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| {{General relativity}}
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| In [[differential geometry]], the '''Einstein tensor''' (also '''trace-reversed Ricci tensor'''), named after [[Albert Einstein]], is used to express the [[curvature]] of a [[Riemannian manifold]]. In [[general relativity]], the Einstein tensor occurs in the [[Einstein field equations]] for [[gravitation]] describing [[spacetime]] curvature in a manner consistent with energy considerations.
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| == Definition ==
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| The Einstein tensor <math>\mathbf{G}</math> is a rank 2 [[tensor]] defined over [[Riemannian manifold]]s. In index-free notation it is defined as
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| ::<math>\mathbf{G}=\mathbf{R}-\frac{1}{2}\mathbf{g}R,</math>
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| where <math>\mathbf{R}</math> is the [[Ricci tensor]], <math>\mathbf{g}</math> is the [[metric tensor]] and <math>R</math> is the [[scalar curvature]]. In component form, the previous equation reads as | |
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| ::<math>G_{\mu\nu} = R_{\mu\nu} - {1\over2} g_{\mu\nu}R.</math>
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| The Einstein tensor is symmetric | |
| ::<math>G_{\mu\nu} = G_{\nu\mu}\,</math>
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| and, like the [[stress–energy tensor]], divergenceless | |
| ::<math>G^{\mu\nu}{}_{; \nu} = 0\,.</math>
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| == Explicit form ==
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| The [[Ricci tensor]] depends only on the [[metric tensor]], so the Einstein tensor can be defined directly with just the metric tensor. However, this expression is complex and rarely quoted in textbooks. The complexity of this expression can be shown using the formula for the Ricci tensor in terms of [[Christoffel symbols]]:
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| ::<math>
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| \begin{align}
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| G_{\alpha\beta} &= R_{\alpha\beta} - \frac{1}{2} g_{\alpha\beta} R \\
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| &= R_{\alpha\beta} - \frac{1}{2} g_{\alpha\beta} g^{\gamma\zeta} R_{\gamma\zeta} \\
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| &= (\delta^\gamma_\alpha \delta^\zeta_\beta - \frac{1}{2} g_{\alpha\beta}g^{\gamma\zeta}) R_{\gamma\zeta} \\
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| &= (\delta^\gamma_\alpha \delta^\zeta_\beta - \frac{1}{2} g_{\alpha\beta}g^{\gamma\zeta})(\Gamma^\epsilon_{\gamma\zeta,\epsilon} - \Gamma^\epsilon_{\gamma\epsilon,\zeta} + \Gamma^\epsilon_{\epsilon\sigma} \Gamma^\sigma_{\gamma\zeta} - \Gamma^\epsilon_{\zeta\sigma} \Gamma^\sigma_{\epsilon\gamma}),
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| \end{align}
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| </math>
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| where <math>\delta^\alpha_\beta</math> is the [[Kronecker tensor]] and the Christoffel symbol <math>\Gamma^\alpha_{\beta\gamma}</math> is defined as
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| ::<math>\Gamma^\alpha_{\beta\gamma} = \frac{1}{2} g^{\alpha\epsilon}(g_{\beta\epsilon,\gamma} + g_{\gamma\epsilon,\beta} - g_{\beta\gamma,\epsilon}).</math>
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| Before cancellations, this formula results in <math>2 \times (6+6+9+9) = 60</math> individual terms. Cancellations bring this number down somewhat. <!-- exactly how much? -->
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| In the special case of a locally [[inertial reference frame]] near a point, the first derivatives of the metric tensor vanish and the component form of the Einstein tensor is considerably simplified:
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| :: <math>\begin{align}G_{\alpha\beta} & = g^{\gamma\mu}\bigl[ g_{\gamma[\beta,\mu]\alpha} + g_{\alpha[\mu,\beta]\gamma} - \frac{1}{2} g_{\alpha\beta} g^{\epsilon\sigma} (g_{\epsilon[\mu,\sigma]\gamma} + g_{\gamma[\sigma,\mu]\epsilon})\bigr] \\ & = g^{\gamma\mu} (\delta^\epsilon_\alpha \delta^\sigma_\beta - \frac{1}{2} g^{\epsilon\sigma}g_{\alpha\beta})(g_{\epsilon[\mu,\sigma]\gamma} + g_{\gamma[\sigma,\mu]\epsilon}),\end{align}</math>
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| where square brackets conventionally denote [[Antisymmetric tensor|antisymmetrization]] over bracketed indices, i.e.
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| :: <math>g_{\alpha[\beta,\gamma]\epsilon} \, = \frac{1}{2} (g_{\alpha\beta,\gamma\epsilon} - g_{\alpha\gamma,\beta\epsilon}).</math>
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| == Trace ==
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| The [[trace (linear algebra)|trace]] of the Einstein tensor can be computed by [[Tensor contraction|contract]]ing the equation in the [[#Definition|definition]] with the [[metric tensor]] <math>g^{\mu\nu}</math>. In <math>n</math> dimensions (of arbitrary signature):
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| ::<math>\begin{align}g^{\mu\nu}G_{\mu\nu} &= g^{\mu\nu}R_{\mu\nu} - {1\over2} g^{\mu\nu}g_{\mu\nu}R \\ G &= R - {1\over2} (nR) \\ G &= {{2-n}\over2}R\end{align}</math>
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| The special case of 4 dimensions in physics (3 space, 1 time) gives <math>G\,</math>, the trace of the Einstein tensor, as the negative of <math>R\,</math>, the [[Ricci tensor]]'s trace. Thus another name for the Einstein tensor is the ''trace-reversed Ricci tensor''.
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| == Use in general relativity ==
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| The Einstein tensor allows the [[Einstein field equations]] (without a [[cosmological constant]]) to be written in the concise form:
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| ::<math>G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}.</math>
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| which becomes in [[geometrized units]],
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| ::<math>G_{\mu\nu} = 8 \pi \, T_{\mu\nu}.</math>
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| From the [[#Explicit form|explicit form of the Einstein tensor]], the Einstein tensor is a [[nonlinear]] function of the metric tensor, but is linear in the second [[partial derivative]]s of the metric. As a symmetric 2nd rank tensor, the Einstein tensor has 10 independent components in a 4-dimensional space. It follows that the Einstein field equations are a set of 10 [[Types of differential equations|quasilinear]] second-order partial differential equations for the metric tensor.
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| The [[Bianchi identities]] can also be easily expressed with the aid of the Einstein tensor:
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| ::<math> \nabla_{\mu} G^{\mu\nu} = 0.</math>
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| The Bianchi identities automatically ensure the covariant conservation of the [[stress–energy tensor]] in curved spacetimes:
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| ::<math>\nabla_{\mu} T^{\mu\nu} = 0.</math>
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| The geometric significance of the Einstein tensor is highlighted by this identity. In coordinate frames respecting the gauge condition
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| ::<math>\Gamma^{\rho}_{\mu\nu} G^{\mu\nu} = 0</math>
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| an ordinary conservation law for the stress tensor density can be stated:
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| ::<math>\partial_{\mu}(\sqrt{g} T^{\mu\nu}) = 0</math>.
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| The Einstein tensor plays the role of distinguishing these frames.
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| ==Uniqueness==
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| David Lovelock has shown that, in a four dimensional [[differentiable manifold]], the Einstein tensor is the only [[tensor]]ial and [[divergence]]-free function of the <math>g_{\mu\nu}</math> and at most their first and second partial derivatives.<ref>{{Cite journal
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| |last=Lovelock |first=D.
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| |title=The Einstein Tensor and Its Generalizations
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| |journal=Journal of Mathematical Physics
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| |year=1971 |volume=12 |issue=3 |pages=498–502
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| |url=http://jmp.aip.org/resource/1/jmapaq/v12/i3/p498_s1?isAuthorized=nof
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| |ref=
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| |bibcode = 1971JMP....12..498L |doi = 10.1063/1.1665613 }}</ref>
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| <ref>{{Cite journal
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| |last=Lovelock |first=D.
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| |title=The uniqueness of the Einstein field equations in a four-dimensional space
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| |journal=Archive for Rational Mechanics and Analysis
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| |year=1969 |volume=33 |issue=1 |pages=54–70
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| |url=http://link.springer.com/article/10.1007%2FBF00248156?LI=true
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| |ref=
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| |bibcode = 1969ArRMA..33...54L |doi = 10.1007/BF00248156 }}</ref>
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| <ref>{{Cite journal
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| |last=Farhoudi |first=M.
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| |title=Lovelock Tensor as Generalized Einstein Tensor
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| |journal=General Relativity and Gravitation
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| |year=2009 |volume=41 |issue=1 |pages=17–29
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| |url=http://arxiv.org/abs/gr-qc/9510060
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| |ref=
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| }}</ref>
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| <ref>{{Cite book
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| | author=[[Wolfgang Rindler|Rindler, Wolfgang]]
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| | title=Relativity: Special, General, and Cosmological
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| | publisher=Oxford University Press
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| | year=2001
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| | isbn=0-19-850836-0
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| | page = 299
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| }}</ref> We also note that the [[Einstein field equation]] is not the only equation which satisfies the three conditions:<ref>{{Cite book
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| | author=[[Bernard Schutz|Schutz, Bernard]]
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| | title=A First Course in General Relativity
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| | publisher=Cambridge University Press
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| | edition=2
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| | date=May 31, 2009
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| | isbn=0-521-88705-4
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| | page = 185
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| }}</ref>
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| # Resemble but generalize [[Gauss's law for gravity|Newton-Poisson gravitational equation]]
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| # Applied to all coordinate systems, and
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| # Guarantee local covariant conservation of energy–momentum for any metric tensor.
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| Many alternatives theory have been proposed, like [[Einstein–Cartan theory]], etc... that also satisfies the above conditions.
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| == See also ==
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| {{Portal|Physics}}
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| * [[Mathematics of general relativity]]
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| * [[General relativity resources]]
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| ==References==
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| {{reflist}}
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| *{{cite book
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| | last = Ohanian
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| | first = Hans C.
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| | coauthors = Remo Ruffini
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| | title = Gravitation and Spacetime
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| | edition = Second edition
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| | publisher = [[W. W. Norton & Company]]
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| | year = 1994
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| | isbn = 0-393-96501-5
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| }}
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| *{{cite book
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| | last = Martin
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| | first = John Legat
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| | title = General Relativity: A First Course for Physicists
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| | edition = Revised edition
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| | series = Prentice Hall International Series in Physics and Applied Physics
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| | year = 1995
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| | publisher = [[Prentice Hall]]
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| | isbn = 0-13-291196-5
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| }}
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| {{tensors}}
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| [[Category:Tensors in general relativity]]
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| [[Category:Albert Einstein|Tensor]]
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