771:
33:
101:
4843:
784:
3199:
wavefronts at distant galaxies which have already encountered the wave, he will see their images undistorted. This must be the case, since he cannot know the wave is coming until it reaches his location, for it is traveling at the speed of light. However, this can be confirmed by direct computation
3377:
Penrose was also the first to understand the strange nature of causality in pp-sandwich wave spacetimes. He showed that some or all of the null geodesics emitted at a given event will be refocused at a later event (or string of events). The details depend upon whether the wave is purely
978:
An example of pp-wave given when gravity is in presence of matter is the gravitational field surrounding a neutral Weyl fermion: the system consists in a gravitational field that is a pp-wave, no electrodynamic radiation, and a massless spinor exhibiting axial symmetry. In the
3241:
The effect of a passing polarized gravitational plane wave on the relative positions of a cloud of (initially static) test particles will be qualitatively very similar. We might mention here that in general, the motion of test particles in pp-wave spacetimes can exhibit
3016:, because they are the second type in the symmetry classification of Sippel and Gönner. A limiting case of certain axisymmetric pp-waves yields the Aichelburg/Sexl ultraboost modeling an ultrarelativistic encounter with an isolated spherically symmetric object.
3238:, he will see circular images alternately squeezed horizontally while expanded vertically, and squeezed vertically while expanded horizontally. This directly exhibits the characteristic effect of a gravitational wave in general relativity on light.
2137:. This radiation can be gravitational, electromagnetic, Weyl fermions, or some hypothetical kind of massless radiation other than these three, or any combination of these. All this radiation is traveling in the same direction, and the null vector
2863:
3038:
of nonflat pp-waves in the sense defined above. They are only a generalization in the sense that although the
Brinkmann metric form is preserved, they are not necessarily the vacuum solutions studied by Ehlers and Kundt, Sippel and Gönner, etc.
1887:
2095:
3348:. To show this, he used techniques imported from algebraic geometry to "blow up" the spacetime so that the given null geodesic becomes the covariantly constant null geodesic congruence of a plane wave. This construction is called a
2997:; all points on a given two-dimensional wavefront are equivalent. This not quite true for more general pp-waves. Plane waves are important for many reasons; to mention just one, they are essential for the beautiful topic of
3257:
them. PP waves provide a rare exception to this rule: if you have two PP waves sharing the same covariantly constant null vector (the same geodesic null congruence, i.e. the same wave vector field), with metric functions
2565:
172:
1973:
1761:
1533:
1146:
3231:
wavefronts at distant galaxies which the wave has not yet reached. Now he sees their optical images sheared and magnified (or demagnified) in a time-dependent manner. If the wave happens to be a
963:
plane wave spacetimes. In particular, the physical experience of an observer who whizzes by a gravitating object (such as a star or a black hole) at nearly the speed of light can be modelled by an
4546:
542:
3374:. If you view the Riemann tensor as a second rank tensor acting on bivectors, the vanishing of invariants is analogous to the fact that a nonzero null vector has vanishing squared length.
3192:
PP-waves enjoy numerous striking properties. Some of their more abstract mathematical properties have already been mentioned. In this section a few additional properties are presented.
1804:
3104:
Since they constitute a very simple and natural class of
Lorentzian manifolds, defined in terms of a null congruence, it is not very surprising that they are also important in other
3195:
Consider an inertial observer in
Minkowski spacetime who encounters a sandwich plane wave. Such an observer will experience some interesting optical effects. If he looks into the
2988:
2928:
3090:
815:
3604:
2764:
2756:
2358:
2168:
1585:
1303:
3225:
2302:
1602:
A Lorentzian manifold is a pp-wave if and only if it admits a one-parameter subgroup of isometries having null orbits, and whose curvature tensor has vanishing eigenvalues.
944:
Furthermore, in general relativity, disturbances in the gravitational field itself can propagate, at the speed of light, as "wrinkles" in the curvature of spacetime. Such
3336:
3296:
1267:
1661:
1691:
1403:
3139:
vacuum solutions in general relativity and in the Brans/Dicke theory are precisely the vacuum pp-waves (but the Brans/Dicke theory admits further wavelike solutions).
194:
1340:
948:
is the gravitational field analogue of electromagnetic radiation. In general relativity, the gravitational analogue of electromagnetic plane waves are precisely the
3561:
2600:
1810:
1984:
4865:
2696:
2646:
2275:
2723:
2670:
2620:
2405:
2378:
2245:
2203:
1364:
1235:
1211:
1169:
399:
4593:
4828:
4617:
4363:
752:
562:
404:
2223:
From the form of Ricci spinor given in the preceding section, it is immediately apparent that a pp-wave spacetime (written in the
Brinkmann chart) is a
517:
3893:
3476:
808:
2413:
1693:(describing any gravitational field which may be present) each have only one nonvanishing component. Specifically, with respect to the NP tetrad
4399:
4353:
430:
264:
770:
4537:
469:
107:
4021:
1898:
949:
54:
3370:. Such statement does not hold in higher-dimensions since there are higher-dimensional pp-waves of algebraic type II with non-vanishing
1587:. But the definition of a pp-wave spacetime does not impose this equation, so it is entirely mathematical and belongs to the study of
1178:
The definition which is now standard in the literature is more sophisticated. It makes no reference to any coordinate chart, so it is a
801:
4645:
4274:
3459:
is an axisymmetric plane wave which models the gravitational field of an infinitely long beam of incoherent electromagnetic radiation.
2182:
Unfortunately, the terminology concerning pp-waves, while fairly standard, is highly confusing and tends to promote misunderstanding.
181:
1699:
1415:
1028:
4279:
3709:
3686:
207:
76:
4599:
3397:. This is an exact result in fully nonlinear general relativity which is analogous to a similar result concerning electromagnetic
487:
100:
3034:
covariantly constant null bivector field. The name is potentially misleading, since as Steele points out, these are nominally a
4312:
3181:
394:
3441:
968:
4415:
4493:
3472:
3172:
It is natural to generalize pp-waves to higher dimensions, where they enjoy similar properties to those we have discussed.
1605:
A Lorentzian manifold with nonvanishing curvature is a (nontrivial) pp-wave if and only if it admits a covariantly constant
4556:
527:
4846:
4588:
4423:
481:
189:
3622:
3498:
1664:
1631:
1627:
929:. In particular, in general relativity, we must take into account the gravitational effects of the energy density of the
547:
4410:
3533:
2327:. It turns out that the most general pp-wave spacetime has only one Killing vector field, the null geodesic congruence
845:
4170:
3389:. In the case of plane waves, these gauge transformations allow us to always regard two colliding plane waves to have
4451:
4185:
788:
279:
199:
47:
41:
4738:
4180:
4133:
3701:
1588:
692:
240:
3371:
3356:
492:
4768:
4394:
4075:
3573:
3382:
2699:
1619:
884:
304:
3366:, yet the curvature is almost never zero. This is because in four-dimension all pp-waves belong to the class of
1272:
This definition was introduced by Ehlers and Kundt in 1962. To relate
Brinkmann's definition to this one, take
567:
58:
4793:
4343:
4103:
4014:
3360:
3120:
3108:
953:
727:
717:
384:
4522:
4444:
3124:
2705:
Equivalently, a plane-wave is a pp-wave with at least a five-dimensional Lie algebra of
Killing vector fields
1767:
446:
3599:
approaching the curvature singularity which is present in many physically important solutions, including the
3580:
4064:
4055:
3254:
890:
712:
4611:
4517:
3506:
3116:
3020:
2998:
2934:
2874:
2384:
918:
512:
4748:
4475:
4149:
4113:
3445:
682:
667:
274:
3696:
Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius & Herlt, Eduard (2003).
3140:
3049:
2858:{\displaystyle X={\frac {\partial }{\partial u}}(px+qy)\,\partial _{v}+p\,\partial _{x}+q\,\partial _{y}}
4753:
4358:
4297:
3994:
2649:
2324:
2116:
2109:
1019:
980:
930:
657:
225:
3510:
3480:
4658:
4584:
4428:
4030:
3960:
3835:
3810:
Ehlers, JĂŒrgen & Kundt, Wolfgang (1962). "Exact solutions of the gravitational field equations".
3790:
3753:
3600:
3518:
3449:
3418:
3386:
3232:
3132:
3105:
2728:
2330:
2308:
2140:
2100:
This means that any pp-wave spacetime can be interpreted, in the context of general relativity, as a
1538:
1275:
1214:
1175:. This was the original definition of Brinkmann, and it has the virtue of being easy to understand.
477:
3679:
Symmetries and
Curvature Structure in General Relativity (World Scientific Lecture Notes in Physics)
3253:
is well known. This implies that if you have two exact solutions, there is almost never any way to
3169:
quantizations of pp-wave spacetimes offers a glimpse into the yet unknown world of quantum gravity.
3043:
987:
4733:
4728:
4723:
4718:
4550:
4499:
4317:
4302:
4128:
4059:
4007:
3592:
3567:
3349:
3203:
3093:
2280:
1183:
1011:
938:
672:
652:
642:
637:
622:
389:
4622:
3899:
H.-J. Schmidt (1998). "A two-dimensional representation of four-dimensional gravitational waves,"
3417:
examples of pp-waves. ("Explicit" means that the metric functions can be written down in terms of
532:
4798:
4698:
4480:
4461:
4455:
4406:
4348:
4257:
4175:
4093:
4070:
4038:
3976:
3950:
3874:
3769:
3617:
3402:
3301:
3261:
2101:
1243:
829:
722:
607:
379:
284:
269:
230:
92:
3173:
1636:
3501:, a gravitational plane wave which, should it collide head-on with a twin, will produce in the
3227:. Now suppose that after the wave passes, our observer turns about face and looks through the
3162:
2277:). Physically, these represent purely gravitational radiation propagating along the null rays
1669:
1663:(describing any matter or nongravitational fields which may be present in a spacetime) and the
1369:
4808:
4678:
4632:
4233:
4190:
3705:
3682:
3577:, which propagates through an initially flat spacetime, progressively destroying the universe,
3422:
2248:
1882:{\displaystyle {\vec {m}}={\frac {1}{\sqrt {2}}}\,\left(\partial _{x}+i\,\partial _{y}\right)}
1412:. The vacuum Einstein equations are very simple for pp waves, and in fact linear: the metric
1306:
971:. The gravitational field of a beam of light is modelled, in general relativity, by a certain
602:
327:
3941:
Cianci, R.; Fabbri, L.; Vignolo S. (2015). "Exact solutions for Weyl fermions with gravity".
2090:{\displaystyle \Psi _{0}={\frac {1}{4}}\,\left(\left(H_{xx}-H_{yy}\right)+2i\,H_{xy}\right).}
1312:
4773:
4713:
4703:
4650:
4627:
4205:
3968:
3866:
3843:
3798:
3761:
3426:
3165:
quantum corrections vanish identically for any pp-wave spacetime. This means that studying
926:
857:
697:
677:
632:
612:
557:
3539:
2573:
4788:
4763:
4688:
4683:
4566:
4527:
4489:
4433:
4307:
4243:
3909:
3166:
3154:
2307:
Ehlers and Kundt and Sippel and Gönner have classified vacuum pp-wave spacetimes by their
2224:
2214:
2210:
2206:
1623:
1179:
1172:
991:
841:
732:
707:
592:
587:
451:
332:
294:
4571:
2675:
2625:
2254:
502:
3964:
3839:
3794:
3757:
3655:
4813:
4485:
4469:
4465:
4368:
4338:
4195:
4098:
2708:
2655:
2605:
2390:
2363:
2230:
2188:
2134:
1408:
Neither of these definitions make any mention of any field equation; in fact, they are
1349:
1220:
1196:
1154:
983:
spacetime, there exists a complete set of exact solutions for both gravity and matter.
906:
radiation associated with some hypothetical distinct type relativistic classical field,
877:
861:
775:
742:
737:
425:
289:
3857:
H. W. Brinkmann (1925). "Einstein spaces which are mapped conformally on each other".
3744:
Tupper, B. O. J. (1974). "Common solutions of the
Einstein and Brans-Dicke theories".
2387:, which were first studied by Baldwin and Jeffery. A plane wave is a pp-wave in which
4859:
4803:
4783:
4778:
4693:
4561:
4389:
4333:
4165:
4118:
3980:
3972:
3878:
3781:
Penrose, Roger (1965). "A remarkable property of plane waves in general relativity".
3773:
3596:
3526:
3487:
3367:
3341:
1015:
972:
702:
662:
617:
597:
522:
420:
299:
4823:
4743:
4708:
4238:
4200:
3927:
3491:
3243:
3158:
3128:
2320:
995:
647:
627:
3641:
Cianci, R.; Fabbri, L.; Vignolo S., Exact solutions for Weyl fermions with gravity
17:
1598:
Ehlers and Kundt gave several more coordinate-free characterizations, including:
4607:
4576:
4123:
3456:
3112:
3009:
2220:
Going in the other direction, pp-waves include several important special cases.
2171:
2105:
1190:
922:
897:
537:
507:
4818:
4384:
4228:
4223:
3802:
3398:
747:
309:
235:
3536:; this spacetime is foliated by spacelike hyperslices which are isometric to
2993:
Intuitively, the distinction is that the wavefronts of plane waves are truly
4215:
3887:
Generating a dynamical M2 brane from super-gravitons in a pp-wave background
3587:(type 11 in the Sippel and Gönner symmetry classification), which exhibit a
3381:
Every pp-wave admits many different
Brinkmann charts. These are related by
3250:
3031:
2560:{\displaystyle H(u,x,y)=a(u)\,(x^{2}-y^{2})+2\,b(u)\,xy+c(u)\,(x^{2}+y^{2})}
2316:
2130:
1626:
of any pp-wave spacetime vanishes identically. Equivalently, we can find a
873:
318:
3848:
3823:
910:
or any combination of these, so long as the radiation is all moving in the
4758:
4108:
3886:
3143:
has reformulated the theory of (four-dimensional) pp-waves in terms of a
1606:
687:
497:
337:
3023:
for a discussion of physically important special cases of plane waves.)
959:
There are physically important examples of pp-wave spacetimes which are
3870:
3765:
167:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
937:
provide the direct generalization of ordinary plane wave solutions in
4438:
3448:
which models the physical experience of an observer who whizzes by a
1968:{\displaystyle \Phi _{00}={\frac {1}{4}}\,\left(H_{xx}+H_{yy}\right)}
990:
in 1925 and have been rediscovered many times since, most notably by
3735:
Penrose, Roger (1976). "Any spacetime has a plane wave as a limit".
3722:
Sippel, R. & Gönner, H. (1986). "Symmetry classes of pp waves".
2383:
The most important class of particularly symmetric pp-waves are the
3955:
2672:
describes the wave profile of any nongravitational radiation. If
921:, provide the most general analogue in general relativity of the
4419:
3479:
solutions, which generalize solutions which are well-known from
2185:
In any pp-wave spacetime, the covariantly constant vector field
4003:
3930:, "Plane Polarized Waves in the General Theory of Relativity,"
1756:{\displaystyle {\vec {\ell }}=\partial _{u}-H/2\,\partial _{v}}
1528:{\displaystyle ds^{2}=H(u,x,y)\,du^{2}+2\,du\,dv+dx^{2}+dy^{2}}
1141:{\displaystyle ds^{2}=H(u,x,y)\,du^{2}+2\,du\,dv+dx^{2}+dy^{2}}
3355:
Penrose also pointed out that in a pp-wave spacetime, all the
2407:
is quadratic, and can hence be transformed to the simple form
26:
3999:
3521:, thereby permitting a classical peek at the local geometry
3012:
Lie algebra of
Killing vector fields. These are also called
2648:
describe the wave profiles of the two linearly independent
1978:
and the only nonvanishing component of the Weyl spinor is
2698:, we have the vacuum plane waves, which are often called
3030:. These are nonflat Lorentzian spacetimes which admit a
3153:
Pp-waves also play an important role in the search for
3046:. These have vanishing curvature except on some range
2652:
of gravitational radiation which may be present, while
2319:, and as usual it is easier to classify the underlying
1892:
the only nonvanishing component of the Ricci spinor is
3115:. In particular, pp-waves are exact solutions in the
3092:, and represent a gravitational wave moving through a
3542:
3304:
3264:
3206:
3052:
2937:
2877:
2767:
2731:
2711:
2678:
2658:
2628:
2608:
2576:
2416:
2393:
2366:
2333:
2283:
2257:
2233:
2191:
2143:
1987:
1901:
1813:
1770:
1702:
1672:
1639:
1541:
1418:
1372:
1352:
1315:
1278:
1246:
1223:
1199:
1157:
1031:
110:
3042:
Another important special class of pp-waves are the
4671:
4536:
4508:
4377:
4326:
4288:
4267:
4256:
4214:
4158:
4142:
4084:
4048:
4037:
3378:gravitational, purely electromagnetic, or neither.
3555:
3346:every Lorentzian spacetime looks like a plane wave
3330:
3290:
3219:
3084:
2982:
2922:
2857:
2750:
2717:
2690:
2664:
2640:
2614:
2594:
2559:
2399:
2372:
2352:
2296:
2269:
2239:
2197:
2162:
2089:
1967:
1881:
1798:
1755:
1685:
1655:
1579:
1527:
1397:
1358:
1334:
1297:
1261:
1229:
1205:
1163:
1140:
166:
952:among the plane wave spacetimes. They are called
3812:Gravitation: an Introduction to Current Research
3385:, which in this context may be considered to be
2122:In other words, pp-waves model various kinds of
1346:notation for tensor equations, the condition on
2213:(the class of Lorentzian manifolds admitting a
3200:of the optical scalars of the null congruence
2380:, there are additional Killing vector fields.
4015:
3698:Exact Solutions of Einstein's Field Equations
1609:. (If so, this bivector is a null bivector.)
1213:is called a pp-wave spacetime. That is, the
854:plane-fronted waves with parallel propagation
809:
8:
3822:Baldwin, O. R. & Jeffery, G. B. (1926).
2178:Relation to other classes of exact solutions
3570:is a gravitational plane wave exhibiting a
3249:The fact that Einstein's field equation is
4264:
4045:
4022:
4008:
4000:
3026:J. D. Steele has introduced the notion of
3008:, which in general have a two-dimensional
1618:It is a purely mathematical fact that the
816:
802:
368:
258:
88:
3954:
3847:
3589:weak nonscalar null curvature singularity
3547:
3541:
3322:
3309:
3303:
3282:
3269:
3263:
3211:
3205:
3076:
3057:
3051:
2939:
2938:
2936:
2879:
2878:
2876:
2849:
2844:
2832:
2827:
2815:
2810:
2774:
2766:
2742:
2730:
2710:
2677:
2657:
2627:
2607:
2575:
2548:
2535:
2527:
2505:
2492:
2477:
2464:
2456:
2415:
2392:
2365:
2360:. However, for various special forms of
2344:
2332:
2288:
2282:
2256:
2251:(with respect to the spatial coordinates
2232:
2190:
2154:
2142:
2070:
2065:
2042:
2026:
2011:
2001:
1992:
1986:
1951:
1935:
1925:
1915:
1906:
1900:
1868:
1863:
1851:
1841:
1829:
1815:
1814:
1812:
1790:
1772:
1771:
1769:
1747:
1742:
1734:
1722:
1704:
1703:
1701:
1677:
1671:
1644:
1638:
1562:
1546:
1540:
1519:
1503:
1486:
1479:
1467:
1459:
1426:
1417:
1377:
1371:
1351:
1326:
1314:
1289:
1277:
1245:
1222:
1198:
1156:
1132:
1116:
1099:
1092:
1080:
1072:
1039:
1030:
917:A special type of pp-wave spacetime, the
155:
146:
134:
115:
109:
77:Learn how and when to remove this message
3477:monochromatic electromagnetic plane wave
3450:spherically symmetric gravitating object
3344:has observed that near a null geodesic,
3004:A more general subclass consists of the
1799:{\displaystyle {\vec {n}}=\partial _{v}}
40:This article includes a list of general
3681:. Singapore: World Scientific Pub. Co.
3634:
438:
412:
371:
317:
91:
3824:"The relativity theory of plane waves"
3473:monochromatic gravitational plane wave
3127:, and certain gravitation theories of
840:for short, are an important family of
4866:Exact solutions in general relativity
3906:(1998) 215â224 (arXiv:gr-qc/9712034).
2983:{\displaystyle {\ddot {q}}=aq-bp-cq.}
2923:{\displaystyle {\ddot {p}}=-ap+bq-cp}
2209:. Therefore, pp-waves belong to the
1535:obeys these equations if and only if
7:
3737:Differential Geometry and Relativity
3393:, and thus the waves can be said to
4594:TolmanâOppenheimerâVolkoff equation
4547:FriedmannâLemaĂźtreâRobertsonâWalker
3499:Schwarzschild generating plane wave
3085:{\displaystyle u_{1}<u<u_{2}}
896:massless radiation associated with
3894:D-branes in the pp-wave background
3534:uniform electromagnetic plane wave
3208:
3180:are essential building blocks for
2846:
2829:
2812:
2780:
2776:
2758:and four more which have the form
2739:
2602:are arbitrary smooth functions of
2341:
2285:
2151:
1989:
1903:
1865:
1848:
1787:
1744:
1719:
1674:
1641:
1628:NewmanâPenrose complex null tetrad
1591:. In the next section we turn to
1286:
1247:
1018:can be described, with respect to
935:purely electromagnetic plane waves
127:
46:it lacks sufficient corresponding
25:
4364:HamiltonâJacobiâEinstein equation
3885:Yi-Fei Chen and J.X. Lu (2004), "
3188:Geometric and physical properties
2217:with vanishing optical scalars).
2205:always has identically vanishing
880:. This radiation may consist of:
4842:
4841:
2115:as may be verified by using the
1309:orthogonal to the hypersurfaces
856:, and was introduced in 1962 by
783:
782:
769:
99:
31:
3182:eleven-dimensional supergravity
2751:{\displaystyle X=\partial _{v}}
2353:{\displaystyle k=\partial _{v}}
2163:{\displaystyle k=\partial _{v}}
1580:{\displaystyle H_{xx}+H_{yy}=0}
1410:entirely independent of physics
1298:{\displaystyle k=\partial _{v}}
1182:definition. It states that any
4171:Massâenergy equivalence (E=mc)
3973:10.1140/epjc/s10052-015-3698-9
3338:gives a third exact solution.
3028:generalised pp-wave spacetimes
2807:
2789:
2554:
2528:
2524:
2518:
2502:
2496:
2483:
2457:
2453:
2447:
2438:
2420:
1820:
1777:
1709:
1456:
1438:
1069:
1051:
1:
3452:at nearly the speed of light,
3220:{\displaystyle \partial _{v}}
2297:{\displaystyle \partial _{v}}
967:pp-wave spacetime called the
872:The pp-waves solutions model
3912:, "On Gravitational Waves,"
3372:polynomial scalar invariants
3357:polynomial scalar invariants
986:Pp-waves were introduced by
4186:Relativistic Doppler effect
3656:"On Generalised P.P. Waves"
3509:solution a region which is
3331:{\displaystyle H_{1}+H_{2}}
3291:{\displaystyle H_{1},H_{2}}
3178:higher-dimensional pp-waves
1262:{\displaystyle \nabla k=0.}
280:Gravitational time dilation
4882:
4657:In computational physics:
4181:Relativity of simultaneity
3702:Cambridge University Press
3442:AichelburgâSexl ultraboost
3413:There are many noteworthy
3383:coordinate transformations
3100:Relation to other theories
1656:{\displaystyle \Phi _{ij}}
1589:pseudo-Riemannian geometry
969:AichelburgâSexl ultraboost
933:itself. When we do this,
400:MathissonâPapapetrouâDixon
241:Pseudo-Riemannian manifold
4839:
4494:LenseâThirring precession
4076:Doubly special relativity
3803:10.1103/RevModPhys.37.215
3601:Schwarzschild black holes
3121:higher curvature theories
3019:(See also the article on
2700:plane gravitational waves
1686:{\displaystyle \Psi _{i}}
1620:characteristic polynomial
1398:{\displaystyle k_{a;b}=0}
1237:must vanish identically:
954:gravitational plane waves
885:electromagnetic radiation
846:Einstein's field equation
4354:Post-Newtonian formalism
4344:Einstein field equations
4280:Mathematical formulation
4104:Hyperbolic orthogonality
3623:NewmanâPenrose formalism
3519:Schwarzschild black hole
3481:weak-field approximation
3236:gravitational plane wave
3109:classical field theories
1593:physical interpretations
925:familiar to students of
405:HamiltonâJacobiâEinstein
385:Einstein field equations
208:Mathematical formulation
4065:Galilean transformation
4056:Principle of relativity
3605:FRW cosmological models
3591:and which arise as the
3581:homogeneous plane waves
3486:exact solutions of the
2622:. Physically speaking,
2133:traveling at the local
1614:Physical interpretation
1595:of pp-wave spacetimes.
1335:{\displaystyle v=v_{0}}
1002:Mathematical definition
946:gravitational radiation
891:gravitational radiation
61:more precise citations.
4150:Lorentz transformation
3892:Bum-Hoon Lee (2005), "
3849:10.1098/rspa.1926.0051
3572:strong nonscalar null
3557:
3421:or perhaps well-known
3332:
3292:
3221:
3086:
2984:
2924:
2859:
2752:
2719:
2692:
2666:
2642:
2616:
2596:
2561:
2401:
2374:
2354:
2298:
2271:
2241:
2199:
2164:
2091:
1969:
1883:
1800:
1757:
1687:
1657:
1581:
1529:
1399:
1360:
1336:
1299:
1263:
1231:
1207:
1165:
1142:
275:Gravitational redshift
168:
4618:WeylâLewisâPapapetrou
4359:Raychaudhuri equation
4298:Equivalence principle
3828:Proc. R. Soc. Lond. A
3677:Hall, Graham (2004).
3574:curvature singularity
3558:
3556:{\displaystyle S^{3}}
3465:plane wave spacetimes
3463:Explicit examples of
3434:axisymmetric pp-waves
3432:Explicit examples of
3387:gauge transformations
3333:
3293:
3222:
3161:has pointed out, all
3125:KaluzaâKlein theories
3087:
3021:plane wave spacetimes
3006:axisymmetric pp-waves
2999:colliding plane waves
2985:
2925:
2860:
2753:
2720:
2693:
2667:
2643:
2617:
2597:
2595:{\displaystyle a,b,c}
2562:
2402:
2385:plane wave spacetimes
2375:
2355:
2325:Killing vector fields
2299:
2272:
2242:
2200:
2165:
2092:
1970:
1884:
1801:
1758:
1688:
1658:
1582:
1530:
1400:
1361:
1337:
1300:
1264:
1232:
1208:
1166:
1143:
1020:Brinkmann coordinates
981:Weyl-Lewis-Papapetrou
931:electromagnetic field
919:plane wave spacetimes
563:WeylâLewisâPapapetrou
518:KerrâNewmanâde Sitter
338:EinsteinâRosen bridge
270:Gravitational lensing
226:Equivalence principle
169:
4659:Numerical relativity
4500:pulsar timing arrays
3995:Pp-wave on arxiv.org
3932:Phys. Z. Sowjetunion
3540:
3507:colliding plane wave
3446:impulsive plane wave
3419:elementary functions
3302:
3262:
3255:linearly superimpose
3204:
3176:has shown that such
3050:
2935:
2875:
2765:
2729:
2709:
2676:
2656:
2626:
2606:
2574:
2414:
2391:
2364:
2331:
2281:
2255:
2231:
2189:
2170:plays the role of a
2141:
1985:
1899:
1811:
1768:
1700:
1670:
1637:
1539:
1416:
1370:
1350:
1313:
1276:
1244:
1221:
1215:covariant derivative
1197:
1188:covariantly constant
1155:
1029:
493:EinsteinâRosen waves
219:Fundamental concepts
108:
4551:Friedmann equations
4445:HulseâTaylor binary
4407:Gravitational waves
4303:Riemannian geometry
4129:Proper acceleration
4114:Maxwell's equations
4060:Galilean relativity
3965:2015EPJC...75..478C
3840:1926RSPSA.111...95B
3795:1965RvMP...37..215P
3758:1974IJTP...11..353T
3746:Int. J. Theor. Phys
3739:. pp. 271â275.
3391:parallel wavefronts
3298:respectively, then
3150:theory of gravity.
3141:Hans-JĂŒrgen Schmidt
3135:has shown that the
3094:Minkowski spacetime
2691:{\displaystyle c=0}
2641:{\displaystyle a,b}
2315:. This is always a
2270:{\displaystyle x,y}
1184:Lorentzian manifold
1012:Lorentzian manifold
447:KaluzaâKlein theory
333:Minkowski spacetime
285:Gravitational waves
4600:ReissnerâNordström
4518:BransâDicke theory
4349:Linearized gravity
4176:Length contraction
4094:Frame of reference
4071:Special relativity
3901:Int. J. Mod. Phys.
3871:10.1007/BF01208647
3814:. pp. 49â101.
3766:10.1007/BF01808090
3618:Gravitational wave
3595:of an appropriate
3553:
3403:special relativity
3364:vanish identically
3328:
3288:
3217:
3117:BransâDicke theory
3082:
2980:
2920:
2855:
2748:
2715:
2688:
2662:
2650:polarization modes
2638:
2612:
2592:
2557:
2397:
2370:
2350:
2294:
2267:
2237:
2195:
2160:
2102:null dust solution
2087:
1965:
1879:
1796:
1753:
1683:
1653:
1577:
1525:
1395:
1356:
1332:
1295:
1259:
1227:
1203:
1161:
1138:
834:pp-wave spacetimes
830:general relativity
776:Physics portal
548:OppenheimerâSnyder
488:ReissnerâNordström
380:Linearized gravity
328:Spacetime diagrams
231:Special relativity
164:
93:General relativity
18:Pp-wave spacetimes
4853:
4852:
4667:
4666:
4646:OzsvĂĄthâSchĂŒcking
4252:
4251:
4234:Minkowski diagram
4191:Thomas precession
4134:Relativistic mass
3914:J. Franklin Inst.
3511:locally isometric
3505:of the resulting
3427:Mathieu functions
3423:special functions
2947:
2887:
2787:
2718:{\displaystyle X}
2665:{\displaystyle c}
2615:{\displaystyle u}
2400:{\displaystyle H}
2373:{\displaystyle H}
2249:harmonic function
2240:{\displaystyle H}
2198:{\displaystyle k}
2009:
1923:
1839:
1838:
1823:
1780:
1712:
1359:{\displaystyle k}
1307:coordinate vector
1230:{\displaystyle k}
1206:{\displaystyle k}
1164:{\displaystyle H}
1008:pp-wave spacetime
826:
825:
459:
458:
345:
344:
87:
86:
79:
16:(Redirected from
4873:
4845:
4844:
4628:van Stockum dust
4400:Two-body problem
4318:Mach's principle
4265:
4206:Terrell rotation
4046:
4024:
4017:
4010:
4001:
3984:
3958:
3882:
3853:
3851:
3815:
3806:
3777:
3740:
3731:
3717:See Section 24.5
3715:
3692:
3673:
3671:
3669:
3660:
3642:
3639:
3585:SG11 plane waves
3562:
3560:
3559:
3554:
3552:
3551:
3503:interaction zone
3337:
3335:
3334:
3329:
3327:
3326:
3314:
3313:
3297:
3295:
3294:
3289:
3287:
3286:
3274:
3273:
3226:
3224:
3223:
3218:
3216:
3215:
3091:
3089:
3088:
3083:
3081:
3080:
3062:
3061:
2989:
2987:
2986:
2981:
2949:
2948:
2940:
2929:
2927:
2926:
2921:
2889:
2888:
2880:
2864:
2862:
2861:
2856:
2854:
2853:
2837:
2836:
2820:
2819:
2788:
2786:
2775:
2757:
2755:
2754:
2749:
2747:
2746:
2724:
2722:
2721:
2716:
2697:
2695:
2694:
2689:
2671:
2669:
2668:
2663:
2647:
2645:
2644:
2639:
2621:
2619:
2618:
2613:
2601:
2599:
2598:
2593:
2566:
2564:
2563:
2558:
2553:
2552:
2540:
2539:
2482:
2481:
2469:
2468:
2406:
2404:
2403:
2398:
2379:
2377:
2376:
2371:
2359:
2357:
2356:
2351:
2349:
2348:
2303:
2301:
2300:
2295:
2293:
2292:
2276:
2274:
2273:
2268:
2246:
2244:
2243:
2238:
2204:
2202:
2201:
2196:
2169:
2167:
2166:
2161:
2159:
2158:
2096:
2094:
2093:
2088:
2083:
2079:
2078:
2077:
2055:
2051:
2050:
2049:
2034:
2033:
2010:
2002:
1997:
1996:
1974:
1972:
1971:
1966:
1964:
1960:
1959:
1958:
1943:
1942:
1924:
1916:
1911:
1910:
1888:
1886:
1885:
1880:
1878:
1874:
1873:
1872:
1856:
1855:
1840:
1834:
1830:
1825:
1824:
1816:
1805:
1803:
1802:
1797:
1795:
1794:
1782:
1781:
1773:
1762:
1760:
1759:
1754:
1752:
1751:
1738:
1727:
1726:
1714:
1713:
1705:
1692:
1690:
1689:
1684:
1682:
1681:
1662:
1660:
1659:
1654:
1652:
1651:
1632:Ricci-NP scalars
1586:
1584:
1583:
1578:
1570:
1569:
1554:
1553:
1534:
1532:
1531:
1526:
1524:
1523:
1508:
1507:
1472:
1471:
1431:
1430:
1404:
1402:
1401:
1396:
1388:
1387:
1365:
1363:
1362:
1357:
1344:index-gymnastics
1341:
1339:
1338:
1333:
1331:
1330:
1304:
1302:
1301:
1296:
1294:
1293:
1268:
1266:
1265:
1260:
1236:
1234:
1233:
1228:
1212:
1210:
1209:
1204:
1170:
1168:
1167:
1162:
1147:
1145:
1144:
1139:
1137:
1136:
1121:
1120:
1085:
1084:
1044:
1043:
950:vacuum solutions
939:Maxwell's theory
927:electromagnetism
818:
811:
804:
791:
786:
785:
778:
774:
773:
558:van Stockum dust
543:RobertsonâWalker
369:
259:
173:
171:
170:
165:
163:
162:
150:
142:
141:
123:
122:
103:
89:
82:
75:
71:
68:
62:
57:this article by
48:inline citations
35:
34:
27:
21:
4881:
4880:
4876:
4875:
4874:
4872:
4871:
4870:
4856:
4855:
4854:
4849:
4835:
4663:
4567:BKL singularity
4557:LemaĂźtreâTolman
4532:
4528:Quantum gravity
4510:
4504:
4490:geodetic effect
4464:(together with
4434:LISA Pathfinder
4373:
4322:
4308:Penrose diagram
4290:
4284:
4259:
4248:
4244:Minkowski space
4210:
4154:
4138:
4086:
4080:
4040:
4033:
4028:
3991:
3943:Eur. Phys. J. C
3940:
3910:Albert Einstein
3856:
3821:
3817:See Section 2-5
3809:
3780:
3743:
3734:
3721:
3712:
3695:
3689:
3676:
3667:
3665:
3658:
3654:
3651:
3646:
3645:
3640:
3636:
3631:
3614:
3543:
3538:
3537:
3513:to part of the
3411:
3395:collide head-on
3318:
3305:
3300:
3299:
3278:
3265:
3260:
3259:
3207:
3202:
3201:
3190:
3155:quantum gravity
3145:two-dimensional
3133:B. O. J. Tupper
3102:
3072:
3053:
3048:
3047:
3014:SG2 plane waves
2933:
2932:
2873:
2872:
2845:
2828:
2811:
2779:
2763:
2762:
2738:
2727:
2726:
2707:
2706:
2674:
2673:
2654:
2653:
2624:
2623:
2604:
2603:
2572:
2571:
2544:
2531:
2473:
2460:
2412:
2411:
2389:
2388:
2362:
2361:
2340:
2329:
2328:
2313:self-isometries
2309:autometry group
2284:
2279:
2278:
2253:
2252:
2229:
2228:
2227:if and only if
2225:vacuum solution
2215:null congruence
2207:optical scalars
2187:
2186:
2180:
2150:
2139:
2138:
2066:
2038:
2022:
2021:
2017:
2016:
2012:
1988:
1983:
1982:
1947:
1931:
1930:
1926:
1902:
1897:
1896:
1864:
1847:
1846:
1842:
1809:
1808:
1786:
1766:
1765:
1743:
1718:
1698:
1697:
1673:
1668:
1667:
1665:Weyl-NP scalars
1640:
1635:
1634:
1624:Einstein tensor
1616:
1558:
1542:
1537:
1536:
1515:
1499:
1463:
1422:
1414:
1413:
1373:
1368:
1367:
1366:can be written
1348:
1347:
1322:
1311:
1310:
1285:
1274:
1273:
1242:
1241:
1219:
1218:
1195:
1194:
1186:which admits a
1180:coordinate-free
1173:smooth function
1153:
1152:
1128:
1112:
1076:
1035:
1027:
1026:
1004:
992:Albert Einstein
870:
842:exact solutions
822:
781:
768:
767:
760:
759:
583:
582:
573:
572:
528:LemaĂźtreâTolman
473:
472:
461:
460:
452:Quantum gravity
439:Advanced theory
366:
365:
364:
347:
346:
295:Geodetic effect
256:
255:
246:
245:
221:
220:
204:
174:
151:
130:
111:
106:
105:
83:
72:
66:
63:
53:Please help to
52:
36:
32:
23:
22:
15:
12:
11:
5:
4879:
4877:
4869:
4868:
4858:
4857:
4851:
4850:
4840:
4837:
4836:
4834:
4833:
4826:
4821:
4816:
4811:
4806:
4801:
4796:
4791:
4786:
4781:
4776:
4771:
4766:
4761:
4756:
4754:Choquet-Bruhat
4751:
4746:
4741:
4736:
4731:
4726:
4721:
4716:
4711:
4706:
4701:
4696:
4691:
4686:
4681:
4675:
4673:
4669:
4668:
4665:
4664:
4662:
4661:
4654:
4653:
4648:
4643:
4636:
4635:
4630:
4625:
4620:
4615:
4606:Axisymmetric:
4603:
4602:
4597:
4591:
4580:
4579:
4574:
4569:
4564:
4559:
4554:
4545:Cosmological:
4542:
4540:
4534:
4533:
4531:
4530:
4525:
4520:
4514:
4512:
4506:
4505:
4503:
4502:
4497:
4486:frame-dragging
4483:
4478:
4473:
4470:Einstein rings
4466:Einstein cross
4459:
4448:
4447:
4442:
4436:
4431:
4426:
4413:
4403:
4402:
4397:
4392:
4387:
4381:
4379:
4375:
4374:
4372:
4371:
4369:Ernst equation
4366:
4361:
4356:
4351:
4346:
4341:
4339:BSSN formalism
4336:
4330:
4328:
4324:
4323:
4321:
4320:
4315:
4310:
4305:
4300:
4294:
4292:
4286:
4285:
4283:
4282:
4277:
4271:
4269:
4262:
4254:
4253:
4250:
4249:
4247:
4246:
4241:
4236:
4231:
4226:
4220:
4218:
4212:
4211:
4209:
4208:
4203:
4198:
4196:Ladder paradox
4193:
4188:
4183:
4178:
4173:
4168:
4162:
4160:
4156:
4155:
4153:
4152:
4146:
4144:
4140:
4139:
4137:
4136:
4131:
4126:
4121:
4116:
4111:
4106:
4101:
4099:Speed of light
4096:
4090:
4088:
4082:
4081:
4079:
4078:
4073:
4068:
4062:
4052:
4050:
4043:
4035:
4034:
4029:
4027:
4026:
4019:
4012:
4004:
3998:
3997:
3990:
3989:External links
3987:
3986:
3985:
3938:
3921:
3920:
3907:
3897:
3890:
3883:
3854:
3819:
3807:
3789:(1): 215â220.
3783:Rev. Mod. Phys
3778:
3752:(5): 353â356.
3741:
3732:
3724:Gen. Rel. Grav
3719:
3710:
3693:
3687:
3674:
3650:
3647:
3644:
3643:
3633:
3632:
3630:
3627:
3626:
3625:
3620:
3613:
3610:
3609:
3608:
3593:Penrose limits
3578:
3564:
3550:
3546:
3530:
3495:
3484:
3461:
3460:
3453:
3410:
3407:
3401:as treated in
3368:VSI spacetimes
3361:Riemann tensor
3325:
3321:
3317:
3312:
3308:
3285:
3281:
3277:
3272:
3268:
3214:
3210:
3189:
3186:
3148:metric-dilaton
3101:
3098:
3079:
3075:
3071:
3068:
3065:
3060:
3056:
3044:sandwich waves
2991:
2990:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2946:
2943:
2930:
2919:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2886:
2883:
2866:
2865:
2852:
2848:
2843:
2840:
2835:
2831:
2826:
2823:
2818:
2814:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2785:
2782:
2778:
2773:
2770:
2745:
2741:
2737:
2734:
2714:
2687:
2684:
2681:
2661:
2637:
2634:
2631:
2611:
2591:
2588:
2585:
2582:
2579:
2568:
2567:
2556:
2551:
2547:
2543:
2538:
2534:
2530:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2504:
2501:
2498:
2495:
2491:
2488:
2485:
2480:
2476:
2472:
2467:
2463:
2459:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2396:
2369:
2347:
2343:
2339:
2336:
2311:, or group of
2291:
2287:
2266:
2263:
2260:
2236:
2194:
2179:
2176:
2157:
2153:
2149:
2146:
2135:speed of light
2098:
2097:
2086:
2082:
2076:
2073:
2069:
2064:
2061:
2058:
2054:
2048:
2045:
2041:
2037:
2032:
2029:
2025:
2020:
2015:
2008:
2005:
2000:
1995:
1991:
1976:
1975:
1963:
1957:
1954:
1950:
1946:
1941:
1938:
1934:
1929:
1922:
1919:
1914:
1909:
1905:
1890:
1889:
1877:
1871:
1867:
1862:
1859:
1854:
1850:
1845:
1837:
1833:
1828:
1822:
1819:
1806:
1793:
1789:
1785:
1779:
1776:
1763:
1750:
1746:
1741:
1737:
1733:
1730:
1725:
1721:
1717:
1711:
1708:
1680:
1676:
1650:
1647:
1643:
1630:such that the
1615:
1612:
1611:
1610:
1603:
1576:
1573:
1568:
1565:
1561:
1557:
1552:
1549:
1545:
1522:
1518:
1514:
1511:
1506:
1502:
1498:
1495:
1492:
1489:
1485:
1482:
1478:
1475:
1470:
1466:
1462:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1429:
1425:
1421:
1394:
1391:
1386:
1383:
1380:
1376:
1355:
1329:
1325:
1321:
1318:
1292:
1288:
1284:
1281:
1270:
1269:
1258:
1255:
1252:
1249:
1226:
1202:
1160:
1149:
1148:
1135:
1131:
1127:
1124:
1119:
1115:
1111:
1108:
1105:
1102:
1098:
1095:
1091:
1088:
1083:
1079:
1075:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1042:
1038:
1034:
1022:, in the form
1003:
1000:
988:Hans Brinkmann
908:
907:
901:
894:
888:
878:speed of light
876:moving at the
869:
866:
862:Wolfgang Kundt
824:
823:
821:
820:
813:
806:
798:
795:
794:
793:
792:
779:
762:
761:
758:
757:
750:
745:
740:
735:
730:
725:
720:
715:
710:
705:
700:
695:
690:
685:
680:
675:
670:
665:
660:
655:
650:
645:
640:
635:
630:
625:
620:
615:
610:
605:
600:
595:
590:
584:
580:
579:
578:
575:
574:
571:
570:
565:
560:
555:
550:
545:
540:
535:
530:
525:
520:
515:
510:
505:
500:
495:
490:
485:
474:
468:
467:
466:
463:
462:
457:
456:
455:
454:
449:
441:
440:
436:
435:
434:
433:
431:Post-Newtonian
428:
423:
415:
414:
410:
409:
408:
407:
402:
397:
392:
387:
382:
374:
373:
367:
363:
362:
359:
355:
354:
353:
352:
349:
348:
343:
342:
341:
340:
335:
330:
322:
321:
315:
314:
313:
312:
307:
302:
297:
292:
290:Frame-dragging
287:
282:
277:
272:
267:
265:Kepler problem
257:
253:
252:
251:
248:
247:
244:
243:
238:
233:
228:
222:
218:
217:
216:
213:
212:
211:
210:
205:
203:
202:
197:
192:
186:
184:
176:
175:
161:
158:
154:
149:
145:
140:
137:
133:
129:
126:
121:
118:
114:
104:
96:
95:
85:
84:
39:
37:
30:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4878:
4867:
4864:
4863:
4861:
4848:
4838:
4832:
4831:
4827:
4825:
4822:
4820:
4817:
4815:
4812:
4810:
4807:
4805:
4802:
4800:
4797:
4795:
4792:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4772:
4770:
4767:
4765:
4762:
4760:
4757:
4755:
4752:
4750:
4747:
4745:
4742:
4740:
4739:Chandrasekhar
4737:
4735:
4732:
4730:
4727:
4725:
4722:
4720:
4717:
4715:
4712:
4710:
4707:
4705:
4702:
4700:
4699:Schwarzschild
4697:
4695:
4692:
4690:
4687:
4685:
4682:
4680:
4677:
4676:
4674:
4670:
4660:
4656:
4655:
4652:
4649:
4647:
4644:
4642:
4638:
4637:
4634:
4631:
4629:
4626:
4624:
4621:
4619:
4616:
4613:
4609:
4605:
4604:
4601:
4598:
4595:
4592:
4590:
4586:
4585:Schwarzschild
4582:
4581:
4578:
4575:
4573:
4570:
4568:
4565:
4563:
4560:
4558:
4555:
4552:
4548:
4544:
4543:
4541:
4539:
4535:
4529:
4526:
4524:
4521:
4519:
4516:
4515:
4513:
4507:
4501:
4498:
4495:
4491:
4487:
4484:
4482:
4481:Shapiro delay
4479:
4477:
4474:
4471:
4467:
4463:
4460:
4457:
4453:
4450:
4449:
4446:
4443:
4440:
4437:
4435:
4432:
4430:
4427:
4425:
4424:collaboration
4421:
4417:
4414:
4412:
4408:
4405:
4404:
4401:
4398:
4396:
4393:
4391:
4390:Event horizon
4388:
4386:
4383:
4382:
4380:
4376:
4370:
4367:
4365:
4362:
4360:
4357:
4355:
4352:
4350:
4347:
4345:
4342:
4340:
4337:
4335:
4334:ADM formalism
4332:
4331:
4329:
4325:
4319:
4316:
4314:
4311:
4309:
4306:
4304:
4301:
4299:
4296:
4295:
4293:
4287:
4281:
4278:
4276:
4273:
4272:
4270:
4266:
4263:
4261:
4255:
4245:
4242:
4240:
4239:Biquaternions
4237:
4235:
4232:
4230:
4227:
4225:
4222:
4221:
4219:
4217:
4213:
4207:
4204:
4202:
4199:
4197:
4194:
4192:
4189:
4187:
4184:
4182:
4179:
4177:
4174:
4172:
4169:
4167:
4166:Time dilation
4164:
4163:
4161:
4157:
4151:
4148:
4147:
4145:
4141:
4135:
4132:
4130:
4127:
4125:
4122:
4120:
4119:Proper length
4117:
4115:
4112:
4110:
4107:
4105:
4102:
4100:
4097:
4095:
4092:
4091:
4089:
4083:
4077:
4074:
4072:
4069:
4066:
4063:
4061:
4057:
4054:
4053:
4051:
4047:
4044:
4042:
4036:
4032:
4025:
4020:
4018:
4013:
4011:
4006:
4005:
4002:
3996:
3993:
3992:
3988:
3982:
3978:
3974:
3970:
3966:
3962:
3957:
3952:
3948:
3944:
3939:
3936:
3933:
3929:
3926:
3925:
3924:
3918:
3915:
3911:
3908:
3905:
3902:
3898:
3895:
3891:
3888:
3884:
3880:
3876:
3872:
3868:
3864:
3860:
3855:
3850:
3845:
3841:
3837:
3833:
3829:
3825:
3820:
3818:
3813:
3808:
3804:
3800:
3796:
3792:
3788:
3784:
3779:
3775:
3771:
3767:
3763:
3759:
3755:
3751:
3747:
3742:
3738:
3733:
3729:
3725:
3720:
3718:
3713:
3711:0-521-46136-7
3707:
3703:
3700:. Cambridge:
3699:
3694:
3690:
3688:981-02-1051-5
3684:
3680:
3675:
3664:
3657:
3653:
3652:
3648:
3638:
3635:
3628:
3624:
3621:
3619:
3616:
3615:
3611:
3606:
3602:
3598:
3597:null geodesic
3594:
3590:
3586:
3582:
3579:
3576:
3575:
3569:
3568:wave of death
3565:
3548:
3544:
3535:
3531:
3528:
3527:event horizon
3524:
3520:
3516:
3512:
3508:
3504:
3500:
3496:
3493:
3489:
3488:gravitational
3485:
3482:
3478:
3474:
3470:
3469:
3468:
3466:
3458:
3454:
3451:
3447:
3443:
3439:
3438:
3437:
3435:
3430:
3428:
3424:
3420:
3416:
3408:
3406:
3404:
3400:
3396:
3392:
3388:
3384:
3379:
3375:
3373:
3369:
3365:
3362:
3358:
3353:
3351:
3350:Penrose limit
3347:
3343:
3342:Roger Penrose
3339:
3323:
3319:
3315:
3310:
3306:
3283:
3279:
3275:
3270:
3266:
3256:
3252:
3247:
3245:
3239:
3237:
3234:
3230:
3212:
3198:
3193:
3187:
3185:
3183:
3179:
3175:
3170:
3168:
3164:
3160:
3157:, because as
3156:
3151:
3149:
3146:
3142:
3138:
3134:
3130:
3126:
3122:
3118:
3114:
3110:
3107:
3099:
3097:
3095:
3077:
3073:
3069:
3066:
3063:
3058:
3054:
3045:
3040:
3037:
3033:
3029:
3024:
3022:
3017:
3015:
3011:
3007:
3002:
3000:
2996:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2944:
2941:
2931:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2884:
2881:
2871:
2870:
2869:
2850:
2841:
2838:
2833:
2824:
2821:
2816:
2804:
2801:
2798:
2795:
2792:
2783:
2771:
2768:
2761:
2760:
2759:
2743:
2735:
2732:
2712:
2703:
2701:
2685:
2682:
2679:
2659:
2651:
2635:
2632:
2629:
2609:
2589:
2586:
2583:
2580:
2577:
2549:
2545:
2541:
2536:
2532:
2521:
2515:
2512:
2509:
2506:
2499:
2493:
2489:
2486:
2478:
2474:
2470:
2465:
2461:
2450:
2444:
2441:
2435:
2432:
2429:
2426:
2423:
2417:
2410:
2409:
2408:
2394:
2386:
2381:
2367:
2345:
2337:
2334:
2326:
2322:
2318:
2314:
2310:
2305:
2289:
2264:
2261:
2258:
2250:
2234:
2226:
2221:
2218:
2216:
2212:
2208:
2192:
2183:
2177:
2175:
2173:
2155:
2147:
2144:
2136:
2132:
2129:
2125:
2120:
2118:
2114:
2111:
2107:
2104:. Also, the
2103:
2084:
2080:
2074:
2071:
2067:
2062:
2059:
2056:
2052:
2046:
2043:
2039:
2035:
2030:
2027:
2023:
2018:
2013:
2006:
2003:
1998:
1993:
1981:
1980:
1979:
1961:
1955:
1952:
1948:
1944:
1939:
1936:
1932:
1927:
1920:
1917:
1912:
1907:
1895:
1894:
1893:
1875:
1869:
1860:
1857:
1852:
1843:
1835:
1831:
1826:
1817:
1807:
1791:
1783:
1774:
1764:
1748:
1739:
1735:
1731:
1728:
1723:
1715:
1706:
1696:
1695:
1694:
1678:
1666:
1648:
1645:
1633:
1629:
1625:
1621:
1613:
1608:
1604:
1601:
1600:
1599:
1596:
1594:
1590:
1574:
1571:
1566:
1563:
1559:
1555:
1550:
1547:
1543:
1520:
1516:
1512:
1509:
1504:
1500:
1496:
1493:
1490:
1487:
1483:
1480:
1476:
1473:
1468:
1464:
1460:
1453:
1450:
1447:
1444:
1441:
1435:
1432:
1427:
1423:
1419:
1411:
1406:
1392:
1389:
1384:
1381:
1378:
1374:
1353:
1345:
1327:
1323:
1319:
1316:
1308:
1290:
1282:
1279:
1256:
1253:
1250:
1240:
1239:
1238:
1224:
1216:
1200:
1192:
1189:
1185:
1181:
1176:
1174:
1158:
1133:
1129:
1125:
1122:
1117:
1113:
1109:
1106:
1103:
1100:
1096:
1093:
1089:
1086:
1081:
1077:
1073:
1066:
1063:
1060:
1057:
1054:
1048:
1045:
1040:
1036:
1032:
1025:
1024:
1023:
1021:
1017:
1016:metric tensor
1013:
1009:
1001:
999:
997:
993:
989:
984:
982:
976:
974:
973:axi-symmetric
970:
966:
962:
957:
955:
951:
947:
942:
940:
936:
932:
928:
924:
920:
915:
913:
905:
902:
899:
898:Weyl fermions
895:
892:
889:
886:
883:
882:
881:
879:
875:
867:
865:
863:
859:
858:JĂŒrgen Ehlers
855:
851:
847:
843:
839:
835:
831:
819:
814:
812:
807:
805:
800:
799:
797:
796:
790:
780:
777:
772:
766:
765:
764:
763:
756:
755:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
699:
696:
694:
693:Chandrasekhar
691:
689:
686:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
629:
626:
624:
621:
619:
616:
614:
611:
609:
608:Schwarzschild
606:
604:
601:
599:
596:
594:
591:
589:
586:
585:
577:
576:
569:
568:HartleâThorne
566:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
539:
536:
534:
531:
529:
526:
524:
521:
519:
516:
514:
511:
509:
506:
504:
501:
499:
496:
494:
491:
489:
486:
483:
479:
478:Schwarzschild
476:
475:
471:
465:
464:
453:
450:
448:
445:
444:
443:
442:
437:
432:
429:
427:
424:
422:
419:
418:
417:
416:
411:
406:
403:
401:
398:
396:
393:
391:
388:
386:
383:
381:
378:
377:
376:
375:
370:
360:
357:
356:
351:
350:
339:
336:
334:
331:
329:
326:
325:
324:
323:
320:
316:
311:
308:
306:
303:
301:
300:Event horizon
298:
296:
293:
291:
288:
286:
283:
281:
278:
276:
273:
271:
268:
266:
263:
262:
261:
260:
250:
249:
242:
239:
237:
234:
232:
229:
227:
224:
223:
215:
214:
209:
206:
201:
198:
196:
193:
191:
188:
187:
185:
183:
180:
179:
178:
177:
159:
156:
152:
147:
143:
138:
135:
131:
124:
119:
116:
112:
102:
98:
97:
94:
90:
81:
78:
70:
60:
56:
50:
49:
43:
38:
29:
28:
19:
4829:
4640:
4523:KaluzaâKlein
4275:Introduction
4201:Twin paradox
3946:
3942:
3934:
3931:
3928:Nathan Rosen
3922:
3916:
3913:
3903:
3900:
3862:
3858:
3831:
3827:
3816:
3811:
3786:
3782:
3749:
3745:
3736:
3730:: 1129â1243.
3727:
3723:
3716:
3697:
3678:
3666:. Retrieved
3663:J. D. Steele
3662:
3637:
3588:
3584:
3571:
3522:
3514:
3502:
3492:Weyl fermion
3464:
3462:
3433:
3431:
3414:
3412:
3394:
3390:
3380:
3376:
3363:
3354:
3345:
3340:
3248:
3240:
3235:
3228:
3196:
3194:
3191:
3177:
3171:
3159:Gary Gibbons
3152:
3147:
3144:
3136:
3129:J. W. Moffat
3106:relativistic
3103:
3096:background.
3041:
3036:special case
3035:
3027:
3025:
3018:
3013:
3005:
3003:
2994:
2992:
2867:
2725:, including
2704:
2569:
2382:
2321:Lie algebras
2312:
2306:
2222:
2219:
2184:
2181:
2127:
2123:
2121:
2117:Bel criteria
2112:
2099:
1977:
1891:
1617:
1597:
1592:
1409:
1407:
1343:
1271:
1187:
1177:
1150:
1007:
1005:
996:Nathan Rosen
985:
977:
964:
960:
958:
945:
943:
934:
916:
911:
909:
903:
871:
853:
849:
837:
833:
827:
753:
713:Raychaudhuri
552:
182:Introduction
73:
64:
45:
4612:KerrâNewman
4583:Spherical:
4452:Other tests
4395:Singularity
4327:Formulation
4289:Fundamental
4143:Formulation
4124:Proper time
4085:Fundamental
3949:(10): 478.
3865:: 119â145.
3834:(757): 95.
3490:field of a
3457:Bonnor beam
3399:plane waves
3113:gravitation
2211:Kundt class
2172:wave vector
2110:Petrov type
2108:always has
2106:Weyl tensor
1191:null vector
923:plane waves
914:direction.
852:stands for
848:. The term
728:van Stockum
658:Oppenheimer
513:KerrâNewman
305:Singularity
59:introducing
4764:Zel'dovich
4672:Scientists
4651:Alcubierre
4458:of Mercury
4456:precession
4385:Black hole
4268:Background
4260:relativity
4229:World line
4224:Light cone
4049:Background
4041:relativity
4031:Relativity
3956:1507.06439
3649:References
3174:C. M. Hull
3167:tree-level
3131:. Indeed,
3119:, various
1342:. In the
581:Scientists
413:Formalisms
361:Formalisms
310:Black hole
236:World line
42:references
4734:Robertson
4719:Friedmann
4714:Eddington
4704:de Sitter
4538:Solutions
4416:detectors
4411:astronomy
4378:Phenomena
4313:Geodesics
4216:Spacetime
4159:Phenomena
3981:119618017
3879:121619009
3859:Math. Ann
3774:122456995
3251:nonlinear
3233:polarized
3229:departing
3209:∂
3163:loop term
3032:self-dual
2969:−
2960:−
2945:¨
2912:−
2894:−
2885:¨
2847:∂
2830:∂
2813:∂
2781:∂
2777:∂
2740:∂
2471:−
2342:∂
2317:Lie group
2286:∂
2152:∂
2131:radiation
2124:classical
2036:−
1990:Ψ
1904:Φ
1866:∂
1849:∂
1821:→
1788:∂
1778:→
1745:∂
1729:−
1720:∂
1710:→
1707:ℓ
1675:Ψ
1642:Φ
1287:∂
1248:∇
998:in 1937.
975:pp-wave.
965:impulsive
874:radiation
673:Robertson
638:Friedmann
633:Eddington
623:Nordström
613:de Sitter
470:Solutions
395:Geodesics
390:Friedmann
372:Equations
358:Equations
319:Spacetime
254:Phenomena
160:ν
157:μ
148:κ
139:ν
136:μ
128:Λ
120:ν
117:μ
67:July 2018
4860:Category
4847:Category
4724:LemaĂźtre
4689:Einstein
4679:Poincaré
4639:Others:
4623:TaubâNUT
4589:interior
4511:theories
4509:Advanced
4476:redshift
4291:concepts
4109:Rapidity
4087:concepts
3668:June 12,
3612:See also
3515:interior
3467:include
3436:include
3425:such as
3415:explicit
3409:Examples
3197:oncoming
2128:massless
1607:bivector
904:massless
868:Overview
838:pp-waves
789:Category
653:LemaĂźtre
618:Reissner
603:Poincaré
588:Einstein
533:TaubâNUT
498:Wormhole
482:interior
195:Timeline
4789:Hawking
4784:Penrose
4769:Novikov
4749:Wheeler
4694:Hilbert
4684:Lorentz
4641:pp-wave
4462:lensing
4258:General
4039:Special
3961:Bibcode
3937:(1937).
3923:43â54.
3919:(1937).
3836:Bibcode
3791:Bibcode
3754:Bibcode
3359:of the
3010:Abelian
1622:of the
1171:is any
1010:is any
708:Hawking
703:Penrose
678:Bardeen
668:Wheeler
598:Hilbert
593:Lorentz
553:pp-wave
190:History
55:improve
4830:others
4819:Thorne
4809:Misner
4794:Taylor
4779:Geroch
4774:Ehlers
4744:Zwicky
4562:Kasner
3979:
3877:
3772:
3708:
3685:
3523:inside
3471:exact
3444:is an
3137:common
2995:planar
2868:where
2570:Here,
1305:, the
1193:field
1151:where
1014:whose
832:, the
787:
754:others
748:Thorne
738:Newman
718:Taylor
698:Ehlers
683:Walker
648:Zwicky
523:Kasner
44:, but
4824:Weiss
4804:Bondi
4799:Hulse
4729:Milne
4633:discs
4577:Milne
4572:Gödel
4429:Virgo
3977:S2CID
3951:arXiv
3875:S2CID
3770:S2CID
3659:(PDF)
3629:Notes
3583:, or
3517:of a
3244:chaos
2247:is a
836:, or
723:Hulse
663:Gödel
643:Milne
538:Milne
503:Gödel
200:Tests
4759:Kerr
4709:Weyl
4608:Kerr
4468:and
4422:and
4420:LIGO
3706:ISBN
3683:ISBN
3670:2005
3603:and
3566:the
3532:the
3525:the
3497:the
3475:and
3455:The
3440:The
3123:and
3070:<
3064:<
2126:and
994:and
912:same
860:and
733:Taub
688:Kerr
628:Weyl
508:Kerr
426:BSSN
4814:Yau
4439:GEO
3969:doi
3917:223
3867:doi
3844:doi
3832:111
3799:doi
3762:doi
3429:.)
3111:of
2323:of
1217:of
961:not
844:of
828:In
743:Yau
421:ADM
4862::
4488:/
4454::
4409::
3975:.
3967:.
3959:.
3947:75
3945:.
3935:12
3904:D7
3873:.
3863:18
3861:.
3842:.
3830:.
3826:.
3797:.
3787:37
3785:.
3768:.
3760:.
3750:11
3748:.
3728:12
3726:.
3704:.
3661:.
3405:.
3352:.
3246:.
3184:.
3001:.
2702:.
2304:.
2174:.
2119:.
1908:00
1405:.
1257:0.
1006:A
956:.
941:.
864:.
850:pp
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4596:)
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3889:"
3881:.
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3838::
3805:.
3801::
3793::
3776:.
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3756::
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3691:.
3672:.
3607:.
3563:,
3549:3
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3324:2
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3276:,
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3067:u
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2805:y
2802:q
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2769:X
2744:v
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2683:=
2680:c
2660:c
2636:b
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2537:2
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2529:(
2525:)
2522:u
2519:(
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2513:+
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2500:u
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2458:(
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2335:k
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2113:N
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1994:0
1962:)
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