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{{Short description|Field-equations in general relativity}}
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In the [[General relativity|general theory of relativity]], the '''Einstein field equations''' ('''EFE'''; also known as '''Einstein's equations''') relate the geometry of [[spacetime]] to the distribution of [[Matter#In general relativity and cosmology|matter]] within it.<ref name="ein">{{cite journal |last=Einstein |first=Albert |title=The Foundation of the General Theory of Relativity |journal=[[Annalen der Physik]] |volume=354 |issue=7 |pages=769 |year=1916 |url=http://www.alberteinstein.info/gallery/science.html |doi=10.1002/andp.19163540702 |format=[[PDF]] |bibcode=1916AnP...354..769E |archive-url=https://web.archive.org/web/20120206225139/http://www.alberteinstein.info/gallery/gtext3.html |archive-date=2012-02-06}}</ref>

The equations were published by [[Albert Einstein]] in 1915 in the form of a [[Tensor|tensor equation]]<ref name=Ein1915>{{cite journal |last=Einstein |first=Albert |author-link=Albert Einstein |date=November 25, 1915 |title=Die Feldgleichungen der Gravitation |journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin |pages=844–847 |url=http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/echo/einstein/sitzungsberichte/6E3MAXK4/index.meta |access-date=2017-08-21}}</ref> which related the local ''{{vanchor|spacetime [[curvature]]|SPACETIME_CURVATURE}}'' (expressed by the [[Einstein tensor]]) with the local energy, [[momentum]] and stress within that spacetime (expressed by the [[stress–energy tensor]]).{{sfnp|Misner|Thorne|Wheeler|1973|p=916 [ch. 34]}}

Analogously to the way that [[electromagnetic field]]s are related to the distribution of [[Charge (physics)|charge]]s and [[Electric current|current]]s via [[Maxwell's equations]], the EFE relate the [[spacetime geometry]] to the distribution of mass–energy, momentum and stress, that is, they determine the [[Metric tensor (general relativity)|metric tensor]] of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear [[partial differential equation]]s when used in this way. The solutions of the EFE are the components of the metric tensor. The [[inertia]]l trajectories of particles and radiation ([[Geodesics in general relativity|geodesics]]) in the resulting geometry are then calculated using the [[geodesic equation]].

As well as implying local energy–momentum conservation, the EFE reduce to [[Newton's law of gravitation]] in the limit of a weak gravitational field and velocities that are much less than the [[speed of light]].<ref name="Carroll">{{cite book |last=Carroll |first=Sean |author-link=Sean M. Carroll |year=2004 |title=Spacetime and Geometry – An Introduction to General Relativity |pages=151–159 |publisher=Addison Wesley |isbn=0-8053-8732-3}}</ref>

Exact solutions for the EFE can only be found under simplifying assumptions such as [[Spacetime symmetries|symmetry]]. Special classes of [[Exact solutions in general relativity|exact solutions]] are most often studied since they model many gravitational phenomena, such as [[rotating black hole]]s and the [[Metric expansion of space|expanding universe]]. Further simplification is achieved in approximating the spacetime as having only small deviations from [[Minkowski space|flat spacetime]], leading to the [[Linearized gravity#Linearized Einstein field equations|linearized EFE]]. These equations are used to study phenomena such as [[gravitational waves]].

== 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 [[Divergence|divergenceless]] 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 form the core of the [[mathematics of general relativity|mathematical formulation]] of [[general relativity]]. They readily imply 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 test objects move through spacetime.<ref>{{Citation |last=Ehlers |first=Jurgen |title=Equation of Motion of Small Bodies in Relativity |date=2003-09-16 |url=http://arxiv.org/abs/gr-qc/0309074 |access-date=2025-07-01 |publisher=arXiv |doi=10.48550/arXiv.gr-qc/0309074 |id=arXiv:gr-qc/0309074 |last2=Geroch |first2=Robert}}</ref> 

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).

== Cosmological constant ==
{{Main|Cosmological constant}}

In the Einstein field equations
<math display="block">G_{\mu \nu} + \Lambda g_{\mu \nu} = \kappa T_{\mu \nu} \,,</math>
the term containing the cosmological constant {{math|Λ}} was absent from the version in which he originally published them.  Einstein then included the term with the cosmological constant to allow for a [[static universe|universe that is not expanding or contracting]]. This effort was unsuccessful because:
* any desired steady state solution described by this equation is unstable, and
* observations by [[Edwin Hubble]] showed that our universe is [[expanding universe|expanding]].

Einstein then abandoned {{math|Λ}}, remarking to [[George Gamow]] "that the introduction of the cosmological term was the biggest blunder of his life".<ref name = gamow>{{cite book| last = Gamow| first = George| author-link = George Gamow| title = My World Line : An Informal Autobiography| publisher = [[Viking Adult]]| date = April 28, 1970| isbn = 0-670-50376-2| url = http://www.jb.man.ac.uk/~jpl/cosmo/blunder.html| access-date = 2007-03-14 }}</ref>

The inclusion of this term does not create inconsistencies. For many years the cosmological constant was almost universally assumed to be zero. More recent [[astronomy|astronomical]] observations have shown an [[accelerating expansion of the universe]], and to explain this a positive value of {{math|Λ}} is needed.<ref name=wahl>{{cite news |last=Wahl |first=Nicolle |date=2005-11-22 |title=Was Einstein's 'biggest blunder' a stellar success? |url=http://www.news.utoronto.ca/bin6/051122-1839.asp |publisher =University of Toronto |work=News@UofT |archive-url=https://web.archive.org/web/20070307191343/http://www.news.utoronto.ca/bin6/051122-1839.asp <!-- Bot retrieved archive --> |archive-date=2007-03-07 }}</ref><ref name= turner>
{{cite journal
 |last=Turner |first=Michael S.
 |s2cid=16669258
 |date=May 2001
 |title=Making Sense of the New Cosmology
 |journal=Int. J. Mod. Phys. A
 |volume=17|issue=S1
 |pages=180–196
 |doi=10.1142/S0217751X02013113
 |arxiv=astro-ph/0202008|bibcode = 2002IJMPA..17S.180T
}}</ref> The effect of the cosmological constant is negligible at the scale of a galaxy or smaller.

Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side and incorporated as part of the stress–energy tensor:
<math display="block">T_{\mu \nu}^\mathrm{(vac)} = - \frac{\Lambda}{\kappa} g_{\mu \nu} \,.</math>

This tensor describes a [[vacuum state]] with an [[vacuum energy|energy density]] {{math|''ρ''{{sub|vac}}}} and isotropic pressure {{math|''p''{{sub|vac}}}} that are fixed constants and given by
<math display="block">\rho_\mathrm{vac} = - p_\mathrm{vac} = \frac{\Lambda}{\kappa},</math>
where it is assumed that {{math|Λ}} has SI unit m{{sup|−2}} and {{math|''κ''}} is defined as above.

The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign. This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity.

== Features ==

=== Conservation of energy and momentum ===
General relativity is consistent with the local conservation of energy and momentum expressed as
<math display="block">\nabla_\beta T^{\alpha\beta} = {T^{\alpha\beta}}_{;\beta} = 0.</math>

{{math proof|title=Derivation of local energy–momentum conservation|proof=
[[Tensor contraction|Contracting]] the [[Riemann curvature tensor#Symmetries and identities|differential Bianchi identity]]
<math display="block">R_{\alpha\beta[\gamma\delta;\varepsilon]} = 0</math>
with {{mvar|g{{sup|αβ}}}} gives, using the fact that the metric tensor is covariantly constant, i.e. {{math|1=''g''{{sup|''αβ''}}{{sub|;''γ''}} = 0}},
<math display="block">{R^\gamma}_{\beta\gamma\delta;\varepsilon} + {R^\gamma}_{\beta\varepsilon\gamma;\delta} + {R^\gamma}_{\beta\delta\varepsilon;\gamma} = 0</math>

The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten:
<math display="block">{R^\gamma}_{\beta\gamma\delta;\varepsilon} - {R^\gamma}_{\beta\gamma\varepsilon;\delta} + {R^\gamma}_{\beta\delta\varepsilon;\gamma} = 0</math>
which is equivalent to
<math display="block">R_{\beta\delta;\varepsilon} - R_{\beta\varepsilon;\delta} + {R^\gamma}_{\beta\delta\varepsilon;\gamma} = 0</math>
using the definition of the [[Ricci tensor]].

Next, contract again with the metric
<math display="block">g^{\beta\delta}\left(R_{\beta\delta;\varepsilon} - R_{\beta\varepsilon;\delta} + {R^\gamma}_{\beta\delta\varepsilon;\gamma}\right) = 0</math>
to get
<math display="block">{R^\delta}_{\delta;\varepsilon} - {R^\delta}_{\varepsilon;\delta} + {R^{\gamma\delta}}_{\delta\varepsilon;\gamma} = 0 .</math>

The definitions of the Ricci curvature tensor and the scalar curvature then show that
<math display="block">R_{;\varepsilon} - 2{R^\gamma}_{\varepsilon;\gamma} = 0 ,</math>
which can be rewritten as
<math display="block">\left({R^\gamma}_{\varepsilon} - \tfrac{1}{2}{g^\gamma}_{\varepsilon}R\right)_{;\gamma} = 0 .</math>

A final contraction with {{math|''g''{{sup|''εδ''}}}} gives
<math display="block">\left(R^{\gamma\delta} - \tfrac{1}{2}g^{\gamma\delta}R\right)_{;\gamma} = 0 ,</math>
which by the symmetry of the bracketed term and the definition of the [[Einstein tensor]], gives, after relabelling the indices,
<math display="block"> {G^{\alpha\beta}}_{;\beta} = 0 .</math>

Using the EFE, this immediately gives,
<math display="block">\nabla_\beta T^{\alpha\beta} = {T^{\alpha\beta}}_{;\beta} = 0</math>
}}

which expresses the local conservation of stress–energy. This conservation law is a physical requirement. With his field equations Einstein ensured that general relativity is consistent with this conservation condition.

=== Nonlinearity ===

The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example, [[Maxwell's equations]] of [[electromagnetism]] are linear in the [[electric field|electric]] and [[magnetic field]]s, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is the [[Schrödinger equation]] of [[quantum mechanics]], which is linear in the [[wavefunction]].

=== Correspondence principle ===
The EFE reduce to [[Newton's law of gravity]] by using both the [[weak-field approximation]] and the [[Classical limit|low-velocity approximation]]. The constant {{mvar|G}} appearing in the EFE is determined by making these two approximations.

{{math proof |title=Derivation of Newton's law of gravity | proof=
Newtonian gravitation can be written as the theory of a scalar field, <math>\Phi</math>, which is the gravitational potential in joules per kilogram of the gravitational field <math>g=-\nabla\Phi</math>, see [[Gauss's law for gravity#Poisson's equation and gravitational potential|Gauss's law for gravity]]
<math display="block">\nabla^2 \Phi \left(\vec{x},t\right) = 4 \pi G \rho \left(\vec{x},t\right)</math>
where {{mvar|ρ}} is the mass density. The orbit of a [[free-fall]]ing particle satisfies
<math display="block">\ddot{\vec{x}}(t) = \vec{g} = - \nabla \Phi \left(\vec{x} (t),t\right) \,.</math>

In tensor notation, these become
<math display="block">\begin{align}
\Phi_{,i i} &= 4 \pi G \rho \\
\frac{d^2 x^i}{d t^2} &= - \Phi_{,i} \,.
\end{align}</math>

In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form
<math display="block">R_{\mu \nu} = K \left(T_{\mu \nu} - \tfrac{1}{2} T g_{\mu \nu}\right)</math>
for some constant, {{mvar|K}}, and the [[geodesic equation]]
<math display="block">\frac{d^2 x^\alpha}{d \tau^2} = - \Gamma^\alpha_{\beta \gamma} \frac{d x^\beta}{d \tau} \frac{d x^\gamma}{d \tau} \,.</math>

To see how the latter reduces to the former, we assume that the test particle's velocity is approximately zero
<math display="block">\frac{d x^\beta}{d \tau} \approx \left(\frac{dt}{d \tau}, 0, 0, 0\right) </math>
and thus
<math display="block">\frac{d}{d t} \left( \frac{dt}{d \tau} \right) \approx 0 </math>
and that the metric and its derivatives are approximately static and that the squares of deviations from the Minkowski metric are negligible. Applying these simplifying assumptions to the spatial components of the geodesic equation gives
<math display="block">\frac{d^2 x^i}{d t^2} \approx - \Gamma^i_{0 0} </math>
where two factors of {{math|{{sfrac|''dt''|''dτ''}}}} have been divided out. This will reduce to its Newtonian counterpart, provided
<math display="block">\Phi_{,i} \approx \Gamma^i_{0 0} = \tfrac{1}{2} g^{i \alpha} \left(g_{\alpha 0 , 0} + g_{0 \alpha , 0} - g_{0 0 , \alpha}\right) \,.</math>

Our assumptions force {{math|1=''α'' = ''i''}} and the time (0) derivatives to be zero. So this simplifies to
<math display="block">2 \Phi_{,i} \approx g^{i j} \left(- g_{0 0 , j}\right) \approx - g_{0 0 , i} \,</math>
which is satisfied by letting
<math display="block">g_{0 0} \approx - c^2 - 2 \Phi \,.</math>

Turning to the Einstein equations, we only need the time-time component
<math display="block">R_{0 0} = K \left(T_{0 0} - \tfrac{1}{2} T g_{0 0}\right)</math>
the low speed and static field assumptions imply that
<math display="block">T_{\mu \nu} \approx \operatorname{diag} \left(T_{0 0}, 0, 0, 0\right) \approx \operatorname{diag} \left(\rho c^4, 0, 0, 0\right) \,.</math>

So
<math display="block">T = g^{\alpha \beta} T_{\alpha \beta} \approx g^{0 0} T_{0 0} \approx -\frac{1}{c^2} \rho c^4 = - \rho c^2 \,</math>
and thus
<math display="block">K \left(T_{0 0} - \tfrac{1}{2} T g_{0 0}\right) \approx K \left(\rho c^4 - \tfrac{1}{2} \left(- \rho c^2\right) \left(- c^2\right)\right) = \tfrac{1}{2} K \rho c^4 \,.</math>

From the definition of the Ricci tensor
<math display="block">R_{0 0} = \Gamma^\rho_{0 0 , \rho} - \Gamma^\rho_{\rho 0 , 0}
+ \Gamma^\rho_{\rho \lambda} \Gamma^\lambda_{0 0}
- \Gamma^\rho_{0 \lambda} \Gamma^\lambda_{\rho 0}.</math>

Our simplifying assumptions make the squares of {{mvar|Γ}} disappear together with the time derivatives
<math display="block">R_{0 0} \approx \Gamma^i_{0 0 , i} \,.</math>

Combining the above equations together
<math display="block">\Phi_{,i i} \approx \Gamma^i_{0 0 , i} \approx R_{0 0} = K \left(T_{0 0} - \tfrac{1}{2} T g_{0 0}\right) \approx \tfrac{1}{2} K \rho c^4 </math>
which reduces to the Newtonian field equation provided
<math display="block">\tfrac{1}{2} K \rho c^4 = 4 \pi G \rho ,</math>
which will occur if
<math display="block">K = \frac{8 \pi G}{c^4} \,.</math>
}}

== Vacuum field equations ==
[[Image:Swiss-Commemorative-Coin-1979b-CHF-5-obverse.png|thumb|A Swiss commemorative coin from 1979, showing the vacuum field equations with zero cosmological constant (top).]]
If the energy–momentum tensor {{mvar|T{{sub|μν}}}} is zero in the region under consideration, then the field equations are also referred to as the [[Field equation#Vacuum field equations|vacuum field equations]]. By setting {{math|1=''T''{{sub|''μν''}} = 0}} in the [[#Equivalent formulations|trace-reversed field equations]], the vacuum field equations, also known as 'Einstein vacuum equations' (EVE), can be written as
<math display="block">R_{\mu \nu} = 0 \,.</math>

In the case of nonzero cosmological constant, the equations are
<math display="block">R_{\mu \nu} = \frac{\Lambda}{\frac{D}{2} -1} g_{\mu \nu} \,.</math>

The solutions to the vacuum field equations are called [[vacuum solution (general relativity)|vacuum solutions]]. Flat [[Minkowski space]] is the simplest example of a vacuum solution. Nontrivial examples include the [[Schwarzschild solution]] and the [[Kerr solution]].

[[Manifold]]s with a vanishing [[Ricci tensor]], {{math|1=''R''{{sub|''μν''}} = 0}}, are referred to as [[Ricci-flat manifold]]s and manifolds with a Ricci tensor proportional to the metric as [[Einstein manifold]]s.

== Einstein–Maxwell equations ==
{{See also|Maxwell's equations in curved spacetime}}
If the energy–momentum tensor {{math|''T''{{sub|''μν''}}}} is that of an [[electromagnetic field]] in [[free space]], i.e. if the [[electromagnetic stress–energy tensor]]
<math display="block">T^{\alpha \beta} = \, -\frac{1}{\mu_0} \left( {F^\alpha}^\psi {F_\psi}^\beta + \tfrac{1}{4} g^{\alpha \beta} F_{\psi\tau} F^{\psi\tau}\right) </math>
is used, then the Einstein field equations are called the ''Einstein–Maxwell equations'' (with [[cosmological constant]] {{math|Λ}}, taken to be zero in conventional relativity theory):
<math display="block">G^{\alpha \beta} + \Lambda g^{\alpha \beta} = \frac{\kappa}{\mu_0} \left( {F^\alpha}^\psi {F_\psi}^\beta + \tfrac{1}{4} g^{\alpha \beta} F_{\psi\tau} F^{\psi\tau}\right).</math>

Additionally, the [[Electromagnetic tensor#The field tensor and relativity|covariant Maxwell equations]] are also applicable in free space:{{refn|name=covariance|The full expression here with partial derivatives, denoted by a comma, is fully covariant, despite the individual terms not being covariant.}}
<math display="block">\begin{align}
{F^{\alpha\beta}}_{;\beta} &= 0 \\
F_{[\alpha\beta;\gamma]}&=\tfrac{1}{3}\left(F_{\alpha\beta;\gamma} + F_{\beta\gamma;\alpha}+F_{\gamma\alpha;\beta}\right)=\tfrac{1}{3}\left(F_{\alpha\beta,\gamma} + F_{\beta\gamma,\alpha}+F_{\gamma\alpha,\beta}\right)= 0 ,
\end{align}</math>
where the semicolon represents a [[covariant derivative]], and the brackets denote [[exterior algebra#Alternating tensor algebra|anti-symmetrization]]. The first equation asserts that the 4-[[divergence]] of the [[2-form]] {{mvar|F}} is zero, and the second that its [[exterior derivative]] is zero. From the latter, it follows by the [[Poincaré lemma]] that in a coordinate chart it is possible to introduce an electromagnetic field potential {{math|''A''<sub>''α''</sub>}} such that{{refn|name=covariance}}
<math display="block">F_{\alpha\beta} = A_{\alpha;\beta} - A_{\beta;\alpha} = A_{\alpha,\beta} - A_{\beta,\alpha}</math>
in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived.<ref>{{cite book|last=Brown|first=Harvey|url=https://books.google.com/books?id=T6IVyWiPQksC&q=Maxwell+and+potential+and+%22generally+covariant%22&pg=PA164| title=Physical Relativity|page=164|publisher=Oxford University Press|year=2005 | isbn=978-0-19-927583-0}}</ref> However, there are global solutions of the equation that may lack a globally defined potential.<ref>{{Cite journal | last1=Trautman | first1=Andrzej | s2cid=123364248 | author-link = Andrzej Trautman| title=Solutions of the Maxwell and Yang–Mills equations associated with Hopf fibrings | year=1977 | journal=[[International Journal of Theoretical Physics]] | volume=16 | issue=9|pages=561–565 | doi=10.1007/BF01811088|bibcode = 1977IJTP...16..561T }}.</ref>

== Solutions ==
{{Main|Solutions of the Einstein field equations}}

The solutions of the Einstein field equations are [[metric tensor (general relativity)|metrics]] of spacetime. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as [[post-Newtonian approximation]]s. Even so, there are several cases where the field equations have been solved completely, and those are called exact solutions.<ref name="Stephani et al" />

The study of exact solutions of Einstein's field equations is one of the activities of [[physical cosmology|cosmology]]. It leads to the prediction of [[black hole]]s and to different models of evolution of the [[universe]].

One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum.<ref>{{cite journal|last1=Ellis|first1=G. F. R.|last2=MacCallum|first2=M.|s2cid=122577276|title=A class of homogeneous cosmological models|journal=Comm. Math. Phys.|volume=12|issue=2|date=1969|pages=108–141|bibcode=1969CMaPh..12..108E |doi=10.1007/BF01645908|url=http://projecteuclid.org/euclid.cmp/1103841345|url-access=subscription}}</ref> In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright,<ref>{{cite journal|last1=Hsu|first1=L.|last2=Wainwright|first2=J|title=Self-similar spatially homogeneous cosmologies: orthogonal perfect fluid and vacuum solutions|journal=Class. Quantum Grav.|volume=3|date=1986|issue=6|pages=1105–1124|doi=10.1088/0264-9381/3/6/011 |bibcode=1986CQGra...3.1105H|s2cid=250907312 }}</ref> self-similar solutions to the Einstein field equations are fixed points of the resulting [[dynamical system]]. New solutions have been discovered using these methods by LeBlanc<ref>{{cite journal|last=LeBlanc|first=V. G.|title=Asymptotic states of magnetic Bianchi I cosmologies|date=1997|journal=Class. Quantum Grav.| volume=14|issue=8 |page=2281 |doi=10.1088/0264-9381/14/8/025|bibcode=1997CQGra..14.2281L|s2cid=250876974 }}</ref> and Kohli and Haslam.<ref>{{cite journal|last1=Kohli |first1=Ikjyot Singh|last2=Haslam|first2=Michael C.|title=Dynamical systems approach to a Bianchi type I viscous magnetohydrodynamic model|journal=Phys. Rev. D|volume=88|page=063518|date=2013|issue=6|doi=10.1103/physrevd.88.063518| arxiv=1304.8042| bibcode=2013PhRvD..88f3518K|s2cid=119178273}}</ref>

== Linearized EFE ==
{{Main|Linearized gravity}}

The nonlinearity of the EFE makes finding exact solutions difficult. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the [[gravitational field]] is very weak and the [[spacetime]] approximates that of Minkowski space. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the [[Minkowski metric]], ignoring higher-power terms. This linearization procedure can be used to investigate the phenomena of [[gravitational radiation]].

== Polynomial form ==
Despite the EFE as written containing the inverse of the metric tensor, they can be arranged in a form that contains the metric tensor in polynomial form and without its inverse. First, the determinant of the metric in 4 dimensions can be written
<math display="block">\det(g) = \tfrac{1}{24} \varepsilon^{\alpha\beta\gamma\delta} \varepsilon^{\kappa\lambda\mu\nu} g_{\alpha\kappa} g_{\beta\lambda} g_{\gamma\mu} g_{\delta\nu}</math>
using the [[Levi-Civita symbol]]; and the inverse of the metric in 4 dimensions can be written as:
<math display="block">g^{\alpha\kappa} = \frac{\tfrac{1}{6} \varepsilon^{\alpha\beta\gamma\delta} \varepsilon^{\kappa\lambda\mu\nu} g_{\beta\lambda} g_{\gamma\mu} g_{\delta\nu} }{ \det(g)}\,.</math>

Substituting this expression of the inverse of the metric into the equations then multiplying both sides by a suitable power of {{math|det(''g'')}} to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives. The [[Einstein–Hilbert action]] from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields.<ref>{{cite journal |last=Katanaev |first=M. O. |s2cid=6263993 |year=2006 |title=Polynomial form of the Hilbert–Einstein action |journal=Gen. Rel. Grav. |volume=38 |issue=8 |pages=1233–1240 |arxiv=gr-qc/0507026 |doi=10.1007/s10714-006-0310-5 |bibcode=2006GReGr..38.1233K }}</ref>

== See also ==
{{div col|colwidth=25em}}
* [[Conformastatic spacetimes]]
* [[Einstein–Hilbert action]]
* [[Equivalence principle]]
* [[Exact solutions in general relativity]]
* [[General relativity resources]]
* [[History of general relativity]]
* [[Hamilton–Jacobi–Einstein equation]]
* [[Mathematics of general relativity]]
* [[Numerical relativity]]
* [[Ricci calculus]]
{{div col end}}

== Notes ==
{{reflist|30em}}

== References ==
See [[General relativity resources]].
* {{cite book|author1-link=Charles Misner|first1=Charles W.|last1=Misner|author2-link=Kip S. Thorne|first2=Kip S.|last2=Thorne |author3-link=John Archibald Wheeler|first3=John Archibald|last3=Wheeler|title=[[Gravitation (book)|Gravitation]] |location=San Francisco|publisher=[[W. H. Freeman]]|year=1973|isbn=978-0-7167-0344-0}}
* {{cite book|last=Weinberg|first=Steven|year=1972|title=Gravitation and Cosmology|isbn=0-471-92567-5|publisher=John Wiley & Sons |author-link=Steven Weinberg|url-access=registration|url=https://archive.org/details/gravitationcosmo00stev_0}}
* {{cite book|last=Peacock|first=John A.|year=1999|title=Cosmological Physics|url=https://archive.org/details/cosmologicalphys0000peac |url-access=registration|isbn=978-0521410724|publisher=Cambridge University Press|author-link=John A. Peacock}}

== External links ==
{{Wikibooks | General Relativity}}
{{Wikiversity | General Relativity}}
* {{springer|title=Einstein equations|id=p/e035210}}
* [https://web.archive.org/web/20110621005940/http://www.black-holes.org/relativity6.html Caltech Tutorial on Relativity] — A simple introduction to Einstein's Field Equations.
* [http://math.ucr.edu/home/baez/einstein/einstein.html The Meaning of Einstein's Equation] — An explanation of Einstein's field equation, its derivation, and some of its consequences
* [https://www.youtube.com/watch?v=8MWNs7Wfk84&feature=PlayList&p=858478F1EC364A2C&index=2 Video Lecture on Einstein's Field Equations] by [[MIT]] Physics Professor Edmund Bertschinger.
* [http://scitation.aip.org/content/aip/magazine/physicstoday/article/68/11/10.1063/PT.3.2979 Arch and scaffold: How Einstein found his field equations] Physics Today November 2015, History of the Development of the Field Equations

=== External images ===
* [https://web.archive.org/web/20180226091926/https://www.ilorentz.org/history/wallformulas/images/pages/page_2.html The Einstein field equation on the wall of the Museum Boerhaave in downtown Leiden]
* [[Suzanne Imber]], [https://imaggeo.egu.eu/view/886/ "The impact of general relativity on the Atacama Desert"], Einstein field equation on the side of a train in Bolivia.

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