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{{Use American English|date=January 2019}}{{Short description|Field equations in general relativity
}}
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In [[General relativity|general theory of relativity]] the '''Einstein field equations''' ('''EFE'''; also known as '''Einstein's equations''') relate the geometry of [[space-time]] with the distribution of [[matter]] within it.<ref name="ein">{{cite journal |last=Einstein |first=Albert |authorlink= |title=The Foundation of the General Theory of Relativity |journal=[[Annalen der Physik]] |volume=354 |issue=7 |pages=769 |year=1916 |publisher= |url=http://www.alberteinstein.info/gallery/science.html |doi=10.1002/andp.19163540702 |format=[[PDF]] |id= |accessdate= |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 first published by Einstein in 1915 in the form of a [[Tensor|tensor equation]]<ref name=Ein1915>{{cite journal |last=Einstein |first=Albert |authorlink=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 |accessdate=2017-08-21}}</ref> which related the local ''{{vanchor|spacetime [[curvature]]|SPACETIME_CURVATURE}}'' (expressed by the [[Einstein tensor]]) with the local energy and [[momentum]] within that spacetime (expressed by the [[stress–energy tensor]]).{{sfnp|Misner|Thorne|Wheeler|1973|p=916 [ch. 34]}}

Similar to the way that [[electromagnetic field]]s are determined using [[Charge (physics)|charge]]s and [[Electric current|current]]s via [[Maxwell's equations]], the EFE are used to determine the [[spacetime geometry]] resulting from the presence of mass–energy and linear momentum, that is, they determine the [[Metric tensor (general relativity)|metric tensor]] of spacetime for a given arrangement of stress–energy in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of non-linear [[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 obeying local energy–momentum conservation, the EFE reduce to [[Newton's law of gravitation]] where the gravitational field is weak and velocities are much less than the [[speed of light]].<ref name="Carroll">{{cite book |last=Carroll |first=Sean |authorlink=Sean M. Carroll |year=2004 |title=Spacetime and Geometry – An Introduction to General Relativity |pages=151–159 |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 as 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 actual spacetime as [[Minkowski space|flat spacetime]] with a small deviation, 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=Mathematics}}
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"/>

{{Equation box 1
|indent =:
|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>
|cellpadding
|border
|border colour = #50C878
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[[File:EinsteinLeiden4.jpg|{{largethumb}}|EFE on a wall in [[Leiden]] ]]
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]].

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]] [[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 | 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.

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>Rendall, Alan D. "Theorems on existence and global dynamics for the Einstein equations." ''Living Reviews in Relativity'' 8.1 (2005): 6.</ref>

One can write the EFE in a more compact form by defining the [[Einstein tensor]]
:<math>G_{\mu \nu} = R_{\mu \nu} - \tfrac{1}{2} R g_{\mu \nu},</math>

which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as
:<math>G_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}.</math>

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).

Using [[geometrized units]] where {{math|''G'' {{=}} ''c'' {{=}} 1}}, this can be rewritten as
:<math>G_{\mu \nu} + \Lambda g_{\mu \nu} = 8 \pi T_{\mu \nu}\,.</math>

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.

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]].

===Sign convention===
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):

:<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 [[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|(− + −)}}.

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 {{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>R - \frac{D}{2} R + D \Lambda = \frac{8 \pi G}{c^4} T \,</math>

where {{mvar|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>

If one adds {{math|−{{sfrac|1|2}}''g{{sub|μν}}''}} 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}} 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 {{mvar|g{{sub|μν}}}} in the expression on the right with the [[Minkowski metric]] without significant loss of accuracy).

==The cosmological constant==
{{main|Cosmological constant}}

Einstein modified his original field equations to include a [[cosmological constant]] term {{mvar|Λ}} proportional to the [[Metric (mathematics)|metric]]
:<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>

Since {{mvar|Λ}} is constant, the energy conservation law is unaffected.

The cosmological constant term was originally introduced by Einstein to allow for a [[static universe|universe that is not expanding or contracting]]. This effort was unsuccessful because:

* the universe described by this theory was unstable, and
* observations by [[Edwin Hubble]] confirmed that our universe is [[expanding universe|expanding]].

So, Einstein abandoned {{mvar|Λ}}, calling it the "biggest blunder [he] ever made".<ref name = gamow>{{cite book| last = Gamow| first = George| authorlink = 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| accessdate = 2007-03-14 }}</ref>

Despite Einstein's motivation for introducing the cosmological constant term, there is nothing inconsistent with the presence of such a term in the equations. For many years the cosmological constant was almost universally considered to be 0. However, recent improved [[astronomy|astronomical]] techniques have found that a positive value of {{mvar|Λ}} is needed to explain the [[accelerating universe]].<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 | access-date=|publisher =University of Toronto|work = News@UofT |archiveurl = https://web.archive.org/web/20070307191343/http://www.news.utoronto.ca/bin6/051122-1839.asp <!-- Bot retrieved archive --> |archivedate = 2007-03-07}}</ref><ref name= turner>{{cite journal
 |last=Turner | first=Michael S.
 |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> However, 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, written as part of the stress–energy tensor:
:<math>T_{\mu \nu}^\mathrm{(vac)} = - \frac{\Lambda c^4}{8 \pi G} g_{\mu \nu} \,.</math>

The resulting [[vacuum energy]] density is constant and given by
:<math>\rho_\mathrm{vac} = \frac{\Lambda c^2}{8 \pi G}</math>

The existence of a cosmological constant is thus equivalent to the existence of a non-zero vacuum energy. Thus, the terms "cosmological constant" and "vacuum energy" are now 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>\nabla_\beta T^{\alpha\beta} = {T^{\alpha\beta}}_{;\beta} = 0</math>.

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Derivation of local energy-momentum conservation
|-
|
Contracting the [[Riemann curvature tensor#Symmetries and identities|differential Bianchi identity]]

:<math>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|''g{{sup|αβ}}{{sub|;γ}}'' {{=}} 0}},

:<math>{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>{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>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>g^{\beta\delta}\left(R_{\beta\delta;\varepsilon} - R_{\beta\varepsilon;\delta} + {R^\gamma}_{\beta\delta\varepsilon;\gamma}\right) = 0</math>

to get

:<math>{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>R_{;\varepsilon} - 2{R^\gamma}_{\varepsilon;\gamma} = 0</math>

which can be rewritten as

:<math>\left({R^\gamma}_{\varepsilon} - \tfrac{1}{2}{g^\gamma}_{\varepsilon}R\right)_{;\gamma} = 0</math>

A final contraction with {{mvar|g{{sup|εδ}}}} gives

:<math>\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> {G^{\alpha\beta}}_{;\beta} = 0 </math>

Using the EFE, this immediately gives,

:<math>\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 [[Schrödinger's equation]] of [[quantum mechanics]] which is linear in the [[wavefunction]].

===The correspondence principle===
The EFE reduce to [[Newton's law of gravity]] by using both the [[weak-field approximation]] and the [[slow-motion approximation]]. In fact, the constant {{mvar|G}} appearing in the EFE is determined by making these two approximations.

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Derivation of Newton's law of gravity
|-
|
Newtonian gravitation can be written as the theory of a scalar field, {{mvar|Φ}}, which is the gravitational potential in joules per kilogram
:<math>\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>\ddot{\vec{x}}[t] = - \nabla \Phi \left[\vec{x} [t],t\right] \,.</math>

In tensor notation, these become
:<math>\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>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>\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>\frac{d x^\beta}{d \tau} \approx \left(\frac{d t}{d \tau}, 0, 0, 0\right) </math>

and thus
:<math>\frac{d}{d t} \left( \frac{d t}{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>\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>\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|''α'' {{=}} ''i''}} and the time (0) derivatives to be zero. So this simplifies to
:<math>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>g_{0 0} \approx - c^2 - 2 \Phi \,.</math>

Turning to the Einstein equations, we only need the time-time component
:<math>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>T_{\mu \nu} \approx \mathrm{diag} \left(T_{0 0}, 0, 0, 0\right) \approx \mathrm{diag} \left(\rho c^4, 0, 0, 0\right) \,.</math>

So
:<math>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>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>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>R_{0 0} \approx \Gamma^i_{0 0 , i} \,.</math>

Combining the above equations together
:<math>\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>\tfrac{1}{2} K \rho c^4 = 4 \pi G \rho \,</math>

which will occur if
:<math>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|''T{{sub|μν}}'' {{=}} 0}} in the [[#Equivalent_formulations|trace-reversed field equations]], the vacuum equations can be written as

:<math>R_{\mu \nu} = 0 \,.</math>

In the case of nonzero cosmological constant, the equations are

:<math>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|''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 {{mvar|T{{sub|μν}}}} is that of an [[electromagnetic field]] in [[free space]], i.e. if the [[electromagnetic stress–energy tensor]]

:<math>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]] {{mvar|Λ}}, taken to be zero in conventional relativity theory):

:<math>R^{\alpha \beta} - \tfrac{1}{2}R g^{\alpha \beta} + \Lambda g^{\alpha \beta} = \frac{8 \pi G}{c^4 \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:

:<math>\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#The alternating tensor algebra|anti-symmetrization]]. The first equation asserts that the 4-[[divergence]] of the [[two-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 {{mvar|A<sub>α</sub>}} such that

:<math>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/?id=T6IVyWiPQksC&pg=PA164&dq=Maxwell+and+potential+and+%22generally+covariant%22| 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 which may lack a globally defined potential.<ref>{{Cite journal | last1=Trautman | first1=Andrzej | authorlink = 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 numerous 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.|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}}</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|pages=1105–1124|doi=10.1088/0264-9381/3/6/011|bibcode=1986CQGra...3.1105H}}</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|page=2281|doi=10.1088/0264-9381/14/8/025|bibcode=1997CQGra..14.2281L}}</ref> and Kohli and Haslam.<ref>{{cite journal|last=Kohli|first=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|url=https://arxiv.org/pdf/1304.8042|doi=10.1103/physrevd.88.063518|arxiv=1304.8042|bibcode=2013PhRvD..88f3518K}}</ref>

==The 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]], with terms that are quadratic in or higher powers of the deviation being ignored. This linearization procedure can be used to investigate the phenomena of [[gravitational radiation]].

==Polynomial form==
One might think that EFE are non-polynomial since they contain the inverse of the metric tensor. However, the equations can be arranged so that they contain only the metric tensor and not its inverse. First, the determinant of the metric in 4 dimensions can be written:

:<math>\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>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 definition of the inverse of the metric into the equations then multiplying both sides by {{math|det(''g'')}} until there are none left in the denominator results in polynomial equations in the metric tensor and its first and second derivatives. The 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.|date=July 7, 2005|title=Polynomial form of the Hilbert–Einstein action|journal=Gen. Rel. Grav.|volume=38|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}}
*[[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|ref=harv}}
*{{cite book|last=Weinberg|first=Steven|year=1972|title=Gravitation and Cosmology|isbn=0-471-92567-5|publisher=John Wiley & Sons|authorlink=Steven Weinberg|ref=harv|url-access=registration|url=https://archive.org/details/gravitationcosmo00stev_0}}
*{{cite book|last=Peacock|first=John A.|year=1994|title=Cosmological Physics|url=https://archive.org/details/cosmologicalphys0000peac|url-access=registration|isbn=978-0521410724|publisher=Cambridge University Press|authorlink=John A. Peacock|ref=harv}}

==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
*[http://ilorentz.org/history/wallformulas/images/pages/page_2.html The Einstein field equation on the wall of the Museum Boerhaave in downtown Leiden]

{{Einstein}}
{{Relativity}}

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[[Category:Albert Einstein]]
[[Category:Equations of physics]]
[[Category:General relativity]]
[[Category:Partial differential equations]]