1419:
20:
1363:, itself a constant of nature, approx. 1.44 solar masses) and an electron approximates to 10, an interesting variation on the 10 and 10 that are typically associated with Dirac and Eddington respectively. (The physics defining the Chandrasekhar mass produces a ratio that is the −3/2 power of the gravitational fine-structure constant, 10.)
1319:: In 1978, G. Blake argued that paleontological data is consistent with the "multiplicative" scenario but not the "additive" scenario. Arguments both for and against LNH are also made from astrophysical considerations. For example, D. Falik argued that LNH is inconsistent with experimental results for
263:
1358:
Various authors have introduced new sets of numbers into the original "coincidence" considered by Dirac and his contemporaries, thus broadening or even departing from Dirac's own conclusions. Jordan (1947) noted that the mass ratio for a typical star (specifically, a star of the
492:
381:
714:
1403:(for example Planck density). This ratio of densities, and other ratios (using four fundamental constants: speed of light in vacuum c, Newtonian constant of gravity G, reduced Planck constant ℏ, and Hubble constant H) computes to an exact number,
1079:
1298:
Dirac's theory has inspired and continues to inspire a significant body of scientific literature in a variety of disciplines, with it sparking off many speculations, arguments and new ideas in terms of applications. In the context of
1281:
that describes the structure of spacetime in terms of a ratio of gravitational and electromagnetic units. He also provided alternative scenarios for the continuous creation of matter, one of the other significant issues in LNH:
910:
162:
566:
806:
1315:
demonstrated in 1962 how a simple revision of the parameters (in this case, the age of the Solar System) can invalidate Teller's conclusions. The debate is further complicated by the choice of LNH
387:
274:
620:
1199:
131:
96:
1548:
1257:
1166:
1139:
1875:
47:
in the present cosmological epoch. According to Dirac's hypothesis, the apparent similarity of these ratios might not be a mere coincidence but instead could imply a
996:
1223:
1105:
937:
2213:
2143:
840:
740:. For Milne, space was not a structured object but simply a system of reference in which relations such as this could accommodate Einstein's conclusions:
258:{\displaystyle {\frac {R_{\text{U}}}{r_{\text{e}}}}\approx {\frac {r_{\text{H}}}{r_{\text{e}}}}\approx 4.1666763\cdot 10^{42}\approx 10^{42.62\ldots },}
145:
LNH was Dirac's personal response to a set of large number "coincidences" that had intrigued other theorists of his time. The "coincidences" began with
1444:
497:
1418:
2453:
1695:
1658:
746:
2428:
1307:
seemed to raise a serious objection to LNH in 1948 when he argued that variations in the strength of gravity are not consistent with
2418:
156:, might also be the hypothetical radius of a particle whose rest energy is equal to the gravitational self-energy of the electron:
2463:
1384:
1432:
487:{\displaystyle r_{\text{H}}={\frac {e^{2}}{4\pi \epsilon _{0}\ m_{\text{H}}c^{2}}}\approx 1.5671987\cdot 10^{27}\,\mathrm {m} }
376:{\displaystyle r_{\text{e}}={\frac {e^{2}}{4\pi \epsilon _{0}\ m_{\text{e}}c^{2}}}\approx 3.7612682\cdot 10^{-16}\mathrm {m} }
1399:
in the universe. Valev (2019) found an equation connecting cosmological parameters (for example density of the universe) and
1320:
2423:
1949:
V. Canuto, S. Hsieh (1978). "The 3 K blackbody radiation, Dirac's Large
Numbers Hypothesis, and scale-covariant cosmology".
736:
a few years before Dirac formulated LNH. Milne was inspired not by large number coincidences but by a dislike of
Einstein's
1339:, it simply states that the large numbers in LNH are a necessary coincidence for intelligent beings since they parametrize
2301:
1372:
44:
737:
709:{\displaystyle {\frac {e^{2}}{4\pi \epsilon _{0}\ Gm_{\text{e}}^{2}}}\approx 4.1666763\cdot 10^{42}\approx {\sqrt {N}}}
2468:
2448:
1711:
J. P.Uzan (2003). "The fundamental constants and their variation, Observational status and theoretical motivations".
1263:
noted that such a temporal variation does not necessarily follow from Dirac's assumptions, a corresponding change of
2299:
C.-G. Shao; J. Shen; B. Wang; R.-K. Su (2006). "Dirac
Cosmology and the Acceleration of the Contemporary Universe".
1683:
1646:
834:
The Weyl and
Eddington ratios above can be rephrased in a variety of ways, as for instance in the context of time:
1407:. This provides evidence of the Dirac large numbers hypothesis by connecting the macro-world and the micro-world.
1371:
Several authors have recently identified and pondered the significance of yet another large number, approximately
1271:
is constant, otherwise the law of conserved energy is violated. Dirac met this difficulty by introducing into the
2088:
1992:
1951:
1914:
1713:
579:
1272:
721:
1625:
1202:
56:
1847:
1387:
identified 10 with the ratio of the universe's volume to the volume of a typical nucleon bounded by its
1380:
1332:
1171:
972:
2473:
2384:
2363:
A. Unzicker (2009). "A Look at the
Abandoned Contributions to Cosmology of Dirac, Sciama and Dicke".
2320:
2271:
2222:
2183:
2152:
2038:
2001:
1960:
1923:
1884:
1816:
1779:
1732:
1557:
1546:
A. Eddington (1931). "Preliminary Note on the Masses of the
Electron, the Proton, and the Universe".
1520:
1479:
103:
66:
2458:
1379:, which Nottale (1993) and Matthews (1997) associated in an LNH context with a scaling law for the
1360:
1336:
733:
725:
60:
1228:
1144:
1117:
2400:
2374:
2365:
2349:
2336:
2310:
2287:
2261:
2238:
2199:
2118:
1748:
1722:
1573:
1511:
1470:
1388:
1289:'multiplicative' creation (new matter is created where there are already concentrations of mass).
968:
48:
1767:
1687:
1595:
2117:
H. Lyre (2003). "C. F. Weizsäcker's
Reconstruction of Physics: Yesterday, Today and Tomorrow".
2252:
1851:
1691:
1654:
1438:
1650:
1640:
136:
Physical constants are actually not constant. Their values depend on the age of the
Universe.
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2174:
2160:
2097:
2046:
2009:
1968:
1931:
1892:
1824:
1787:
1740:
1565:
1528:
1487:
1074:{\displaystyle {\frac {e^{2}}{4\pi \epsilon _{0}Gm_{\text{p}}m_{\text{e}}}}\approx 10^{40}.}
607:
1807:
1085:
2388:
2324:
2275:
2226:
2187:
2156:
2042:
2005:
1964:
1927:
1888:
1820:
1783:
1736:
1561:
1524:
1483:
1676:
1424:
1340:
1208:
1090:
940:
922:
614:, the estimated number of charged particles in the universe, with the following ratio:
43:
to that of force scales. The ratios constitute very large, dimensionless numbers: some
2332:
2250:
G. A. Mena
Marugan; S. Carneiro (2002). "Holography and the large number hypothesis".
2065:
1375:. This is for example the ratio of the theoretical and observational estimates of the
2442:
2340:
2291:
2242:
1752:
1642:
Cosmology and
Controversy: The historical development of two theories of the universe
1577:
1376:
1327:
consistent. One argument that has created significant controversy was put forward by
1304:
976:
905:{\displaystyle {\frac {c\,t}{r_{\text{e}}}}\approx 3.47\cdot 10^{41}\approx 10^{42},}
2433:
2404:
1912:
D. Falik (1979). "Primordial Nucleosynthesis and Dirac's Large Numbers Hypothesis".
720:
In addition to the examples of Weyl and Eddington, Dirac was also influenced by the
2203:
1400:
1328:
1312:
1308:
1260:
146:
2348:
S. Ray; U. Mukhopadhyay; P. P. Ghosh (2007). "Large Number Hypothesis: A Review".
1396:
2283:
2211:
P. A. M. Dirac (1974). "Cosmological Models and the Large Numbers Hypothesis".
2102:
2083:
2066:"Mach's Principle, Dirac's Large Numbers and the Cosmological Constant Problem"
19:
1897:
1870:
1791:
1744:
1569:
1414:
1300:
100:
The mass of the universe is proportional to the square of the universe's age:
36:
2396:
2050:
1532:
1491:
1316:
2234:
2164:
1828:
1766:
Saibal, Ray; Mukhopadhyay, Utpal; Ray, Soham; Bhattacharjee, Arjak (2019).
561:{\displaystyle m_{\text{H}}c^{2}={\frac {Gm_{\text{e}}^{2}}{r_{\text{e}}}}}
1286:'additive' creation (new matter is created uniformly throughout space) and
728:, who lectured on the topic in Cambridge in 1933. The notion of a varying-
2123:
1447: – Hypothetical conflict with the laws of physics as currently known
1344:
1108:
987:
40:
1768:"Dirac's large number hypothesis: A journey from concept to implication"
1727:
2315:
2266:
967:, the age of the universe is about 10 units of time. This is the same
2195:
983:
1506:
1465:
1391:, and he identified this ratio with the sum of elementary events or
2014:
1987:
1972:
1935:
2379:
2354:
979:
801:{\displaystyle G=\left(\!{\frac {c^{3}}{M_{\text{U}}}}\!\right)t,}
18:
1988:"Primordial nucleosynthesis and Dirac's large numbers hypothesis"
149:(1919), who speculated that the observed radius of the universe,
1855:
1352:
1348:
1112:
829:
1392:
1267:
has not been found. According to general relativity, however,
1276:
16:
Hypothesis relating age of the universe to physical constants
1871:"The Large Numbers Hypothesis and the rotation of the Earth"
2419:
Audio of Dirac talking about the large numbers hypothesis
1805:
E. Teller (1948). "On the change of physical constants".
1205:
is approximately 10. Dirac interpreted this to mean that
1168:
of the proton and electron, and the permittivity factor
950:
is the classical electron radius. Hence, in units where
822:
is the age of the universe. According to this relation,
830:
Dirac's interpretation of the large number coincidences
2172:
P. A. M. Dirac (1937). "The Cosmological Constants".
1231:
1211:
1174:
1147:
1120:
1093:
999:
925:
843:
749:
623:
500:
390:
277:
165:
106:
69:
2429:
Robert Matthews: Dirac's coincidences sixty years on
2141:
P. A. M. Dirac (1938). "A New Basis for Cosmology".
596:denotes the mass of the hypothetical particle, and
1675:
1549:Proceedings of the Cambridge Philosophical Society
1251:
1217:
1193:
1160:
1133:
1099:
1073:
931:
904:
800:
708:
560:
486:
375:
257:
125:
90:
1876:Monthly Notices of the Royal Astronomical Society
786:
761:
1507:"Eine neue Erweiterung der Relativitätstheorie"
1201:in atomic units (equal to 1), the value of the
55:The strength of gravity, as represented by the
39:in 1937 relating ratios of size scales in the
2029:P. Jordan (1947). "Die Herkunft der Sterne".
8:
2214:Proceedings of the Royal Society of London A
2144:Proceedings of the Royal Society of London A
1596:"Evidence of Dirac large numbers hypothesis"
2378:
2353:
2314:
2265:
2122:
2101:
2013:
1896:
1772:International Journal of Modern Physics D
1726:
1241:
1230:
1210:
1185:
1173:
1152:
1146:
1125:
1119:
1092:
1062:
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1036:
1023:
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1000:
998:
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893:
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844:
842:
778:
768:
762:
748:
699:
690:
668:
663:
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630:
624:
622:
606:The coincidence was further developed by
550:
539:
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505:
499:
479:
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472:
450:
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410:
404:
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368:
359:
337:
327:
314:
297:
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243:
230:
209:
199:
193:
182:
172:
166:
164:
117:
105:
87:
79:
68:
1323:whereas Canuto and Hsieh argued that it
1622:Relativity, Gravity and World Structure
1456:
1445:Time-variation of fundamental constants
1435: – Physical constant with no units
732:cosmology first appears in the work of
610:(1931) who related the above ratios to
1294:Later developments and interpretations
2434:The Mysterious Eddington–Dirac Number
2084:"Dirac's coincidences sixty years on"
7:
1589:
1587:
1603:Proceedings of the Romanian Academy
1441: – Unsolved problem in physics
59:, is inversely proportional to the
2424:Full transcript of Dirac's speech.
1194:{\displaystyle 4\pi \epsilon _{0}}
480:
369:
14:
1417:
818:is the mass of the universe and
1433:Dimensionless physical constant
1321:microwave background radiation
126:{\displaystyle M\propto t^{2}}
91:{\displaystyle G\propto 1/t\,}
29:Dirac large numbers hypothesis
1:
2302:Classical and Quantum Gravity
1794:– via World Scientific.
1678:Dirac: A Scientific Biography
1385:Carl Friedrich von Weizsäcker
603:is its electrostatic radius.
589:is the mass of the electron,
51:with these unusual features:
2454:Obsolete scientific theories
1986:V. Canuto, S. Hsieh (1980).
1377:energy density of the vacuum
1252:{\displaystyle G\approx 1/t}
1161:{\displaystyle m_{\text{e}}}
1134:{\displaystyle m_{\text{p}}}
919:is the age of the universe,
738:general theory of relativity
35:) is an observation made by
2333:10.1088/0264-9381/23/11/003
1355:would not arise otherwise.
2490:
2284:10.1103/PhysRevD.65.087303
2089:Astronomy & Geophysics
1684:Cambridge University Press
1647:Princeton University Press
1277:
2031:Astronomische Nachrichten
1993:The Astrophysical Journal
1952:The Astrophysical Journal
1915:The Astrophysical Journal
1792:10.1142/S0218271819300143
1745:10.1103/RevModPhys.75.403
1714:Reviews of Modern Physics
1570:10.1017/S0305004100009269
1466:"Zur Gravitationstheorie"
580:classical electron radius
2397:10.1002/andp.20095210108
2103:10.1093/astrog/39.6.6.19
2051:10.1002/asna.19472751012
1533:10.1002/andp.19193641002
1492:10.1002/andp.19173591804
1273:Einstein field equations
1084:Hence, interpreting the
722:primeval-atom hypothesis
2464:Astronomical hypotheses
1898:10.1093/mnras/185.2.399
1626:Oxford University Press
1373:120 orders of magnitude
1351:and hence carbon-based
2235:10.1098/rspa.1974.0095
2165:10.1098/rspa.1938.0053
1829:10.1103/PhysRev.73.801
1778:(8): 1930014–1930096.
1331:in 1961. Known as the
1253:
1219:
1203:gravitational constant
1195:
1162:
1135:
1101:
1075:
933:
906:
802:
710:
562:
488:
377:
259:
127:
92:
57:gravitational constant
45:40 orders of magnitude
24:
1381:cosmological constant
1333:anthropic coincidence
1254:
1220:
1196:
1163:
1136:
1102:
1076:
934:
907:
826:increases over time.
803:
711:
563:
489:
378:
260:
128:
93:
22:
2082:R. Matthews (1998).
1850:. pp. 138–141.
1620:E. A. Milne (1935).
1229:
1225:varies with time as
1209:
1172:
1145:
1118:
1091:
997:
971:as the ratio of the
923:
841:
747:
621:
498:
388:
275:
163:
104:
67:
2389:2009AnP...521...57U
2325:2006CQGra..23.3707S
2276:2002PhRvD..65h7303M
2227:1974RSPSA.338..439D
2188:1937Natur.139..323D
2157:1938RSPSA.165..199D
2043:1947dhds.book.....J
2006:1980ApJ...239L..91C
1965:1978ApJ...224..302C
1928:1979ApJ...231L...1F
1889:1978MNRAS.185..399B
1821:1948PhRv...73..801T
1784:2019IJMPD..2830014R
1737:2003RvMP...75..403U
1562:1931PCPS...27...15E
1525:1919AnP...364..101W
1484:1917AnP...359..117W
1337:fine-tuned universe
734:Edward Arthur Milne
673:
544:
61:age of the universe
2469:1937 introductions
2449:Physical cosmology
2366:Annalen der Physik
1512:Annalen der Physik
1471:Annalen der Physik
1389:Compton wavelength
1361:Chandrasekhar mass
1249:
1215:
1191:
1158:
1131:
1097:
1071:
969:order of magnitude
929:
902:
798:
706:
659:
558:
530:
484:
373:
255:
123:
88:
25:
2309:(11): 3707–3720.
2253:Physical Review D
2221:(1615): 439–446.
1869:G. Blake (1978).
1842:G. Gamow (1962).
1697:978-0-521-38089-8
1674:H. Kragh (1990).
1660:978-0-691-02623-7
1639:H. Kragh (1996).
1594:D. Valev (2019).
1439:Hierarchy problem
1275:a gauge function
1218:{\displaystyle G}
1155:
1128:
1100:{\displaystyle e}
1053:
1049:
1039:
932:{\displaystyle c}
865:
862:
784:
781:
704:
675:
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553:
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508:
457:
443:
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215:
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202:
188:
185:
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2408:
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2357:
2344:
2318:
2295:
2269:
2246:
2207:
2196:10.1038/139323a0
2168:
2151:(921): 199–208.
2129:
2128:
2126:
2124:quant-ph/0309183
2114:
2108:
2107:
2105:
2079:
2073:
2072:
2070:
2061:
2055:
2054:
2026:
2020:
2019:
2017:
1983:
1977:
1976:
1946:
1940:
1939:
1909:
1903:
1902:
1900:
1866:
1860:
1859:
1839:
1833:
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1802:
1796:
1795:
1763:
1757:
1756:
1730:
1708:
1702:
1701:
1681:
1671:
1665:
1664:
1636:
1630:
1629:
1617:
1611:
1610:
1600:
1591:
1582:
1581:
1543:
1537:
1536:
1505:H. Weyl (1919).
1502:
1496:
1495:
1464:H. Weyl (1917).
1461:
1427:
1422:
1421:
1406:
1303:, for instance,
1280:
1279:
1258:
1256:
1255:
1250:
1245:
1224:
1222:
1221:
1216:
1200:
1198:
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1167:
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1156:
1153:
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1137:
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1130:
1129:
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1103:
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1080:
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1050:
1047:
1041:
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1028:
1027:
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966:
956:
938:
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935:
930:
911:
909:
908:
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897:
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863:
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845:
807:
805:
804:
799:
791:
787:
785:
783:
782:
779:
773:
772:
763:
726:Georges Lemaître
715:
713:
712:
707:
705:
700:
695:
694:
676:
674:
672:
667:
664:
653:
652:
651:
635:
634:
625:
608:Arthur Eddington
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216:
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194:
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177:
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132:
130:
129:
124:
122:
121:
97:
95:
94:
89:
83:
2489:
2488:
2484:
2483:
2482:
2480:
2479:
2478:
2439:
2438:
2415:
2362:
2347:
2298:
2249:
2210:
2171:
2140:
2137:
2135:Further reading
2132:
2116:
2115:
2111:
2081:
2080:
2076:
2068:
2063:
2062:
2058:
2028:
2027:
2023:
1985:
1984:
1980:
1948:
1947:
1943:
1911:
1910:
1906:
1868:
1867:
1863:
1841:
1840:
1836:
1808:Physical Review
1804:
1803:
1799:
1765:
1764:
1760:
1710:
1709:
1705:
1698:
1673:
1672:
1668:
1661:
1638:
1637:
1633:
1619:
1618:
1614:
1598:
1593:
1592:
1585:
1545:
1544:
1540:
1519:(10): 101–133.
1504:
1503:
1499:
1478:(18): 117–145.
1463:
1462:
1458:
1454:
1423:
1416:
1413:
1404:
1369:
1311:data. However,
1309:paleontological
1296:
1227:
1226:
1207:
1206:
1181:
1170:
1169:
1148:
1143:
1142:
1121:
1116:
1115:
1089:
1088:
1058:
1042:
1032:
1019:
1012:
1002:
995:
994:
964:
958:
951:
949:
921:
920:
889:
876:
855:
846:
839:
838:
832:
817:
774:
764:
760:
756:
745:
744:
686:
643:
636:
626:
619:
618:
602:
595:
588:
577:
546:
526:
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1469:
1459:
1401:Planck units
1370:
1357:
1329:Robert Dicke
1324:
1313:George Gamow
1297:
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1261:George Gamow
1083:
959:
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144:
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2474:Coincidence
1649:. pp.
1397:information
1317:cosmologies
1259:. Although
2459:Paul Dirac
2443:Categories
1721:(2): 403.
1686:. p.
1452:References
1301:geophysics
982:between a
973:electrical
141:Background
37:Paul Dirac
23:Paul Dirac
2380:0708.3518
2355:0705.1836
2341:119339090
2292:119452710
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49:cosmology
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1856:62008840
1411:See also
1345:hydrogen
1109:electron
988:electron
41:Universe
2385:Bibcode
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2002:Bibcode
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1844:Gravity
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