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| − | Einstein field equations | + | Einstein field equations (EFE), |
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| − | {{#tag:math|R_{\mu \nu} - {1 \over 2}R \, g_{\mu \nu} + \Lambda g_{\mu \nu}= {8 \pi G \over c^4} T_{\mu \nu} }} | + | {{#tag:math|R_{\mu \nu} - {1 \over 2}R \, g_{\mu \nu} + \Lambda g_{\mu \nu}= {8 \pi G \over c^4} T_{\mu \nu} }}, |
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| − | {{#tag:math|R_{\mu \nu} }} Ricci curvature tensor | + | where {{#tag:math|R_{\mu \nu} }} is the Ricci curvature tensor, {{#tag:math|R}} is the scalar curvature, {{#tag:math|g_{\mu \nu} }} is the metric tensor (general relativity), {{#tag:math|Λ }} is the cosmological constant, {{#tag:math|G}} is Newton's gravitational constant, {{#tag:math|c}} is the speed of light in vacuum, and {{#tag:math|T_{\mu \nu} }} is the stress–energy tensor. |
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| − | ==Mathematical form==
| + | The above form of the EFE is the standard established by Gravitation by Misner, Thorne, and Wheeler. The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3): |
| − | {{Spacetime|cTopic=Mathematics}}
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| − | The Einstein field equations (EFE) may be written in the form:<ref>{{cite book |title=Einstein's General Theory of Relativity: With Modern Applications in Cosmology |edition=illustrated |first1=Øyvind |last1=Grøn |first2=Sigbjorn |last2=Hervik |publisher=Springer Science & Business Media |year=2007 |isbn=978-0-387-69200-5 |page=180 |url=https://books.google.com/books?id=IyJhCHAryuUC&pg=PA180}}</ref><ref name="ein"/>
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| − | {{Equation box 1
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| − | |indent =:
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| − | |equation = <math>R_{\mu \nu} - \tfrac{1}{2}R g_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G }{c^4} T_{\mu \nu}</math>
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| − | [[File:EinsteinLeiden4.jpg|{{largethumb}}|EFE on a wall in [[Leiden]] ]]
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| − | where {{mvar|R{{sub|μν}}}} is the [[Ricci curvature|Ricci curvature tensor]], {{mvar|R}} is the [[scalar curvature]], {{mvar|g{{sub|μν}}}} is the [[metric tensor (general relativity)|metric tensor]], {{mvar|Λ}} is the [[cosmological constant]], {{mvar|G}} is [[gravitational constant|Newton's gravitational constant]], {{mvar|c}} is the [[speed of light]] in vacuum, and {{mvar|T{{sub|μν}}}} is the [[stress–energy tensor]].
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| − | The EFE is a tensor equation relating a set of [[symmetric tensor|symmetric 4 × 4 tensors]]. Each tensor has 10 independent components. The four [[Bianchi identities]] reduce the number of independent equations from 10 to 6, leaving the metric with four [[gauge fixing]] [[Degrees of freedom (physics and chemistry)|degrees of freedom]], which correspond to the freedom to choose a coordinate system.
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| − | Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in {{mvar|n}} dimensions.<ref name="Stephani et al">{{cite book | last = Stephani | first = Hans |first2=D. |last2=Kramer |first3=M. |last3=MacCallum |first4=C. |last4=Hoenselaers |first5=E. |last5=Herlt | title = Exact Solutions of Einstein's Field Equations | publisher = [[Cambridge University Press]] | year = 2003 | isbn = 0-521-46136-7 }}</ref> The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when {{mvar|T}} is identically zero) define [[Einstein manifold]]s.
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| − | Despite the simple appearance of the equations they are actually quite complicated. Given a specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor {{mvar|g{{sub|μν}}}}, as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. In fact, when fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic [[partial differential equation]]s.<ref>Alan D. Rendall,“Theorems on Existence and Global Dynamics for the Einstein Equations”,Living Rev. Relativity,8, (2005), 6. [Online Article]: cited [2019-12-10],http://www.livingreviews.org/lrr-2005-6</ref>
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| − | One can write the EFE in a more compact form by defining the [[Einstein tensor]]
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| − | :<math>G_{\mu \nu} = R_{\mu \nu} - \tfrac{1}{2} R g_{\mu \nu},</math>
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| − | which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as
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| − | :<math>G_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}.</math>
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| − | In standard units, each term on the left has units of 1/length<sup>2</sup>. With this choice of [[Einstein constant]] as 8πG/c<sup>4</sup>, then the stress-energy tensor on the right side of the equation must be written with each component in units of energy-density (i.e., energy per volume = pressure).
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| − | Using [[geometrized units]] where {{math|''G'' {{=}} ''c'' {{=}} 1}}, this can be rewritten as
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| − | :<math>G_{\mu \nu} + \Lambda g_{\mu \nu} = 8 \pi T_{\mu \nu}\,.</math>
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| − | The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of equations dictating how matter/energy determines the curvature of spacetime.
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| − | These equations, together with the [[geodesic (general relativity)|geodesic equation]],<ref name="SW1993">{{cite book|last=Weinberg |first=Steven|title=Dreams of a Final Theory: the search for the fundamental laws of nature|year=1993|publisher=Vintage Press|pages=107, 233|isbn=0-09-922391-0}}</ref> which dictates how freely-falling matter moves through space-time, form the core of the [[mathematics of general relativity|mathematical formulation]] of [[general relativity]].
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| − | ===Sign convention===
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| − | The above form of the EFE is the standard established by [[Gravitation (book)|Misner, Thorne, and Wheeler]].{{sfnp|Misner|Thorne|Wheeler|1973|p=501ff}} The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3): | |
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| | :<math> | | :<math> |
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| | :<math>R_{\mu \nu}=[S2]\times [S3] \times {R^\alpha}_{\mu\alpha\nu} </math> | | :<math>R_{\mu \nu}=[S2]\times [S3] \times {R^\alpha}_{\mu\alpha\nu} </math> |
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| − | With these definitions [[Gravitation (book)|Misner, Thorne, and Wheeler]] classify themselves as {{math|(+ + +)}}, whereas Weinberg (1972){{sfnp|Weinberg|1972}} and Peacock (1994){{sfnp|Peacock|1994}} are {{math|(+ − −)}}, Peebles (1980)<ref>{{cite book |last=Peebles |first=Phillip James Edwin |title=The Large-scale Structure of the Universe |location= |publisher=Princeton University Press |year=1980 |isbn=0-691-08239-1 }}</ref> and Efstathiou et al. (1990)<ref>{{cite journal |last=Efstathiou |first=G. |first2=W. J. |last2=Sutherland |first3=S. J. |last3=Maddox |title=The cosmological constant and cold dark matter |journal=[[Nature (journal)|Nature]] |volume=348 |issue=6303 |year=1990 |pages=705 |doi=10.1038/348705a0 }}</ref> are {{math|(− + +)}}, Rindler (1977){{citation needed|date=October 2014}}, Atwater (1974){{citation needed|date=October 2014}}, Collins Martin & Squires (1989)<ref>{{cite book |last=Collins |first=P. D. B. |last2=Martin |first2=A. D. |last3=Squires |first3=E. J. |year=1989 |title=Particle Physics and Cosmology |location=New York |publisher=Wiley |isbn=0-471-60088-1 }}</ref> are {{math|(− + −)}}.
| + | Taking the trace with respect to the metric of both sides of the EFE one gets |
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| − | Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative
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| − | :<math>R_{\mu \nu} - \tfrac{1}{2} R g_{\mu \nu} - \Lambda g_{\mu \nu} = -\frac{8 \pi G}{c^4} T_{\mu \nu}.</math>
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| − | The sign of the (very small) cosmological term would change in both these versions, if the {{math|(+ − − −)}} metric [[sign convention]] is used rather than the MTW {{math|(− + + +)}} metric sign convention adopted here.
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| − | ===Equivalent formulations===
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| − | Taking the [[Scalar curvature#Definition|trace with respect to the metric]] of both sides of the EFE one gets | |
| | :<math>R - \frac{D}{2} R + D \Lambda = \frac{8 \pi G}{c^4} T \,</math> | | :<math>R - \frac{D}{2} R + D \Lambda = \frac{8 \pi G}{c^4} T \,</math> |
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| − | where {{mvar|D}} is the spacetime dimension. This expression can be rewritten as | + | where {{#tag:math|D}} is the spacetime dimension. This expression can be rewritten as |
| | :<math>-R + \frac{D \Lambda}{\frac{D}{2} -1} = \frac{8 \pi G}{c^4} \frac{T}{\frac{D}{2}-1} \,.</math> | | :<math>-R + \frac{D \Lambda}{\frac{D}{2} -1} = \frac{8 \pi G}{c^4} \frac{T}{\frac{D}{2}-1} \,.</math> |
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| − | If one adds {{math|−{{sfrac|1|2}}''g{{sub|μν}}''}} times this to the EFE, one gets the following equivalent "trace-reversed" form | + | If one adds {{#tag:math|−\frac{1}{2} g_{\mu \nu} }} times this to the EFE, one gets the following equivalent "trace-reversed" form |
| | :<math>R_{\mu \nu} - \frac{ \Lambda g_{\mu \nu}}{\frac{D}{2}-1} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \frac{1}{D-2}Tg_{\mu \nu}\right) .\,</math> | | :<math>R_{\mu \nu} - \frac{ \Lambda g_{\mu \nu}}{\frac{D}{2}-1} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \frac{1}{D-2}Tg_{\mu \nu}\right) .\,</math> |
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| − | For example, in {{math|''D'' {{=}} 4}} dimensions this reduces to | + | For example, in {{#tag:math|D = 4 }} dimensions this reduces to |
| | :<math>R_{\mu \nu} - \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \tfrac{1}{2}T\,g_{\mu \nu}\right) .\,</math> | | :<math>R_{\mu \nu} - \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \tfrac{1}{2}T\,g_{\mu \nu}\right) .\,</math> |
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| − | Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace {{mvar|g{{sub|μν}}}} in the expression on the right with the [[Minkowski metric]] without significant loss of accuracy). | + | Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace {{#tag:math|g_{\mu \nu} }} in the expression on the right with the Minkowski metric without significant loss of accuracy). |
Einstein field equations (EFE),
[math]R_{\mu \nu} - {1 \over 2}R \, g_{\mu \nu} + \Lambda g_{\mu \nu}= {8 \pi G \over c^4} T_{\mu \nu} [/math],
where [math]R_{\mu \nu} [/math] is the Ricci curvature tensor, [math]R[/math] is the scalar curvature, [math]g_{\mu \nu} [/math] is the metric tensor (general relativity), [math]Λ [/math] is the cosmological constant, [math]G[/math] is Newton's gravitational constant, [math]c[/math] is the speed of light in vacuum, and [math]T_{\mu \nu} [/math] is the stress–energy tensor.
The above form of the EFE is the standard established by Gravitation by Misner, Thorne, and Wheeler. The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):
- [math]
\begin{align}
g_{\mu \nu} & = [S1] \times \operatorname{diag}(-1,+1,+1,+1) \\[6pt]
{R^\mu}_{\alpha \beta \gamma} & = [S2] \times \left(\Gamma^\mu_{\alpha \gamma,\beta}-\Gamma^\mu_{\alpha \beta,\gamma}+\Gamma^\mu_{\sigma \beta}\Gamma^\sigma_{\gamma \alpha}-\Gamma^\mu_{\sigma \gamma}\Gamma^\sigma_{\beta \alpha}\right) \\[6pt]
G_{\mu \nu} & = [S3] \times \frac{8 \pi G}{c^4} T_{\mu \nu}
\end{align}
[/math]
The third sign above is related to the choice of convention for the Ricci tensor:
- [math]R_{\mu \nu}=[S2]\times [S3] \times {R^\alpha}_{\mu\alpha\nu} [/math]
Taking the trace with respect to the metric of both sides of the EFE one gets
- [math]R - \frac{D}{2} R + D \Lambda = \frac{8 \pi G}{c^4} T \,[/math]
where [math]D[/math] is the spacetime dimension. This expression can be rewritten as
- [math]-R + \frac{D \Lambda}{\frac{D}{2} -1} = \frac{8 \pi G}{c^4} \frac{T}{\frac{D}{2}-1} \,.[/math]
If one adds [math]−\frac{1}{2} g_{\mu \nu} [/math] times this to the EFE, one gets the following equivalent "trace-reversed" form
- [math]R_{\mu \nu} - \frac{ \Lambda g_{\mu \nu}}{\frac{D}{2}-1} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \frac{1}{D-2}Tg_{\mu \nu}\right) .\,[/math]
For example, in [math]D = 4 [/math] dimensions this reduces to
- [math]R_{\mu \nu} - \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \tfrac{1}{2}T\,g_{\mu \nu}\right) .\,[/math]
Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace [math]g_{\mu \nu} [/math] in the expression on the right with the Minkowski metric without significant loss of accuracy).
