36:
552:, so we must expect the gravitational field itself to possess energy, and it does. However, determining the precise location of this gravitational field energy is technically problematical in general relativity, by its very nature of the clean separation into a universal gravitational interaction and "all the rest".
555:
The fact that the gravitational field itself possesses energy yields a way to understand the nonlinearity of the
Einstein field equation: this gravitational field energy itself produces more gravity. (This is described as "the gravity of gravity", or by saying that "gravity gravitates".) This means
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Several of the families mentioned here, members of which are obtained by solving an appropriate linear or nonlinear, real or complex partial differential equation, turn out to be very closely related, in perhaps surprising ways.
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609:(Robert M. Kerns and Walter J. Wild 1982) (a Schwarzschild object immersed in an ambient "almost uniform" gravitational field),
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vanishes. This follows from the fact that these two second rank tensors stand in a kind of dual relationship; they are the
832:. Cambridge monographs on mathematical physics (2nd ed.). Cambridge, UK ; New York: Cambridge University Press.
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603:(a famous counterexample describing the exterior gravitational field of an isolated object with strange properties),
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also vanishes identically, so that no matter or non-gravitational fields are present. These are distinct from the
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in a vacuum region, it might seem that according to general relativity, vacuum regions must contain no
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332:{\displaystyle G_{ab}=R_{ab}-{\frac {R}{2}}\,g_{ab},\;\;R_{ab}=G_{ab}-{\frac {G}{2}}\,g_{ab}}
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645:(An anisotropic solution, used to study gravitational chaos in three or more dimensions).
639:(the circularly polarized sinusoidal gravitational wave, another famous counterexample).
679:(Frederick J. Ernst 1968) (the family of all stationary axisymmetric vacuum solutions),
615:(two Kerr objects sharing the same axis of rotation, but held apart by unphysical zero
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in addition to the gravitational field. Vacuum solutions are also distinct from the
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1925) (the family of all cylindrically symmetric nonrotating vacuum solutions),
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It is a mathematical fact that the
Einstein tensor vanishes if and only if the
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708:(Robert H. Gowdy) (cosmological models constructed using gravitational waves),
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plus terms built out of the Ricci tensor: the Weyl and
Riemann tensors agree,
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in a
Lorentzian manifold is a region in which the Einstein tensor vanishes.
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according to general relativity than it is according to Newton's theory.
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term (and thus, the lambdavacuums can be taken as cosmological models).
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591:(which describes the spacetime geometry around a spherical mass),
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690:) (the family of all cylindrically symmetric vacuum solutions),
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These all belong to one or more general families of solutions:
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619:"cables" going out to suspension points infinitely removed),
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Well-known examples of explicit vacuum solutions include:
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597:(which describes the geometry around a rotating object),
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Vacuum solutions are a special case of the more general
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that the gravitational field outside the Sun is a bit
734:
Introduction to the mathematics of general relativity
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60:. Unsourced material may be challenged and removed.
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420:{\displaystyle R={R^{a}}_{a},\;\;G={G^{a}}_{a}=-R}
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27:Lorentzian manifold with vanishing Einstein tensor
769:Max Planck Institute for Gravitational Physics
716:In addition to these, we also have the vacuum
661:) (the family of all static vacuum solutions),
430:A third equivalent condition follows from the
826:Exact solutions of Einstein's field equations
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120:Learn how and when to remove this message
69:"Vacuum solution" general relativity
782:"Zur Theorie binärer Gravitationsfelder"
148:vanishes identically. According to the
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548:. But the gravitational field can do
857:Exact solutions in general relativity
575:(which describes empty space with no
184:exact solutions in general relativity
7:
58:adding citations to reliable sources
25:
493:{\displaystyle R_{abcd}=C_{abcd}}
34:
45:needs additional citations for
160:, which take into account the
1:
823:Stephani, Hans, ed. (2003).
873:
780:Beck, Guido (1925-12-01).
722:gravitational plane waves
617:active gravitational mass
759:"The gravity of gravity"
637:Oszváth–Schücking vacuum
537:{\displaystyle T^{ab}=0}
436:Riemann curvature tensor
158:electrovacuum solutions
150:Einstein field equation
786:Zeitschrift fĂĽr Physik
757:Markus Pössel (2007),
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166:lambdavacuum solutions
152:, this means that the
577:cosmological constant
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440:Weyl curvature tensor
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190:Equivalent conditions
170:cosmological constant
162:electromagnetic field
720:, which include the
631:colliding plane wave
589:Schwarzschild vacuum
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504:Gravitational energy
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154:stress–energy tensor
54:improve this article
623:Khan–Penrose vacuum
573:Minkowski spacetime
432:Ricci decomposition
142:Lorentzian manifold
798:10.1007/BF01328358
739:Topological defect
718:pp-wave spacetimes
613:double Kerr vacuum
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175:More generally, a
134:general relativity
839:978-0-521-46136-8
607:Kerns–Wild vacuum
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16:(Redirected from
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792:(1): 713–728.
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71: –
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65:Find sources:
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43:This article
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659:Hermann Weyl
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196:Ricci tensor
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110:October 2023
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52:Please help
47:verification
44:
706:Gowdy vacua
677:Ernst vacua
595:Kerr vacuum
583:Milne model
745:References
670:Guido Beck
666:Beck vacua
655:Weyl vacua
342:where the
80:newspapers
806:0044-3328
412:−
303:−
242:−
851:Category
728:See also
564:Examples
558:stronger
817:Sources
633:model),
434:of the
94:scholar
836:
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546:energy
508:Since
344:traces
144:whose
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830:(PDF)
140:is a
101:JSTOR
87:books
834:ISBN
802:ISSN
704:the
693:the
682:the
675:the
664:the
653:the
550:work
346:are
136:, a
73:news
794:doi
132:In
56:by
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796::
697:(
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532:0
529:=
524:b
521:a
517:T
486:d
483:c
480:b
477:a
473:C
469:=
464:d
461:c
458:b
455:a
451:R
415:R
409:=
404:a
397:a
393:G
387:=
384:G
379:,
374:a
367:a
363:R
357:=
354:R
325:b
322:a
318:g
311:2
308:G
298:b
295:a
291:G
287:=
282:b
279:a
275:R
269:,
264:b
261:a
257:g
250:2
247:R
237:b
234:a
230:R
226:=
221:b
218:a
214:G
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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