4180:
4177:
4175:
4172:
4170:
4167:
4165:
4162:
4160:
4157:
4155:
4152:
4150:
4147:
4145:
4142:
4140:
4137:
4135:
4134:Schwarzschild
4132:
4130:
4127:
4125:
4122:
4120:
4117:
4115:
4112:
4111:
4103:
4102:
4095:
4094:Hartle–Thorne
4092:
4090:
4087:
4085:
4082:
4080:
4077:
4075:
4072:
4070:
4067:
4065:
4062:
4060:
4057:
4055:
4052:
4050:
4047:
4045:
4042:
4040:
4037:
4035:
4032:
4030:
4027:
4025:
4022:
4020:
4017:
4015:
4012:
4009:
4005:
4004:Schwarzschild
4002:
4001:
3997:
3991:
3990:
3979:
3976:
3974:
3971:
3970:
3969:
3968:
3964:
3963:
3958:
3955:
3953:
3950:
3948:
3945:
3944:
3943:
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3897:
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3852:
3851:
3850:
3849:
3846:
3843:
3842:
3837:
3834:
3832:
3829:
3827:
3826:Event horizon
3824:
3822:
3819:
3817:
3814:
3812:
3809:
3807:
3804:
3802:
3799:
3797:
3794:
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3722:
3719:
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3714:
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3711:
3709:
3706:
3705:
3704:
3703:
3685:
3682:
3678:
3673:
3669:
3664:
3661:
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3645:
3642:
3638:
3628:
3624:
3623:
3620:
3617:
3616:
3594:
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3580:
3572:
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3513:
3510:
3500:
3491:
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3477:
3468:
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3443:
3439:
3434:
3431:
3424:
3423:
3422:
3402:
3396:
3391:
3387:
3380:
3377:
3374:
3365:
3357:
3350:
3347:
3340:
3337:
3331:
3322:
3319:
3310:
3304:
3296:
3293:
3289:
3280:
3276:
3272:
3267:
3264:
3260:
3252:
3235:
3232:
3229:
3220:
3212:
3205:
3202:
3195:
3192:
3186:
3177:
3174:
3165:
3159:
3151:
3148:
3144:
3135:
3131:
3127:
3122:
3119:
3115:
3107:
3088:
3084:
3080:
3077:
3074:
3069:
3066:
3063:
3054:
3046:
3039:
3036:
3029:
3026:
3020:
3011:
3008:
2999:
2993:
2985:
2982:
2978:
2971:
2966:
2963:
2959:
2951:
2937:
2932:
2927:
2924:
2917:
2914:
2911:
2906:
2903:
2899:
2891:
2890:
2889:
2867:
2863:
2853:
2845:
2841:
2832:
2824:
2817:
2814:
2804:
2795:
2789:
2781:
2772:
2769:
2759:
2753:
2750:
2746:
2742:
2739:
2736:
2729:
2728:
2727:
2705:
2698:
2695:
2688:
2685:
2679:
2676:
2670:
2662:
2659:
2655:
2651:
2646:
2643:
2639:
2635:
2630:
2627:
2623:
2619:
2614:
2611:
2607:
2602:
2597:
2592:
2589:
2582:
2579:
2576:
2571:
2568:
2564:
2556:
2555:
2554:
2552:
2533:
2530:
2527:
2505:
2503:
2484:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2449:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2402:
2388:
2384:
2381:
2378:
2375:
2372:
2369:
2362:
2361:
2360:
2356:
2336:
2331:
2327:
2318:
2313:
2309:
2300:
2295:
2291:
2282:
2277:
2258:
2257:
2256:
2254:
2244:
2242:
2238:
2220:
2210:
2209:sinc function
2191:
2183:
2178:
2156:
2153:
2147:
2139:
2135:
2127:
2126:
2125:
2108:
2105:
2100:
2094:
2090:
2084:
2080:
2073:
2068:
2062:
2058:
2054:
2048:
2045:
2042:
2036:
2030:
2027:
2024:
2021:
2011:
2008:
2005:
2002:
1998:
1992:
1988:
1982:
1974:
1971:
1955:
1952:
1949:
1945:
1941:
1935:
1927:
1923:
1915:
1914:
1913:
1912:
1908:
1904:
1900:
1896:
1888:
1884:
1881:. The letter
1880:
1876:
1872:
1866:
1861:
1857:
1853:
1849:
1845:
1839:
1834:
1830:
1826:
1822:
1818:
1815:
1784:
1780:
1774:
1770:
1765:
1761:
1757:
1753:
1749:
1745:
1742:
1741:
1740:
1716:
1713:
1710:
1705:
1692:
1682:
1676:
1673:
1668:
1665:
1653:
1638:
1635:
1632:
1627:
1624:
1617:
1614:
1611:
1606:
1598:
1593:
1587:
1584:
1579:
1576:
1569:
1560:
1555:
1549:
1541:
1537:
1529:
1528:
1527:
1481:
1463:
1455:
1447:
1443:
1439:
1434:
1430:
1421:
1416:
1397:
1396:
1395:
1393:
1389:
1385:
1375:
1373:
1364:
1360:
1356:
1352:
1348:
1342:
1337:
1333:
1329:
1321:
1317:
1314:
1283:
1279:
1275:
1271:
1263:
1259:
1258:circumference
1255:
1249:
1245:
1240:
1236:
1232:
1228:
1224:
1220:
1217:
1216:
1215:
1213:
1195:
1190:
1186:
1176:
1173:
1168:
1164:
1160:
1155:
1151:
1142:
1137:
1116:
1111:
1094:
1090:
1086:
1078:
1074:
1070:
1067:
1064:
1057:
1053:
1041:
1036:
1017:
1016:
1015:
1007:
1005:
1001:
997:
993:
962:
958:
954:
913:
908:
891:
883:
877:
872:
867:
863:
852:
848:
844:
841:
836:
832:
821:
817:
813:
806:
805:
804:
802:
798:
788:
786:
782:
778:
774:
770:
766:
762:
758:
754:
750:
746:
742:
738:
734:
730:
726:
722:
718:
714:
710:
706:
702:
698:
694:
690:
684:
621:
617:
605:
600:
598:
593:
591:
586:
585:
583:
582:
577:
567:
565:
556:
555:
554:
553:
546:
543:
541:
538:
535:
529:
528:
525:
520:
519:
512:
509:
508:
504:
501:
499:
496:
494:
491:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
464:
461:
459:
456:
454:
451:
449:
446:
444:
441:
439:
436:
434:
431:
429:
426:
424:
421:
419:
416:
414:
411:
409:
406:
404:
401:
399:
396:
394:
391:
389:
386:
384:
381:
379:
376:
374:
371:
369:
366:
364:
361:
359:
356:
355:
348:
347:
339:
333:
331:
328:
326:
323:
321:
318:
316:
313:
311:
308:
306:
303:
301:
298:
297:
294:
289:
288:
277:
274:
270:
267:
265:
262:
260:
257:
255:
252:
248:
245:
243:
240:
239:
238:
237:
233:
232:
227:
224:
220:
217:
215:
212:
211:
210:
209:
205:
204:
198:
192:
191:
184:
181:
179:
176:
174:
171:
169:
166:
162:
159:
157:
154:
152:
149:
145:
142:
141:
137:
131:
130:
119:
116:
112:
109:
107:
104:
103:
102:
101:
97:
96:
91:
88:
84:
81:
80:
79:
78:
70:
69:
64:
61:
59:
56:
54:
51:
47:
44:
43:
42:
41:
37:
33:
32:
29:
26:
25:
21:
20:
8333:Solar System
8230:
7947:
7924:Kaluza–Klein
7676:Introduction
7602:Twin paradox
7395:
7377:
7343:(1): 69–79.
7340:
7336:
7312:
7258:
7254:
7248:
7203:
7199:
7189:
7146:
7142:
7135:
7108:
7104:
7082:
7070:
7019:
7015:
7009:
6997:. Retrieved
6951:
6947:
6937:
6925:. Retrieved
6919:
6909:
6876:2203.06142v1
6866:
6862:
6825:
6824:, Series 2,
6819:
6810:
6784:
6778:
6769:
6743:
6737:
6728:
6704:
6698:
6689:
6673:
6669:
6660:
6644:
6640:
6632:
6610:
6604:
6595:
6561:
6557:
6548:
6515:
6511:
6502:
6477:
6473:
6467:
6416:
6412:
6405:
6354:
6350:
6344:
6333:the original
6320:
6301:
6295:
6276:
6270:
6251:
6245:
6225:
6218:
6181:
6175:
6132:
6126:
6120:
6112:
6107:
6042:
5935:
5872:
5815:static model
5809:of space of
5802:
5801:
5791:
5787:
5783:
5779:
5777:
5766:
5759:
5752:Edwin Hubble
5747:
5737:
5732:
5722:
5716:
5703:Planck epoch
5700:
5693:
5686:
5677:
5675:
5429:
5420:quintessence
5417:
5359:scalar field
5356:
5349:
5304:
5297:
5244:
5140:
5087:
5076:
5049:
4829:
4807:
4792:
4427:
4359:
4279:
4239:Raychaudhuri
4068:
3708:Introduction
3420:
2887:
2725:
2551:Ricci tensor
2516:
2501:
2499:
2359:by defining
2354:
2351:
2252:
2250:
2240:
2236:
2206:
2123:
1911:power series
1906:
1902:
1894:
1892:
1886:
1882:
1878:
1874:
1870:
1864:
1859:
1855:
1847:
1843:
1837:
1832:
1828:
1824:
1820:
1782:
1778:
1772:
1768:
1762:is then the
1759:
1755:
1751:
1747:
1743:
1738:
1498:
1391:
1387:
1383:
1381:
1368:
1362:
1358:
1350:
1346:
1340:
1335:
1331:
1327:
1319:
1281:
1277:
1269:
1261:
1253:
1247:
1243:
1237:is then the
1234:
1230:
1226:
1222:
1218:
1211:
1210:
1013:
1004:scale factor
999:
995:
991:
928:
794:
776:
772:
768:
764:
760:
756:
752:
748:
729:scale factor
691:based on an
619:
615:
613:
338:Probe (WMAP)
272:
269:Reionization
250:
222:
196:
164:
160:
147:
144:Hubble's law
135:
114:
86:
49:
8321:Outer space
8309:Spaceflight
8013:Kerr–Newman
7984:Spherical:
7853:Other tests
7796:Singularity
7728:Formulation
7690:Fundamental
7544:Formulation
7525:Proper time
7486:Fundamental
6787:: 257–271,
6746:: 187–201,
6707:: 284–301,
6326:"Cosmology"
6002:light years
5701:During the
5300:dark energy
5119:gravitation
4499:giving the
4254:van Stockum
4184:Oppenheimer
4039:Kerr–Newman
3831:Singularity
1324:{−1, 0, +1}
1122:where
797:homogeneity
705:homogeneous
293:Experiments
226:Dark matter
219:Dark energy
161:FLRW metric
98:Backgrounds
8343:Categories
8165:Zel'dovich
8073:Scientists
8052:Alcubierre
7859:of Mercury
7857:precession
7786:Black hole
7669:Background
7661:relativity
7630:World line
7625:Light cone
7450:Background
7442:relativity
7432:Relativity
7268:2105.09790
7213:2009.14826
7206:(2): L51.
7156:2010.08366
7029:2105.09790
6961:2004.03305
6869:: 49–211.
6426:1502.07033
6099:References
6012:See also:
4421:such as a
4107:Scientists
3939:Formalisms
3887:Formalisms
3836:Black hole
3762:World line
1372:elliptical
1268:(like the
779:of modern
679: ...
373:Copernicus
351:Scientists
206:Components
8285:Astronomy
8135:Robertson
8120:Friedmann
8115:Eddington
8105:de Sitter
7939:Solutions
7817:detectors
7812:astronomy
7779:Phenomena
7714:Geodesics
7617:Spacetime
7560:Phenomena
7367:0035-8711
7293:234790314
7240:222066749
7181:223953708
7062:234790314
7054:0264-9381
6994:215238834
6986:0004-6361
6927:March 24,
6901:247411131
6676:: 51–85,
6647:: 49–56,
6586:120551579
6540:125190902
6480:: 73–80.
6451:0217-7323
6397:119496918
6389:0020-7748
6167:119500217
6034:universe?
5988:ρ
5916:ρ
5910:π
5849:¨
5834:˙
5635:−
5618:ρ
5595:κ
5589:−
5572:˙
5525:˙
5494:ρ
5488:˙
5460:˙
5457:ρ
5441:−
5384:ρ
5378:−
5325:ρ
5269:ρ
5266:−
5225:κ
5222:Λ
5211:→
5174:κ
5170:Λ
5165:−
5162:ρ
5159:→
5156:ρ
5102:˙
5019:Λ
4980:ρ
4956:κ
4950:−
4937:¨
4884:ρ
4869:˙
4856:−
4847:˙
4844:ρ
4762:κ
4759:−
4743:Λ
4740:−
4688:˙
4663:¨
4632:ρ
4613:κ
4588:Λ
4582:−
4530:˙
4503:when the
4485:ν
4482:μ
4474:κ
4466:ν
4463:μ
4455:Λ
4447:ν
4444:μ
4397:ρ
4199:Robertson
4164:Friedmann
4159:Eddington
4149:Nordström
4139:de Sitter
3996:Solutions
3921:Geodesics
3916:Friedmann
3898:Equations
3884:Equations
3845:Spacetime
3780:Phenomena
3686:ν
3683:μ
3674:κ
3665:ν
3662:μ
3654:Λ
3646:ν
3643:μ
3611:Solutions
3514:˙
3469:¨
3403:θ
3397:
3351:˙
3323:¨
3294:−
3268:ϕ
3265:ϕ
3206:˙
3178:¨
3149:−
3123:θ
3120:θ
3078:−
3040:˙
3012:¨
2983:−
2928:¨
2915:−
2818:˙
2773:¨
2751:−
2699:˙
2680:¨
2660:−
2593:¨
2580:−
2508:Curvature
2481:ϕ
2478:
2472:θ
2469:
2434:ϕ
2431:
2425:θ
2422:
2385:θ
2382:
2273:Σ
2109:⋯
2106:−
2049:−
1972:−
1961:∞
1946:∑
1905:and
1850:) is the
1812:measures
1799:Σ
1677:
1666:−
1588:
1577:−
1513:Ω
1477:Ω
1412:Σ
1353:) is the
1311:measures
1298:Σ
1187:ϕ
1177:θ
1174:
1152:θ
1133:Ω
1107:Ω
1068:−
1032:Σ
977:Σ
938:Σ
904:Σ
845:−
833:τ
814:−
781:cosmology
749:Friedmann
713:expanding
709:isotropic
503:Zeldovich
403:Friedmann
378:de Sitter
305:BOOMERanG
234:Structure
199:Structure
83:Inflation
8248:Category
8125:Lemaître
8090:Einstein
8080:Poincaré
8040:Others:
8024:Taub–NUT
7990:interior
7912:theories
7910:Advanced
7877:redshift
7692:concepts
7510:Rapidity
7488:concepts
7080:(1973),
6999:24 March
6459:51948117
4795:Big Bang
4315:Category
4179:Lemaître
4144:Reissner
4129:Poincaré
4114:Einstein
4059:Taub–NUT
4024:Wormhole
4008:interior
3721:Timeline
2517:In flat
1901:of both
801:isotropy
719:that is
717:universe
564:Category
483:Suntzeff
443:Lemaître
393:Einstein
358:Aaronson
151:Redshift
53:Universe
46:Big Bang
8273:Physics
8259:Portals
8190:Hawking
8185:Penrose
8170:Novikov
8150:Wheeler
8095:Hilbert
8085:Lorentz
8042:pp-wave
7863:lensing
7659:General
7440:Special
7345:Bibcode
7273:Bibcode
7218:Bibcode
7161:Bibcode
7113:Bibcode
7111:: 1–5,
7034:Bibcode
6966:Bibcode
6881:Bibcode
6830:Bibcode
6789:Bibcode
6748:Bibcode
6709:Bibcode
6678:Bibcode
6649:Bibcode
6615:Bibcode
6566:Bibcode
6520:Bibcode
6482:Bibcode
6431:Bibcode
6369:Bibcode
6196:Bibcode
6147:Bibcode
6113:assumes
5976:is the
5805:is the
5707:quantum
4234:Hawking
4229:Penrose
4204:Bardeen
4194:Wheeler
4124:Hilbert
4119:Lorentz
4079:pp-wave
3716:History
695:of the
687:) is a
681:
488:Sunyaev
473:Schmidt
463:Penzias
458:Penrose
433:Huygens
423:Hawking
408:Galileo
8231:others
8220:Thorne
8210:Misner
8195:Taylor
8180:Geroch
8175:Ehlers
8145:Zwicky
7963:Kasner
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7320:
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7090:
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6165:
6053:Planck
5980:, and
5936:where
5784:models
5780:metric
5125:. The
4313:
4280:others
4274:Thorne
4264:Newman
4244:Taylor
4224:Ehlers
4209:Walker
4174:Zwicky
4049:Kasner
2500:where
2124:or as
1897:is an
1499:where
929:where
767:), or
689:metric
562:
498:Wilson
493:Tolman
453:Newton
448:Mather
438:Kepler
428:Hubble
388:Ehlers
368:Alpher
363:Alfvén
271:
249:
221:
163:
146:
138:Future
113:
85:
48:
8297:Stars
8225:Weiss
8205:Bondi
8200:Hulse
8130:Milne
8034:discs
7978:Milne
7973:Gödel
7830:Virgo
7289:S2CID
7263:arXiv
7236:S2CID
7208:arXiv
7177:S2CID
7151:arXiv
7058:S2CID
7024:arXiv
6990:S2CID
6956:arXiv
6897:S2CID
6871:arXiv
6582:S2CID
6536:S2CID
6455:S2CID
6421:arXiv
6393:S2CID
6359:arXiv
6336:(PDF)
6329:(PDF)
6186:arXiv
6163:S2CID
6137:arXiv
5050:with
4249:Hulse
4189:Gödel
4169:Milne
4064:Milne
4029:Gödel
3726:Tests
2447:, and
2251:When
1775:) = 1
1250:) = 1
959:, or
478:Smoot
468:Rubin
413:Gamow
398:Ellis
383:Dicke
8160:Kerr
8110:Weyl
8009:Kerr
7869:and
7823:and
7821:LIGO
7384:ISBN
7363:ISSN
7318:ISBN
7088:ISBN
7050:ISSN
7001:2022
6982:ISSN
6929:2022
6447:ISSN
6385:ISSN
6306:ISBN
6281:ISBN
6256:ISBN
6231:ISBN
6204:ISBN
6051:and
6049:WMAP
5694:The
5375:<
5141:The
4820:WMAP
4818:and
4816:COBE
4799:ΛCDM
4259:Taub
4214:Kerr
4154:Weyl
4034:Kerr
3952:BSSN
2553:are
2211:and
1877:and
1867:= +1
1840:= ±1
1714:<
1674:sinh
1615:>
1343:= ±1
799:and
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620:FLRW
614:The
418:Guth
8215:Yau
7840:GEO
7353:doi
7341:137
7281:doi
7226:doi
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7169:doi
7147:653
7121:doi
7042:doi
6974:doi
6952:636
6889:doi
6838:doi
6797:doi
6756:doi
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6645:A47
6623:doi
6574:doi
6528:doi
6490:doi
6439:doi
6377:doi
6155:doi
6133:254
4269:Yau
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2466:sin
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2419:sin
2379:cos
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87:·
50:·
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