65:(beyond the Einstein field equations) for the evolution of the metric tensor. The Einstein field equations alone do not fully determine the evolution of the metric relative to the coordinate system. It might seem that they would since there are ten equations to determine the ten components of the metric. However, due to the second Bianchi identity of the
28:
form. In other words, the description of the world as given by the laws of physics does not depend on our choice of coordinate systems. However, it is often useful to fix upon a particular coordinate system, in order to solve actual problems or make actual predictions. A coordinate condition selects
396:
everywhere. (However, since the
Riemann and hence the Ricci tensor for Minkowski coordinates is identically zero, the Einstein equations give zero energy/matter for Minkowski coordinates; so Minkowski coordinates cannot be an acceptable final answer.) Unlike the harmonic and synchronous coordinate
64:
Naively, one might think that coordinate conditions would take the form of equations for the evolution of the four coordinates, and indeed in some cases (e.g. the harmonic coordinate condition) they can be put in that form. However, it is more usual for them to appear as four additional equations
400:
An example of an under-determinative condition is the algebraic statement that the determinant of the metric tensor is −1, which still leaves considerable gauge freedom. This condition would have to be supplemented by other conditions in order to remove the ambiguity in the metric tensor.
412:
form of the metric. This Kerr-Schild condition goes well beyond removing coordinate ambiguity, and thus also prescribes a type of physical space-time structure. The determinant of the metric tensor in a Kerr-Schild metric is negative one, which by itself is an under-determinative coordinate
391:
Many other coordinate conditions have been employed by physicists, though none as pervasively as those described above. Almost all coordinate conditions used by physicists, including the harmonic and synchronous coordinate conditions, would be satisfied by a metric tensor that equals the
641:
536:
73:
is zero which means that four of the ten equations are redundant, leaving four degrees of freedom which can be associated with the choice of the four coordinates. The same result can be derived from a
Kramers-Moyal-van-Kampen expansion of the
155:
427:, then one should try to choose a coordinate condition which will make the expansion converge as quickly as possible (or at least prevent it from diverging). Similarly, for numerical methods one needs to avoid
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at a surface that is separate from the point-source, but that singularity is merely an artifact of the choice of coordinate conditions, rather than arising from actual physical reality.
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293:
449:
381:
216:
443:, then one gets a theory which is in some sense consistent with both special and general relativity. Among the simplest examples of such coordinate conditions are these:
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An example of an over-determinative condition is the algebraic statement that the difference between the metric tensor and the
Minkowski tensor is simply a
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683:
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The synchronous coordinate condition is neither generally covariant nor
Lorentz covariant. This coordinate condition resolves the ambiguity of the
53:. Thus, coordinate conditions are a type of gauge condition. No coordinate condition is generally covariant, but many coordinate conditions are
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102:
780:
page 302 (Oxford
University Press, 1998). Generalizations of the Kerr-Schild conditions have been suggested; e.g. see Hildebrandt, Sergi.
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When choosing coordinate conditions, it is important to beware of illusions or artifacts that can be created by that choice. For example,
813:
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If one combines a coordinate condition which is
Lorentz covariant, such as the harmonic coordinate condition mentioned above, with the
42:
79:
698:
C.-P. Ma and E. Bertschinger (1995). "Cosmological perturbation theory in the synchronous and conformal
Newtonian gauges".
91:
636:{\displaystyle g^{\alpha \beta }{}_{,\beta }=k\,g^{\mu \nu }{}_{,\gamma }\eta _{\mu \nu }\eta ^{\alpha \gamma }\,.}
754:
409:
440:
424:
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38:
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conditions, some commonly used coordinate conditions may be either under-determinative or over-determinative.
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A particularly useful coordinate condition is the harmonic condition (also known as the "de Donder gauge"):
531:{\displaystyle g_{\alpha \beta ,\gamma }\eta ^{\beta \gamma }=k\,g_{\mu \nu ,\alpha }\eta ^{\mu \nu }\,.}
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Synchronous coordinates are also known as
Gaussian coordinates. They are frequently used in
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If one is going to solve the
Einstein field equations using approximate methods such as the
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Lorentz covariant. This coordinate condition resolves the ambiguity of the metric tensor
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21:
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by providing four additional differential equations that the metric tensor must satisfy.
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equals everywhere at an initial time. This situation is analogous to the failure of the
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to determine the potentials uniquely. In both cases, the ambiguity can be removed by
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Another particularly useful coordinate condition is the synchronous condition:
172:. This harmonic condition is frequently used by physicists when working with
164:(also known as the "affine connection"), and the "g" with superscripts is the
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by providing four algebraic equations that the metric tensor must satisfy.
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Although the harmonic coordinate condition is not generally covariant, it
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150:{\displaystyle 0=\Gamma _{\beta \gamma }^{\alpha }g^{\beta \gamma }\!.}
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do not determine the metric uniquely, even if one knows what the
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176:. This condition is also frequently used to derive the
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820:, page 26 (Institute of Mathematical Sciences 2005).
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811:“Lectures on Introduction to General Relativity”
758:Indian Journal of Pure and Applied Mathematics
796:Exact Solutions of Einstein's Field Equations
782:“Kerr-Schild and Generalized Metric Motions,”
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800:page 485 (Cambridge University Press 2003).
688:page 20 (Cambridge University Press 1990).
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288:{\displaystyle g_{01}=g_{02}=g_{03}=0\!}
836:Coordinate charts in general relativity
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435:Lorentz covariant coordinate conditions
776:The Mathematical Theory of Black Holes
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755:“On a Generalized Peres Space-Time,”
681:Stephani, Hans and Stewart, John.
33:Indeterminacy in general relativity
408:times itself, which is known as a
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82:for decomposing tensor products).
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672:page 391 (World Scientific 1994).
646:where one can fix the constant
668:Selected Papers of Abdus Salam
376:{\displaystyle g_{\mu \nu }\!}
211:{\displaystyle g_{\mu \nu }\!}
1:
92:Harmonic coordinate condition
650:to be any convenient value.
431:(coordinate singularities).
178:post-Newtonian approximation
331:{\displaystyle g_{00}=-1\!}
80:Clebsch–Gordan coefficients
29:such coordinate system(s).
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784:page 22 (Arxiv.org 2002).
760:(1975) citing Moller, C.
762:The Theory of Relativity
441:Einstein field equations
425:post-Newtonian expansion
69:, the divergence of the
67:Riemann curvature tensor
39:Einstein field equations
793:Stephani, Hans et al.
764:(Clarendon Press 1972).
228:Synchronous coordinates
222:Synchronous coordinates
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24:can be expressed in a
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809:Date, Ghanashyam.
773:Chandrasekhar, S.
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174:gravitational waves
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26:generally covariant
816:2011-07-20 at the
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51:gauge fixing
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413:condition.
410:Kerr-Schild
78:(using the
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654:Footnotes
625:γ
622:α
618:η
612:ν
609:μ
605:η
599:γ
587:ν
584:μ
567:β
555:β
552:α
520:ν
517:μ
513:η
507:α
501:ν
498:μ
481:γ
478:β
474:η
468:γ
462:β
459:α
368:ν
365:μ
344:cosmology
322:−
203:ν
200:μ
139:γ
136:β
126:α
121:γ
118:β
114:Γ
830:Category
814:Archived
706:: 7–25.
429:caustics
718:Bibcode
168:of the
166:inverse
736:
20:, the
734:S2CID
708:arXiv
298:and
37:The
726:doi
704:455
57:or
16:In
832::
746:^
732:.
724:.
716:.
702:.
346:.
314:00
274:03
261:02
248:01
185:is
180:.
61:.
798:,
778:,
740:.
728::
720::
710::
686:,
670:,
648:k
631:.
596:,
580:g
575:k
572:=
564:,
548:g
526:.
504:,
494:g
489:k
486:=
465:,
455:g
361:g
338:.
325:1
319:=
310:g
282:0
279:=
270:g
266:=
257:g
253:=
244:g
196:g
145:.
132:g
110:=
107:0
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