1074:
1031:
487:
296:
248:
529:
409:
83:
109:
328:
167:
57:
766:
367:
129:
369:. Particles with a very small mass do not need much energy to travel at a speed close to c. The latter is used to derive orbits of massless particles such as the
433:
187:
1115:
374:
777:
441:
1057:
337:
This can result from holding the mass fixed and increasing the kinetic energy to very large values or by holding the energy
1108:
1134:
1144:
1101:
253:
195:
1139:
493:
1042:
1081:
388:
62:
1047:
580:
88:
1052:
301:
138:
36:
579:
is proper time, and travel starts at rest and without changing direction of acceleration (see
1085:
751:
352:
114:
192:
The energy of an ultrarelativistic particle is almost completely due to its kinetic energy
768:
707:
667:
418:
172:
132:
1128:
1073:
699:
586:
Fixed target collision with ultrarelativistic motion of the center of mass:
412:
331:
20:
370:
1026:{\displaystyle E_{k}=(\gamma -1)mc^{2}={\frac {1}{2}}mv^{2}+\left}
748:. Its kinetic energy can be approximated by first term of the
632:
are energies of the particle and the target respectively (so
482:{\displaystyle 1-\beta \approx {\frac {1}{2\gamma ^{2}}}}
1089:
780:
754:
496:
444:
421:
391:
355:
304:
256:
198:
175:
141:
117:
91:
65:
39:
385:
Below are few ultrarelativistic approximations when
349:to very small values which also imply a very large
27:when its speed is very close to the speed of light
1025:
760:
666:For calculations of the energy of a particle, the
523:
481:
427:
403:
361:
322:
290:
242:
181:
161:
123:
103:
77:
51:
250:. The total energy can also be approximated as
1109:
8:
722:), the approximation is essentially exact.
670:of the ultrarelativistic limit for a speed
1116:
1102:
535:Motion with constant proper acceleration:
995:
982:
976:
967:
945:
921:
915:
885:
875:
869:
856:
842:
825:
816:
785:
779:
753:
495:
470:
457:
443:
420:
390:
354:
303:
291:{\displaystyle E=\gamma mc^{2}\approx pc}
273:
255:
234:
203:
197:
174:
151:
140:
116:
90:
64:
38:
744:, where its speed is much smaller than
243:{\displaystyle E_{k}=(\gamma -1)mc^{2}}
657:is energy in the center of mass frame.
524:{\displaystyle w\approx \ln(2\gamma )}
373:from those of massive particles (cf.
7:
1070:
1068:
375:Kepler problem in general relativity
718:practically indistinguishable from
1088:. You can help Knowledge (XXG) by
14:
1072:
381:Ultrarelativistic approximations
16:Very close to the speed of light
404:{\displaystyle \beta \approx 1}
78:{\displaystyle \beta \approx 1}
964:
954:
933:
924:
806:
794:
518:
509:
224:
212:
33:. Notations commonly used are
1:
662:Accuracy of the approximation
568:is proper acceleration (with
343:fixed and shrinking the mass
104:{\displaystyle \gamma \gg 1}
323:{\displaystyle p=\gamma mv}
1161:
1067:
1058:Aichelburg–Sexl ultraboost
698:%. For particles such as
554:is the distance traveled,
162:{\displaystyle \beta =v/c}
52:{\displaystyle v\approx c}
330:is the Lorentz invariant
761:{\displaystyle \gamma }
362:{\displaystyle \gamma }
189:is the speed of light.
124:{\displaystyle \gamma }
23:, a particle is called
1084:-related article is a
1027:
762:
525:
483:
429:
405:
363:
324:
292:
244:
183:
163:
125:
105:
79:
53:
1043:Relativistic particle
1028:
763:
526:
484:
430:
406:
364:
325:
293:
245:
184:
164:
126:
106:
80:
54:
778:
752:
710:) are usually above
494:
442:
419:
389:
353:
302:
254:
196:
173:
139:
115:
89:
63:
37:
1048:Classical mechanics
730:The opposite case (
581:proper acceleration
1135:Special relativity
1053:Special relativity
1023:
758:
742:classical particle
583:for more details).
521:
479:
425:
401:
359:
320:
288:
240:
179:
159:
121:
101:
75:
49:
1097:
1096:
1004:
974:
891:
864:
833:
740:) is a so-called
477:
428:{\displaystyle w}
182:{\displaystyle c}
25:ultrarelativistic
1152:
1145:Relativity stubs
1118:
1111:
1104:
1076:
1069:
1032:
1030:
1029:
1024:
1022:
1018:
1005:
1003:
1002:
990:
989:
977:
975:
973:
972:
971:
953:
952:
939:
922:
920:
919:
892:
890:
889:
880:
879:
870:
865:
857:
847:
846:
834:
826:
821:
820:
790:
789:
767:
765:
764:
759:
747:
739:
721:
717:
713:
705:
697:
693:
683:
679:
656:
647:
631:
622:
613:
612:
611:
578:
574:
567:
553:
549:
530:
528:
527:
522:
488:
486:
485:
480:
478:
476:
475:
474:
458:
434:
432:
431:
426:
410:
408:
407:
402:
368:
366:
365:
360:
348:
342:
329:
327:
326:
321:
297:
295:
294:
289:
278:
277:
249:
247:
246:
241:
239:
238:
208:
207:
188:
186:
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180:
168:
166:
165:
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155:
130:
128:
127:
122:
110:
108:
107:
102:
84:
82:
81:
76:
58:
56:
55:
50:
32:
1160:
1159:
1155:
1154:
1153:
1151:
1150:
1149:
1125:
1124:
1123:
1122:
1066:
1039:
991:
978:
963:
941:
940:
923:
911:
881:
871:
855:
851:
838:
812:
781:
776:
775:
769:binomial series
750:
749:
745:
731:
728:
719:
715:
711:
703:
695:
685:
681:
671:
664:
655:
649:
646:
639:
633:
630:
624:
621:
615:
610:
604:
597:
595:
593:
587:
576:
569:
555:
551:
536:
492:
491:
466:
462:
440:
439:
417:
416:
387:
386:
383:
351:
350:
344:
338:
300:
299:
269:
252:
251:
230:
199:
194:
193:
171:
170:
137:
136:
113:
112:
87:
86:
61:
60:
35:
34:
28:
17:
12:
11:
5:
1158:
1156:
1148:
1147:
1142:
1140:Approximations
1137:
1127:
1126:
1121:
1120:
1113:
1106:
1098:
1095:
1094:
1077:
1065:
1062:
1061:
1060:
1055:
1050:
1045:
1038:
1035:
1034:
1033:
1021:
1017:
1014:
1011:
1008:
1001:
998:
994:
988:
985:
981:
970:
966:
962:
959:
956:
951:
948:
944:
938:
935:
932:
929:
926:
918:
914:
910:
907:
904:
901:
898:
895:
888:
884:
878:
874:
868:
863:
860:
854:
850:
845:
841:
837:
832:
829:
824:
819:
815:
811:
808:
805:
802:
799:
796:
793:
788:
784:
757:
727:
724:
708:Lorentz factor
668:relative error
663:
660:
659:
658:
653:
644:
637:
628:
619:
608:
602:
591:
584:
532:
531:
520:
517:
514:
511:
508:
505:
502:
499:
489:
473:
469:
465:
461:
456:
453:
450:
447:
424:
400:
397:
394:
382:
379:
358:
319:
316:
313:
310:
307:
287:
284:
281:
276:
272:
268:
265:
262:
259:
237:
233:
229:
226:
223:
220:
217:
214:
211:
206:
202:
178:
158:
154:
150:
147:
144:
133:Lorentz factor
120:
100:
97:
94:
74:
71:
68:
48:
45:
42:
15:
13:
10:
9:
6:
4:
3:
2:
1157:
1146:
1143:
1141:
1138:
1136:
1133:
1132:
1130:
1119:
1114:
1112:
1107:
1105:
1100:
1099:
1093:
1091:
1087:
1083:
1078:
1075:
1071:
1063:
1059:
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1040:
1036:
1019:
1015:
1012:
1009:
1006:
999:
996:
992:
986:
983:
979:
968:
960:
957:
949:
946:
942:
936:
930:
927:
916:
912:
908:
905:
902:
899:
896:
893:
886:
882:
876:
872:
866:
861:
858:
852:
848:
843:
839:
835:
830:
827:
822:
817:
813:
809:
803:
800:
797:
791:
786:
782:
774:
773:
772:
770:
755:
743:
738:
734:
725:
723:
709:
701:
692:
688:
678:
674:
669:
661:
652:
643:
636:
627:
618:
607:
601:
590:
585:
582:
572:
566:
562:
558:
547:
543:
539:
534:
533:
515:
512:
506:
503:
500:
497:
490:
471:
467:
463:
459:
454:
451:
448:
445:
438:
437:
436:
422:
414:
398:
395:
392:
380:
378:
376:
372:
356:
347:
341:
335:
333:
317:
314:
311:
308:
305:
285:
282:
279:
274:
270:
266:
263:
260:
257:
235:
231:
227:
221:
218:
215:
209:
204:
200:
190:
176:
156:
152:
148:
145:
142:
134:
118:
98:
95:
92:
72:
69:
66:
46:
43:
40:
31:
26:
22:
1090:expanding it
1079:
741:
736:
732:
729:
726:Other limits
690:
686:
676:
672:
665:
650:
641:
634:
625:
616:
605:
599:
588:
570:
564:
560:
556:
545:
541:
537:
384:
345:
339:
336:
191:
29:
24:
18:
694:it is just
684:%, and for
415:is denoted
1129:Categories
1082:relativity
1064:References
801:−
798:γ
756:γ
700:neutrinos
680:is about
516:γ
507:
501:≈
468:γ
455:≈
452:β
449:−
396:≈
393:β
357:γ
312:γ
280:≈
264:γ
219:−
216:γ
143:β
119:γ
96:≫
93:γ
70:≈
67:β
44:≈
1037:See also
702:, whose
550:, where
413:rapidity
332:momentum
648:), and
596:√
131:is the
21:physics
689:= 0.99
675:= 0.95
614:where
411:. The
371:photon
298:where
111:where
1080:This
1086:stub
623:and
377:).
169:and
575:),
573:≫ 1
544:/(2
85:or
59:or
19:In
1131::
771::
735:≪
712:10
682:10
654:CM
640:≫
594:≈
592:CM
571:aτ
565:dτ
561:dφ
559:=
540:≈
504:ln
435::
334:.
135:,
1117:e
1110:t
1103:v
1092:.
1020:]
1016:.
1013:.
1010:.
1007:+
1000:n
997:2
993:c
987:n
984:2
980:v
969:2
965:)
961:!
958:n
955:(
950:n
947:2
943:2
937:!
934:)
931:n
928:2
925:(
917:2
913:c
909:m
906:+
903:.
900:.
897:.
894:+
887:2
883:c
877:4
873:v
867:m
862:8
859:3
853:[
849:+
844:2
840:v
836:m
831:2
828:1
823:=
818:2
814:c
810:m
807:)
804:1
795:(
792:=
787:k
783:E
746:c
737:c
733:v
720:c
716:v
714:(
706:(
704:γ
696:2
691:c
687:v
677:c
673:v
651:E
645:2
642:E
638:1
635:E
629:2
626:E
620:1
617:E
609:2
606:E
603:1
600:E
598:2
589:E
577:τ
563:/
557:a
552:d
548:)
546:a
542:e
538:d
519:)
513:2
510:(
498:w
472:2
464:2
460:1
446:1
423:w
399:1
346:m
340:E
318:v
315:m
309:=
306:p
286:c
283:p
275:2
271:c
267:m
261:=
258:E
236:2
232:c
228:m
225:)
222:1
213:(
210:=
205:k
201:E
177:c
157:c
153:/
149:v
146:=
99:1
73:1
47:c
41:v
30:c
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