20:
4216:
225:
667:
1611:(time as measured by the object undergoing acceleration itself). Therefore, the rapidity of an object in a given frame can be viewed simply as the velocity of that object as would be calculated non-relativistically by an inertial guidance system on board the object itself if it accelerated from rest in that frame to its given speed.
2771:
469:
298:
established the natural logarithm as the area of a hyperbolic sector or an equivalent area against an asymptote. In spacetime theory, the connection of events by light divides the universe into Past, Future, or
Elsewhere based on a Here and Now . On any line in space, a light beam may be directed
2478:
2359:
1950:
1815:
2585:
2874:
is the component of momentum along the beam axis. This is the rapidity of the boost along the beam axis which takes an observer from the lab frame to a frame in which the particle moves only perpendicular to the beam. Related to this is the concept of
2578:
446:. There the axes are measured by clock and meter-stick, more familiar benchmarks, and the basis of spacetime theory. So the delineation of rapidity as a hyperbolic parameter of beam-space is a reference to the seventeenth-century origin of our precious
2205:
1600:
1085:
2363:
247:, i.e., a rotation through an imaginary angle. This angle therefore represents (in one spatial dimension) a simple additive measure of the velocity between frames. The rapidity parameter replacing velocity was introduced in 1910 by
916:
2990:
2255:
662:{\displaystyle {\begin{pmatrix}ct'\\x'\end{pmatrix}}={\begin{pmatrix}\cosh w&-\sinh w\\-\sinh w&\cosh w\end{pmatrix}}{\begin{pmatrix}ct\\x\end{pmatrix}}=\mathbf {\Lambda } (w){\begin{pmatrix}ct\\x\end{pmatrix}}.}
1820:
1688:
1269:
1384:
2861:
733:
1452:
1157:
2250:
840:
2483:
2123:
1676:
2488:
2128:
3919:
1457:
991:
2051:
986:
2766:{\displaystyle w=\operatorname {artanh} {\frac {|\mathbf {p} |c}{E}}={\frac {1}{2}}\ln {\frac {E+|\mathbf {p} |c}{E-|\mathbf {p} |c}}=\ln {\frac {E+|\mathbf {p} |c}{mc^{2}}}~.}
385:
3156:
3090:
66:. For low speeds, rapidity and velocity are almost exactly proportional but, for higher velocities, rapidity takes a larger value, with the rapidity of light being infinite.
774:
with one-dimensional Lie algebra spanned by the anti-diagonal unit matrix, showing that the rapidity is the coordinate on this Lie algebra. This action may be depicted in a
440:
2094:
3187:
851:
2885:
1196:
3966:
1293:
4201:
3990:
3736:
2473:{\displaystyle \sinh w=\sinh \left(\operatorname {artanh} {\frac {v}{c}}\right)={\frac {\frac {v}{c}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}=\beta \gamma ,}
2778:
1389:
1106:
3772:
3726:
2213:
3910:
3394:
1625:
4018:
3647:
3266:
2354:{\displaystyle \cosh w=\cosh \left(\operatorname {artanh} {\frac {v}{c}}\right)={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}=\gamma }
1945:{\displaystyle e^{-w}=\gamma (1-\beta )=\gamma \left(1-{\frac {v}{c}}\right)={\sqrt {\frac {1-{\tfrac {v}{c}}}{1+{\tfrac {v}{c}}}}}.}
3652:
3363:
3258:
1810:{\displaystyle e^{w}=\gamma (1+\beta )=\gamma \left(1+{\frac {v}{c}}\right)={\sqrt {\frac {1+{\tfrac {v}{c}}}{1-{\tfrac {v}{c}}}}},}
685:
3972:
3043:
3685:
1955:
796:
3788:
3343:
3866:
3929:
3068:
4219:
3961:
3796:
3212:
3783:
295:
3543:
4238:
3824:
3558:
2056:
771:
89:
4111:
3553:
3506:
4141:
3767:
3448:
1288:
1177:
63:
4166:
3716:
3476:
3387:
3296:
Rhodes, John A.; Semon, Mark D. (2004). "Relativistic velocity space, Wigner rotation, and Thomas precession".
2573:{\displaystyle {\begin{aligned}E&=mc^{2}\cosh w\\\left|\mathbf {p} \right|&=mc\,\sinh w.\end{aligned}}}
284:
3895:
3817:
2775:
However, experimental particle physicists often use a modified definition of rapidity relative to a beam axis
2200:{\displaystyle {\begin{aligned}E&=\gamma mc^{2}\\\left|\mathbf {p} \right|&=\gamma mv.\end{aligned}}}
3437:
3428:
3240:
1607:(the acceleration 'felt' by the object being accelerated) is the rate of change of rapidity with respect to
923:
447:
3984:
3890:
4121:
3848:
3522:
3486:
3005:
1276:
333:
236:
4126:
3731:
3670:
1595:{\displaystyle \tanh w={\frac {\tanh w_{1}+\tanh w_{2}}{1+\tanh w_{1}\tanh w_{2}}}=\tanh(w_{1}+w_{2})}
1080:{\displaystyle \mathbf {\Lambda } (w_{1}+w_{2})=\mathbf {\Lambda } (w_{1})\mathbf {\Lambda } (w_{2}).}
338:
4031:
3957:
3801:
3403:
3355:
3315:
3020:
182:
4106:
4101:
4096:
4091:
3923:
3872:
3690:
3675:
3501:
3432:
3380:
3208:
3145:
3141:
1604:
1100:
248:
240:
190:
74:
3995:
4171:
4071:
3853:
3834:
3828:
3779:
3721:
3630:
3548:
3466:
3443:
3411:
3331:
3305:
3195:
779:
405:
264:
47:
4181:
4051:
4013:
4005:
3606:
3563:
3359:
3254:
3039:
775:
443:
392:
232:
3123:
2066:
4146:
4086:
4076:
4023:
4000:
3578:
3348:
3323:
3232:
3064:
3000:
272:
70:
19:
3277:
4161:
4136:
4061:
4056:
3939:
3900:
3862:
3806:
3680:
3616:
3151:
3086:
3015:
252:
244:
3944:
3199:
3169:
3319:
4186:
3858:
3842:
3838:
3741:
3711:
3568:
3471:
3251:
3178:
3010:
2876:
1184:
767:
400:
59:
328:
can be used to gauge velocities (in the first quadrant). Zero velocity corresponds to
4232:
4176:
4156:
4151:
4066:
3934:
3762:
3706:
3538:
3491:
463:
3335:
1176:. The simplicity of this formula contrasts with the complexity of the corresponding
4196:
4116:
4081:
3611:
3573:
3218:
268:
3980:
3949:
3496:
3226:
3211:(1936), "Restricted relativity in terms of hyperbolic functions of rapidities",
3165:
3102:
1608:
260:
224:
4191:
3757:
3601:
3596:
3075:
911:{\displaystyle \mathbf {Z} ={\begin{pmatrix}0&-1\\-1&0\end{pmatrix}}.}
2985:{\displaystyle y=\ln {\frac {E+p_{z}c}{\sqrt {m^{2}c^{4}+p_{T}^{2}c^{2}}}}~.}
3588:
186:
4131:
1264:{\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}\equiv \cosh w,}
78:
51:
3327:
3310:
1379:{\displaystyle u={\frac {u_{1}+u_{2}}{1+{\frac {u_{1}u_{2}}{c^{2}}}}}}
3811:
263:(1911) and this term was adopted by many subsequent authors, such as
3044:
Fundamental
Equations for Electromagnetic Processes in Moving Bodies
2582:
So, rapidity can be calculated from measured energy and momentum by
3171:
Optical geometry of motion, a new view of the theory of relativity
18:
3069:
Application of
Lobachevskian Geometry in the Theory of Relativity
2856:{\displaystyle y={\frac {1}{2}}\ln {\frac {E+p_{z}c}{E-p_{z}c}},}
3792:
308:
82:
3376:
1087:
This establishes the useful additive property of rapidity: if
728:{\displaystyle {\begin{pmatrix}p&q\\q&p\end{pmatrix}}}
3372:
3191:, Paris: volume 156, pages 215-218; volume 157, pages 703-705
223:
1447:{\displaystyle \beta _{i}={\frac {u_{i}}{c}}=\tanh {w_{i}}}
3276:. Oxford University Press. pp. 91â127. Archived from
2882:
Rapidity relative to a beam axis can also be expressed as
1152:{\displaystyle w_{\text{AC}}=w_{\text{AB}}+w_{\text{BC}},}
1183:
As we can see from the
Lorentz transformation above, the
307:
as the events of the left beam. Then a resting frame has
2245:{\displaystyle w=\operatorname {artanh} {\frac {v}{c}},}
835:{\displaystyle \mathbf {\Lambda } (w)=e^{\mathbf {Z} w}}
399:
to these coordinates. Many authors refer instead to the
177:. The inverse hyperbolic tangent has the unit interval
1924:
1904:
1789:
1769:
868:
694:
632:
586:
520:
478:
2888:
2781:
2588:
2486:
2366:
2258:
2216:
2126:
2069:
1958:
1823:
1691:
1628:
1460:
1392:
1296:
1199:
1109:
994:
926:
854:
799:
688:
472:
442:, using rapidity for a parameter, as in the standard
408:
341:
77:
in relative motion, each frame being associated with
1671:{\displaystyle \beta \gamma =\tanh w\cosh w=\sinh w}
1622:
appears frequently, and is from the above arguments
4044:
3909:
3881:
3750:
3699:
3661:
3640:
3629:
3587:
3531:
3515:
3457:
3421:
3410:
3267:"The non-Euclidean style of Minkowskian relativity"
3157:
A History of the
Theories of Aether and Electricity
3091:
A History of the
Theories of Aether and Electricity
3347:
2984:
2855:
2765:
2572:
2480:the energy and scalar momentum can be written as:
2472:
2353:
2244:
2199:
2088:
2045:
1944:
1809:
1670:
1594:
1446:
1378:
1263:
1151:
1079:
980:
910:
834:
727:
661:
434:
379:
848:is the negative of the anti-diagonal unit matrix
391:is the rapidity, and is equal to the area of the
1275:is implicitly used as a hyperbolic angle in the
303:as the events passed by the right beam and the
69:Mathematically, rapidity can be defined as the
3237:Relativity: Special, General, and Cosmological
3388:
3183:La théorie de la relativité et la cinématique
1168:denotes the rapidity of a frame of reference
450:, and a supplement to spacetime diagramming.
8:
920:A key property of the matrix exponential is
62:. Velocities must be combined by Einstein's
3274:The Symbolic Universe: Geometry and Physics
3637:
3418:
3395:
3381:
3373:
122:is the velocity of light. For low speeds,
3309:
3188:Comptes rendus de l'Académie des Sciences
2967:
2957:
2952:
2939:
2929:
2914:
2901:
2887:
2838:
2817:
2804:
2788:
2780:
2748:
2731:
2726:
2721:
2712:
2692:
2687:
2682:
2666:
2661:
2656:
2647:
2631:
2614:
2609:
2604:
2601:
2587:
2553:
2531:
2508:
2487:
2485:
2447:
2437:
2431:
2414:
2396:
2365:
2334:
2324:
2318:
2306:
2288:
2257:
2229:
2215:
2165:
2151:
2127:
2125:
2080:
2068:
2046:{\displaystyle w=\ln \left=-\ln \left\,.}
2039:
1957:
1923:
1903:
1893:
1875:
1828:
1822:
1788:
1768:
1758:
1740:
1696:
1690:
1627:
1583:
1570:
1545:
1529:
1505:
1486:
1473:
1459:
1437:
1432:
1412:
1406:
1397:
1391:
1365:
1354:
1344:
1337:
1323:
1310:
1303:
1295:
1237:
1228:
1222:
1206:
1198:
1140:
1127:
1114:
1108:
1065:
1053:
1044:
1032:
1020:
1007:
995:
993:
969:
956:
931:
925:
863:
855:
853:
822:
821:
800:
798:
689:
687:
627:
613:
581:
515:
473:
471:
462:arises in the linear representation of a
426:
413:
407:
365:
349:
340:
3248:Linear Algebra and Group Representations
3032:
58:to accommodate the limit determined by
2116:of a particle of non-zero (rest) mass
981:{\displaystyle e^{Xs+Xt}=e^{Xs}e^{Xt}}
1681:Exponential and logarithmic relations
7:
988:from which immediately follows that
3967:TolmanâOppenheimerâVolkoff equation
3920:FriedmannâLemaĂźtreâRobertsonâWalker
1685:From the above expressions we have
136:. Since in relativity any velocity
772:indefinite orthogonal group O(1,1)
14:
3737:HamiltonâJacobiâEinstein equation
1172:relative to a frame of reference
332:. Any point on the hyperbola has
4215:
4214:
3174:. Cambridge: Heffner & Sons.
3124:"The Review of Particle Physics"
2727:
2688:
2662:
2610:
2532:
2166:
2100:In experimental particle physics
2059:factor associated with rapidity
1054:
1033:
996:
856:
823:
801:
614:
380:{\displaystyle (e^{w},\ e^{-w})}
16:Measure of relativistic velocity
140:is constrained to the interval
3544:Massâenergy equivalence (E=mc)
2732:
2722:
2693:
2683:
2667:
2657:
2615:
2605:
2031:
2019:
1991:
1979:
1855:
1843:
1720:
1708:
1589:
1563:
1287:. We relate rapidities to the
1071:
1058:
1050:
1037:
1026:
1000:
811:
805:
624:
618:
374:
342:
1:
3272:. In Jeremy John Gray (ed.).
3213:American Mathematical Monthly
3204:. London: Macmillan & Co.
3146:Vladimir VariÄak#Publications
321:. The rectangular hyperbola
3559:Relativistic Doppler effect
3298:American Journal of Physics
3229:, volume 15, pages 200-209.
466:as a vector-matrix product
435:{\displaystyle x^{2}-y^{2}}
279:Area of a hyperbolic sector
90:inverse hyperbolic function
50:, the classical concept of
4255:
4030:In computational physics:
3554:Relativity of simultaneity
3221:(1936), "When and Where",
3107:Optical Geometry of Motion
255:. The parameter was named
239:could be seen as simply a
99:corresponding to velocity
4212:
3867:LenseâThirring precession
3449:Doubly special relativity
3350:Classical Electrodynamics
3072:Physikalische Zeitschrift
1289:velocity-addition formula
1178:velocity-addition formula
770:. Such matrices form the
296:Grégoire de Saint-Vincent
64:velocity-addition formula
23:Rapidity is the value of
3727:Post-Newtonian formalism
3717:Einstein field equations
3653:Mathematical formulation
3477:Hyperbolic orthogonality
3201:The Theory of Relativity
3152:Whittaker, Edmund Taylor
3144:(1910, 1912, 1924), see
3133:(2008) 1, Section 38.5.2
3055:Sommerfeld, Phys. Z 1909
448:transcendental functions
299:left or right. Take the
193:; that is, the interval
73:that differentiates two
3438:Galilean transformation
3429:Principle of relativity
3346:(1999) . "Chapter 11".
3292:(see page 17 of e-link)
3241:Oxford University Press
2207:With the definition of
2089:{\displaystyle k=e^{w}}
3523:Lorentz transformation
3265:Walter, Scott (1999).
3250:, volume 1, page 229,
3006:Lorentz transformation
2986:
2857:
2767:
2574:
2474:
2355:
2246:
2201:
2090:
2047:
1946:
1811:
1672:
1596:
1448:
1380:
1277:Lorentz transformation
1265:
1153:
1081:
982:
912:
836:
729:
663:
436:
381:
334:light-cone coordinates
237:Lorentz transformation
228:
43:
3991:WeylâLewisâPapapetrou
3732:Raychaudhuri equation
3671:Equivalence principle
3356:John Wiley & Sons
3215:, volume 43, page 70.
2987:
2858:
2768:
2575:
2475:
2356:
2247:
2202:
2091:
2048:
1947:
1812:
1673:
1597:
1449:
1381:
1266:
1154:
1082:
983:
913:
837:
730:
664:
437:
382:
245:spacetime coordinates
227:
22:
4032:Numerical relativity
3873:pulsar timing arrays
3246:Shaw, Ronald (1982)
3227:Thomas Stearns Eliot
3027:Notes and references
3021:Theory of relativity
2886:
2779:
2586:
2484:
2364:
2256:
2214:
2124:
2108:and scalar momentum
2067:
1956:
1821:
1689:
1626:
1458:
1390:
1294:
1197:
1107:
992:
924:
852:
797:
793:can be expressed as
686:
470:
406:
339:
3924:Friedmann equations
3818:HulseâTaylor binary
3780:Gravitational waves
3676:Riemannian geometry
3502:Proper acceleration
3487:Maxwell's equations
3433:Galilean relativity
3344:Jackson, John David
3320:2004AmJPh..72..943R
3209:Vladimir Karapetoff
3196:Silberstein, Ludwik
2962:
1605:Proper acceleration
1101:frames of reference
311:along the diagonal
241:hyperbolic rotation
75:frames of reference
39:and speed of light
4239:Special relativity
3973:ReissnerâNordström
3891:BransâDicke theory
3722:Linearized gravity
3549:Length contraction
3467:Frame of reference
3444:Special relativity
2982:
2948:
2853:
2763:
2570:
2568:
2470:
2351:
2242:
2197:
2195:
2086:
2043:
1942:
1933:
1913:
1807:
1798:
1778:
1668:
1592:
1444:
1376:
1279:expressions using
1261:
1149:
1077:
978:
908:
899:
832:
780:matrix exponential
725:
719:
659:
650:
604:
575:
506:
432:
377:
265:Ludwik Silberstein
235:explained how the
229:
48:special relativity
44:
4226:
4225:
4040:
4039:
4019:OzsvĂĄthâSchĂŒcking
3625:
3624:
3607:Minkowski diagram
3564:Thomas precession
3507:Relativistic mass
3328:10.1119/1.1652040
3128:Physics Letters B
3040:Hermann Minkowski
2978:
2974:
2973:
2848:
2796:
2759:
2755:
2701:
2639:
2626:
2456:
2455:
2453:
2423:
2404:
2343:
2342:
2340:
2296:
2237:
1937:
1936:
1932:
1912:
1883:
1802:
1801:
1797:
1777:
1748:
1552:
1421:
1374:
1371:
1244:
1243:
1143:
1130:
1117:
776:spacetime diagram
444:spacetime diagram
393:hyperbolic sector
360:
287:of the hyperbola
233:Hermann Minkowski
126:is approximately
4246:
4218:
4217:
4001:van Stockum dust
3773:Two-body problem
3691:Mach's principle
3638:
3579:Terrell rotation
3419:
3397:
3390:
3383:
3374:
3369:
3354:(3rd ed.).
3353:
3339:
3313:
3291:
3289:
3288:
3282:
3271:
3233:Wolfgang Rindler
3205:
3175:
3161:
3142:Vladimir VariÄak
3134:
3116:
3110:
3100:
3094:
3084:
3078:
3065:Vladimir Varicak
3062:
3056:
3053:
3047:
3037:
3001:Bondi k-calculus
2991:
2989:
2988:
2983:
2976:
2975:
2972:
2971:
2961:
2956:
2944:
2943:
2934:
2933:
2924:
2923:
2919:
2918:
2902:
2873:
2862:
2860:
2859:
2854:
2849:
2847:
2843:
2842:
2826:
2822:
2821:
2805:
2797:
2789:
2772:
2770:
2769:
2764:
2757:
2756:
2754:
2753:
2752:
2739:
2735:
2730:
2725:
2713:
2702:
2700:
2696:
2691:
2686:
2674:
2670:
2665:
2660:
2648:
2640:
2632:
2627:
2622:
2618:
2613:
2608:
2602:
2579:
2577:
2576:
2571:
2569:
2539:
2535:
2513:
2512:
2479:
2477:
2476:
2471:
2457:
2454:
2452:
2451:
2442:
2441:
2432:
2424:
2416:
2415:
2410:
2406:
2405:
2397:
2360:
2358:
2357:
2352:
2344:
2341:
2339:
2338:
2329:
2328:
2319:
2311:
2307:
2302:
2298:
2297:
2289:
2251:
2249:
2248:
2243:
2238:
2230:
2210:
2206:
2204:
2203:
2198:
2196:
2173:
2169:
2156:
2155:
2119:
2115:
2107:
2095:
2093:
2092:
2087:
2085:
2084:
2062:
2052:
2050:
2049:
2044:
2038:
2034:
1998:
1994:
1951:
1949:
1948:
1943:
1938:
1935:
1934:
1925:
1915:
1914:
1905:
1895:
1894:
1889:
1885:
1884:
1876:
1836:
1835:
1816:
1814:
1813:
1808:
1803:
1800:
1799:
1790:
1780:
1779:
1770:
1760:
1759:
1754:
1750:
1749:
1741:
1701:
1700:
1677:
1675:
1674:
1669:
1621:
1617:
1601:
1599:
1598:
1593:
1588:
1587:
1575:
1574:
1553:
1551:
1550:
1549:
1534:
1533:
1511:
1510:
1509:
1491:
1490:
1474:
1453:
1451:
1450:
1445:
1443:
1442:
1441:
1422:
1417:
1416:
1407:
1402:
1401:
1385:
1383:
1382:
1377:
1375:
1373:
1372:
1370:
1369:
1360:
1359:
1358:
1349:
1348:
1338:
1329:
1328:
1327:
1315:
1314:
1304:
1286:
1282:
1274:
1271:so the rapidity
1270:
1268:
1267:
1262:
1245:
1242:
1241:
1232:
1227:
1226:
1211:
1207:
1193:
1187:identifies with
1175:
1171:
1167:
1158:
1156:
1155:
1150:
1145:
1144:
1141:
1132:
1131:
1128:
1119:
1118:
1115:
1098:
1094:
1090:
1086:
1084:
1083:
1078:
1070:
1069:
1057:
1049:
1048:
1036:
1025:
1024:
1012:
1011:
999:
987:
985:
984:
979:
977:
976:
964:
963:
948:
947:
917:
915:
914:
909:
904:
903:
859:
847:
841:
839:
838:
833:
831:
830:
826:
804:
792:
765:
753:
742:
738:
734:
732:
731:
726:
724:
723:
681:
668:
666:
665:
660:
655:
654:
617:
609:
608:
580:
579:
511:
510:
503:
491:
461:
441:
439:
438:
433:
431:
430:
418:
417:
398:
390:
386:
384:
383:
378:
373:
372:
358:
354:
353:
331:
327:
320:
306:
302:
293:
273:Wolfgang Rindler
249:Vladimir VariÄak
215:
207:
180:
176:
164:
154:
139:
135:
125:
121:
117:
102:
98:
94:
71:hyperbolic angle
54:is converted to
42:
38:
34:
4254:
4253:
4249:
4248:
4247:
4245:
4244:
4243:
4229:
4228:
4227:
4222:
4208:
4036:
3940:BKL singularity
3930:LemaĂźtreâTolman
3905:
3901:Quantum gravity
3883:
3877:
3863:geodetic effect
3837:(together with
3807:LISA Pathfinder
3746:
3695:
3681:Penrose diagram
3663:
3657:
3632:
3621:
3617:Minkowski space
3583:
3527:
3511:
3459:
3453:
3413:
3406:
3401:
3366:
3342:
3295:
3286:
3284:
3280:
3269:
3264:
3194:
3164:
3150:
3138:
3137:
3117:
3113:
3101:
3097:
3087:E. T. Whittaker
3085:
3081:
3063:
3059:
3054:
3050:
3038:
3034:
3029:
3016:Proper velocity
2997:
2963:
2935:
2925:
2910:
2903:
2884:
2883:
2872:
2864:
2834:
2827:
2813:
2806:
2777:
2776:
2744:
2740:
2714:
2675:
2649:
2603:
2584:
2583:
2567:
2566:
2540:
2527:
2524:
2523:
2504:
2494:
2482:
2481:
2443:
2433:
2389:
2385:
2362:
2361:
2330:
2320:
2281:
2277:
2254:
2253:
2212:
2211:
2208:
2194:
2193:
2174:
2161:
2158:
2157:
2147:
2134:
2122:
2121:
2117:
2109:
2105:
2102:
2076:
2065:
2064:
2060:
2015:
2011:
1975:
1971:
1954:
1953:
1916:
1896:
1868:
1864:
1824:
1819:
1818:
1781:
1761:
1733:
1729:
1692:
1687:
1686:
1683:
1624:
1623:
1619:
1615:
1614:The product of
1579:
1566:
1541:
1525:
1512:
1501:
1482:
1475:
1456:
1455:
1433:
1408:
1393:
1388:
1387:
1386:by recognizing
1361:
1350:
1340:
1339:
1330:
1319:
1306:
1305:
1292:
1291:
1284:
1280:
1272:
1233:
1218:
1195:
1194:
1188:
1173:
1169:
1166:
1160:
1136:
1123:
1110:
1105:
1104:
1096:
1092:
1088:
1061:
1040:
1016:
1003:
990:
989:
965:
952:
927:
922:
921:
898:
897:
892:
883:
882:
874:
864:
850:
849:
843:
817:
795:
794:
783:
755:
744:
740:
736:
718:
717:
712:
706:
705:
700:
690:
684:
683:
682:is of the type
672:
649:
648:
642:
641:
628:
603:
602:
596:
595:
582:
574:
573:
562:
547:
546:
532:
516:
505:
504:
496:
493:
492:
484:
474:
468:
467:
459:
456:
422:
409:
404:
403:
396:
388:
361:
345:
337:
336:
329:
322:
312:
304:
300:
288:
281:
253:E. T. Whittaker
222:
209:
194:
178:
166:
156:
141:
137:
127:
123:
119:
104:
100:
96:
95:, the rapidity
92:
40:
36:
24:
17:
12:
11:
5:
4252:
4250:
4242:
4241:
4231:
4230:
4224:
4223:
4213:
4210:
4209:
4207:
4206:
4199:
4194:
4189:
4184:
4179:
4174:
4169:
4164:
4159:
4154:
4149:
4144:
4139:
4134:
4129:
4127:Choquet-Bruhat
4124:
4119:
4114:
4109:
4104:
4099:
4094:
4089:
4084:
4079:
4074:
4069:
4064:
4059:
4054:
4048:
4046:
4042:
4041:
4038:
4037:
4035:
4034:
4027:
4026:
4021:
4016:
4009:
4008:
4003:
3998:
3993:
3988:
3979:Axisymmetric:
3976:
3975:
3970:
3964:
3953:
3952:
3947:
3942:
3937:
3932:
3927:
3918:Cosmological:
3915:
3913:
3907:
3906:
3904:
3903:
3898:
3893:
3887:
3885:
3879:
3878:
3876:
3875:
3870:
3859:frame-dragging
3856:
3851:
3846:
3843:Einstein rings
3839:Einstein cross
3832:
3821:
3820:
3815:
3809:
3804:
3799:
3786:
3776:
3775:
3770:
3765:
3760:
3754:
3752:
3748:
3747:
3745:
3744:
3742:Ernst equation
3739:
3734:
3729:
3724:
3719:
3714:
3712:BSSN formalism
3709:
3703:
3701:
3697:
3696:
3694:
3693:
3688:
3683:
3678:
3673:
3667:
3665:
3659:
3658:
3656:
3655:
3650:
3644:
3642:
3635:
3627:
3626:
3623:
3622:
3620:
3619:
3614:
3609:
3604:
3599:
3593:
3591:
3585:
3584:
3582:
3581:
3576:
3571:
3569:Ladder paradox
3566:
3561:
3556:
3551:
3546:
3541:
3535:
3533:
3529:
3528:
3526:
3525:
3519:
3517:
3513:
3512:
3510:
3509:
3504:
3499:
3494:
3489:
3484:
3479:
3474:
3472:Speed of light
3469:
3463:
3461:
3455:
3454:
3452:
3451:
3446:
3441:
3435:
3425:
3423:
3416:
3408:
3407:
3402:
3400:
3399:
3392:
3385:
3377:
3371:
3370:
3364:
3340:
3293:
3262:
3252:Academic Press
3244:
3230:
3216:
3206:
3192:
3176:
3162:
3160:. p. 441.
3148:
3136:
3135:
3111:
3095:
3079:
3057:
3048:
3046:via Wikisource
3031:
3030:
3028:
3025:
3024:
3023:
3018:
3013:
3011:Pseudorapidity
3008:
3003:
2996:
2993:
2981:
2970:
2966:
2960:
2955:
2951:
2947:
2942:
2938:
2932:
2928:
2922:
2917:
2913:
2909:
2906:
2900:
2897:
2894:
2891:
2877:pseudorapidity
2868:
2852:
2846:
2841:
2837:
2833:
2830:
2825:
2820:
2816:
2812:
2809:
2803:
2800:
2795:
2792:
2787:
2784:
2762:
2751:
2747:
2743:
2738:
2734:
2729:
2724:
2720:
2717:
2711:
2708:
2705:
2699:
2695:
2690:
2685:
2681:
2678:
2673:
2669:
2664:
2659:
2655:
2652:
2646:
2643:
2638:
2635:
2630:
2625:
2621:
2617:
2612:
2607:
2600:
2597:
2594:
2591:
2565:
2562:
2559:
2556:
2552:
2549:
2546:
2543:
2541:
2538:
2534:
2530:
2526:
2525:
2522:
2519:
2516:
2511:
2507:
2503:
2500:
2497:
2495:
2493:
2490:
2489:
2469:
2466:
2463:
2460:
2450:
2446:
2440:
2436:
2430:
2427:
2422:
2419:
2413:
2409:
2403:
2400:
2395:
2392:
2388:
2384:
2381:
2378:
2375:
2372:
2369:
2350:
2347:
2337:
2333:
2327:
2323:
2317:
2314:
2310:
2305:
2301:
2295:
2292:
2287:
2284:
2280:
2276:
2273:
2270:
2267:
2264:
2261:
2252:and thus with
2241:
2236:
2233:
2228:
2225:
2222:
2219:
2192:
2189:
2186:
2183:
2180:
2177:
2175:
2172:
2168:
2164:
2160:
2159:
2154:
2150:
2146:
2143:
2140:
2137:
2135:
2133:
2130:
2129:
2120:are given by:
2101:
2098:
2083:
2079:
2075:
2072:
2042:
2037:
2033:
2030:
2027:
2024:
2021:
2018:
2014:
2010:
2007:
2004:
2001:
1997:
1993:
1990:
1987:
1984:
1981:
1978:
1974:
1970:
1967:
1964:
1961:
1952:or explicitly
1941:
1931:
1928:
1922:
1919:
1911:
1908:
1902:
1899:
1892:
1888:
1882:
1879:
1874:
1871:
1867:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1834:
1831:
1827:
1806:
1796:
1793:
1787:
1784:
1776:
1773:
1767:
1764:
1757:
1753:
1747:
1744:
1739:
1736:
1732:
1728:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1699:
1695:
1682:
1679:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1591:
1586:
1582:
1578:
1573:
1569:
1565:
1562:
1559:
1556:
1548:
1544:
1540:
1537:
1532:
1528:
1524:
1521:
1518:
1515:
1508:
1504:
1500:
1497:
1494:
1489:
1485:
1481:
1478:
1472:
1469:
1466:
1463:
1440:
1436:
1431:
1428:
1425:
1420:
1415:
1411:
1405:
1400:
1396:
1368:
1364:
1357:
1353:
1347:
1343:
1336:
1333:
1326:
1322:
1318:
1313:
1309:
1302:
1299:
1260:
1257:
1254:
1251:
1248:
1240:
1236:
1231:
1225:
1221:
1217:
1214:
1210:
1205:
1202:
1185:Lorentz factor
1164:
1148:
1139:
1135:
1126:
1122:
1113:
1076:
1073:
1068:
1064:
1060:
1056:
1052:
1047:
1043:
1039:
1035:
1031:
1028:
1023:
1019:
1015:
1010:
1006:
1002:
998:
975:
972:
968:
962:
959:
955:
951:
946:
943:
940:
937:
934:
930:
907:
902:
896:
893:
891:
888:
885:
884:
881:
878:
875:
873:
870:
869:
867:
862:
858:
829:
825:
820:
816:
813:
810:
807:
803:
768:unit hyperbola
722:
716:
713:
711:
708:
707:
704:
701:
699:
696:
695:
693:
658:
653:
647:
644:
643:
640:
637:
634:
633:
631:
626:
623:
620:
616:
612:
607:
601:
598:
597:
594:
591:
588:
587:
585:
578:
572:
569:
566:
563:
561:
558:
555:
552:
549:
548:
545:
542:
539:
536:
533:
531:
528:
525:
522:
521:
519:
514:
509:
502:
499:
495:
494:
490:
487:
483:
480:
479:
477:
455:
452:
429:
425:
421:
416:
412:
401:unit hyperbola
376:
371:
368:
364:
357:
352:
348:
344:
280:
277:
221:
218:
185:and the whole
60:speed of light
15:
13:
10:
9:
6:
4:
3:
2:
4251:
4240:
4237:
4236:
4234:
4221:
4211:
4205:
4204:
4200:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4180:
4178:
4175:
4173:
4170:
4168:
4165:
4163:
4160:
4158:
4155:
4153:
4150:
4148:
4145:
4143:
4140:
4138:
4135:
4133:
4130:
4128:
4125:
4123:
4120:
4118:
4115:
4113:
4112:Chandrasekhar
4110:
4108:
4105:
4103:
4100:
4098:
4095:
4093:
4090:
4088:
4085:
4083:
4080:
4078:
4075:
4073:
4072:Schwarzschild
4070:
4068:
4065:
4063:
4060:
4058:
4055:
4053:
4050:
4049:
4047:
4043:
4033:
4029:
4028:
4025:
4022:
4020:
4017:
4015:
4011:
4010:
4007:
4004:
4002:
3999:
3997:
3994:
3992:
3989:
3986:
3982:
3978:
3977:
3974:
3971:
3968:
3965:
3963:
3959:
3958:Schwarzschild
3955:
3954:
3951:
3948:
3946:
3943:
3941:
3938:
3936:
3933:
3931:
3928:
3925:
3921:
3917:
3916:
3914:
3912:
3908:
3902:
3899:
3897:
3894:
3892:
3889:
3888:
3886:
3880:
3874:
3871:
3868:
3864:
3860:
3857:
3855:
3854:Shapiro delay
3852:
3850:
3847:
3844:
3840:
3836:
3833:
3830:
3826:
3823:
3822:
3819:
3816:
3813:
3810:
3808:
3805:
3803:
3800:
3798:
3797:collaboration
3794:
3790:
3787:
3785:
3781:
3778:
3777:
3774:
3771:
3769:
3766:
3764:
3763:Event horizon
3761:
3759:
3756:
3755:
3753:
3749:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3725:
3723:
3720:
3718:
3715:
3713:
3710:
3708:
3707:ADM formalism
3705:
3704:
3702:
3698:
3692:
3689:
3687:
3684:
3682:
3679:
3677:
3674:
3672:
3669:
3668:
3666:
3660:
3654:
3651:
3649:
3646:
3645:
3643:
3639:
3636:
3634:
3628:
3618:
3615:
3613:
3612:Biquaternions
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3594:
3592:
3590:
3586:
3580:
3577:
3575:
3572:
3570:
3567:
3565:
3562:
3560:
3557:
3555:
3552:
3550:
3547:
3545:
3542:
3540:
3539:Time dilation
3537:
3536:
3534:
3530:
3524:
3521:
3520:
3518:
3514:
3508:
3505:
3503:
3500:
3498:
3495:
3493:
3492:Proper length
3490:
3488:
3485:
3483:
3480:
3478:
3475:
3473:
3470:
3468:
3465:
3464:
3462:
3456:
3450:
3447:
3445:
3442:
3439:
3436:
3434:
3430:
3427:
3426:
3424:
3420:
3417:
3415:
3409:
3405:
3398:
3393:
3391:
3386:
3384:
3379:
3378:
3375:
3367:
3365:0-471-30932-X
3361:
3357:
3352:
3351:
3345:
3341:
3337:
3333:
3329:
3325:
3321:
3317:
3312:
3311:gr-qc/0501070
3307:
3303:
3299:
3294:
3283:on 2013-10-16
3279:
3275:
3268:
3263:
3260:
3259:0-12-639201-3
3256:
3253:
3249:
3245:
3242:
3238:
3234:
3231:
3228:
3224:
3223:The Criterion
3220:
3217:
3214:
3210:
3207:
3203:
3202:
3197:
3193:
3190:
3189:
3185:(in French),
3184:
3180:
3177:
3173:
3172:
3167:
3163:
3159:
3158:
3153:
3149:
3147:
3143:
3140:
3139:
3132:
3129:
3125:
3121:
3115:
3112:
3108:
3104:
3099:
3096:
3092:
3088:
3083:
3080:
3077:
3073:
3070:
3066:
3061:
3058:
3052:
3049:
3045:
3041:
3036:
3033:
3026:
3022:
3019:
3017:
3014:
3012:
3009:
3007:
3004:
3002:
2999:
2998:
2994:
2992:
2979:
2968:
2964:
2958:
2953:
2949:
2945:
2940:
2936:
2930:
2926:
2920:
2915:
2911:
2907:
2904:
2898:
2895:
2892:
2889:
2880:
2878:
2871:
2867:
2850:
2844:
2839:
2835:
2831:
2828:
2823:
2818:
2814:
2810:
2807:
2801:
2798:
2793:
2790:
2785:
2782:
2773:
2760:
2749:
2745:
2741:
2736:
2718:
2715:
2709:
2706:
2703:
2697:
2679:
2676:
2671:
2653:
2650:
2644:
2641:
2636:
2633:
2628:
2623:
2619:
2598:
2595:
2592:
2589:
2580:
2563:
2560:
2557:
2554:
2550:
2547:
2544:
2542:
2536:
2528:
2520:
2517:
2514:
2509:
2505:
2501:
2498:
2496:
2491:
2467:
2464:
2461:
2458:
2448:
2444:
2438:
2434:
2428:
2425:
2420:
2417:
2411:
2407:
2401:
2398:
2393:
2390:
2386:
2382:
2379:
2376:
2373:
2370:
2367:
2348:
2345:
2335:
2331:
2325:
2321:
2315:
2312:
2308:
2303:
2299:
2293:
2290:
2285:
2282:
2278:
2274:
2271:
2268:
2265:
2262:
2259:
2239:
2234:
2231:
2226:
2223:
2220:
2217:
2190:
2187:
2184:
2181:
2178:
2176:
2170:
2162:
2152:
2148:
2144:
2141:
2138:
2136:
2131:
2113:
2099:
2097:
2081:
2077:
2073:
2070:
2058:
2057:Doppler-shift
2053:
2040:
2035:
2028:
2025:
2022:
2016:
2012:
2008:
2005:
2002:
1999:
1995:
1988:
1985:
1982:
1976:
1972:
1968:
1965:
1962:
1959:
1939:
1929:
1926:
1920:
1917:
1909:
1906:
1900:
1897:
1890:
1886:
1880:
1877:
1872:
1869:
1865:
1861:
1858:
1852:
1849:
1846:
1840:
1837:
1832:
1829:
1825:
1804:
1794:
1791:
1785:
1782:
1774:
1771:
1765:
1762:
1755:
1751:
1745:
1742:
1737:
1734:
1730:
1726:
1723:
1717:
1714:
1711:
1705:
1702:
1697:
1693:
1680:
1678:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1612:
1610:
1606:
1602:
1584:
1580:
1576:
1571:
1567:
1560:
1557:
1554:
1546:
1542:
1538:
1535:
1530:
1526:
1522:
1519:
1516:
1513:
1506:
1502:
1498:
1495:
1492:
1487:
1483:
1479:
1476:
1470:
1467:
1464:
1461:
1438:
1434:
1429:
1426:
1423:
1418:
1413:
1409:
1403:
1398:
1394:
1366:
1362:
1355:
1351:
1345:
1341:
1334:
1331:
1324:
1320:
1316:
1311:
1307:
1300:
1297:
1290:
1278:
1258:
1255:
1252:
1249:
1246:
1238:
1234:
1229:
1223:
1219:
1215:
1212:
1208:
1203:
1200:
1192:
1186:
1181:
1179:
1163:
1146:
1137:
1133:
1124:
1120:
1111:
1102:
1074:
1066:
1062:
1045:
1041:
1029:
1021:
1017:
1013:
1008:
1004:
973:
970:
966:
960:
957:
953:
949:
944:
941:
938:
935:
932:
928:
918:
905:
900:
894:
889:
886:
879:
876:
871:
865:
860:
846:
827:
818:
814:
808:
790:
786:
781:
777:
773:
769:
763:
759:
751:
747:
720:
714:
709:
702:
697:
691:
679:
675:
669:
656:
651:
645:
638:
635:
629:
621:
610:
605:
599:
592:
589:
583:
576:
570:
567:
564:
559:
556:
553:
550:
543:
540:
537:
534:
529:
526:
523:
517:
512:
507:
500:
497:
488:
485:
481:
475:
465:
464:Lorentz boost
458:The rapidity
454:Lorentz boost
453:
451:
449:
445:
427:
423:
419:
414:
410:
402:
394:
369:
366:
362:
355:
350:
346:
335:
325:
319:
315:
310:
297:
291:
286:
278:
276:
274:
270:
266:
262:
258:
254:
250:
246:
242:
238:
234:
226:
219:
217:
213:
206:
202:
198:
192:
188:
184:
174:
170:
163:
159:
153:
149:
145:
134:
130:
115:
111:
107:
91:
86:
85:coordinates.
84:
80:
76:
72:
67:
65:
61:
57:
53:
49:
35:for velocity
32:
28:
21:
4202:
3896:KaluzaâKlein
3648:Introduction
3574:Twin paradox
3481:
3349:
3304:(7): 90â93.
3301:
3297:
3285:. Retrieved
3278:the original
3273:
3247:
3236:
3225:, edited by
3222:
3219:Frank Morley
3200:
3186:
3182:
3170:
3166:Robb, Alfred
3155:
3130:
3127:
3119:
3114:
3106:
3098:
3082:
3071:
3060:
3051:
3035:
2881:
2869:
2865:
2774:
2581:
2111:
2103:
2054:
1684:
1613:
1603:
1190:
1182:
1161:
919:
844:
788:
784:
766:lies on the
761:
757:
749:
745:
677:
673:
670:
457:
323:
317:
313:
289:
282:
269:Frank Morley
256:
230:
211:
204:
200:
196:
172:
168:
161:
157:
151:
147:
143:
132:
128:
113:
109:
105:
87:
68:
55:
45:
30:
26:
3985:KerrâNewman
3956:Spherical:
3825:Other tests
3768:Singularity
3700:Formulation
3662:Fundamental
3516:Formulation
3497:Proper time
3458:Fundamental
3239:, page 53,
3179:Ămile Borel
3118:Amsler, C.
3103:Alfred Robb
3093:, page 441.
2104:The energy
1609:proper time
743:satisfying
671:The matrix
271:(1936) and
261:Alfred Robb
4137:Zel'dovich
4045:Scientists
4024:Alcubierre
3831:of Mercury
3829:precession
3758:Black hole
3641:Background
3633:relativity
3602:World line
3597:Light cone
3422:Background
3414:relativity
3404:Relativity
3287:2009-01-08
3076:Wikisource
782:notation,
754:, so that
285:quadrature
208:maps onto
165:satisfies
155:the ratio
88:Using the
4107:Robertson
4092:Friedmann
4087:Eddington
4077:de Sitter
3911:Solutions
3789:detectors
3784:astronomy
3751:Phenomena
3686:Geodesics
3589:Spacetime
3532:Phenomena
2899:
2832:−
2802:
2710:
2680:−
2645:
2599:
2558:
2518:
2465:γ
2462:β
2429:−
2394:
2383:
2371:
2349:γ
2316:−
2286:
2275:
2263:
2227:
2182:γ
2142:γ
2029:β
2026:−
2017:γ
2009:
2003:−
1989:β
1977:γ
1969:
1901:−
1873:−
1862:γ
1853:β
1850:−
1841:γ
1830:−
1817:and thus
1786:−
1727:γ
1718:β
1706:γ
1663:
1651:
1642:
1633:γ
1630:β
1561:
1539:
1523:
1499:
1480:
1465:
1430:
1395:β
1253:
1247:≡
1216:−
1201:γ
1055:Λ
1034:Λ
997:Λ
887:−
877:−
802:Λ
615:Λ
568:
557:
551:−
541:
535:−
527:
420:−
367:−
187:real line
108:= artanh(
4233:Category
4220:Category
4097:LemaĂźtre
4062:Einstein
4052:Poincaré
4012:Others:
3996:TaubâNUT
3962:interior
3884:theories
3882:Advanced
3849:redshift
3664:concepts
3482:Rapidity
3460:concepts
3336:14764378
3198:(1914).
3181:(1913),
3168:(1911).
3154:(1910).
2995:See also
842:, where
501:′
489:′
275:(2001).
267:(1914),
257:rapidity
231:In 1908
210:ââ <
189:for its
181:for its
167:â1 <
79:distance
56:rapidity
52:velocity
4162:Hawking
4157:Penrose
4142:Novikov
4122:Wheeler
4067:Hilbert
4057:Lorentz
4014:pp-wave
3835:lensing
3631:General
3412:Special
3316:Bibcode
3235:(2001)
3105:(1911)
3089:(1910)
3067:(1910)
3042:(1908)
1454:and so
1103:, then
251:and by
243:of the
220:History
179:(â1, 1)
25:artanh(
4203:others
4192:Thorne
4182:Misner
4167:Taylor
4152:Geroch
4147:Ehlers
4117:Zwicky
3935:Kasner
3362:
3334:
3257:
3120:et al.
2977:
2863:where
2758:
2596:artanh
2391:artanh
2283:artanh
2224:artanh
1159:where
397:(1, 1)
387:where
359:
330:(1, 1)
305:y-axis
301:x-axis
214:< â
183:domain
175:< 1
118:where
93:artanh
4197:Weiss
4177:Bondi
4172:Hulse
4102:Milne
4006:discs
3950:Milne
3945:Gödel
3802:Virgo
3332:S2CID
3306:arXiv
3281:(PDF)
3270:(PDF)
1189:cosh
778:. In
735:with
395:from
203:<
199:<
191:image
150:<
146:<
4132:Kerr
4082:Weyl
3981:Kerr
3841:and
3795:and
3793:LIGO
3360:ISBN
3255:ISBN
3074:via
2555:sinh
2515:cosh
2380:sinh
2368:sinh
2272:cosh
2260:cosh
2055:The
1660:sinh
1648:cosh
1639:tanh
1618:and
1558:tanh
1536:tanh
1520:tanh
1496:tanh
1477:tanh
1462:tanh
1427:tanh
1283:and
1250:cosh
1099:are
1095:and
739:and
565:cosh
554:sinh
538:sinh
524:cosh
309:time
283:The
83:time
81:and
4187:Yau
3812:GEO
3324:doi
3131:667
3109:p.9
2063:is
752:= 1
326:= 1
294:by
292:= 1
259:by
103:is
46:In
4235::
3861:/
3827::
3782::
3358:.
3330:.
3322:.
3314:.
3302:72
3300:.
3126:,
3122:,
2896:ln
2879:.
2799:ln
2707:ln
2642:ln
2096:.
2006:ln
1966:ln
1180:.
1165:PQ
1142:BC
1129:AB
1116:AC
1091:,
760:,
748:â
324:xy
316:=
290:xy
216:.
171:/
160:/
131:/
112:/
29:/
3987:)
3983:(
3969:)
3960:(
3926:)
3922:(
3869:)
3865:(
3845:)
3814:)
3791:(
3440:)
3431:(
3396:e
3389:t
3382:v
3368:.
3338:.
3326::
3318::
3308::
3290:.
3261:.
3243:.
2980:.
2969:2
2965:c
2959:2
2954:T
2950:p
2946:+
2941:4
2937:c
2931:2
2927:m
2921:c
2916:z
2912:p
2908:+
2905:E
2893:=
2890:y
2870:z
2866:p
2851:,
2845:c
2840:z
2836:p
2829:E
2824:c
2819:z
2815:p
2811:+
2808:E
2794:2
2791:1
2786:=
2783:y
2761:.
2750:2
2746:c
2742:m
2737:c
2733:|
2728:p
2723:|
2719:+
2716:E
2704:=
2698:c
2694:|
2689:p
2684:|
2677:E
2672:c
2668:|
2663:p
2658:|
2654:+
2651:E
2637:2
2634:1
2629:=
2624:E
2620:c
2616:|
2611:p
2606:|
2593:=
2590:w
2564:.
2561:w
2551:c
2548:m
2545:=
2537:|
2533:p
2529:|
2521:w
2510:2
2506:c
2502:m
2499:=
2492:E
2468:,
2459:=
2449:2
2445:c
2439:2
2435:v
2426:1
2421:c
2418:v
2412:=
2408:)
2402:c
2399:v
2387:(
2377:=
2374:w
2346:=
2336:2
2332:c
2326:2
2322:v
2313:1
2309:1
2304:=
2300:)
2294:c
2291:v
2279:(
2269:=
2266:w
2240:,
2235:c
2232:v
2221:=
2218:w
2209:w
2191:.
2188:v
2185:m
2179:=
2171:|
2167:p
2163:|
2153:2
2149:c
2145:m
2139:=
2132:E
2118:m
2114:|
2112:p
2110:|
2106:E
2082:w
2078:e
2074:=
2071:k
2061:w
2041:.
2036:]
2032:)
2023:1
2020:(
2013:[
2000:=
1996:]
1992:)
1986:+
1983:1
1980:(
1973:[
1963:=
1960:w
1940:.
1930:c
1927:v
1921:+
1918:1
1910:c
1907:v
1898:1
1891:=
1887:)
1881:c
1878:v
1870:1
1866:(
1859:=
1856:)
1847:1
1844:(
1838:=
1833:w
1826:e
1805:,
1795:c
1792:v
1783:1
1775:c
1772:v
1766:+
1763:1
1756:=
1752:)
1746:c
1743:v
1738:+
1735:1
1731:(
1724:=
1721:)
1715:+
1712:1
1709:(
1703:=
1698:w
1694:e
1666:w
1657:=
1654:w
1645:w
1636:=
1620:Îł
1616:ÎČ
1590:)
1585:2
1581:w
1577:+
1572:1
1568:w
1564:(
1555:=
1547:2
1543:w
1531:1
1527:w
1517:+
1514:1
1507:2
1503:w
1493:+
1488:1
1484:w
1471:=
1468:w
1439:i
1435:w
1424:=
1419:c
1414:i
1410:u
1404:=
1399:i
1367:2
1363:c
1356:2
1352:u
1346:1
1342:u
1335:+
1332:1
1325:2
1321:u
1317:+
1312:1
1308:u
1301:=
1298:u
1285:ÎČ
1281:Îł
1273:w
1259:,
1256:w
1239:2
1235:c
1230:/
1224:2
1220:v
1213:1
1209:1
1204:=
1191:w
1174:P
1170:Q
1162:w
1147:,
1138:w
1134:+
1125:w
1121:=
1112:w
1097:C
1093:B
1089:A
1075:.
1072:)
1067:2
1063:w
1059:(
1051:)
1046:1
1042:w
1038:(
1030:=
1027:)
1022:2
1018:w
1014:+
1009:1
1005:w
1001:(
974:t
971:X
967:e
961:s
958:X
954:e
950:=
945:t
942:X
939:+
936:s
933:X
929:e
906:.
901:)
895:0
890:1
880:1
872:0
866:(
861:=
857:Z
845:Z
828:w
824:Z
819:e
815:=
812:)
809:w
806:(
791:)
789:w
787:(
785:Î
764:)
762:q
758:p
756:(
750:q
746:p
741:q
737:p
721:)
715:p
710:q
703:q
698:p
692:(
680:)
678:w
676:(
674:Î
657:.
652:)
646:x
639:t
636:c
630:(
625:)
622:w
619:(
611:=
606:)
600:x
593:t
590:c
584:(
577:)
571:w
560:w
544:w
530:w
518:(
513:=
508:)
498:x
486:t
482:c
476:(
460:w
428:2
424:y
415:2
411:x
389:w
375:)
370:w
363:e
356:,
351:w
347:e
343:(
318:y
314:x
212:w
205:c
201:v
197:c
195:â
173:c
169:v
162:c
158:v
152:c
148:v
144:c
142:â
138:v
133:c
129:v
124:w
120:c
116:)
114:c
110:v
106:w
101:v
97:w
41:c
37:v
33:)
31:c
27:v
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