36:
1544:. He added that Alan Guth, one of the co-authors of the theorem, disagrees with Vilenkin and believes that the universe had no beginning. Vilenkin argues that the Carroll-Chen model constructed by Carroll and Jennie Chen, and supported by Guth, to elude the BGV theorem's conclusions persists to indicate a singularity in the history of the universe as it has a reversal of the
1120:
204:, a spacetime is said to be geodesically complete if all its geodesics can be extended indefinitely without encountering any singularities or boundaries. On the contrary, a spacetime that is geodesically past-incomplete features geodesics that reach a boundary or a singularity within a finite amount of
1593:
and are geodesically complete. The authors claim that the geodesic incompleteness of inflationary spacetime is still an open issue. Furthermore, there are examples of infinite cyclic models solving the problem of unbounded entropy growth which are geodesically complete. In both of these studies, the
1610:
movement regarding the existence of God. Vilenkin stated "What causes the universe to pop out of nothing? No cause is needed." Regarding the BGV theorem itself, Vilenkin told Craig: "I think you represented what I wrote about the BGV theorem in my papers and to you personally very accurately."
860:
334:
1408:
1252:
1530:. Vilenkin and Audrey Mithani have argued that none of these models escape the implications of the theorem. In 2017, Vilenkin stated that he does not think there are any viable cosmological models that escape the scenario.
1115:{\displaystyle \int _{t_{\rm {i}}}^{t_{\rm {f}}}H(\tau )d\tau =\int _{a(t_{\rm {i}})}^{a(t_{\rm {f}})}{\frac {mda}{\sqrt {m^{2}a^{2}+p^{2}a(t_{\rm {f}})}}}=F(\gamma _{\rm {f}})-F(\gamma _{\rm {i}})\leq F(\gamma _{\rm {f}})}
1460:
Current astronomical observations, show that the universe is expanding, thus the BGV implies that there must be a boundary or singularity in the history of the universe. This singularity has often been associated to the
686:
838:
459:
217:
550:
736:
1591:
1508:
1465:. However the theorem does not tell if it is associated to any other event in the past. The theorem also does not allow to tell when the singularity takes place, or if it is a
1450:
1316:
421:
1472:
Some physical theories do not discard the possibility of a non-accelerated expansion before a certain moment in time. For example, the expansion rate could be different from
1167:
1324:
430:
The theorem only applies to classical spacetime, but it does not assume any specific mass content of the universe and it does not require gravity to be described by
2106:
1522:
Alternative models, where the average expansion of the universe throughout its history does not hold, have been proposed under the notions of emergent spacetime,
1594:
authors argue that the previous investigations often did not use mathematically precise formulations of the BGV theorem and thus reached incomplete conclusions.
1175:
1625:
1927:
172:, who developed its mathematical formulation in 2003. The BGV theorem is also popular outside physics, especially in religious and philosophical debates.
100:
72:
1635:
455:, the current model of cosmology. However, this derivation can be generalized to an arbitrary space-time with no appeal to homogeneity or isotropy.
79:
1858:
1828:
2151:
1602:
Vilenkin has also written about the religious significance of the BGV theorem. In
October 2015, Vilenkin responded to arguments made by theist
1868:
86:
68:
1752:
1663:
447:
Here is an example of derivation of the BGV theorem for an expanding homogeneous isotropic flat universe (in units of speed of light
2132:
1806:
1779:
119:
57:
50:
2107:
https://www.reasonablefaith.org/writings/question-answer/honesty-transparency-full-disclosure-and-the-borde-guth-vilenkin-theorem
1679:
Borde, Arvind; Guth, Alan H.; Vilenkin, Alexander (15 April 2003). "Inflationary space-times are incomplete in past directions".
614:
754:
185:
760:
93:
329:{\displaystyle H_{\rm {av}}={\frac {1}{\tau _{\rm {f}}-\tau _{\rm {i}}}}\int _{t_{\rm {i}}}^{t_{\rm {f}}}H(\tau )d\tau }
2156:
1620:
1630:
468:
2122:
46:
1466:
157:
1919:
1796:
1769:
577:
431:
691:
2022:
Lesnefsky, J. E.; Easson, D. A.; Davies, P. C. W. (2023). "Past-completeness of inflationary spacetimes".
1558:
1511:
1475:
1417:
1283:
388:
590:, we can consider the universe to be filled with comoving particles. For an observer with proper time
1988:
1840:
1698:
1129:
2068:
2031:
2004:
1978:
1949:
1885:
1722:
1688:
1603:
181:
169:
149:
2059:
Pavlović, Petar; Sossich, Marko (2024). "Geodesically complete cyclic cosmologies and entropy".
1771:
The Kalam
Cosmological Argument, Volume 2: Scientific Evidence for the Beginning of the Universe
1403:{\displaystyle \tau _{\rm {f}}-\tau _{\rm {i}}\leq {\frac {F(\gamma _{\rm {f}})}{H_{\rm {av}}}}}
2128:
1864:
1802:
1775:
1748:
1714:
1659:
1537:
1523:
196:. These paths are the equivalent of the shortest path (straight lines) between two points in
2078:
2041:
1996:
1884:
Mithani, Audrey; Vilenkin, Alexander (20 April 2012). "Did the universe have a beginning?".
1706:
452:
376:
17:
1541:
1533:
197:
1992:
1702:
1247:{\displaystyle F(\gamma )={\frac {1}{2}}\ln \left({\frac {\gamma +1}{\gamma -1}}\right)}
1262:
2082:
2145:
2008:
1545:
1726:
1527:
1710:
1414:
Thereby any non-comoving past-directed timelike geodesic satisfying the condition
2045:
1969:
Vilenkin, Alexander (2013). "Arrows of time and the beginning of the universe".
1607:
1552:
205:
35:
2000:
1948:
Carroll, Sean M. (2018-06-04). "Why Is There
Something, Rather Than Nothing?".
609:
595:
1555:
constructed an uncountable infinite class of classical solutions which have
201:
193:
189:
165:
161:
1718:
160:
throughout its history cannot be infinite in the past but must have a past
1462:
153:
1540:
spacetime, and may not hold under consideration of a complete theory of
145:
1693:
1452:, must have a finite proper length, and so must be past-incomplete.
2073:
2036:
1954:
2095:
1983:
1890:
29:
211:
In this context, we can define the average expansion rate as
681:{\displaystyle P^{\mu }=mdx^{\mu }/d\tau =(E,\mathbf {p} )}
1904:
Alexander
Vilenkin, "The Beginning of the Universe" in
164:
boundary. It is named after the authors Arvind Borde,
1561:
1478:
1420:
1327:
1286:
1178:
1132:
863:
833:{\displaystyle p(t)=p_{\rm {f}}\;a(t_{\rm {f}})/a(t)}
763:
694:
617:
471:
391:
220:
1743:
Perlov, Delia; Vilenkin, Alexander (7 August 2017).
1654:
Perlov, Delia; Vilenkin, Alexander (7 August 2017).
427:
then the spacetime is geodesically past-incomplete.
382:
The BGV theorem states that for any spacetime where
1798:
Scientific
Approaches to the Philosophy of Religion
1585:
1502:
1444:
1402:
1310:
1246:
1161:
1114:
832:
730:
680:
544:
415:
328:
1747:. Cham, Switzerland: Springer. pp. 330–31.
1658:. Cham, Switzerland: Springer. pp. 330–31.
1768:Copan, Paul; Craig, William Lane (2017-11-16).
1738:
1736:
545:{\displaystyle ds=dt^{2}-a^{2}(t)dx_{i}dx^{i}}
8:
375:is the expansion parameter, also called the
1551:Joseph E. Lesnefsky, Damien A. Easson and
791:
460:Friedmann–Lemaître–Robertson–Walker metric
2072:
2035:
1982:
1953:
1906:The Kalam Cosmological Argument: Volume 2
1889:
1692:
1567:
1566:
1560:
1484:
1483:
1477:
1469:or any other kind of boundary condition.
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265:
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250:
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226:
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219:
120:Learn how and when to remove this message
1536:argues that the theorem only applies to
568:=1,2,3) are the spatial coordinates and
1646:
731:{\displaystyle E={\sqrt {p^{2}+m^{2}}}}
192:particles or objects follow in curved
56:Please improve this article by adding
7:
1852:
1850:
1822:
1820:
1818:
1636:Penrose–Hawking singularity theorems
750:| the magnitude of the 3-momentum.
1827:Vilenkin, Alexander (2015-10-23).
1626:Gibbons–Hawking–York boundary term
1586:{\displaystyle H_{\rm {av}}\geq 0}
1571:
1568:
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1392:
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25:
2121:Calcagni, Gianluca (2017-01-06).
1857:Calcagni, Gianluca (2017-01-06).
1503:{\displaystyle H_{\rm {av}}>0}
1445:{\displaystyle H_{\rm {av}}>0}
1311:{\displaystyle H_{\rm {av}}>0}
416:{\displaystyle H_{\rm {av}}>0}
1265:. For any non-comoving observer
671:
34:
2124:Classical and Quantum Cosmology
2096:https://youtube.com/IbOh6aJrIYI
2083:10.1140/epjc/s10052-024-12621-z
1930:from the original on 2014-02-25
1908:, Bloomsbury, 2017, pp. 150–158
1860:Classical and Quantum Cosmology
1829:"The Beginning of the Universe"
371:is the proper final time), and
1379:
1364:
1188:
1182:
1162:{\displaystyle H={\dot {a}}/a}
1109:
1094:
1085:
1070:
1061:
1046:
1034:
1019:
964:
949:
941:
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847:is the final momentum at time
827:
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767:
675:
661:
513:
507:
451:=1). Which is consistent with
317:
311:
1:
2152:Eponymous theorems of physics
1774:. Bloomsbury Publishing USA.
1711:10.1103/PhysRevLett.90.151301
1169:is the Hubble parameter, and
355:is the proper initial time),
69:"Borde–Guth–Vilenkin theorem"
58:secondary or tertiary sources
27:Theorem in physical cosmology
1918:Carroll, Sean (2014-02-24).
580:. Along a timeline geodesic
2061:European Physical Journal C
2046:10.1103/PhysRevD.107.044024
1795:Nagasawa, Y. (2012-07-25).
1621:Kalam cosmological argument
755:geodesic equation of motion
156:that has, on average, been
18:Borde-Guth-Vilenkin theorem
2173:
2001:10.1103/PhysRevD.88.043516
1920:"Post-Debate Reflections"
1745:Cosmology for the Curious
1656:Cosmology for the Curious
1518:Limitations and criticism
1467:gravitational singularity
188:represent the paths that
432:Einstein field equations
1681:Physical Review Letters
152:which deduces that any
1631:Gibbons–Hawking effect
1587:
1504:
1446:
1404:
1312:
1248:
1163:
1116:
834:
732:
682:
546:
417:
330:
45:relies excessively on
1588:
1505:
1447:
1405:
1313:
1249:
1164:
1117:
835:
733:
683:
547:
418:
331:
1843:on 27 December 2023.
1559:
1510:up to the period of
1476:
1418:
1325:
1318:it is follows that
1284:
1176:
1130:
861:
761:
692:
615:
469:
389:
346:is an initial time (
218:
1993:2013PhRvD..88d3516V
1839:(4). Archived from
1703:2003PhRvL..90o1301B
968:
896:
307:
134:Borde–Guth–Vilenkin
2157:Physical cosmology
1604:William Lane Craig
1583:
1500:
1442:
1400:
1308:
1244:
1159:
1112:
918:
864:
830:
757:, it follows that
728:
678:
542:
413:
326:
275:
182:general relativity
170:Alexander Vilenkin
150:physical cosmology
2024:Physical Review D
1971:Physical Review D
1924:Sean Carroll Blog
1870:978-3-319-41127-9
1524:eternal inflation
1398:
1238:
1202:
1148:
1038:
1037:
726:
273:
130:
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122:
104:
16:(Redirected from
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878:
839:
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790:
789:
788:
742:is the mass and
737:
735:
734:
729:
727:
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724:
712:
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702:
687:
685:
684:
679:
674:
651:
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528:
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377:Hubble parameter
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2135:
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2115:Further reading
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2090:
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1648:
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1600:
1598:Use in theology
1562:
1557:
1556:
1542:quantum gravity
1534:Sean M. Carroll
1520:
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883:
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738:is the energy,
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703:
690:
689:
637:
618:
613:
612:
598:
586:
581:
532:
519:
497:
484:
467:
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443:For FLRW metric
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369:
363:
361:
353:
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294:
280:
260:
245:
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221:
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208:into the past.
198:Euclidean space
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109:
106:
63:
61:
55:
51:primary sources
39:
28:
23:
22:
15:
12:
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1940:
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1869:
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1814:
1807:
1787:
1780:
1760:
1754:978-3319570402
1753:
1732:
1687:(15): 151301.
1671:
1665:978-3319570402
1664:
1645:
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1623:
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1357:
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1298:
1295:
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1263:Lorentz factor
1256:
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1233:
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1219:
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594:following the
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403:
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367:
362:a final time (
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26:
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9:
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4:
3:
2:
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2134:9783319411279
2130:
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2030:(4): 044024.
2029:
2025:
2018:
2015:
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2002:
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1994:
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1977:(4): 043516.
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1548:in the past.
1547:
1546:arrow of time
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1528:cyclic models
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71: –
70:
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65:Find sources:
59:
53:
52:
48:
43:This article
41:
37:
32:
31:
19:
2127:. Springer.
2123:
2102:
2091:
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2027:
2023:
2017:
1974:
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1964:
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1932:. Retrieved
1923:
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1905:
1900:
1879:
1863:. Springer.
1859:
1841:the original
1836:
1832:
1801:. Springer.
1797:
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1680:
1674:
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1601:
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1471:
1459:
1456:Implications
1413:
1279:
1274:
1270:
1266:
1258:
1257:
1125:
848:
841:
752:
747:
743:
739:
603:
599:
591:
582:
578:scale factor
573:
569:
565:
561:
557:
555:
462:is given by
457:
448:
446:
429:
426:
381:
372:
364:
356:
348:
340:
338:
210:
190:free-falling
179:
141:
137:
133:
131:
116:
107:
97:
90:
83:
76:
64:
44:
1608:New Atheism
1553:Paul Davies
206:proper time
2146:Categories
2074:2305.06719
2037:2207.00955
1955:1802.02231
1934:2019-11-19
1642:References
1269:>1 and
1261:being the
610:4-momentum
596:world line
588:= constant
453:ΛCDM model
438:Derivation
176:Definition
110:March 2021
80:newspapers
47:references
2067:(3) 242.
2009:119213877
1984:1305.3836
1891:1204.4658
1833:Inference
1578:≥
1538:classical
1512:inflation
1369:γ
1356:≤
1345:τ
1341:−
1330:τ
1280:Assuming
1232:−
1229:γ
1218:γ
1208:
1186:γ
1146:˙
1099:γ
1089:≤
1075:γ
1065:−
1051:γ
920:∫
913:τ
904:τ
866:∫
753:From the
656:τ
643:μ
624:μ
576:) is the
560:is time,
495:−
324:τ
315:τ
277:∫
262:τ
258:−
247:τ
202:cosmology
194:spacetime
186:geodesics
166:Alan Guth
162:spacetime
158:expanding
1928:Archived
1727:46902994
1719:12732026
1615:See also
1606:and the
1463:Big Bang
1277:)>0.
688:, where
608:, has a
154:universe
1989:Bibcode
1699:Bibcode
1526:, and
1126:where
854:. Thus
146:theorem
142:theorem
94:scholar
2131:
2007:
1867:
1805:
1778:
1751:
1725:
1717:
1662:
1275:γ
1267:γ
1259:γ
840:where
604:τ
592:τ
556:where
365:τ
349:τ
339:where
184:, the
96:
89:
82:
75:
67:
2069:arXiv
2032:arXiv
2005:S2CID
1979:arXiv
1950:arXiv
1886:arXiv
1723:S2CID
1689:arXiv
200:. In
144:is a
101:JSTOR
87:books
2129:ISBN
1865:ISBN
1803:ISBN
1776:ISBN
1749:ISBN
1715:PMID
1660:ISBN
1495:>
1437:>
1303:>
458:The
408:>
168:and
132:The
73:news
2079:doi
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2028:107
1997:doi
1707:doi
180:In
148:in
138:BGV
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1987:.
1975:88
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1926:.
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