8267:
4303:
3633:
5697:(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.
8243:
4316:
559:
36:
8303:
8327:
8279:
571:
8315:
1206:
8291:
6066:
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:
4807:
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
6065:
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)
5780:
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
4808:
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:
4648:
3611:
4794:
5051:
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:
5677:
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
5558:
1532:
4925:
3104:
3249:
6039:
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
5686:) 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 (
5203:
3704:
5741:, 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.
4503:
1918:
4519:
3427:
2202:
5246:
4654:
4931:
4366:
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
5937:
2948:
5419:
1810:
1524:
1309:
988:
2495:
1400:
2445:
6860:
Abdalla, Elcio; et al. (June 2022). "Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies".
5874:
809:
4820:. 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:
6101:, 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:
5299:
2261:
5564:
3255:
5351:
7096:. 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:
5121:
4425:
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}}}
7013:
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?".
5442:
6091:
6004:
2547:
4393:
2888:
In more general FLRW space using spherical coordinates (called "reduced-circumference polar coordinates" above), the surviving components of the Ricci tensor are
4842:
2954:
6602:(1931), "Expansion of the universe, A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulæ",
5980:
5960:
5074:
3110:
7252:
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
4347:
5436:
This is due to McCrea and Milne, although sometimes incorrectly ascribed to
Friedmann. The Friedmann equations are equivalent to this pair of equations:
531:
6604:
5766:
8347:
3726:
5693:
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)
8352:
6819:
6638:(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 }
6035:
and assumed by all models that use the
Friedmann–Lemaître–Robertson–Walker metric, including the current version of ΛCDM, or is the universe
5749:
5157:
3639:
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
4440:
3931:
601:
7993:
5756:(Annals of the Scientific Society of Brussels). In the face of the observational evidence for the expansion of the universe obtained by
4643:{\displaystyle \left({\frac {\dot {a}}{a}}\right)^{2}+{\frac {kc^{2}}{a^{2}}}-{\frac {\Lambda c^{2}}{3}}={\frac {\kappa c^{4}}{3}}\rho }
8228:
4284:
8017:
7763:
7140:
Siewert, Thilo M.; Schmidt-Rubart, Matthias; Schwarz, Dominik J. (2021). "Cosmic radio dipole: Estimators and frequency dependence".
5255:
can be interpreted as arising from a form of energy that has negative pressure, equal in magnitude to its (positive) energy density:
4094:
3936:
3606:{\displaystyle R=6\left({\frac {{\ddot {a}}(t)}{c^{2}a(t)}}+{\frac {{\dot {a}}^{2}(t)}{c^{2}a^{2}(t)}}+{\frac {k}{a^{2}(t)}}\right).}
963:. It is normally written as a function of three spatial coordinates, but there are several conventions for doing so, detailed below.
4789:{\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.}
4340:
4049:
6235:. Springer Praxis books in astronomy and planetary science (2. ed., reprinted ed.). Chichester, UK: Praxis Publ. p. 61.
5046:{\displaystyle {\frac {\ddot {a}}{a}}=-{\frac {\kappa c^{4}}{6}}\left(\rho +{\frac {3p}{c^{2}}}\right)+{\frac {\Lambda c^{2}}{3}}}
2130:
7080:
5209:
7799:
7753:
6410:
Rosu, Haret C.; Mancas, S. C.; Chen, Pisin (2015-05-05). "Barotropic FRW cosmologies with
Chiellini damping in comoving time".
3962:
3796:
7937:
7386:
7320:
7090:
6315:
6290:
6265:
6240:
5800:) that assume that the only contributions to stress–energy are cold matter ("dust"), radiation, and a cosmological constant.
4001:
731:
of the universe as a function of time. Depending on geographical or historical preferences, the set of the four scientists –
692:
7421:
7193:
Secrest, Nathan J.; Hausegger, Sebastian von; Rameez, Mohamed; Mohayaee, Roya; Sarkar, Subir; Colin, Jacques (2021-02-25).
6054:
6043:
5885:
335:
4333:
6121:
multiple connectedness in the positive curvature case and says that "we are still free to restore" simple connectedness.
7674:
6213:
6062:
5733:
published his work, it remained relatively unnoticed by his contemporaries. Friedmann was in direct communication with
3713:
2894:
8045:
7679:
5812:
3739:
1489:{\displaystyle \mathrm {d} \mathbf {\Sigma } ^{2}=\mathrm {d} r^{2}+S_{k}(r)^{2}\,\mathrm {d} \mathbf {\Omega } ^{2}}
1354:
544:
5373:
1788:
1502:
1287:
966:
796:
704:
594:
8266:
7999:
5094:
The second equation states that both the energy density and the pressure cause the expansion rate of the universe
4302:
4019:
3632:
2452:
7712:
6914:
6186:
Ellis, G. F. R.; van Elst, H. (1999). "Cosmological models (Cargèse lectures 1998)". In Marc Lachièze-Rey (ed.).
3926:
2405:
919:{\displaystyle -c^{2}\mathrm {d} \tau ^{2}=-c^{2}\mathrm {d} t^{2}+{a(t)}^{2}\mathrm {d} \mathbf {\Sigma } ^{2},}
177:
7815:
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:
5829:
5363:
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
8246:
7988:
7893:
7823:
4428:
4013:
3721:
182:
105:
7956:
5727:
first derived the main results of the FLRW model in 1922 and 1924. Although the prestigious physics journal
4059:
7810:
6941:
Migkas, K.; Schellenberger, G.; Reiprich, T. H.; Pacaud, F.; Ramos-Ceja, M. E.; Lovisari, L. (April 2020).
5682:
The first equation says that the decrease in the mass contained in a fixed cube (whose side is momentarily
5127:, with pressure playing a similar role to that of energy (or mass) density, according to the principles of
4079:
2873:{\displaystyle R=6c^{-2}\left({\frac {{\ddot {a}}(t)}{a(t)}}+{\frac {{\dot {a}}^{2}(t)}{a^{2}(t)}}\right).}
8357:
7851:
7585:
7570:
7099:
7076:
6006:
is the density of space of this universe. The numerical value of
Einstein's radius is of the order of 10
5687:
5084:
4320:
3811:
3731:
2342:{\displaystyle \mathrm {d} \mathbf {\Sigma } ^{2}=\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}.}
587:
563:
110:
5672:{\displaystyle {\frac {{\dot {a}}^{2}}{2}}-{\frac {\kappa c^{4}a^{3}\rho }{6a}}=-{\frac {kc^{2}}{2}}\,.}
5123:
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:
8257:
8138:
7580:
7533:
4821:
4510:
4224:
3772:
523:
329:
309:
117:
62:
5261:
8168:
7794:
7475:
6943:"Probing cosmic isotropy with a new X-ray galaxy cluster sample through the L X – T scaling relation"
4024:
3836:
1273:
712:
155:
5879:
in the
Friedmann equation, the radius of curvature of space of this universe (Einstein's radius) is
5729:
5083:
The first equation can be derived also from thermodynamical considerations and is equivalent to the
1563:
8193:
7743:
7503:
7414:
5317:
5091:(which is implicitly assumed in the derivation of the Friedmann–Lemaître–Robertson–Walker metric).
4435:
4259:
4249:
4099:
3916:
1003:
728:
696:
324:
89:
5704:
term is assumed to be treated as dark energy and thus merged into the density and pressure terms.
8362:
7922:
7844:
7464:
7455:
6036:
5077:
4817:
4244:
3978:
2214:
292:
172:
5097:
8148:
8011:
7917:
7875:
7549:
7513:
6814:
6032:
5983:
5816:
5788:
since they proved its generic properties, is different from the dynamical "Friedmann–Lemaître"
5777:
5553:{\displaystyle -a^{3}{\dot {\rho }}=3a^{2}{\dot {a}}\rho +{\frac {3a^{2}p{\dot {a}}}{c^{2}}}\,}
5425:
4398:
4214:
4199:
4044:
3806:
744:
724:
1214:
is a constant representing the curvature of the space. There are two common unit conventions:
8153:
7758:
7697:
6778:
6737:
6698:
6230:
6019:
5701:
5252:
5148:
5132:
4189:
3757:
497:
299:
241:
7163:
7102:(2007), "Proving Almost-Homogeneity of the Universe: An Almost Ehlers-Geren-Sachs Theorem",
6968:
8058:
7984:
7828:
7430:
7343:
7271:
7216:
7159:
7111:
7032:
6964:
6879:
6828:
6787:
6746:
6707:
6676:
6647:
6613:
6564:
6518:
6480:
6429:
6367:
6201:
6152:
6069:
5989:
5752:, arrived independently at results similar to those of Friedmann and published them in the
5694:
4920:{\displaystyle {\dot {\rho }}=-3{\frac {\dot {a}}{a}}\left(\rho +{\frac {p}{c^{2}}}\right)}
4009:
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:
7381:(Repr ed.). Oxford : New York: Clarendon Press ; Oxford University Press.
4369:
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:
8331:
8133:
8128:
8118:
7950:
7899:
7717:
7702:
7528:
7459:
7407:
6919:
6773:
6732:
6693:
6552:
6506:
6058:
5773:
5724:
5364:
4809:
4506:
4361:
4204:
4174:
4169:
3921:
1851:
1371:
740:
732:
402:
372:
319:
275:
167:
57:
8123:
7347:
7275:
7220:
7115:
7036:
6883:
6832:
6791:
6750:
6711:
6680:
6664:
6651:
6635:
6617:
6599:
6568:
6555:(1924). "Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes".
6522:
6484:
6433:
6371:
6205:
6156:
6053:, uses the FLRW metric. By combining the observation data from some experiments such as
5745:
4395:
does require
Einstein's field equations together with a way of calculating the density,
4184:
4154:
736:
442:
8319:
8307:
8198:
8098:
8022:
7880:
7861:
7855:
7806:
7748:
7657:
7575:
7493:
7470:
7438:
7375:
7287:
7261:
7234:
7206:
7175:
7149:
7056:
7022:
6988:
6954:
6895:
6869:
6580:
6534:
6453:
6419:
6391:
6357:
6348:
Rosu, H. C.; Ojeda-May, P. (June 2006). "Supersymmetry of FRW Barotropic Cosmologies".
6191:
6168:
6142:
5965:
5945:
5357:
5128:
5059:
4254:
4139:
4064:
3911:
3816:
3801:
3762:
3624:
1763:
1238:
780:
700:
462:
432:
397:
367:
314:
258:
27:
7395:(See Chapter 23 for a particularly clear and concise introduction to the FLRW models.)
6471:
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
8208:
8040:
8032:
7633:
7590:
7382:
7361:
7316:
7291:
7238:
7179:
7086:
7060:
7048:
6992:
6980:
6899:
6584:
6538:
6445:
6395:
6383:
6311:
6286:
6261:
6236:
6209:
6172:
6164:
5712:
5088:
4084:
3859:
1898:
1813:
1312:
688:
492:
8078:
6942:
6457:
6031:
Is the universe homogeneous and isotropic at large enough scales, as claimed by the
4134:
8283:
8113:
8103:
8050:
8027:
7605:
7351:
7279:
7224:
7167:
7119:
7040:
6972:
6887:
6836:
6795:
6754:
6715:
6621:
6572:
6526:
6488:
6437:
6375:
6160:
6050:
5820:
5764:, and in 1930–31 Lemaître's paper was translated into English and published in the
5761:
4836:
The pair of equations given above is equivalent to the following pair of equations
4804:
4513:
is similarly assumed to be isotropic and homogeneous. The resulting equations are:
4209:
4164:
4144:
4089:
960:
952:
784:
625:
575:
377:
213:
82:
8173:
7171:
6976:
4229:
387:
362:
8188:
8163:
8088:
8083:
7966:
7927:
7889:
7833:
7707:
7643:
7310:
6133:
5734:
4264:
4239:
4124:
4119:
3983:
3864:
3826:
956:
502:
437:
422:
407:
392:
382:
246:
143:
7971:
6891:
4034:
1260:
because it is equal to the measured circumference of a circle (at that value of
8271:
8213:
7885:
7869:
7865:
7768:
7738:
7595:
7498:
7283:
7229:
7194:
7044:
4307:
4274:
4269:
3957:
3821:
720:
487:
447:
6840:
6492:
6441:
6379:
6331:
8341:
8203:
8183:
8178:
8093:
7961:
7789:
7733:
7565:
7518:
7365:
7356:
7331:
7052:
6984:
6626:
6449:
6387:
4234:
4149:
4129:
4054:
3952:
3831:
2208:
1257:
472:
457:
357:
4194:
783:, although such a description is also associated with the further developed
8295:
8223:
8143:
8108:
7638:
7600:
5757:
5708:
4179:
4159:
2550:
1910:
1014:
In reduced-circumference polar coordinates the spatial metric has the form
477:
452:
427:
412:
268:
5781:
general relativity, which were always assumed by Friedmann and Lemaître).
8007:
7976:
7523:
5305:
5136:
5124:
4069:
4039:
225:
218:
5748:, a Belgian priest, astronomer and periodic professor of physics at the
2549:
FLRW space using Cartesian coordinates, the surviving components of the
8218:
7784:
7628:
7623:
7251:
6589:
English trans. in 'General Relativity and Gravitation' 1999 vol.31, 31–
6576:
6530:
6007:
5823:
that was supposed to represent our universe in idealized form. Putting
5198:{\displaystyle \rho \rightarrow \rho -{\frac {\Lambda }{\kappa c^{2}}}}
4279:
3841:
3767:
3699:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
467:
6362:
6196:
6147:
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
7615:
7012:
6874:
5715:
effects. So they may cause a deviation from the Friedmann equations.
4498:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }}
3850:
708:
417:
5760:
in the late 1920s, Lemaître's results were noticed in particular by
8158:
7508:
7266:
7211:
7154:
7124:
7027:
6959:
6800:
6759:
6719:
6424:
6023:
4800:
4219:
4029:
3869:
800:
716:
150:
52:
45:
6940:
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
7838:
7192:
637:
7399:
6283:
Spacetime and geometry: an introduction to general relativity
7139:
6131:
Lachieze-Rey, M.; Luminet, J.-P. (1995). "Cosmic Topology".
2197:{\displaystyle S_{k}(r)=r\;\mathrm {sinc} \,(r{\sqrt {k}}),}
35:
7819:
5241:{\displaystyle p\rightarrow p+{\frac {\Lambda }{\kappa }}.}
4825:
1722:
675:
666:
657:
646:
8290:
7312:
The measure of the universe: a history of modern cosmology
6285:. New York: Cambridge University Press. pp. 329–333.
5151:
term can be omitted if we make the following replacements
1009:
6190:. NATO Science Series C. Vol. 541. pp. 1–116.
6117:
For an early reference, see Robertson (1935); Robertson
4816:, i.e., a model that follows the FLRW metric apart from
2235:
is one of the imaginary, zero or real square roots of
8255:
6260:. Chicago: University of Chicago Press. p. 116.
6072:
5992:
5968:
5948:
5932:{\displaystyle R_{\text{E}}=c/{\sqrt {4\pi G\rho }},}
5888:
5832:
5567:
5445:
5376:
5320:
5264:
5212:
5160:
5100:
5062:
4934:
4845:
4657:
4522:
4443:
4401:
4372:
3642:
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:
7097:
6310:. Chicago: University of Chicago Press. p. 97.
5803:
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:
7195:"A Test of the Cosmological Principle with Quasars"
7138:See Siewert et al. for a recent summary of results
6130:
663:
660:
643:
628:
7374:
6085:
5998:
5974:
5954:
5931:
5868:
5671:
5552:
5413:
5345:
5293:
5240:
5197:
5115:
5068:
5045:
4919:
4788:
4642:
4497:
4419:
4387:
3698:
3605:
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:
7336:Monthly Notices of the Royal Astronomical Society
6605:Monthly Notices of the Royal Astronomical Society
5784:This solution, often called the Robertson–Walker
5767:Monthly Notices of the Royal Astronomical Society
1893:Though it is usually defined piecewise as above,
8339:
6817:(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
7332:"Classification of Uniform Cosmological Models"
6669:Annales de la Société Scientifique de Bruxelles
6640:Annales de la Société Scientifique de Bruxelles
6228:
5754:Annales de la Société Scientifique de Bruxelles
5087:, assuming the expansion of the universe is an
1394:is proportional to radial distance; this gives
7006:
6820:Proceedings of the London Mathematical Society
6776:(1936), "Kinematics and world structure III",
6409:
5414:{\displaystyle p<-{\frac {\rho c^{2}}{3}}.}
4799: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
7415:
7372:
6855:
6853:
6851:
6849:
6735:(1936), "Kinematics and world structure II",
6347:
6185:
6049:The current standard model of cosmology, the
5304:which is an equation of state of vacuum with
4341:
1377:
595:
7074:
6350:International Journal of Theoretical Physics
5076:, the spatial curvature index, serving as a
2490:{\displaystyle z=r\sin \theta \sin \phi \,,}
6470:
6025:
5690:) contained within a part of the universe.
5431:
2440:{\displaystyle y=r\sin \theta \cos \phi \,}
2239:. These definitions are valid for all
7422:
7408:
6846:
6696:(1935), "Kinematics and world structure",
4348:
4334:
2512:
2159:
616:Friedmann–Lemaître–Robertson–Walker metric
602:
588:
34:
7355:
7265:
7228:
7210:
7153:
7123:
7026:
6958:
6873:
6799:
6772:
6758:
6731:
6692:
6625:
6551:
6505:
6423:
6361:
6195:
6146:
5665:
5549:
5287:
4803:cosmological model including the current
2483:
2436:
2387:
2174:
1664:
1575:
1468:
1179:
7329:
6912:
6663:
6634:
6598:
6509:(1922). "Über die Krümmung des Raumes".
5869:{\displaystyle {\dot {a}}={\ddot {a}}=0}
5142:
4434:This metric has an analytic solution to
2883:
2246:
1873:is essentially a third angle along with
8348:Coordinate charts in general relativity
7082:The large scale structure of space-time
6859:
6280:
6229:Bergström, Lars; Goobar, Ariel (2008).
6188:Theoretical and Observational Cosmology
5137:causes an acceleration in the expansion
1264:), centered at the origin, divided by 2
1010:Reduced-circumference polar coordinates
8340:
6813:
6179:
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
8353:Exact solutions in general relativity
7403:
7308:
6222:
6124:
6915:"Do We Live in a Lopsided Universe?"
6305:
6255:
7994:Tolman–Oppenheimer–Volkoff equation
7947:Friedmann–Lemaître–Robertson–Walker
6862:Journal of High Energy Astrophysics
6232:Cosmology and particle astrophysics
6044:(more unsolved problems in physics)
5792:, which are specific solutions for
5718:
13:
7302:
6667:(1933), "l'Univers en expansion",
5227:
5175:
5024:
4748:
4593:
4460:
3659:
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:
8374:
7764:Hamilton–Jacobi–Einstein equation
7377:Introducing Einstein's relativity
6013:
5984:Newtonian constant of gravitation
5809:Einstein's radius of the universe
5804:Einstein's radius of the universe
5744:Friedmann died in 1925. In 1927,
5424:Such a field is sometimes called
5311:An attempt to generalize this to
4831:
790:
545:Timeline of cosmological theories
310:Cosmic Background Explorer (COBE)
8325:
8313:
8301:
8289:
8277:
8265:
8242:
8241:
7315:. New York: Dover Publications.
6913:Billings, Lee (April 15, 2020).
6473:Quarterly Journal of Mathematics
6408:Their solutions can be found in
5294:{\displaystyle p=-\rho c^{2}\,,}
4315:
4314:
4301:
3631:
2272:
1798:
1512:
1476:
1411:
1297:
1256:is sometimes called the reduced
1132:
1106:
1031:
976:
937:
903:
624:
569:
558:
557:
7245:
7186:
7132:
7067:
6934:
6906:
6807:
6766:
6725:
6686:
6657:
6592:
6545:
6499:
6464:
6402:
4818:primordial density fluctuations
2605:
2207:where sinc is the unnormalized
1119:
325:Sloan Digital Sky Survey (SDSS)
178:Future of an expanding universe
7571:Mass–energy equivalence (E=mc)
7429:
7085:, Cambridge University Press,
6341:
6324:
6299:
6274:
6249:
6111:
5695:gravitational potential energy
5360:without further modification.
5216:
5164:
4429:cosmological equation of state
4411:
4405:
4382:
4376:
3589:
3583:
3558:
3552:
3527:
3521:
3490:
3484:
3466:
3460:
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:
7254:Classical and Quantum Gravity
7098:Stoeger, W. R.; Maartens, R;
7015:Classical and Quantum Gravity
6104:
6010:, or 10 billion light years.
5750:Catholic University of Leuven
5346:{\displaystyle p=w\rho c^{2}}
532:Discovery of cosmic microwave
183:Ultimate fate of the universe
7142:Astronomy & Astrophysics
6947:Astronomy & Astrophysics
6165:10.1016/0370-1573(94)00085-H
6061:with theoretical results of
3616:
2507:
1885:may be used instead of
1835:) has units of length. When
1338:) has units of length. When
715:(or otherwise, contracting)
7:
7586:Relativistic Doppler effect
7172:10.1051/0004-6361/202039840
6977:10.1051/0004-6361/201936602
6892:10.1016/j.jheap.2022.04.002
6026:Unsolved problem in physics
5688:first law of thermodynamics
5085:first law of thermodynamics
4812:is well approximated by an
3812:Gravitational time dilation
2228:{\displaystyle {\sqrt {k}}}
1390:coordinates the coordinate
747:– are variously grouped as
300:Black Hole Initiative (BHI)
10:
8379:
8057:In computational physics:
7581:Relativity of simultaneity
7309:North, John David (1990).
6063:Ehlers–Geren–Sachs theorem
6017:
5116:{\displaystyle {\dot {a}}}
4436:Einstein's field equations
4359:
3932:Mathisson–Papapetrou–Dixon
3773: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
8239:
8071:
7936:
7908:
7894:Lense–Thirring precession
7777:
7726:
7688:
7667:
7656:
7614:
7558:
7542:
7484:
7476:Doubly special relativity
7448:
7437:
7199:The Astrophysical Journal
6442:10.1142/S021773231550100x
6380:10.1007/s10773-006-9123-2
6281:Carroll, Sean M. (2019).
5723:The Soviet mathematician
4420:{\displaystyle \rho (t),}
2255:= 0 one may write simply
1274:Schwarzschild coordinates
703:. The metric describes a
156:Expansion of the universe
7754:Post-Newtonian formalism
7744:Einstein field equations
7680:Mathematical formulation
7504:Hyperbolic orthogonality
7330:Harrison, E. R. (1967).
7284:10.1088/1361-6382/ac1a81
7230:10.3847/2041-8213/abdd40
7045:10.1088/1361-6382/ac1a81
6557:Zeitschrift für Physik A
6511:Zeitschrift für Physik A
6412:Modern Physics Letters A
6306:Wald, Robert M. (1984).
6256:Wald, Robert M. (1984).
5432:Newtonian interpretation
5080:for the first equation.
3937:Hamilton–Jacobi–Einstein
3917:Einstein field equations
3740: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)
7465:Galilean transformation
7456:Principle of relativity
7373:D'Inverno, Ray (1992).
7164:2021A&A...653A...9S
6969:2020A&A...636A..15M
6841:10.1112/plms/s2-42.1.90
6493:10.1093/qmath/os-5.1.73
5962:is the speed of light,
5078:constant of integration
173:Inhomogeneous cosmology
7550:Lorentz transformation
7357:10.1093/mnras/137.1.69
6627:10.1093/mnras/91.5.483
6087:
6033:cosmological principle
6000:
5976:
5956:
5933:
5870:
5778:Arthur Geoffrey Walker
5739:Zeitschrift für Physik
5730:Zeitschrift für Physik
5673:
5554:
5415:
5347:
5295:
5242:
5199:
5117:
5070:
5047:
4921:
4790:
4644:
4511:energy–momentum tensor
4499:
4421:
4389:
3807:Gravitational redshift
3700:
3607:
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
8018:Weyl−Lewis−Papapetrou
7759:Raychaudhuri equation
7698:Equivalence principle
7075:Hawking, Stephen W.;
6779:Astrophysical Journal
6738:Astrophysical Journal
6699:Astrophysical Journal
6088:
6086:{\displaystyle H_{0}}
6020:Shape of the universe
6001:
5999:{\displaystyle \rho }
5977:
5957:
5934:
5871:
5711:, one cannot neglect
5702:cosmological constant
5674:
5555:
5416:
5348:
5296:
5253:cosmological constant
5243:
5200:
5149:cosmological constant
5143:Cosmological constant
5135:, on the other hand,
5133:cosmological constant
5118:
5071:
5048:
4922:
4791:
4645:
4500:
4422:
4390:
4095:Weyl−Lewis−Papapetrou
4050:Kerr–Newman–de Sitter
3870:Einstein–Rosen bridge
3802:Gravitational lensing
3758:Equivalence principle
3701:
3608:
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
8059:Numerical relativity
7900:pulsar timing arrays
6553:Friedmann, Alexander
6507:Friedmann, Alexander
6070:
5990:
5966:
5946:
5886:
5830:
5737:, who, on behalf of
5565:
5443:
5374:
5318:
5262:
5210:
5158:
5098:
5060:
4932:
4843:
4655:
4520:
4441:
4399:
4388:{\displaystyle a(t)}
4370:
4025:Einstein–Rosen waves
3751:Fundamental concepts
3640:
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
7951:Friedmann equations
7845:Hulse–Taylor binary
7807:Gravitational waves
7703:Riemannian geometry
7529:Proper acceleration
7514:Maxwell's equations
7460:Galilean relativity
7348:1967MNRAS.137...69H
7276:2021CQGra..38r4001K
7221:2021ApJ...908L..51S
7116:1995ApJ...443....1S
7077:Ellis, George F. R.
7037:2021CQGra..38r4001K
6953:(April 2020): A15.
6920:Scientific American
6884:2022JHEAp..34...49A
6833:1937PLMS...42...90W
6792:1936ApJ....83..257R
6751:1936ApJ....83..187R
6712:1935ApJ....82..284R
6681:1933ASSB...53...51L
6652:1927ASSB...47...49L
6618:1931MNRAS..91..483L
6569:1924ZPhy...21..326F
6523:1922ZPhy...10..377F
6485:1934QJMat...5...73M
6434:2015MPLA...3050100R
6372:2006IJTP...45.1152R
6206:1999ASIC..541....1E
6157:1995PhR...254..135L
5819:, a long-abandoned
5817:Einstein's universe
5813:radius of curvature
5774:Howard P. Robertson
5725:Alexander Friedmann
4810:observable universe
4507:Friedmann equations
4362:Friedmann equations
3979:Kaluza–Klein theory
3865:Minkowski spacetime
3817: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
8358:Physical cosmology
8000:Reissner–Nordström
7918:Brans–Dicke theory
7749:Linearized gravity
7576:Length contraction
7494:Frame of reference
7471:Special relativity
7073:See pp. 351ff. in
6577:10.1007/BF01328280
6531:10.1007/BF01332580
6308:General relativity
6258:General relativity
6083:
5996:
5972:
5952:
5929:
5866:
5669:
5550:
5411:
5358:general invariance
5343:
5291:
5238:
5195:
5129:general relativity
5113:
5066:
5043:
4917:
4786:
4640:
4495:
4417:
4385:
4308:Physics portal
4080:Oppenheimer–Snyder
4020:Reissner–Nordström
3912:Linearized gravity
3860:Spacetime diagrams
3763:Special relativity
3696:
3625:General relativity
3603:
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
8253:
8252:
8067:
8066:
8046:Ozsváth–Schücking
7652:
7651:
7634:Minkowski diagram
7591:Thomas precession
7534:Relativistic mass
7388:978-0-19-859686-8
7322:978-0-486-66517-7
7092:978-0-521-09906-6
6665:Lemaître, Georges
6636:Lemaître, Georges
6600:Lemaître, Georges
6317:978-0-226-87032-8
6292:978-1-108-48839-6
6267:978-0-226-87032-8
6242:978-3-540-32924-4
5975:{\displaystyle G}
5955:{\displaystyle c}
5924:
5896:
5857:
5842:
5663:
5635:
5592:
5580:
5547:
5533:
5496:
5468:
5406:
5233:
5193:
5139:of the universe.
5110:
5089:adiabatic process
5069:{\displaystyle k}
5041:
5011:
4978:
4950:
4945:
4910:
4882:
4877:
4855:
4814:almost FLRW model
4743:
4701:
4696:
4676:
4671:
4635:
4610:
4585:
4543:
4538:
4358:
4357:
3991:
3990:
3877:
3876:
3593:
3562:
3512:
3494:
3457:
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:
8370:
8330:
8329:
8328:
8318:
8317:
8316:
8306:
8305:
8304:
8294:
8293:
8282:
8281:
8280:
8270:
8269:
8261:
8245:
8244:
8028:van Stockum dust
7800:Two-body problem
7718:Mach's principle
7665:
7664:
7606:Terrell rotation
7446:
7445:
7424:
7417:
7410:
7401:
7400:
7392:
7380:
7369:
7359:
7326:
7296:
7295:
7269:
7249:
7243:
7242:
7232:
7214:
7190:
7184:
7183:
7157:
7136:
7130:
7128:
7127:
7095:
7071:
7065:
7064:
7030:
7010:
7004:
7003:
7001:
6999:
6962:
6938:
6932:
6931:
6929:
6927:
6910:
6904:
6903:
6877:
6857:
6844:
6843:
6811:
6805:
6804:
6803:
6774:Robertson, H. P.
6770:
6764:
6763:
6762:
6733:Robertson, H. P.
6729:
6723:
6722:
6694:Robertson, H. P.
6690:
6684:
6683:
6661:
6655:
6654:
6630:
6629:
6596:
6590:
6588:
6549:
6543:
6542:
6503:
6497:
6496:
6468:
6462:
6461:
6427:
6406:
6400:
6399:
6365:
6356:(6): 1152–1157.
6345:
6339:
6338:
6336:
6328:
6322:
6321:
6303:
6297:
6296:
6278:
6272:
6271:
6253:
6247:
6246:
6226:
6220:
6219:
6199:
6183:
6177:
6176:
6150:
6128:
6122:
6115:
6100:
6098:
6092:
6090:
6089:
6084:
6082:
6081:
6051:Lambda-CDM model
6027:
6005:
6003:
6002:
5997:
5981:
5979:
5978:
5973:
5961:
5959:
5958:
5953:
5938:
5936:
5935:
5930:
5925:
5911:
5909:
5898:
5897:
5894:
5875:
5873:
5872:
5867:
5859:
5858:
5850:
5844:
5843:
5835:
5776:from the US and
5762:Arthur Eddington
5746:Georges Lemaître
5719:Name and history
5678:
5676:
5675:
5670:
5664:
5659:
5658:
5657:
5644:
5636:
5634:
5626:
5622:
5621:
5612:
5611:
5598:
5593:
5588:
5587:
5582:
5581:
5573:
5569:
5559:
5557:
5556:
5551:
5548:
5546:
5545:
5536:
5535:
5534:
5526:
5520:
5519:
5506:
5498:
5497:
5489:
5486:
5485:
5470:
5469:
5461:
5458:
5457:
5420:
5418:
5417:
5412:
5407:
5402:
5401:
5400:
5387:
5352:
5350:
5349:
5344:
5342:
5341:
5300:
5298:
5297:
5292:
5286:
5285:
5247:
5245:
5244:
5239:
5234:
5226:
5204:
5202:
5201:
5196:
5194:
5192:
5191:
5190:
5174:
5122:
5120:
5119:
5114:
5112:
5111:
5103:
5075:
5073:
5072:
5067:
5052:
5050:
5049:
5044:
5042:
5037:
5036:
5035:
5022:
5017:
5013:
5012:
5010:
5009:
5000:
4992:
4979:
4974:
4973:
4972:
4959:
4951:
4946:
4938:
4936:
4926:
4924:
4923:
4918:
4916:
4912:
4911:
4909:
4908:
4896:
4883:
4878:
4870:
4868:
4857:
4856:
4848:
4795:
4793:
4792:
4787:
4779:
4778:
4760:
4759:
4744:
4742:
4741:
4732:
4731:
4730:
4717:
4712:
4711:
4706:
4702:
4697:
4689:
4687:
4677:
4672:
4664:
4662:
4649:
4647:
4646:
4641:
4636:
4631:
4630:
4629:
4616:
4611:
4606:
4605:
4604:
4591:
4586:
4584:
4583:
4574:
4573:
4572:
4559:
4554:
4553:
4548:
4544:
4539:
4531:
4529:
4504:
4502:
4501:
4496:
4494:
4493:
4475:
4474:
4456:
4455:
4426:
4424:
4423:
4418:
4394:
4392:
4391:
4386:
4350:
4343:
4336:
4323:
4318:
4317:
4310:
4306:
4305:
4090:van Stockum dust
4075:Robertson–Walker
3901:
3900:
3791:
3790:
3705:
3703:
3702:
3697:
3695:
3694:
3682:
3674:
3673:
3655:
3654:
3635:
3621:
3620:
3612:
3610:
3609:
3604:
3599:
3595:
3594:
3592:
3582:
3581:
3568:
3563:
3561:
3551:
3550:
3541:
3540:
3530:
3520:
3519:
3514:
3513:
3505:
3500:
3495:
3493:
3480:
3479:
3469:
3459:
3458:
3450:
3446:
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:
8378:
8377:
8373:
8372:
8371:
8369:
8368:
8367:
8338:
8337:
8336:
8326:
8324:
8314:
8312:
8302:
8300:
8288:
8278:
8276:
8264:
8256:
8254:
8249:
8235:
8063:
7967:BKL singularity
7957:Lemaître–Tolman
7932:
7928:Quantum gravity
7910:
7904:
7890:geodetic effect
7864:(together with
7834:LISA Pathfinder
7773:
7722:
7708:Penrose diagram
7690:
7684:
7659:
7648:
7644:Minkowski space
7610:
7554:
7538:
7486:
7480:
7440:
7433:
7428:
7389:
7323:
7305:
7303:Further reading
7300:
7299:
7250:
7246:
7191:
7187:
7137:
7133:
7093:
7072:
7068:
7011:
7007:
6997:
6995:
6939:
6935:
6925:
6923:
6911:
6907:
6858:
6847:
6812:
6808:
6771:
6767:
6730:
6726:
6691:
6687:
6662:
6658:
6632:translated from
6597:
6593:
6550:
6546:
6504:
6500:
6469:
6465:
6418:(20): 1550100.
6407:
6403:
6346:
6342:
6334:
6330:
6329:
6325:
6318:
6304:
6300:
6293:
6279:
6275:
6268:
6254:
6250:
6243:
6227:
6223:
6216:
6184:
6180:
6134:Physics Reports
6129:
6125:
6116:
6112:
6107:
6099:1 km/s/Mpc
6096:
6094:
6077:
6073:
6071:
6068:
6067:
6047:
6046:
6041:
6029:
6022:
6016:
5991:
5988:
5987:
5967:
5964:
5963:
5947:
5944:
5943:
5910:
5905:
5893:
5889:
5887:
5884:
5883:
5849:
5848:
5834:
5833:
5831:
5828:
5827:
5806:
5735:Albert Einstein
5721:
5653:
5649:
5645:
5643:
5627:
5617:
5613:
5607:
5603:
5599:
5597:
5583:
5572:
5571:
5570:
5568:
5566:
5563:
5562:
5541:
5537:
5525:
5524:
5515:
5511:
5507:
5505:
5488:
5487:
5481:
5477:
5460:
5459:
5453:
5449:
5444:
5441:
5440:
5434:
5396:
5392:
5388:
5386:
5375:
5372:
5371:
5367:that satisfies
5356:would not have
5337:
5333:
5319:
5316:
5315:
5281:
5277:
5263:
5260:
5259:
5251:Therefore, the
5225:
5211:
5208:
5207:
5186:
5182:
5178:
5173:
5159:
5156:
5155:
5145:
5102:
5101:
5099:
5096:
5095:
5061:
5058:
5057:
5031:
5027:
5023:
5021:
5005:
5001:
4993:
4991:
4984:
4980:
4968:
4964:
4960:
4958:
4937:
4935:
4933:
4930:
4929:
4904:
4900:
4895:
4888:
4884:
4869:
4867:
4847:
4846:
4844:
4841:
4840:
4834:
4774:
4770:
4755:
4751:
4737:
4733:
4726:
4722:
4718:
4716:
4707:
4688:
4686:
4682:
4681:
4663:
4661:
4656:
4653:
4652:
4625:
4621:
4617:
4615:
4600:
4596:
4592:
4590:
4579:
4575:
4568:
4564:
4560:
4558:
4549:
4530:
4528:
4524:
4523:
4521:
4518:
4517:
4486:
4482:
4467:
4463:
4448:
4444:
4442:
4439:
4438:
4400:
4397:
4396:
4371:
4368:
4367:
4364:
4354:
4313:
4300:
4299:
4292:
4291:
4115:
4114:
4105:
4104:
4060:Lemaître–Tolman
4005:
4004:
3993:
3992:
3984:Quantum gravity
3971:Advanced theory
3898:
3897:
3896:
3879:
3878:
3827:Geodetic effect
3788:
3787:
3778:
3777:
3753:
3752:
3736:
3706:
3687:
3683:
3678:
3666:
3662:
3647:
3643:
3641:
3638:
3637:
3619:
3577:
3573:
3572:
3567:
3546:
3542:
3536:
3532:
3531:
3515:
3504:
3503:
3502:
3501:
3499:
3475:
3471:
3470:
3449:
3448:
3447:
3445:
3444:
3440:
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:
8376:
8366:
8365:
8363:Metric tensors
8360:
8355:
8350:
8335:
8334:
8322:
8310:
8298:
8286:
8274:
8251:
8250:
8240:
8237:
8236:
8234:
8233:
8226:
8221:
8216:
8211:
8206:
8201:
8196:
8191:
8186:
8181:
8176:
8171:
8166:
8161:
8156:
8154:Choquet-Bruhat
8151:
8146:
8141:
8136:
8131:
8126:
8121:
8116:
8111:
8106:
8101:
8096:
8091:
8086:
8081:
8075:
8073:
8069:
8068:
8065:
8064:
8062:
8061:
8054:
8053:
8048:
8043:
8036:
8035:
8030:
8025:
8020:
8015:
8006:Axisymmetric:
8003:
8002:
7997:
7991:
7980:
7979:
7974:
7969:
7964:
7959:
7954:
7945:Cosmological:
7942:
7940:
7934:
7933:
7931:
7930:
7925:
7920:
7914:
7912:
7906:
7905:
7903:
7902:
7897:
7886:frame-dragging
7883:
7878:
7873:
7870:Einstein rings
7866:Einstein cross
7859:
7848:
7847:
7842:
7836:
7831:
7826:
7813:
7803:
7802:
7797:
7792:
7787:
7781:
7779:
7775:
7774:
7772:
7771:
7769:Ernst equation
7766:
7761:
7756:
7751:
7746:
7741:
7739:BSSN formalism
7736:
7730:
7728:
7724:
7723:
7721:
7720:
7715:
7710:
7705:
7700:
7694:
7692:
7686:
7685:
7683:
7682:
7677:
7671:
7669:
7662:
7654:
7653:
7650:
7649:
7647:
7646:
7641:
7636:
7631:
7626:
7620:
7618:
7612:
7611:
7609:
7608:
7603:
7598:
7596:Ladder paradox
7593:
7588:
7583:
7578:
7573:
7568:
7562:
7560:
7556:
7555:
7553:
7552:
7546:
7544:
7540:
7539:
7537:
7536:
7531:
7526:
7521:
7516:
7511:
7506:
7501:
7499:Speed of light
7496:
7490:
7488:
7482:
7481:
7479:
7478:
7473:
7468:
7462:
7452:
7450:
7443:
7435:
7434:
7427:
7426:
7419:
7412:
7404:
7398:
7397:
7387:
7370:
7327:
7321:
7304:
7301:
7298:
7297:
7260:(18): 184001.
7244:
7185:
7131:
7125:10.1086/175496
7091:
7066:
7021:(18): 184001.
7005:
6933:
6905:
6845:
6806:
6801:10.1086/143726
6765:
6760:10.1086/143716
6724:
6720:10.1086/143681
6685:
6656:
6612:(5): 483–490,
6591:
6563:(1): 326–332.
6544:
6517:(1): 377–386.
6498:
6463:
6401:
6340:
6323:
6316:
6298:
6291:
6273:
6266:
6248:
6241:
6221:
6215:978-0792359463
6214:
6178:
6141:(3): 135–214.
6123:
6109:
6108:
6106:
6103:
6080:
6076:
6042:
6030:
6024:
6015:
6014:Current status
6012:
5995:
5971:
5951:
5940:
5939:
5928:
5923:
5920:
5917:
5914:
5908:
5904:
5901:
5892:
5877:
5876:
5865:
5862:
5856:
5853:
5847:
5841:
5838:
5805:
5802:
5720:
5717:
5680:
5679:
5668:
5662:
5656:
5652:
5648:
5642:
5639:
5633:
5630:
5625:
5620:
5616:
5610:
5606:
5602:
5596:
5591:
5586:
5579:
5576:
5560:
5544:
5540:
5532:
5529:
5523:
5518:
5514:
5510:
5504:
5501:
5495:
5492:
5484:
5480:
5476:
5473:
5467:
5464:
5456:
5452:
5448:
5433:
5430:
5422:
5421:
5410:
5405:
5399:
5395:
5391:
5385:
5382:
5379:
5354:
5353:
5340:
5336:
5332:
5329:
5326:
5323:
5302:
5301:
5290:
5284:
5280:
5276:
5273:
5270:
5267:
5249:
5248:
5237:
5232:
5229:
5224:
5221:
5218:
5215:
5205:
5189:
5185:
5181:
5177:
5172:
5169:
5166:
5163:
5144:
5141:
5109:
5106:
5065:
5054:
5053:
5040:
5034:
5030:
5026:
5020:
5016:
5008:
5004:
4999:
4996:
4990:
4987:
4983:
4977:
4971:
4967:
4963:
4957:
4954:
4949:
4944:
4941:
4927:
4915:
4907:
4903:
4899:
4894:
4891:
4887:
4881:
4876:
4873:
4866:
4863:
4860:
4854:
4851:
4833:
4832:Interpretation
4830:
4797:
4796:
4785:
4782:
4777:
4773:
4769:
4766:
4763:
4758:
4754:
4750:
4747:
4740:
4736:
4729:
4725:
4721:
4715:
4710:
4705:
4700:
4695:
4692:
4685:
4680:
4675:
4670:
4667:
4660:
4650:
4639:
4634:
4628:
4624:
4620:
4614:
4609:
4603:
4599:
4595:
4589:
4582:
4578:
4571:
4567:
4563:
4557:
4552:
4547:
4542:
4537:
4534:
4527:
4492:
4489:
4485:
4481:
4478:
4473:
4470:
4466:
4462:
4459:
4454:
4451:
4447:
4416:
4413:
4410:
4407:
4404:
4384:
4381:
4378:
4375:
4360:Main article:
4356:
4355:
4353:
4352:
4345:
4338:
4330:
4327:
4326:
4325:
4324:
4311:
4294:
4293:
4290:
4289:
4282:
4277:
4272:
4267:
4262:
4257:
4252:
4247:
4242:
4237:
4232:
4227:
4222:
4217:
4212:
4207:
4202:
4197:
4192:
4187:
4182:
4177:
4172:
4167:
4162:
4157:
4152:
4147:
4142:
4137:
4132:
4127:
4122:
4116:
4112:
4111:
4110:
4107:
4106:
4103:
4102:
4097:
4092:
4087:
4082:
4077:
4072:
4067:
4062:
4057:
4052:
4047:
4042:
4037:
4032:
4027:
4022:
4017:
4006:
4000:
3999:
3998:
3995:
3994:
3989:
3988:
3987:
3986:
3981:
3973:
3972:
3968:
3967:
3966:
3965:
3963:Post-Newtonian
3960:
3955:
3947:
3946:
3942:
3941:
3940:
3939:
3934:
3929:
3924:
3919:
3914:
3906:
3905:
3899:
3895:
3894:
3891:
3887:
3886:
3885:
3884:
3881:
3880:
3875:
3874:
3873:
3872:
3867:
3862:
3854:
3853:
3847:
3846:
3845:
3844:
3839:
3834:
3829:
3824:
3822:Frame-dragging
3819:
3814:
3809:
3804:
3799:
3797:Kepler problem
3789:
3785:
3784:
3783:
3780:
3779:
3776:
3775:
3770:
3765:
3760:
3754:
3750:
3749:
3748:
3745:
3744:
3743:
3742:
3737:
3735:
3734:
3729:
3724:
3718:
3716:
3708:
3707:
3693:
3690:
3686:
3681:
3677:
3672:
3669:
3665:
3661:
3658:
3653:
3650:
3646:
3636:
3628:
3627:
3618:
3615:
3614:
3613:
3602:
3598:
3591:
3588:
3585:
3580:
3576:
3571:
3566:
3560:
3557:
3554:
3549:
3545:
3539:
3535:
3529:
3526:
3523:
3518:
3511:
3508:
3498:
3492:
3489:
3486:
3483:
3478:
3474:
3468:
3465:
3462:
3456:
3453:
3443:
3439:
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:
8375:
8364:
8361:
8359:
8356:
8354:
8351:
8349:
8346:
8345:
8343:
8333:
8323:
8321:
8311:
8309:
8299:
8297:
8292:
8287:
8285:
8275:
8273:
8268:
8263:
8262:
8259:
8248:
8238:
8232:
8231:
8227:
8225:
8222:
8220:
8217:
8215:
8212:
8210:
8207:
8205:
8202:
8200:
8197:
8195:
8192:
8190:
8187:
8185:
8182:
8180:
8177:
8175:
8172:
8170:
8167:
8165:
8162:
8160:
8157:
8155:
8152:
8150:
8147:
8145:
8142:
8140:
8139:Chandrasekhar
8137:
8135:
8132:
8130:
8127:
8125:
8122:
8120:
8117:
8115:
8112:
8110:
8107:
8105:
8102:
8100:
8099:Schwarzschild
8097:
8095:
8092:
8090:
8087:
8085:
8082:
8080:
8077:
8076:
8074:
8070:
8060:
8056:
8055:
8052:
8049:
8047:
8044:
8042:
8038:
8037:
8034:
8031:
8029:
8026:
8024:
8021:
8019:
8016:
8013:
8009:
8005:
8004:
8001:
7998:
7995:
7992:
7990:
7986:
7985:Schwarzschild
7982:
7981:
7978:
7975:
7973:
7970:
7968:
7965:
7963:
7960:
7958:
7955:
7952:
7948:
7944:
7943:
7941:
7939:
7935:
7929:
7926:
7924:
7921:
7919:
7916:
7915:
7913:
7907:
7901:
7898:
7895:
7891:
7887:
7884:
7882:
7881:Shapiro delay
7879:
7877:
7874:
7871:
7867:
7863:
7860:
7857:
7853:
7850:
7849:
7846:
7843:
7840:
7837:
7835:
7832:
7830:
7827:
7825:
7824:collaboration
7821:
7817:
7814:
7812:
7808:
7805:
7804:
7801:
7798:
7796:
7793:
7791:
7790:Event horizon
7788:
7786:
7783:
7782:
7780:
7776:
7770:
7767:
7765:
7762:
7760:
7757:
7755:
7752:
7750:
7747:
7745:
7742:
7740:
7737:
7735:
7734:ADM formalism
7732:
7731:
7729:
7725:
7719:
7716:
7714:
7711:
7709:
7706:
7704:
7701:
7699:
7696:
7695:
7693:
7687:
7681:
7678:
7676:
7673:
7672:
7670:
7666:
7663:
7661:
7655:
7645:
7642:
7640:
7639:Biquaternions
7637:
7635:
7632:
7630:
7627:
7625:
7622:
7621:
7619:
7617:
7613:
7607:
7604:
7602:
7599:
7597:
7594:
7592:
7589:
7587:
7584:
7582:
7579:
7577:
7574:
7572:
7569:
7567:
7566:Time dilation
7564:
7563:
7561:
7557:
7551:
7548:
7547:
7545:
7541:
7535:
7532:
7530:
7527:
7525:
7522:
7520:
7519:Proper length
7517:
7515:
7512:
7510:
7507:
7505:
7502:
7500:
7497:
7495:
7492:
7491:
7489:
7483:
7477:
7474:
7472:
7469:
7466:
7463:
7461:
7457:
7454:
7453:
7451:
7447:
7444:
7442:
7436:
7432:
7425:
7420:
7418:
7413:
7411:
7406:
7405:
7402:
7396:
7390:
7384:
7379:
7378:
7371:
7367:
7363:
7358:
7353:
7349:
7345:
7341:
7337:
7333:
7328:
7324:
7318:
7314:
7313:
7307:
7306:
7293:
7289:
7285:
7281:
7277:
7273:
7268:
7263:
7259:
7255:
7248:
7240:
7236:
7231:
7226:
7222:
7218:
7213:
7208:
7204:
7200:
7196:
7189:
7181:
7177:
7173:
7169:
7165:
7161:
7156:
7151:
7147:
7143:
7135:
7126:
7121:
7117:
7113:
7109:
7105:
7104:Astrophys. J.
7101:
7100:Ellis, George
7094:
7088:
7084:
7083:
7078:
7070:
7062:
7058:
7054:
7050:
7046:
7042:
7038:
7034:
7029:
7024:
7020:
7016:
7009:
6994:
6990:
6986:
6982:
6978:
6974:
6970:
6966:
6961:
6956:
6952:
6948:
6944:
6937:
6922:
6921:
6916:
6909:
6901:
6897:
6893:
6889:
6885:
6881:
6876:
6871:
6867:
6863:
6856:
6854:
6852:
6850:
6842:
6838:
6834:
6830:
6827:(1): 90–127,
6826:
6822:
6821:
6816:
6815:Walker, A. G.
6810:
6802:
6797:
6793:
6789:
6785:
6781:
6780:
6775:
6769:
6761:
6756:
6752:
6748:
6744:
6740:
6739:
6734:
6728:
6721:
6717:
6713:
6709:
6705:
6701:
6700:
6695:
6689:
6682:
6678:
6674:
6670:
6666:
6660:
6653:
6649:
6645:
6641:
6637:
6633:
6628:
6623:
6619:
6615:
6611:
6607:
6606:
6601:
6595:
6586:
6582:
6578:
6574:
6570:
6566:
6562:
6559:(in German).
6558:
6554:
6548:
6540:
6536:
6532:
6528:
6524:
6520:
6516:
6512:
6508:
6502:
6494:
6490:
6486:
6482:
6478:
6474:
6467:
6459:
6455:
6451:
6447:
6443:
6439:
6435:
6431:
6426:
6421:
6417:
6413:
6405:
6397:
6393:
6389:
6385:
6381:
6377:
6373:
6369:
6364:
6363:gr-qc/0510004
6359:
6355:
6351:
6344:
6337:. p. 23.
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6037:inhomogeneous
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4140:Schwarzschild
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4100:Hartle–Thorne
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3832:Event horizon
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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:
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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:
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1784:
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1774:
1770:
1765:
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1753:
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1740:
1716:
1713:
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1705:
1692:
1682:
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1627:
1624:
1617:
1614:
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1606:
1598:
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1579:
1576:
1569:
1560:
1555:
1549:
1541:
1537:
1529:
1528:
1527:
1481:
1463:
1455:
1447:
1443:
1439:
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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:
8332:Solar System
8229:
7946:
7923:Kaluza–Klein
7675:Introduction
7601:Twin paradox
7394:
7376:
7342:(1): 69–79.
7339:
7335:
7311:
7257:
7253:
7247:
7202:
7198:
7188:
7145:
7141:
7134:
7107:
7103:
7081:
7069:
7018:
7014:
7008:
6996:. Retrieved
6950:
6946:
6936:
6924:. Retrieved
6918:
6908:
6875:2203.06142v1
6865:
6861:
6824:
6823:, Series 2,
6818:
6809:
6783:
6777:
6768:
6742:
6736:
6727:
6703:
6697:
6688:
6672:
6668:
6659:
6643:
6639:
6631:
6609:
6603:
6594:
6560:
6556:
6547:
6514:
6510:
6501:
6476:
6472:
6466:
6415:
6411:
6404:
6353:
6349:
6343:
6326:
6307:
6301:
6282:
6276:
6257:
6251:
6231:
6224:
6187:
6181:
6138:
6132:
6126:
6118:
6113:
6048:
5941:
5878:
5821:static model
5815:of space of
5808:
5807:
5797:
5793:
5789:
5785:
5783:
5772:
5765:
5758:Edwin Hubble
5753:
5743:
5738:
5728:
5722:
5709:Planck epoch
5706:
5699:
5692:
5683:
5681:
5435:
5426:quintessence
5423:
5365:scalar field
5362:
5355:
5310:
5303:
5250:
5146:
5093:
5082:
5055:
4835:
4813:
4798:
4433:
4365:
4285:
4245:Raychaudhuri
4074:
3714: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:
8320:Outer space
8308:Spaceflight
8012:Kerr–Newman
7983:Spherical:
7852:Other tests
7795:Singularity
7727:Formulation
7689:Fundamental
7543:Formulation
7524:Proper time
7485:Fundamental
6786:: 257–271,
6745:: 187–201,
6706:: 284–301,
6332:"Cosmology"
6008:light years
5707:During the
5306:dark energy
5125:gravitation
4505:giving the
4260:van Stockum
4190:Oppenheimer
4045:Kerr–Newman
3837:Singularity
1324:{−1, 0, +1}
1122:where
797:homogeneity
705:homogeneous
293:Experiments
226:Dark matter
219:Dark energy
161:FLRW metric
98:Backgrounds
8342:Categories
8164:Zel'dovich
8072:Scientists
8051:Alcubierre
7858:of Mercury
7856:precession
7785:Black hole
7668:Background
7660:relativity
7629:World line
7624:Light cone
7449:Background
7441:relativity
7431:Relativity
7267:2105.09790
7212:2009.14826
7205:(2): L51.
7155:2010.08366
7028:2105.09790
6960:2004.03305
6868:: 49–211.
6425:1502.07033
6105:References
6018:See also:
4427:such as a
4113:Scientists
3945:Formalisms
3893:Formalisms
3842:Black hole
3768:World line
1372:elliptical
1268:(like the
779:of modern
679: ...
373:Copernicus
351:Scientists
206:Components
8284:Astronomy
8134:Robertson
8119:Friedmann
8114:Eddington
8104:de Sitter
7938:Solutions
7816:detectors
7811:astronomy
7778:Phenomena
7713:Geodesics
7616:Spacetime
7559:Phenomena
7366:0035-8711
7292:234790314
7239:222066749
7180:223953708
7061:234790314
7053:0264-9381
6993:215238834
6985:0004-6361
6926:March 24,
6900:247411131
6675:: 51–85,
6646:: 49–56,
6585:120551579
6539:125190902
6479:: 73–80.
6450:0217-7323
6396:119496918
6388:0020-7748
6173:119500217
6040:universe?
5994:ρ
5922:ρ
5916:π
5855:¨
5840:˙
5641:−
5624:ρ
5601:κ
5595:−
5578:˙
5531:˙
5500:ρ
5494:˙
5466:˙
5463:ρ
5447:−
5390:ρ
5384:−
5331:ρ
5275:ρ
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5231:κ
5228:Λ
5217:→
5180:κ
5176:Λ
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5168:ρ
5165:→
5162:ρ
5108:˙
5025:Λ
4986:ρ
4962:κ
4956:−
4943:¨
4890:ρ
4875:˙
4862:−
4853:˙
4850:ρ
4768:κ
4765:−
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4746:−
4694:˙
4669:¨
4638:ρ
4619:κ
4594:Λ
4588:−
4536:˙
4509:when the
4491:ν
4488:μ
4480:κ
4472:ν
4469:μ
4461:Λ
4453:ν
4450:μ
4403:ρ
4205:Robertson
4170:Friedmann
4165:Eddington
4155:Nordström
4145:de Sitter
4002:Solutions
3927:Geodesics
3922:Friedmann
3904:Equations
3890:Equations
3851:Spacetime
3786:Phenomena
3692:ν
3689:μ
3680:κ
3671:ν
3668:μ
3660:Λ
3652:ν
3649:μ
3617:Solutions
3510:˙
3455:¨
3403:θ
3397:
3351:˙
3323:¨
3294:−
3268:ϕ
3265:ϕ
3206:˙
3178:¨
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3123:θ
3120:θ
3078:−
3040:˙
3012:¨
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2928:¨
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2773:¨
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2660:−
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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
8247:Category
8124:Lemaître
8089:Einstein
8079:Poincaré
8039:Others:
8023:Taub–NUT
7989:interior
7911:theories
7909:Advanced
7876:redshift
7691:concepts
7509:Rapidity
7487:concepts
7079:(1973),
6998:24 March
6458:51948117
4801:Big Bang
4321:Category
4185:Lemaître
4150:Reissner
4135:Poincaré
4120:Einstein
4065:Taub–NUT
4030:Wormhole
4014:interior
3727: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
8272:Physics
8258:Portals
8189:Hawking
8184:Penrose
8169:Novikov
8149:Wheeler
8094:Hilbert
8084:Lorentz
8041:pp-wave
7862:lensing
7658:General
7439:Special
7344:Bibcode
7272:Bibcode
7217:Bibcode
7160:Bibcode
7112:Bibcode
7110:: 1–5,
7033:Bibcode
6965:Bibcode
6880:Bibcode
6829:Bibcode
6788:Bibcode
6747:Bibcode
6708:Bibcode
6677:Bibcode
6648:Bibcode
6614:Bibcode
6565:Bibcode
6519:Bibcode
6481:Bibcode
6430:Bibcode
6368:Bibcode
6202:Bibcode
6153:Bibcode
6119:assumes
5982:is the
5811:is the
5713:quantum
4240:Hawking
4235:Penrose
4210:Bardeen
4200:Wheeler
4130:Hilbert
4125:Lorentz
4085:pp-wave
3722:History
695:of the
687:) is a
681:
488:Sunyaev
473:Schmidt
463:Penzias
458:Penrose
433:Huygens
423:Hawking
408:Galileo
8230:others
8219:Thorne
8209:Misner
8194:Taylor
8179:Geroch
8174:Ehlers
8144:Zwicky
7962:Kasner
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5986:, and
5942:where
5790:models
5786:metric
5131:. The
4319:
4286:others
4280:Thorne
4270:Newman
4250:Taylor
4230:Ehlers
4215:Walker
4180:Zwicky
4055: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:
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138:Future
113:
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8296:Stars
8224:Weiss
8204:Bondi
8199:Hulse
8129:Milne
8033:discs
7977:Milne
7972:Gödel
7829:Virgo
7288:S2CID
7262:arXiv
7235:S2CID
7207:arXiv
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7023:arXiv
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6955:arXiv
6896:S2CID
6870:arXiv
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6454:S2CID
6420:arXiv
6392:S2CID
6358:arXiv
6335:(PDF)
6192:arXiv
6169:S2CID
6143:arXiv
5056:with
4255:Hulse
4195:Gödel
4175:Milne
4070:Milne
4035:Gödel
3732:Tests
2447:, and
2251:When
1775:) = 1
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959:, or
478:Smoot
468:Rubin
413:Gamow
398:Ellis
383:Dicke
8159:Kerr
8109:Weyl
8008:Kerr
7868:and
7822:and
7820:LIGO
7383:ISBN
7362:ISSN
7317:ISBN
7087:ISBN
7049:ISSN
7000:2022
6981:ISSN
6928:2022
6446:ISSN
6384:ISSN
6312:ISBN
6287:ISBN
6262:ISBN
6237:ISBN
6210:ISBN
6057:and
6055:WMAP
5700:The
5381:<
5147:The
4826:WMAP
4824:and
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4805:ΛCDM
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