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| | {{#tag:math|R_{\mu \nu} }} Ricci curvature tensor | | {{#tag:math|R_{\mu \nu} }} Ricci curvature tensor |
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| − | ==Mathematical form==
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| − | {{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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| − | |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=== | | ===Sign convention=== |
Einstein field equations
[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]
[math]R_{\mu \nu} [/math] Ricci curvature tensor
Sign convention
The above form of the EFE is the standard established by Misner, Thorne, and Wheeler.Template:Sfnp 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]
With these definitions Misner, Thorne, and Wheeler classify themselves as Template:Math, whereas Weinberg (1972)Template:Sfnp and Peacock (1994)Template:Sfnp are Template:Math, Peebles (1980)[1] and Efstathiou et al. (1990)[2] are Template:Math, Rindler (1977)Template:Citation needed, Atwater (1974)Template:Citation needed, Collins Martin & Squires (1989)[3] are Template:Math.
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
- [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]
The sign of the (very small) cosmological term would change in both these versions, if the Template:Math metric sign convention is used rather than the MTW Template:Math metric sign convention adopted here.
Equivalent formulations
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 Template:Mvar 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 Template: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 Template: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 Template:Mvar in the expression on the right with the Minkowski metric without significant loss of accuracy).
