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| {{General relativity|cTopic=Equations}}
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| The '''Einstein field equations''' ('''EFE''') or '''Einstein's equations''' are a set of 10 [[equation]]s in [[Albert Einstein|Albert Einstein's]] [[general relativity|general theory of relativity]] which describe the [[fundamental interaction]] of [[gravitation]] as a result of [[spacetime]] being [[curvature|curved]] by [[matter]] and [[energy]].<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/gtext3.html | doi = 10.1002/andp.19163540702| format = [[PDF]] | id = | accessdate = |bibcode = 1916AnP...354..769E }}</ref> First published by Einstein in 1915<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://nausikaa2.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.x.cgi?dir=6E3MAXK4&step=thumb | accessdate=2006-09-12}}</ref> as a [[tensor equation]], the EFE equate local spacetime [[curvature]] (expressed by the [[Einstein tensor]]) with the local energy and [[momentum]] within that spacetime (expressed by the [[stress–energy tensor]]).<ref>{{Cite book
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| | last1=Misner |first1=Charles W. |authorlink1=Charles W. Misner
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| | last2=Thorne |first2=Kip S. |authorlink2=Kip Thorne
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| | last3=Wheeler |first3=John Archibald |authorlink3=John Archibald Wheeler
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| | year=1973
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| | title=[[Gravitation (book)|Gravitation]]
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| | url=
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| | publisher=[[W. H. Freeman]] |location=San Francisco
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| | isbn=978-0-7167-0344-0
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| | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}
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| }} Chapter 34, p 916</ref>
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| Similar to the way that [[electromagnetic field]]s are determined using [[charge (physics)|charge]]s and [[Electric current|currents]] 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 ([[Geodesic (general relativity)|geodesics]]) in the resulting geometry are then calculated using the [[geodesic equation]].
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| 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>
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| 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#Linearised Einstein field equations|linearised EFE]]. These equations are used to study phenomena such as [[gravitational waves]].
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| ==Mathematical form==
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| The Einstein field equations (EFE) may be written in the form:<ref name="ein"/>
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| {{Equation box 1
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| |indent =:
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| |equation = <math>R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}</math>
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| |cellpadding
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| |border
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| where <math>R_{\mu \nu}\,</math> is the [[Ricci curvature tensor]], <math>R\,</math> the [[scalar curvature]], <math>g_{\mu \nu}\,</math> the [[metric tensor (general relativity)|metric tensor]], <math>\Lambda\,</math> is the [[cosmological constant]], <math>G\,</math> is [[gravitational constant|Newton's gravitational constant]], <math>c\,</math> the [[speed of light]] in vacuum, and <math>T_{\mu \nu}\,</math> 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 ''n'' dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when ''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{{by who|date=January 2014}} to be equations for the metric tensor <math>g_{\mu \nu}</math>, 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 10 coupled, nonlinear, hyperbolic-elliptic [[partial differential equation]]s{{fact|date=January 2014}}.
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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} - {1 \over 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 | |
| :<math>G_{\mu \nu} + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}.</math>
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| Using [[geometrized units]] where ''G'' = ''c'' = 1, this can be rewritten as
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| :<math>G_{\mu \nu} + g_{\mu \nu} \Lambda = 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]]. The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):
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| :<math>
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| \begin{align}
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| g_{\mu \nu} & = [S1] \times \operatorname{diag}(-1,+1,+1,+1) \\[6pt]
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| {R^\mu}_{a \beta \gamma} & = [S2] \times (\Gamma^\mu_{a \gamma,\beta}-\Gamma^\mu_{a \beta,\gamma}+\Gamma^\mu_{\sigma \beta}\Gamma^\sigma_{\gamma a}-\Gamma^\mu_{\sigma \gamma}\Gamma^\sigma_{\beta a}) \\[6pt]
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| G_{\mu \nu} & = [S3] \times {8 \pi G \over c^4} T_{\mu \nu}
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| \end{align}
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| </math>
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| The third sign above is related to the choice of convention for the Ricci tensor:
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| :<math>R_{\mu \nu}=[S2]\times [S3] \times {R^a}_{\mu a \nu} </math>
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| With these definitions [[Gravitation (book)|Misner, Thorne, and Wheeler]] classify themselves as <math>(+++)\,</math>, whereas Weinberg (1972) is <math>(+--)\,</math>, Peebles (1980) and Efstathiou (1990) are <math>(-++)\,</math> while Peacock (1994), Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) are <math>(-+-)\,</math>.
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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} - {1 \over 2}g_{\mu \nu}\,R - g_{\mu \nu} \Lambda = -{8 \pi G \over 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 +−−− metric [[sign convention]] is used rather than the MTW −+++ metric sign convention adopted here.
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| ===Equivalent formulations===
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| Taking the [[Trace (linear algebra)|trace]] of both sides of the EFE one gets
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| :<math>R - 2 R + 4 \Lambda = {8 \pi G \over c^4} T \,</math>
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| which simplifies to
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| :<math>-R + 4 \Lambda = {8 \pi G \over c^4} T \,.</math>
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| If one adds <math>- {1 \over 2} g_{\mu \nu} \,</math> times this to the EFE, one gets the following equivalent "trace-reversed" form
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| :<math>R_{\mu \nu} - g_{\mu \nu} \Lambda = {8 \pi G \over c^4} \left(T_{\mu \nu} - {1 \over 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 <math>g_{\mu\nu} \,</math> in the expression on the right with the [[Minkowski metric]] without significant loss of accuracy).
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| ==The cosmological constant==
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| Einstein modified his original field equations to include a cosmological term proportional to the [[Metric (mathematics)|metric]]
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| :<math>R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu} \,.</math>
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| The constant <math>\Lambda</math> is the [[cosmological constant]]. Since <math>\Lambda</math> is constant, the energy conservation law is unaffected.
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| The cosmological constant term was originally introduced by Einstein to allow for a static universe (i.e., one that is not expanding or contracting). This effort was unsuccessful for two reasons: the static universe described by this theory was unstable, and observations of distant galaxies by [[Edwin Hubble|Hubble]] a decade later confirmed that our universe is, in fact, not static but [[expanding universe|expanding]]. So <math>\Lambda</math> was abandoned, with Einstein 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> For many years the cosmological constant was almost universally considered to be 0.
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| Despite [[Einstein]]'s misguided motivation for introducing the cosmological constant term, there is nothing inconsistent with the presence of such a term in the equations. Indeed, recent improved [[astronomy|astronomical]] techniques have found that a positive value of <math>\Lambda</math> 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 | accessdate=2007-03-14 |archiveurl = http://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
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| |last=Turner | first=Michael S.
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| |date=May 2001| title=Making Sense of the New Cosmology
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| |journal=Int.J.Mod.Phys. A17S1
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| |volume=17
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| |pages=180–196
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| |doi=10.1142/S0217751X02013113
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| |arxiv=astro-ph/0202008|bibcode = 2002IJMPA..17S.180T }}</ref>
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| 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:
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| :<math>T_{\mu \nu}^{\mathrm{(vac)}} = - \frac{\Lambda c^4}{8 \pi G} g_{\mu \nu} \,.</math>
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| The resulting [[vacuum energy]] is constant and given by
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| :<math>\rho_{\mathrm{vac}} = \frac{\Lambda c^2}{8 \pi G}</math>
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| The existence of a cosmological constant is thus equivalent to the existence of a non-zero vacuum energy. The terms are now used interchangeably in general relativity.
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| ==Features==
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| ===Conservation of energy and momentum===
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| General relativity is consistent with the local conservation of energy and momentum expressed as
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| :<math>\nabla_\beta T^{\alpha\beta} \, = T^{\alpha\beta}{}_{;\beta} \, = 0</math>.
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| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
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| !Derivation of local energy-momentum conservation
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| |-
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| Contracting the [[Riemann curvature tensor#Symmetries and identities|differential Bianchi identity]]
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| :<math>R_{\alpha\beta[\gamma\delta;\varepsilon]} = \, 0</math>
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| with <math>g^{\alpha\gamma}</math> gives, using the fact that the metric tensor is covariantly constant, i.e. <math>g^{\alpha\beta}{}_{;\gamma}=0</math>,
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| :<math>R^\gamma{}_{\beta\gamma\delta;\varepsilon} + \, R^\gamma{}_{\beta\varepsilon\gamma;\delta} + \, R^\gamma{}_{\beta\delta\varepsilon;\gamma} = \, 0</math>
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| The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten:
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| :<math>R^\gamma{}_{\beta\gamma\delta;\varepsilon} \, - R^\gamma{}_{\beta\gamma\varepsilon;\delta} \, + R^\gamma{}_{\beta\delta\varepsilon;\gamma} \, = 0</math>
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| which is equivalent to
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| :<math>R_{\beta\delta;\varepsilon} \, - R_{\beta\varepsilon;\delta} \, + R^\gamma{}_{\beta\delta\varepsilon;\gamma} \, = 0</math>
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| using the definition of the [[Ricci tensor]].
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| Next, contract again with the metric
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| :<math>g^{\beta\delta}(R_{\beta\delta;\varepsilon} \, - R_{\beta\varepsilon;\delta} \, + R^\gamma{}_{\beta\delta\varepsilon;\gamma}) \, = 0</math>
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| to get
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| :<math>R^\delta{}_{\delta;\varepsilon} \, - R^\delta{}_{\varepsilon;\delta} \, + R^{\gamma\delta}{}_{\delta\varepsilon;\gamma} \, = 0</math>
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| The definitions of the Ricci curvature tensor and the scalar curvature then show that
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| :<math>R_{;\varepsilon} \, - 2R^\gamma{}_{\varepsilon;\gamma} \, = 0</math>
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| which can be rewritten as | |
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| :<math>(R^\gamma{}_{\varepsilon} \, - \frac{1}{2}g^\gamma{}_{\varepsilon}R)_{;\gamma} \, = 0</math>
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| A final contraction with <math>g^{\varepsilon\delta}</math> gives
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| :<math>(R^{\gamma\delta} \, - \frac{1}{2}g^{\gamma\delta}R)_{;\gamma} \, = 0</math>
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| which by the symmetry of the bracketed term and the definition of the [[Einstein tensor]], gives, after relabelling the indices,
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| :<math> G^{\alpha\beta}{}_{;\beta} \, = 0 </math>
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| Using the EFE, this immediately gives,
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| :<math>\nabla_\beta T^{\alpha\beta} \, = T^{\alpha\beta}{}_{;\beta} \, = 0</math>
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| |}
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| 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.
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| ===Nonlinearity===
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| 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]].
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| ===The correspondence principle===
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| 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 ''G'' appearing in the EFE is determined by making these two approximations.
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| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
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| !Derivation of Newton's law of gravity
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| Newtonian gravitation can be written as the theory of a scalar field, <math>\Phi \!</math>, which is the gravitational potential in joules per kilogram
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| :<math>\nabla^2 \Phi [\vec{x},t] = 4 \pi G \rho [\vec{x},t]</math>
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| where <math>\rho \!</math> is the mass density. The orbit of a [[free-fall]]ing particle satisfies
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| :<math>\ddot{\vec{x}}[t] = - \nabla \Phi [\vec{x} [t],t] \,.</math>
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| In tensor notation, these become
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| :<math>\Phi_{,i i} = 4 \pi G \rho \,</math>
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| :<math>\frac{d^2 x^i}{{d t}^2} = - \Phi_{,i} \,.</math>
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| In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form
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| :<math>R_{\mu \nu} = K \left(T_{\mu \nu} - {1 \over 2} T g_{\mu \nu}\right)</math>
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| for some constant, ''K'', and the [[geodesic equation]]
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| :<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>
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| To see how the latter reduce to the former, we assume that the test particle's velocity is approximately zero
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| :<math>\frac{d x^\beta}{d \tau} \approx \left(\frac{d t}{d \tau}, 0, 0, 0\right) </math>
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| and thus
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| :<math>\frac{d}{d t} \left( \frac{d t}{d \tau} \right) \approx 0 </math>
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| 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
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| :<math>\frac{d^2 x^i}{{d t}^2} \approx - \Gamma^i_{0 0} </math>
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| where two factors of <math>\frac{d t}{d \tau}</math> have been divided out. This will reduce to its Newtonian counterpart, provided
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| :<math>\Phi_{,i} \approx \Gamma^i_{0 0} = {1 \over 2} g^{i \alpha} (g_{\alpha 0 , 0} + g_{0 \alpha , 0} - g_{0 0 , \alpha}) \,.</math>
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| Our assumptions force α=i and the time (0) derivatives to be zero. So this simplifies to
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| :<math>2 \Phi_{,i} \approx g^{i j} (- g_{0 0 , j}) \approx - g_{0 0 , i} \,</math>
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| which is satisfied by letting
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| :<math>g_{0 0} \approx - c^2 - 2 \Phi \,.</math>
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| Turning to the Einstein equations, we only need the time-time component
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| :<math>R_{0 0} = K \left(T_{0 0} - {1 \over 2} T g_{0 0}\right)</math>
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| the low speed and static field assumptions imply that
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| :<math>T_{\mu \nu} \approx \mathrm{diag} (T_{0 0}, 0, 0, 0) \approx \mathrm{diag} (\rho c^4, 0, 0, 0) \,.</math>
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| So
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| :<math>T = g^{\alpha \beta} T_{\alpha \beta} \approx g^{0 0} T_{0 0} \approx {-1 \over c^2} \rho c^4 = - \rho c^2 \,</math>
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| and thus
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| :<math>K \left(T_{0 0} - {1 \over 2} T g_{0 0}\right) \approx K \left(\rho c^4 - {1 \over 2} (- \rho c^2) (- c^2)\right) = {1 \over 2} K \rho c^4 \,.</math>
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| From the definition of the Ricci tensor
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| :<math>
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| R_{0 0} = \Gamma^\rho_{0 0 , \rho} - \Gamma^\rho_{\rho 0 , 0}
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| + \Gamma^\rho_{\rho \lambda} \Gamma^\lambda_{0 0}
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| - \Gamma^\rho_{0 \lambda} \Gamma^\lambda_{\rho 0}
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| .</math>
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| Our simplifying assumptions make the squares of Γ disappear together with the time derivatives
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| :<math>R_{0 0} \approx \Gamma^i_{0 0 , i} \,.</math>
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| Combining the above equations together
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| :<math>\Phi_{,i i} \approx \Gamma^i_{0 0 , i} \approx R_{0 0} = K \left(T_{0 0} - {1 \over 2} T g_{0 0}\right) \approx {1 \over 2} K \rho c^4 \,</math>
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| which reduces to the Newtonian field equation provided
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| :<math>{1 \over 2} K \rho c^4 = 4 \pi G \rho \,</math>
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| which will occur if
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| :<math>K = \frac{8 \pi G}{c^4} \,.</math>
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| |}
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| ==Vacuum field equations==
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| [[Image:Swiss-Commemorative-Coin-1979b-CHF-5-obverse.png|right | thumb| A Swiss commemorative coin showing the vacuum field equations with zero cosmological constant (top).]]
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| If the energy-momentum tensor <math>T_{\mu \nu}</math> 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_{\mu \nu} = 0</math> in the [[#Equivalent_formulations|trace-reversed field equations]], the vacuum equations can be written as
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| :<math>R_{\mu \nu} = 0 \,.</math>
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| In the case of nonzero cosmological constant, the equations are
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| :<math>R_{\mu \nu} = \Lambda g_{\mu \nu} \,.</math>
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| 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]]. | |
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| [[Manifold]]s with a vanishing [[Ricci tensor]], <math> R_{\mu \nu}=0 </math>, are referred to as [[Ricci-flat manifold]]s and manifolds with a Ricci tensor proportional to the metric as [[Einstein manifold]]s.
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| ==Einstein–Maxwell equations==
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| {{see also|Maxwell's equations in curved spacetime}}
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| If the energy-momentum tensor <math>T_{\mu \nu}</math> is that of an [[electromagnetic field]] in [[free space]], i.e. if the [[electromagnetic stress–energy tensor]]
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| :<math>T^{\alpha \beta} = \, -\frac{1}{\mu_0} \left( F^{\alpha}{}^{\psi} F_{\psi}{}^{\beta} + {1 \over 4} g^{\alpha \beta} F_{\psi\tau} F^{\psi\tau}\right) </math>
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| is used, then the Einstein field equations are called the ''Einstein–Maxwell equations'' (with [[cosmological constant]] Λ, taken to be zero in conventional relativity theory):
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| :<math>R^{\alpha \beta} - {1 \over 2}R g^{\alpha \beta} + g^{\alpha \beta} \Lambda = \frac{8 \pi G}{c^4 \mu_0} \left( F^{\alpha}{}^{\psi} F_{\psi}{}^{\beta} + {1 \over 4} g^{\alpha \beta} F_{\psi\tau} F^{\psi\tau}\right).</math>
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| Additionally, the [[Electromagnetic tensor#The field tensor and relativity|covariant Maxwell Equations]] are also applicable in free space:
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| :<math>F^{\alpha\beta}{}_{;\beta} \, = 0</math>
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| :<math>F_{[\alpha\beta;\gamma]}=\frac{1}{3}\left(F_{\alpha\beta;\gamma} + F_{\beta\gamma;\alpha}+F_{\gamma\alpha;\beta}\right)=\frac{1}{3}\left(F_{\alpha\beta,\gamma} + F_{\beta\gamma,\alpha}+F_{\gamma\alpha,\beta}\right)= 0. \!</math>
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| 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]] ''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 ''A''<sub>α</sub> such that
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| :<math>F_{\alpha\beta} = A_{\alpha;\beta} - A_{\beta;\alpha} = A_{\alpha,\beta} - A_{\beta,\alpha}\!</math>
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| 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=http://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>
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| ==Solutions==
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| {{main|Solutions of the Einstein field equations}}
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| 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>{{cite book | last = Stephani | first = Hans | coauthors = D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt | title = Exact Solutions of Einstein's Field Equations | publisher = [[Cambridge University Press]] | year = 2003 | isbn = 0-521-46136-7 }}</ref>
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| 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]].
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| ==The linearised EFE==
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| {{Main|Linearized Einstein field equations|Linearized gravity}}
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| 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 linearisation procedure can be used to investigate the phenomena of [[gravitational radiation]].
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| ==Polynomial form==
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| 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:
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| :<math>
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| \det(g) = \frac{1}{24} \varepsilon^{\alpha\beta\gamma\delta} \varepsilon^{\kappa\lambda\mu\nu} g_{\alpha\kappa} g_{\beta\lambda} g_{\gamma\mu} g_{\delta\nu}
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| \,</math>
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| using the [[Levi-Civita symbol]]; and the inverse of the metric in 4 dimensions can be written as:
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| :<math>
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| g^{\alpha\kappa} = \frac{1}{6} \varepsilon^{\alpha\beta\gamma\delta} \varepsilon^{\kappa\lambda\mu\nu} g_{\beta\lambda} g_{\gamma\mu} g_{\delta\nu} / \det(g)
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| \,.</math>
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| Substituting this definition of the inverse of the metric into the equations then multiplying both sides by 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>Einstein's Field Equations in Polynomial Form|http://arxiv.org/pdf/gr-qc/0507026.pdf</ref>
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| == See also ==
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| {{Div col|colwidth=25em}}
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| *[[Einstein–Hilbert action]]
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| *[[Equivalence principle]]
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| *[[General relativity]]
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| *[[General relativity resources]]
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| *[[History of general relativity]]
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| *[[Hamilton–Jacobi–Einstein equation]]
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| *[[Mathematics of general relativity]]
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| *[[Ricci calculus]]
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| *[[Solutions of the Einstein field equations]]
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| **[[Exact solutions of Einstein's field equations]]
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| {{Div col end}}
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| == References ==
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| See [[General relativity resources]].
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| <references/>
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| * Aczel, Amir D., 1999. ''God's Equation: Einstein, Relativity, and the Expanding Universe''. Delta Science. A popular account.
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| * [[Charles Misner]], [[Kip Thorne]], and [[John Archibald Wheeler|John Wheeler]], 1973. ''[[Gravitation (book)|Gravitation]]''. W H Freeman.
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| == External links ==
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| * {{springer|title=Einstein equations|id=p/e035210}}
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| * [http://www.black-holes.org/relativity6.html Caltech Tutorial on Relativity] — A simple introduction to Einstein's Field Equations.
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| * [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
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| *[http://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.
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| {{Einstein}}
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| {{Relativity}}
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| {{DEFAULTSORT:Einstein Field Equations}}
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| [[Category:General relativity]]
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| [[Category:Partial differential equations]]
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| [[Category:Equations of physics]]
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| [[Category:Albert Einstein]]
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