31:
2321:
62:
to define simultaneous events. Two events will be simultaneous when they are on a line hyperbolically orthogonal to a particular timeline. This dependence on a certain timeline is determined by velocity, and is the basis for the
1094:
1207:
863:
where the asymptote is invariant. Hyperbolically orthogonal lines lie in different sectors of the plane, determined by the asymptotes of the hyperbola, thus the relation of hyperbolic orthogonality is a
1232:
wrote in 1910, " hyperbola is unaltered when any pair of conjugate diameters are taken as new axes, and a new unit of length is taken proportional to the length of either of these diameters." On this
194:
The relation of hyperbolic orthogonality actually applies to classes of parallel lines in the plane, where any particular line can represent the class. Thus, for a given hyperbola and asymptote
2024:
779:
834:
697:
619:
462:
353:
253:
1387:
1297:
1273:
Edwin B. Wilson & Gilbert N. Lewis (1912) "The Space-time
Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics" Proceedings of the
497:
385:
844:
the ellipse is a circle and the conjugate diameters are perpendicular while the hyperbola is rectangular and the conjugate diameters are hyperbolic-orthogonal.
2071:
2306:
2095:
1841:
962:
1877:
1831:
2015:
1274:
1499:
1128:
281:
is used to describe orthogonality in analytic geometry, with two elements orthogonal when their bilinear form vanishes. In the plane of
2123:
1752:
1757:
1468:
1439:
1254:
in 1912. They note "in our plane no pair of perpendicular lines is better suited to serve as coordinate axes than any other pair"
2077:
1294:
1790:
856:
542:
The bilinear form may be computed as the real part of the complex product of one number with the conjugate of the other. Then
1893:
1971:
2034:
2324:
2066:
1901:
2343:
1888:
1648:
1929:
1663:
626:
548:
38:
is preserved by rotation in the left diagram; hyperbolic orthogonality with respect to hyperbola (B) is preserved by
2216:
1658:
1611:
893:
64:
884:
study in 1908, the concept of points in a spacetime plane being hyperbolic-orthogonal to a timeline (tangent to a
393:
2246:
1872:
1553:
726:
287:
784:
2271:
1821:
1492:
1410:
860:
852:
80:
2000:
1922:
1542:
1533:
1391:
1233:
2089:
1995:
217:
2226:
1953:
1627:
1591:
865:
186:
2231:
1836:
1775:
1228:. The directions indicated by conjugate diameters are taken for space and time axes in relativity. As
2136:
2062:
1906:
1508:
1421:
1243:
2211:
2206:
2201:
2196:
2028:
1977:
1795:
1780:
1606:
1537:
1485:
1225:
1221:
848:
712:
39:
2100:
2276:
2176:
1958:
1939:
1933:
1884:
1826:
1735:
1653:
1571:
1548:
1516:
1251:
59:
185:). Therefore the reflected line has slope 1/m and the slopes of hyperbolic orthogonal lines are
2286:
2156:
2118:
2110:
1711:
1668:
1464:
1435:
877:
708:
388:
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1314:
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1683:
1452:
1427:
1278:
1247:
467:
124:
30:
1445:
358:
2266:
2241:
2166:
2161:
2044:
2005:
1967:
1911:
1785:
1721:
1442:
1383:
1301:
1229:
855:. Suppose the slope of a vertical line is denoted â so that all lines have a slope in the
2049:
2348:
2291:
1963:
1947:
1943:
1846:
1816:
1673:
1576:
1402:
1295:
Through the
Looking Glass â A glimpse of Euclidâs twin geometry, the Minkowski geometry
282:
2337:
2281:
2261:
2256:
2171:
2039:
1867:
1811:
1643:
1596:
1368:
896:. In Minkowski's development the hyperbola of type (B) above is in use. Two vectors (
278:
35:
17:
1453:
2301:
2221:
2186:
1716:
1678:
274:, a radius to a hyperbola is hyperbolic orthogonal to a tangent to the hyperbola.
1331:
2085:
2054:
1601:
1335:
859:. Then whichever hyperbola (A) or (B) is used, the operation is an example of a
2296:
1862:
1706:
1701:
885:
1089:{\displaystyle c^{2}\ t_{1}\ t_{2}-x_{1}\ x_{2}-y_{1}\ y_{2}-z_{1}\ z_{2}=0.}
1693:
881:
84:
55:
1349:
2236:
1586:
1237:
47:
271:
1434:, chapter 1: A Trip on Einstein's Train, Universitext Springer-Verlag
1916:
1282:
1236:, he then wrote the Lorentz transformation in the modern form using
1202:{\displaystyle {\frac {c\ t_{1}}{x_{1}}}={\frac {x_{2}}{c\ t_{2}}}}
120:
1897:
1416:
Francesco Catoni, Dino
Boccaletti, & Roberto Cannata (2008)
1481:
851:, the operation of taking the hyperbolic orthogonal line is an
87:. Two particular hyperbolas are frequently used in the plane:
1477:
1224:. Any diameter of the original hyperbola is reflected to a
530:
in the hyperbolic number plane are said to be respectively
119:
In this case the lines are hyperbolic orthogonal if their
1334:, ellipse §33, page 38 and hyperbola §41, page 49, from
723:âČ represent the slopes of the conjugate diameters, then
270:
Similar to the perpendularity of a circle radius to the
1348:
1131:
965:
787:
729:
629:
551:
470:
396:
361:
290:
220:
621:
entails perpendicularity in the complex plane, while
2149:
2014:
1986:
1855:
1804:
1766:
1745:
1734:
1692:
1636:
1620:
1562:
1526:
1515:
1388:
A History of the
Theories of Aether and Electricity
54:between two lines separated by the asymptotes of a
1201:
1088:
828:
773:
691:
613:
491:
456:
379:
347:
247:
707:The notion of hyperbolic orthogonality arose in
206:) are hyperbolic orthogonal if there is a pair (
692:{\displaystyle w_{1}w_{2}^{*}+w_{1}^{*}w_{2}=0}
614:{\displaystyle z_{1}z_{2}^{*}+z_{1}^{*}z_{2}=0}
27:Relation of space and time in relativity theory
1493:
1451:J.A. Wheeler; C. Misner; K.S. Thorne (1973).
538:if their respective inner products are zero.
8:
1367:Various English translations on Wikisource:
457:{\displaystyle w_{1}=u+jv,\quad w_{2}=x+jy,}
239:
224:
83:of each other over the asymptote of a given
1424:, Basel. See page 38, Pseudo-orthogonality.
774:{\displaystyle gg'=-{\frac {b^{2}}{a^{2}}}}
348:{\displaystyle z_{1}=u+iv,\quad z_{2}=x+iy}
1742:
1523:
1500:
1486:
1478:
829:{\displaystyle gg'={\frac {b^{2}}{a^{2}}}}
1269:
1267:
1190:
1173:
1167:
1156:
1145:
1132:
1130:
1074:
1061:
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1035:
1022:
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747:
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599:
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584:
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566:
556:
550:
469:
430:
401:
395:
360:
324:
295:
289:
219:
29:
1263:
892:of events relative to the timeline, or
1304:, ICME-10 Copenhagen; pages 6 & 7.
1275:American Academy of Arts and Sciences
956:(meaning hyperbolic orthogonal) when
248:{\displaystyle a\rVert c,\ b\rVert d}
173:, 1) on the line is reflected across
101:When reflected in the x-axis, a line
7:
1432:Orthogonality and Spacetime Geometry
2072:TolmanâOppenheimerâVolkoff equation
2025:FriedmannâLemaĂźtreâRobertsonâWalker
25:
1842:HamiltonâJacobiâEinstein equation
1459:. W.H. Freeman & Co. p.
1212:Given a hyperbola with asymptote
836:in the case of a hyperbola. When
516:in the complex number plane, and
2320:
2319:
868:on sets of lines in the plane.
857:projectively extended real line
715:of ellipses and hyperbolas. If
425:
319:
1649:Massâenergy equivalence (E=mc)
1418:Mathematics of Minkowski Space
781:in the case of an ellipse and
1:
1407:Relativity and Modern Physics
1250:developed the concept within
703:s are hyperbolic orthogonal.
1664:Relativistic Doppler effect
1347:Minkowski, Hermann (1909),
1319:College Mathematics Journal
2365:
2135:In computational physics:
1659:Relativity of simultaneity
894:relativity of simultaneity
888:) has been used to define
65:relativity of simultaneity
2317:
1972:LenseâThirring precession
1554:Doubly special relativity
1355:Physikalische Zeitschrift
1832:Post-Newtonian formalism
1822:Einstein field equations
1758:Mathematical formulation
1582:Hyperbolic orthogonality
1411:Harvard University Press
387:, while in the plane of
145:as asymptote. For lines
52:hyperbolic orthogonality
1543:Galilean transformation
1534:Principle of relativity
1392:Longmans, Green and Co.
1315:Hyperbolic Number Plane
1293:BjĂžrn Felsager (2004),
1234:principle of relativity
355:, the bilinear form is
1628:Lorentz transformation
1203:
1090:
866:heterogeneous relation
847:In the terminology of
830:
775:
693:
615:
493:
492:{\displaystyle xu-yv.}
458:
381:
349:
249:
43:
2096:WeylâLewisâPapapetrou
1837:Raychaudhuri equation
1776:Equivalence principle
1350:"Raum und Zeit"
1277:48:387â507, esp. 415
1204:
1091:
861:hyperbolic involution
831:
776:
694:
616:
536:hyperbolic orthogonal
494:
464:the bilinear form is
459:
382:
380:{\displaystyle xu+yv}
350:
259:is the reflection of
250:
77:hyperbolic orthogonal
58:is a concept used in
42:in the right diagram.
33:
18:Hyperbolic-orthogonal
2137:Numerical relativity
1978:pulsar timing arrays
1317:, also published in
1244:Edwin Bidwell Wilson
1216:, its reflection in
1129:
963:
785:
727:
711:in consideration of
627:
549:
532:Euclidean orthogonal
468:
394:
359:
288:
218:
2344:Minkowski spacetime
2029:Friedmann equations
1923:HulseâTaylor binary
1885:Gravitational waves
1781:Riemannian geometry
1607:Proper acceleration
1592:Maxwell's equations
1538:Galilean relativity
1330:Barry Spain (1957)
1222:conjugate hyperbola
849:projective geometry
713:conjugate diameters
672:
654:
594:
576:
198:, a pair of lines (
153:with −1 <
40:hyperbolic rotation
2078:ReissnerâNordström
1996:BransâDicke theory
1827:Linearized gravity
1654:Length contraction
1572:Frame of reference
1549:Special relativity
1313:Sobczyk, G.(1995)
1300:2011-07-16 at the
1252:synthetic geometry
1226:conjugate diameter
1199:
1086:
880:'s foundation for
826:
771:
689:
658:
640:
611:
580:
562:
489:
454:
389:hyperbolic numbers
377:
345:
245:
169:= 1. The point (1/
99:= 0 as asymptote.
60:special relativity
50:, the relation of
44:
2331:
2330:
2145:
2144:
2124:OzsvĂĄthâSchĂŒcking
1730:
1729:
1712:Minkowski diagram
1669:Thomas precession
1612:Relativistic mass
1422:BirkhÀuser Verlag
1332:Analytical Conics
1197:
1185:
1162:
1140:
1069:
1043:
1017:
991:
978:
878:Hermann Minkowski
824:
769:
709:analytic geometry
235:
125:additive inverses
16:(Redirected from
2356:
2323:
2322:
2106:van Stockum dust
1878:Two-body problem
1796:Mach's principle
1743:
1684:Terrell rotation
1524:
1502:
1495:
1488:
1479:
1474:
1458:
1428:Robert Goldblatt
1395:
1381:
1375:
1362:
1352:
1344:
1338:
1328:
1322:
1311:
1305:
1291:
1285:
1283:10.2307/20022840
1271:
1248:Gilbert N. Lewis
1208:
1206:
1205:
1200:
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1196:
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1183:
1178:
1177:
1168:
1163:
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1149:
1138:
1133:
1125:≠ 0, then
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2327:
2313:
2141:
2045:BKL singularity
2035:LemaĂźtreâTolman
2010:
2006:Quantum gravity
1988:
1982:
1968:geodetic effect
1942:(together with
1912:LISA Pathfinder
1851:
1800:
1786:Penrose diagram
1768:
1762:
1737:
1726:
1722:Minkowski space
1688:
1632:
1616:
1564:
1558:
1518:
1511:
1506:
1471:
1450:
1399:
1398:
1384:E. T. Whittaker
1382:
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1346:
1345:
1341:
1329:
1325:
1312:
1308:
1302:Wayback Machine
1292:
1288:
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1230:E. T. Whittaker
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283:complex numbers
216:
215:
192:
73:
28:
23:
22:
15:
12:
11:
5:
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2232:Choquet-Bruhat
2229:
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2179:
2174:
2169:
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2159:
2153:
2151:
2147:
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2139:
2132:
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2126:
2121:
2114:
2113:
2108:
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2098:
2093:
2084:Axisymmetric:
2081:
2080:
2075:
2069:
2058:
2057:
2052:
2047:
2042:
2037:
2032:
2023:Cosmological:
2020:
2018:
2012:
2011:
2009:
2008:
2003:
1998:
1992:
1990:
1984:
1983:
1981:
1980:
1975:
1964:frame-dragging
1961:
1956:
1951:
1948:Einstein rings
1944:Einstein cross
1937:
1926:
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1849:
1847:Ernst equation
1844:
1839:
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1829:
1824:
1819:
1817:BSSN formalism
1814:
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1676:
1674:Ladder paradox
1671:
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1661:
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1589:
1584:
1579:
1577:Speed of light
1574:
1568:
1566:
1560:
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1497:
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1476:
1475:
1469:
1448:
1425:
1414:
1409:, pages 62,3,
1403:G. D. Birkhoff
1397:
1396:
1394:(see page 441)
1376:
1374:
1373:
1372:
1371:
1369:Space and Time
1339:
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1286:
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1261:
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1189:
1182:
1176:
1172:
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1159:
1155:
1148:
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1137:
1122:
1119:
1115:
1112:
1108:
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1100:
1097:
1096:
1085:
1082:
1077:
1073:
1064:
1060:
1056:
1051:
1047:
1038:
1034:
1030:
1025:
1021:
1012:
1008:
1004:
999:
995:
986:
982:
973:
969:
949:
946:
942:
939:
935:
932:
928:
925:
921:
918:
914:
911:
907:
904:
900:
897:
873:
870:
821:
817:
811:
807:
801:
797:
794:
790:
766:
762:
756:
752:
746:
743:
739:
736:
732:
705:
704:
688:
685:
680:
676:
670:
665:
661:
657:
652:
647:
643:
637:
633:
622:
610:
607:
602:
598:
592:
587:
583:
579:
574:
569:
565:
559:
555:
540:
539:
527:
520:
513:
506:
488:
485:
482:
479:
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453:
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447:
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438:
433:
429:
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412:
409:
404:
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376:
373:
370:
367:
364:
344:
341:
338:
335:
332:
327:
323:
318:
315:
312:
309:
306:
303:
298:
294:
244:
241:
238:
232:
229:
226:
223:
191:
190:
189:of each other.
128:
89:
79:when they are
75:Two lines are
72:
69:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2361:
2350:
2347:
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2316:
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2300:
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2255:
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2245:
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2240:
2238:
2235:
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2230:
2228:
2225:
2223:
2220:
2218:
2217:Chandrasekhar
2215:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2195:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2177:Schwarzschild
2175:
2173:
2170:
2168:
2165:
2163:
2160:
2158:
2155:
2154:
2152:
2148:
2138:
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2133:
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2122:
2120:
2116:
2115:
2112:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2091:
2087:
2083:
2082:
2079:
2076:
2073:
2070:
2068:
2064:
2063:Schwarzschild
2060:
2059:
2056:
2053:
2051:
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2030:
2026:
2022:
2021:
2019:
2017:
2013:
2007:
2004:
2002:
1999:
1997:
1994:
1993:
1991:
1985:
1979:
1976:
1973:
1969:
1965:
1962:
1960:
1959:Shapiro delay
1957:
1955:
1952:
1949:
1945:
1941:
1938:
1935:
1931:
1928:
1927:
1924:
1921:
1918:
1915:
1913:
1910:
1908:
1905:
1903:
1902:collaboration
1899:
1895:
1892:
1890:
1886:
1883:
1882:
1879:
1876:
1874:
1871:
1869:
1868:Event horizon
1866:
1864:
1861:
1860:
1858:
1854:
1848:
1845:
1843:
1840:
1838:
1835:
1833:
1830:
1828:
1825:
1823:
1820:
1818:
1815:
1813:
1812:ADM formalism
1810:
1809:
1807:
1803:
1797:
1794:
1792:
1789:
1787:
1784:
1782:
1779:
1777:
1774:
1773:
1771:
1765:
1759:
1756:
1754:
1751:
1750:
1748:
1744:
1741:
1739:
1733:
1723:
1720:
1718:
1717:Biquaternions
1715:
1713:
1710:
1708:
1705:
1703:
1700:
1699:
1697:
1695:
1691:
1685:
1682:
1680:
1677:
1675:
1672:
1670:
1667:
1665:
1662:
1660:
1657:
1655:
1652:
1650:
1647:
1645:
1644:Time dilation
1642:
1641:
1639:
1635:
1629:
1626:
1625:
1623:
1619:
1613:
1610:
1608:
1605:
1603:
1600:
1598:
1597:Proper length
1595:
1593:
1590:
1588:
1585:
1583:
1580:
1578:
1575:
1573:
1570:
1569:
1567:
1561:
1555:
1552:
1550:
1547:
1544:
1541:
1539:
1535:
1532:
1531:
1529:
1525:
1522:
1520:
1514:
1510:
1503:
1498:
1496:
1491:
1489:
1484:
1483:
1480:
1472:
1470:0-7167-0344-0
1466:
1462:
1457:
1456:
1449:
1447:
1444:
1441:
1440:0-387-96519-X
1437:
1433:
1429:
1426:
1423:
1419:
1415:
1412:
1408:
1404:
1401:
1400:
1393:
1389:
1385:
1380:
1377:
1370:
1366:
1365:
1364:
1363:
1360:
1356:
1351:
1343:
1340:
1337:
1333:
1327:
1324:
1320:
1316:
1310:
1307:
1303:
1299:
1296:
1290:
1287:
1284:
1280:
1276:
1270:
1268:
1264:
1257:
1255:
1253:
1249:
1245:
1241:
1239:
1235:
1231:
1227:
1223:
1220:produces the
1219:
1215:
1210:
1191:
1187:
1180:
1174:
1170:
1164:
1157:
1153:
1146:
1142:
1135:
1083:
1080:
1075:
1071:
1062:
1058:
1054:
1049:
1045:
1036:
1032:
1028:
1023:
1019:
1010:
1006:
1002:
997:
993:
984:
980:
971:
967:
959:
958:
957:
955:
895:
891:
887:
883:
879:
871:
869:
867:
862:
858:
854:
850:
845:
843:
839:
819:
815:
809:
805:
799:
795:
792:
788:
764:
760:
754:
750:
744:
741:
737:
734:
730:
722:
718:
714:
710:
702:
686:
683:
678:
674:
668:
663:
659:
655:
650:
645:
641:
635:
631:
623:
608:
605:
600:
596:
590:
585:
581:
577:
572:
567:
563:
557:
553:
545:
544:
543:
537:
533:
526:
519:
512:
505:
501:
500:
499:
486:
483:
480:
477:
474:
471:
451:
448:
445:
442:
439:
436:
431:
427:
422:
419:
416:
413:
410:
407:
402:
398:
390:
374:
371:
368:
365:
362:
342:
339:
336:
333:
330:
325:
321:
316:
313:
310:
307:
304:
301:
296:
292:
284:
280:
279:bilinear form
275:
273:
268:
266:
262:
258:
242:
236:
230:
227:
221:
213:
209:
205:
201:
197:
188:
184:
180:
176:
172:
168:
164:
160:
157:< 1, when
156:
152:
148:
144:
140:
136:
132:
129:
126:
122:
118:
116:
112:
108:
104:
98:
94:
91:
90:
88:
86:
82:
78:
70:
68:
66:
61:
57:
53:
49:
41:
37:
36:orthogonality
32:
19:
2307:
2001:KaluzaâKlein
1753:Introduction
1679:Twin paradox
1581:
1454:
1431:
1417:
1406:
1379:
1358:
1354:
1342:
1326:
1318:
1309:
1289:
1242:
1217:
1213:
1211:
1111:s are zero,
1103:= 1 and the
1098:
953:
890:simultaneity
889:
875:
872:Simultaneity
846:
841:
837:
720:
716:
706:
700:
699:implies the
541:
535:
531:
524:
517:
510:
503:
502:The vectors
276:
269:
264:
260:
256:
214:) such that
211:
207:
203:
199:
195:
193:
182:
178:
174:
170:
166:
162:
158:
154:
150:
146:
142:
138:
134:
130:
114:
110:
106:
102:
100:
96:
92:
76:
74:
51:
45:
2090:KerrâNewman
2061:Spherical:
1930:Other tests
1873:Singularity
1805:Formulation
1767:Fundamental
1621:Formulation
1602:Proper time
1563:Fundamental
1455:Gravitation
1336:Hathi Trust
1118:≠ 0,
187:reciprocals
81:reflections
2338:Categories
2242:Zel'dovich
2150:Scientists
2129:Alcubierre
1936:of Mercury
1934:precession
1863:Black hole
1746:Background
1738:relativity
1707:World line
1702:Light cone
1527:Background
1519:relativity
1509:Relativity
1321:26:268â80.
1258:References
886:world line
853:involution
137:= 1 with
95:= 1 with
34:Euclidean
2212:Robertson
2197:Friedmann
2192:Eddington
2182:de Sitter
2016:Solutions
1894:detectors
1889:astronomy
1856:Phenomena
1791:Geodesics
1694:Spacetime
1637:Phenomena
1055:−
1029:−
1003:−
882:spacetime
745:−
669:∗
651:∗
591:∗
573:∗
478:−
240:‖
225:‖
181:to (1, 1/
85:hyperbola
56:hyperbola
2325:Category
2202:LemaĂźtre
2167:Einstein
2157:Poincaré
2117:Others:
2101:TaubâNUT
2067:interior
1989:theories
1987:Advanced
1954:redshift
1769:concepts
1587:Rapidity
1565:concepts
1390:Dublin:
1298:Archived
1238:rapidity
796:′
738:′
109:becomes
71:Geometry
48:geometry
2267:Hawking
2262:Penrose
2247:Novikov
2227:Wheeler
2172:Hilbert
2162:Lorentz
2119:pp-wave
1940:lensing
1736:General
1517:Special
1446:0888161
1430:(1987)
1405:(1923)
1386:(1910)
1361:: 75â88
924:) and (
272:tangent
263:across
165:, then
2308:others
2297:Thorne
2287:Misner
2272:Taylor
2257:Geroch
2252:Ehlers
2222:Zwicky
2040:Kasner
1467:
1438:
1184:
1139:
1107:s and
1068:
1042:
1016:
990:
977:
954:normal
952:) are
876:Since
255:, and
234:
121:slopes
2349:Angle
2302:Weiss
2282:Bondi
2277:Hulse
2207:Milne
2111:discs
2055:Milne
2050:Gödel
1907:Virgo
1099:When
2237:Kerr
2187:Weyl
2086:Kerr
1946:and
1900:and
1898:LIGO
1465:ISBN
1436:ISBN
1246:and
719:and
523:and
509:and
161:= 1/
123:are
2292:Yau
1917:GEO
1279:doi
534:or
113:= â
46:In
2340::
1966:/
1932::
1887::
1463:.
1461:58
1443:MR
1420:,
1359:10
1357:,
1353:,
1266:^
1240:.
1209:.
1084:0.
945:,
938:,
931:,
917:,
910:,
903:,
840:=
701:w'
277:A
267:.
210:,
202:,
177:=
151:mx
149:=
141:=
133:â
117:.
115:mx
107:mx
105:=
93:xy
67:.
2092:)
2088:(
2074:)
2065:(
2031:)
2027:(
1974:)
1970:(
1950:)
1919:)
1896:(
1545:)
1536:(
1501:e
1494:t
1487:v
1473:.
1413:.
1281::
1218:A
1214:A
1192:2
1188:t
1181:c
1175:2
1171:x
1165:=
1158:1
1154:x
1147:1
1143:t
1136:c
1123:2
1120:t
1116:1
1113:x
1109:z
1105:y
1101:c
1081:=
1076:2
1072:z
1063:1
1059:z
1050:2
1046:y
1037:1
1033:y
1024:2
1020:x
1011:1
1007:x
998:2
994:t
985:1
981:t
972:2
968:c
950:2
947:t
943:2
940:z
936:2
933:y
929:2
926:x
922:1
919:t
915:1
912:z
908:1
905:y
901:1
898:x
842:b
838:a
820:2
816:a
810:2
806:b
800:=
793:g
789:g
765:2
761:a
755:2
751:b
742:=
735:g
731:g
721:g
717:g
687:0
684:=
679:2
675:w
664:1
660:w
656:+
646:2
642:w
636:1
632:w
609:0
606:=
601:2
597:z
586:1
582:z
578:+
568:2
564:z
558:1
554:z
528:2
525:w
521:1
518:w
514:2
511:z
507:1
504:z
487:.
484:v
481:y
475:u
472:x
452:,
449:y
446:j
443:+
440:x
437:=
432:2
428:w
423:,
420:v
417:j
414:+
411:u
408:=
403:1
399:w
375:v
372:y
369:+
366:u
363:x
343:y
340:i
337:+
334:x
331:=
326:2
322:z
317:,
314:v
311:i
308:+
305:u
302:=
297:1
293:z
265:A
261:d
257:c
243:d
237:b
231:,
228:c
222:a
212:d
208:c
204:b
200:a
196:A
183:m
179:x
175:y
171:m
167:y
163:m
159:x
155:m
147:y
143:x
139:y
135:y
131:x
127:.
111:y
103:y
97:y
20:)
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