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Events connected by light have zero separation. The plenum of spacetime in the plane is split into four quadrants, each of which has the topology of R. The dividing lines are the trajectory of inbound and outbound photons at (0,0). The planar-cosmology topological segmentation is the future F, the
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thus become important in analysing local as well as global aspects of spacetime. The study of spacetime topology is especially important in
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Nowadays, the correct mathematical term for the
Alexandrov topology on spacetime (which goes back to
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on a partial order is usually taken to be the coarsest topology in which only the upper sets
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past P, space left L, and space right D. The homeomorphism of F with R amounts to
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This topology coincides with the manifold topology if and only if the manifold is
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of the plane does not mingle the quadrants, in fact, each one is an
1138:{\displaystyle z=\exp(a+jb)=e^{a}(\cosh b+j\sinh b)\to (a,b),}
1328:
Zeeman, E.C. (1967). "The topology of
Minkowski space".
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As with any manifold, a spacetime possesses a natural
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Hawking, S. W.; King, A. R.; McCarthy, P. J. (1976).
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There are two main types of topology for a spacetime
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are required to be open. This topology goes back to
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1359:Techniques of Differential Topology in Relativity
1185:is the split-complex logarithm and the required
1378:(1964). "Causality Implies the Lorentz Group".
642:{\displaystyle Y^{+}(p,U)\cup Y^{-}(p,U)\cup p}
749:The Alexandrov topology on spacetime, is the
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1205:with each of P, L, and D under the mappings
554:than the manifold topology. It is therefore
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909:{\displaystyle Y^{+}(x)\cap Y^{-}(y)}
675:and some convex normal neighbourhood
573:for the topology is sets of the form
148:Newton's law of universal gravitation
7:
1197:parameter for relative motion in F.
522:which induces the same topology as
477:in the manifold topology such that
308:(a spacetime) and the concepts of
113:Introduction to general relativity
25:
193:Mathematics of general relativity
118:Mathematics of general relativity
366:{\displaystyle \mathbb {R} ^{4}}
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1416:Journal of Mathematical Physics
1380:Journal of Mathematical Physics
508:{\displaystyle E\cap c=O\cap c}
285:, a topic studied primarily in
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344:are the image of open sets in
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959:Note that in mathematics, an
733:chronological past and future
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143:Introduction to gravitation
1474:
1178:{\displaystyle z\to (a,b)}
942:{\displaystyle \,x,y\in M}
844:{\displaystyle E\subset M}
742:
694:{\displaystyle U\subset M}
423:{\displaystyle E\subset M}
18:Global spacetime structure
1276:Gravitational singularity
825:are open for all subsets
542:does on timelike curves.
173:Derivations of relativity
1281:Hantzsche–Wendt_manifold
1203:bijective correspondence
992:{\displaystyle Y^{+}(E)}
818:{\displaystyle Y^{-}(E)}
782:{\displaystyle Y^{+}(E)}
724:{\displaystyle Y^{\pm }}
123:Einstein field equations
1357:Penrose, Roger (1972),
377:Path or Zeeman topology
88:Lorentz transformations
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1008:Alexandr D. Alexandrov
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668:{\displaystyle p\in M}
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1261:Closed timelike curve
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743:Further information:
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397:{\displaystyle \rho }
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279:topological structure
183:Differential geometry
83:Equivalence principle
1458:Lorentzian manifolds
1239:unit hyperbola group
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161:Relevant mathematics
1428:1976JMP....17..174H
1392:1964JMP.....5..490Z
1286:Spacetime curvature
1256:Clifford-Klein form
1231:Hyperbolic rotation
1025:polar decomposition
961:Alexandrov topology
745:Alexandrov topology
739:Alexandrov topology
340:topology. Here the
306:Lorentzian manifold
30:Part of a series on
1453:General relativity
1314:2010-06-16 at the
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404:in which a subset
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314:physical cosmology
287:general relativity
275:Spacetime topology
221:Physics portal
198:Spacetime topology
178:Spacetime diagrams
106:General relativity
78:Spacetime manifold
71:Spacetime concepts
60:General relativity
55:Special relativity
1400:10.1063/1.1704140
1266:Complex spacetime
1189:F → R, Note that
1012:interval topology
751:coarsest topology
535:{\displaystyle M}
470:{\displaystyle O}
450:{\displaystyle c}
332:Manifold topology
320:Types of topology
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136:Classical gravity
16:(Redirected from
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384:: The topology
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731:denote the
295:gravitation
168:Four-vector
1447:Categories
1369:References
1251:4-manifold
1237:under the
1229:on (0,0).
546:Properties
518:It is the
382:Definition
1227:null cone
1158:→
1115:→
1106:
1091:
1051:
934:∈
893:−
885:∩
854:Here the
836:⊂
802:−
717:±
686:⊂
660:∈
634:∪
614:−
606:∪
560:separable
556:Hausdorff
550:Strictly
500:∩
488:∩
415:⊂
392:ρ
342:open sets
299:curvature
283:spacetime
36:Spacetime
1331:Topology
1312:Archived
1291:Wormhole
1245:See also
1201:F is in
1195:rapidity
562:but not
338:manifold
310:topology
234:Category
1424:Bibcode
1388:Bibcode
1193:is the
1145:so that
297:as the
293:models
289:. This
277:is the
232:
1412:(PDF)
1297:Notes
552:finer
301:of a
1103:sinh
1088:cosh
856:base
789:and
571:base
432:open
1432:doi
1396:doi
1340:doi
1217:→ j
1209:→ –
1048:exp
1027:of
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430:is
281:of
1449::
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1418:.
1414:.
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1382:.
1334:.
1241:.
1213:,
1031::
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701:.
569:A
566:.
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373:.
328:.
316:.
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1426::
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1398::
1390::
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1346:.
1342::
1336:6
1326:*
1223:z
1219:z
1215:z
1211:z
1207:z
1191:b
1173:)
1170:b
1167:,
1164:a
1161:(
1155:z
1133:,
1130:)
1127:b
1124:,
1121:a
1118:(
1112:)
1109:b
1100:j
1097:+
1094:b
1085:(
1080:a
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1066:b
1063:j
1060:+
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1045:=
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937:M
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901:y
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879:x
876:(
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839:M
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810:E
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774:E
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766:+
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689:M
683:U
663:M
657:p
637:p
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628:U
625:,
622:p
619:(
610:Y
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600:U
597:,
594:p
591:(
586:+
582:Y
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503:c
497:O
494:=
491:c
485:E
465:O
445:c
418:M
412:E
359:4
354:R
326:M
263:e
256:t
249:v
20:)
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