378:, and then quietly measure the distance between the marks. We can even consider such measurement as a possible operational definition of proper length. From the viewpoint of the experimental physics, the requirement that the marks be made simultaneously is redundant for a stationary object with constant shape and size, and can in this case be dropped from such definition. Since the rod is stationary in
96:
of an object is the length of the object measured by an observer which is at rest relative to it, by applying standard measuring rods on the object. The measurement of the object's endpoints doesn't have to be simultaneous, since the endpoints are constantly at rest at the same positions in the
635:
146:
However, in relatively moving frames the object's endpoints have to be measured simultaneously, since they are constantly changing their position. The resulting length is shorter than the rest length, and is given by the formula for
696:
in any spacetime, curved or flat. In a flat spacetime, the proper distance between two events is the proper distance along a straight path between the two events. In a curved spacetime, there may be more than one straight path
495:
80:. The difference is that the proper distance is defined between two spacelike-separated events (or along a spacelike path), while the proper time is defined between two timelike-separated events (or along a timelike path).
281:
800:
59:. In classical mechanics, lengths are measured based on the assumption that the locations of all points involved are measured simultaneously. But in the theory of relativity, the notion of
203:
536:
688:
The above formula for the proper distance between two events assumes that the spacetime in which the two events occur is flat. Hence, the above formula cannot in general be used in
141:
422:
961:
881:
316:, measured at the endpoints of the same object, only agree with each other when the measurement events were simultaneous in the object's rest frame so that Δ
720:
701:) between two events, so the proper distance along a straight path between two events would not uniquely define the proper distance between the two events.
531:
The definition can be given equivalently with respect to any inertial frame of reference (without requiring the events to be simultaneous in that frame) by
350:
of the rod, and you want to measure its length, you can do it by first marking its endpoints. And it is not necessary that you mark them simultaneously in
214:
1072:
1042:
945:
815:
336:
of an object whose end points happen to be respectively coincident with these events. Consider a solid rod of constant proper length
208:
In comparison, the invariant proper distance between two arbitrary events happening at the endpoints of the same object is given by:
1097:
698:
413:, the proper distance between two spacelike-separated events is the distance between the two events, as measured in an
414:
911:
60:
978:
Franklin, Jerrold (2010). "Lorentz contraction, Bell's spaceships, and rigid body motion in special relativity".
165:
857:
instead of a distance. The − sign in the equation should be dropped with a metric tensor that instead uses the
692:, in which curved spacetimes are considered. It is, however, possible to define the proper distance along a
884:
524:
676:
Two events are spacelike-separated if and only if the above formula gives a real, non-zero value for Δ
997:
107:
52:
389:
of the rod regardless of the time lapse between the two markings. On the other hand, it is not the
56:
630:{\displaystyle \Delta \sigma ={\sqrt {\Delta x^{2}+\Delta y^{2}+\Delta z^{2}-c^{2}\Delta t^{2}}},}
1013:
987:
955:
896:
689:
666:
410:
148:
1079:
1068:
1049:
1038:
941:
906:
31:
1005:
850:
883:
should be dropped with a metric tensor that is normalized to use a distance, or that uses
693:
417:
in which the events are simultaneous. In such a specific frame, the distance is given by
1001:
866:
658:
156:
1091:
1009:
713:
1017:
1062:
1032:
901:
490:{\displaystyle \Delta \sigma ={\sqrt {\Delta x^{2}+\Delta y^{2}+\Delta z^{2}}},}
77:
17:
833:
823:
520:
45:
819:
70:, provides an invariant measure whose value is the same for all observers.
276:{\displaystyle \Delta \sigma ={\sqrt {\Delta x^{2}-c^{2}\Delta t^{2}}}.}
393:
between the marking events if the marks are not made simultaneously in
795:{\displaystyle L=c\int _{P}{\sqrt {-g_{\mu \nu }dx^{\mu }dx^{\nu }}},}
709:
516:
992:
1037:(illustrated ed.). Cambridge University Press. p. 191.
673:= 0 exactly when the events are simultaneous in the given frame.
854:
648:
844:
In the equation above, the metric tensor is assumed to use the
1061:
Kopeikin, Sergei; Efroimsky, Michael; Kaplan, George (2011).
665:
The two formulae are equivalent because of the invariance of
294:, whereas (as explained above) the object's rest length
51:
The measurement of lengths is more complicated in the
869:
836:
separation between neighboring events along the path
723:
539:
425:
217:
168:
110:
1064:
Relativistic
Celestial Mechanics of the Solar System
30:
For the cosmological notion of proper distance, see
875:
794:
629:
489:
275:
197:
135:
1034:Gravity: Newtonian, Post-Newtonian, Relativistic
405:Proper distance between two events in flat space
853:, and is assumed to be normalized to return a
97:object's rest frame, so it is independent of Δ
27:Length of an object in the object's rest frame
8:
198:{\displaystyle L={\frac {L_{0}}{\gamma }}.}
44:is the length of an object in the object's
973:
971:
960:: CS1 maint: location missing publisher (
931:
929:
927:
1031:Poisson, Eric; Will, Clifford M. (2014).
991:
868:
781:
768:
752:
743:
737:
722:
616:
603:
590:
574:
558:
549:
538:
476:
460:
444:
435:
424:
262:
249:
236:
227:
216:
181:
175:
167:
115:
109:
385:, the distance between the marks is the
357:. You can mark one end now (at a moment
923:
364:) and the other end later (at a moment
1067:. John Wiley & Sons. p. 136.
953:
7:
938:Special Relativity and How it Works
320:is zero. As explained by Fayngold:
708:, the proper distance is given in
704:Along an arbitrary spacelike path
651:coordinates of the two events, and
609:
583:
567:
551:
540:
469:
453:
437:
426:
301:can be measured independently of Δ
255:
229:
218:
124:
25:
328:between two events is generally
101:. This length is thus given by:
343:. If you are in the rest frame
136:{\displaystyle L_{0}=\Delta x.}
63:is dependent on the observer.
527:coordinates of the two events.
1:
863:metric signature. Also, the
684:Proper distance along a path
84:Proper length or rest length
980:European Journal of Physics
415:inertial frame of reference
1114:
1010:10.1088/0143-0807/31/2/006
912:Relativity of simultaneity
29:
940:. John Wiley & Sons.
647:is the difference in the
936:Moses Fayngold (2009).
515:are differences in the
324:p. 407: "Note that the
877:
796:
631:
491:
277:
199:
137:
878:
797:
632:
492:
278:
200:
138:
1098:Theory of relativity
867:
721:
537:
423:
215:
166:
108:
53:theory of relativity
1080:Extract of page 136
1050:Extract of page 191
1002:2010EJPh...31..291F
667:spacetime intervals
305:. It follows that Δ
57:classical mechanics
897:Invariant interval
873:
792:
690:general relativity
627:
487:
411:special relativity
273:
195:
149:length contraction
133:
66:A different term,
1074:978-3-527-63457-6
1044:978-1-107-03286-6
907:Comoving distance
885:geometrized units
876:{\displaystyle c}
787:
622:
482:
268:
190:
32:Comoving distance
16:(Redirected from
1105:
1082:
1078:
1058:
1052:
1048:
1028:
1022:
1021:
995:
975:
966:
965:
959:
951:
933:
882:
880:
879:
874:
861:
851:metric signature
848:
818:for the current
801:
799:
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786:
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773:
772:
760:
759:
744:
742:
741:
636:
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623:
621:
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579:
578:
563:
562:
550:
496:
494:
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488:
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481:
480:
465:
464:
449:
448:
436:
332:the same as the
282:
280:
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228:
204:
202:
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196:
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120:
119:
76:is analogous to
21:
1113:
1112:
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1103:
1102:
1088:
1087:
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1085:
1075:
1060:
1059:
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1030:
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1025:
977:
976:
969:
952:
948:
935:
934:
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893:
865:
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777:
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733:
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391:proper distance
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342:
326:proper distance
315:
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258:
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177:
164:
163:
111:
106:
105:
86:
74:Proper distance
68:proper distance
35:
28:
23:
22:
18:Proper distance
15:
12:
11:
5:
1111:
1109:
1101:
1100:
1090:
1089:
1084:
1083:
1073:
1053:
1043:
1023:
986:(2): 291–298.
967:
947:978-3527406074
946:
922:
921:
919:
916:
915:
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909:
904:
899:
892:
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842:
841:
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791:
784:
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758:
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747:
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736:
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712:syntax by the
685:
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663:
662:
659:speed of light
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619:
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569:
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528:
486:
479:
475:
471:
468:
463:
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452:
447:
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401:
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382:
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272:
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257:
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174:
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157:Lorentz factor
144:
143:
132:
129:
126:
123:
118:
114:
85:
82:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1110:
1099:
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1095:
1093:
1081:
1076:
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1066:
1065:
1057:
1054:
1051:
1046:
1040:
1036:
1035:
1027:
1024:
1019:
1015:
1011:
1007:
1003:
999:
994:
989:
985:
981:
974:
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968:
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957:
949:
943:
939:
932:
930:
928:
924:
917:
913:
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908:
905:
903:
900:
898:
895:
894:
890:
888:
886:
870:
862:
856:
852:
849:
839:
835:
831:
828:
825:
821:
817:
816:metric tensor
813:
807:
806:
805:
802:
789:
782:
778:
774:
769:
765:
761:
756:
753:
749:
745:
738:
734:
730:
727:
724:
716:
715:
714:line integral
711:
707:
702:
700:
695:
691:
683:
681:
679:
674:
672:
669:, and since Δ
668:
660:
656:
653:
650:
646:
642:
641:
640:
637:
624:
617:
613:
604:
600:
596:
591:
587:
580:
575:
571:
564:
559:
555:
546:
543:
532:
526:
522:
518:
514:
510:
506:
502:
501:
500:
497:
484:
477:
473:
466:
461:
457:
450:
445:
441:
432:
429:
418:
416:
412:
404:
396:
392:
388:
387:proper length
381:
374:
367:
360:
353:
346:
339:
335:
334:proper length
331:
327:
323:
322:
321:
319:
312:
308:
304:
297:
293:
289:
270:
263:
259:
250:
246:
242:
237:
233:
224:
221:
211:
210:
209:
192:
187:
182:
178:
172:
169:
162:
161:
160:
158:
154:
150:
130:
127:
121:
116:
112:
104:
103:
102:
100:
95:
91:
90:proper length
83:
81:
79:
75:
71:
69:
64:
62:
58:
54:
49:
47:
43:
39:
38:Proper length
33:
19:
1063:
1056:
1033:
1026:
983:
979:
937:
858:
845:
843:
837:
829:
826:mapping, and
808:
803:
717:
705:
703:
687:
677:
675:
670:
664:
654:
644:
638:
533:
530:
512:
508:
504:
498:
419:
408:
394:
390:
386:
379:
372:
365:
358:
351:
344:
337:
333:
329:
325:
317:
310:
306:
302:
295:
291:
290:depends on Δ
287:
285:
207:
152:
145:
98:
93:
89:
87:
73:
72:
67:
65:
61:simultaneity
50:
41:
37:
36:
902:Proper time
94:rest length
78:proper time
42:rest length
918:References
834:coordinate
824:coordinate
521:orthogonal
155:being the
46:rest frame
993:0906.1919
956:cite book
820:spacetime
783:ν
770:μ
757:ν
754:μ
746:−
735:∫
610:Δ
597:−
584:Δ
568:Δ
552:Δ
544:σ
541:Δ
470:Δ
454:Δ
438:Δ
430:σ
427:Δ
256:Δ
243:−
230:Δ
222:σ
219:Δ
188:γ
125:Δ
1092:Category
1018:18059490
891:See also
699:geodesic
649:temporal
55:than in
998:Bibcode
832:is the
814:is the
657:is the
639:where
525:spatial
511:, and Δ
499:where
1071:
1041:
1016:
944:
804:where
710:tensor
517:linear
151:(with
1014:S2CID
988:arXiv
371:) in
1069:ISBN
1039:ISBN
962:link
942:ISBN
860:−+++
855:time
847:+−−−
822:and
694:path
309:and
286:So Δ
88:The
1006:doi
507:, Δ
409:In
330:not
159:):
92:or
40:or
1094::
1012:.
1004:.
996:.
984:31
982:.
970:^
958:}}
954:{{
926:^
887:.
830:dx
811:μν
680:.
523:,
519:,
400:."
48:.
1077:.
1047:.
1020:.
1008::
1000::
990::
964:)
950:.
871:c
840:.
838:P
809:g
790:,
779:x
775:d
766:x
762:d
750:g
739:P
731:c
728:=
725:L
706:P
697:(
678:σ
671:t
661:.
655:c
645:t
643:Δ
625:,
618:2
614:t
605:2
601:c
592:2
588:z
581:+
576:2
572:y
565:+
560:2
556:x
547:=
513:z
509:y
505:x
503:Δ
485:,
478:2
474:z
467:+
462:2
458:y
451:+
446:2
442:x
433:=
398:0
395:K
383:0
380:K
376:0
373:K
369:2
366:t
362:1
359:t
355:0
352:K
348:0
345:K
341:0
338:l
318:t
314:0
311:L
307:σ
303:t
299:0
296:L
292:t
288:σ
271:.
264:2
260:t
251:2
247:c
238:2
234:x
225:=
193:.
183:0
179:L
173:=
170:L
153:γ
131:.
128:x
122:=
117:0
113:L
99:t
34:.
20:)
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