Editing Einstein field equations (section)

== Mathematical form ==
{{Spacetime|cTopic=Relevante mathematics}}
[[File:EinsteinLeiden4.jpg|upright=1.35|thumb|EFE on the wall of the Rijksmuseum Boerhaave in [[Leiden]], Netherlands]]

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"/>
: <math>G_{\mu \nu} + \Lambda g_{\mu \nu} = \kappa T_{\mu \nu} ,</math>
where {{math|''G''{{sub|''μν''}}}} is the Einstein tensor, {{math|''g''{{sub|''μν''}}}} is the metric tensor, {{math|''T''{{sub|''μν''}}}} is the [[stress–energy tensor]], {{math|Λ}} is the [[cosmological constant]] and {{math|''κ''}} is the Einstein gravitational constant.

The Einstein tensor is defined as
: <math>G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} ,</math>
where {{math|''R''{{sub|''μν''}}}} is the [[Ricci curvature|Ricci curvature tensor]], and {{math|''R''}} is the [[scalar curvature]].  This is a symmetric second-degree tensor that depends on only the metric tensor and its first and second derivatives.

The '''Einstein gravitational constant''' is defined as<ref>With the choice of the Einstein gravitational constant as given here, {{math|1=''κ'' = 8''πG''/''c''{{i sup|4}}}}, 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, equivalently pressure).  In Einstein's original publication, the choice is {{math|1=''κ'' = 8''πG''/''c''{{i sup|2}}}}, in which case the stress–energy tensor components have units of mass density.</ref><ref>{{cite book |last1=Adler |first1=Ronald |last2=Bazin |first2=Maurice |last3=Schiffer |first3=Menahem |url=https://www.worldcat.org/oclc/1046135 |title=Introduction to general relativity |date=1975 |publisher=McGraw-Hill |isbn=0-07-000423-4 |edition=2nd |location=New York |oclc=1046135 }}</ref>
: <math>\kappa = \frac{8 \pi G}{c^4} \approx 2.07665\times10^{-43} \, \textrm{N}^{-1} ,</math>
where {{mvar|G}} is the [[gravitational constant|Newtonian constant of gravitation]] and {{mvar|c}} is the [[speed of light]] in [[vacuum]].

The EFE can thus also be written as
: <math>R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} + \Lambda g_{\mu \nu} = \kappa T_{\mu \nu} .</math>

In standard units, each term on the left has quantity dimension of [[Length|L]]<sup>−2</sup>.

The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime.

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 spacetime, form the core of the [[mathematics of general relativity|mathematical formulation]] of [[general relativity]].

The EFE is a tensor equation relating a set of [[symmetric tensor|symmetric 4&nbsp;×&nbsp;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|gauge-fixing]] [[Degrees of freedom (physics and chemistry)|degrees of freedom]], which correspond to the freedom to choose a coordinate system.

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 | last1 = Stephani | first1 = 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 {{math|''T''{{sub|''μν''}}}} is everywhere zero) define [[Einstein manifold]]s.

The equations are more complex than they appear. 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 {{math|''g''{{sub|''μν''}}}}, since both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic [[partial differential equation]]s.<ref>{{cite journal |first=Alan D. |last=Rendall |title=Theorems on Existence and Global Dynamics for the Einstein Equations |journal=Living Rev. Relativ. |volume=8 |year=2005 |issue=1 |at=Article number: 6 |doi=10.12942/lrr-2005-6 |pmid=28179868 |pmc=5256071 |arxiv=gr-qc/0505133 |bibcode=2005LRR.....8....6R |doi-access=free }}</ref>

=== Sign convention ===
The above form of the EFE is the standard established by [[Gravitation (book)|Misner, Thorne, and Wheeler]] (MTW).{{sfnp|Misner|Thorne|Wheeler|1973|p=501ff}} The authors analyzed conventions that exist and classified these according to three signs ([S1] [S2] [S3]):
<math display="block">\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 \kappa T_{\mu \nu}
\end{align}</math>

The third sign above is related to the choice of convention for the Ricci tensor:
<math display="block">R_{\mu \nu} = [S2] \times [S3] \times {R^\alpha}_{\mu\alpha\nu} </math>

With these definitions [[Gravitation (book)|Misner, Thorne, and Wheeler]] classify themselves as {{math|(+ + +)}}, whereas Weinberg (1972){{sfnp|Weinberg|1972}} is {{math|(+ − −)}}, Peebles (1980)<ref>{{cite book |last=Peebles |first=Phillip James Edwin |title=The Large-scale Structure of the Universe |publisher=Princeton University Press |year=1980 |isbn=0-691-08239-1 }}</ref> and Efstathiou et al. (1990)<ref>{{cite journal |last1=Efstathiou |first1=G. |first2=W. J. |last2=Sutherland |first3=S. J. |last3=Maddox |s2cid=12988317 |title=The cosmological constant and cold dark matter |journal=[[Nature (journal)|Nature]] |volume=348 |issue=6303 |year=1990 |pages=705 |doi=10.1038/348705a0 |bibcode=1990Natur.348..705E }}</ref> are {{math|(− + +)}},  Rindler (1977),{{citation needed|date=October 2014}} Atwater (1974),{{citation needed|date=October 2014}} Collins Martin & Squires (1989)<ref>{{cite book |last1=Collins |first1=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> and Peacock (1999){{sfnp|Peacock|1999}} are {{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 display="block">R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} - \Lambda g_{\mu \nu} = -\kappa T_{\mu \nu}.</math>

The sign of the 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.

=== Equivalent formulations ===
Taking the [[Scalar curvature#Definition|trace with respect to the metric]] of both sides of the EFE one gets
<math display="block">R - \frac{D}{2} R + D \Lambda = \kappa T ,</math>
where {{mvar|D}} is the spacetime dimension. Solving for {{math|''R''}} and substituting this in the original EFE, one gets the following equivalent "trace-reversed" form:
<math display="block">R_{\mu \nu} - \frac{2}{D-2} \Lambda g_{\mu \nu} = \kappa \left(T_{\mu \nu} - \frac{1}{D-2}Tg_{\mu \nu}\right) .</math>

In {{math|1=''D'' = 4}} dimensions this reduces to
<math display="block">R_{\mu \nu} - \Lambda g_{\mu \nu} = \kappa \left(T_{\mu \nu} - \frac{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''{{sub|''μν''}}}} in the expression on the right with the [[Minkowski metric]] without significant loss of accuracy).