General relativity
${\displaystyle G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\kappa T_{\mu \nu }}$
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Indifferential geometry, theEinstein tensor (named afterAlbert Einstein; also known as thetrace-reversedRicci tensor) is used to express thecurvature of apseudo-Riemannian manifold. Ingeneral relativity, it occurs in theEinstein field equations forgravitation that describespacetime curvature in a manner that is consistent with conservation of energy and momentum.

The Einstein tensor${\displaystyle \mathit{G}}$ is atensor of order 2 defined overpseudo-Riemannian manifolds. In index-free notation it is defined as$${\displaystyle \mathit{G}=\mathit{R}-\frac{1}{2}\mathit{g}R,}$$where${\displaystyle \mathit{R}}$ is theRicci tensor,${\displaystyle \mathit{g}}$ is themetric tensor and${\displaystyle R}$ is thescalar curvature, which is computed as thetrace of the Ricci tensor${\displaystyle R_{\mu \nu }}$ by⁠${\displaystyle R=g^{\mu \nu }{R}_{\mu \nu }}$⁠. In component form, the previous equation reads as$${\displaystyle G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R.}$$

The Einstein tensor is symmetric$${\displaystyle G_{\mu \nu }=G_{\nu \mu }}$$and, like theon shellstress–energy tensor, has zerodivergence:$${\displaystyle {\mathrm{\nabla }}_{\mu }{G}^{\mu \nu }=0.}$$

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 ofChristoffel symbols:where${\displaystyle {\delta }_{\beta }^{\alpha }}$ is theKronecker tensor and the Christoffel symbol${\displaystyle {\mathrm{\Gamma }}^{\alpha }{}_{\beta \gamma }}$ is defined as$${\displaystyle {\mathrm{\Gamma }}^{\alpha }{}_{\beta \gamma }=\frac{1}{2}{g}^{\alpha ϵ}\left(g_{\beta ϵ,\gamma }+g_{\gamma ϵ,\beta }-g_{\beta \gamma ,ϵ}\right).}$$and terms of the form${\displaystyle {\mathrm{\Gamma }}_{\beta \gamma ,\mu }^{\alpha }}$ or${\displaystyle g_{\beta \gamma ,\mu }}$ represent partial derivatives in theμ-direction, e.g.:$${\displaystyle {\mathrm{\Gamma }}^{\alpha }{}_{\beta \gamma ,\mu }={\mathrm{\partial }}_{\mu }{\mathrm{\Gamma }}^{\alpha }{}_{\beta \gamma }=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\mu }}{\mathrm{\Gamma }}^{\alpha }{}_{\beta \gamma }}$$

Before cancellations, this formula results in${\displaystyle 2\times (6+6+9+9)=60}$ individual terms. Cancellations bring this number down somewhat.

In the special case of a locallyinertial reference frame near a point, the first derivatives of the metric tensor vanish and the component form of the Einstein tensor is considerably simplified:

where square brackets conventionally denoteantisymmetrization over bracketed indices, i.e.$${\displaystyle g_{\alpha [\beta ,\gamma ]ϵ}=\frac{1}{2}\left(g_{\alpha \beta ,\gamma ϵ}-g_{\alpha \gamma ,\beta ϵ}\right).}$$

Thetrace of the Einstein tensor can be computed bycontracting the equation in thedefinition with themetric tensor⁠${\displaystyle g^{\mu \nu }}$⁠. In${\displaystyle n}$ dimensions (of arbitrary signature):

Therefore, in the special case of⁠${\displaystyle n=4}$⁠ dimensions,⁠${\displaystyle G=-R}$⁠. That is, the trace of the Einstein tensor is the negative of theRicci tensor's trace. Thus, another name for the Einstein tensor is thetrace-reversed Ricci tensor. This${\displaystyle n=4}$ case is especially relevant in thetheory of general relativity.

The Einstein tensor allows theEinstein field equations to be written in the concise form:$${\displaystyle G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\kappa T_{\mu \nu },}$$where${\displaystyle \mathrm{\Lambda }}$ is thecosmological constant and${\displaystyle \kappa }$ is theEinstein gravitational constant.

From theexplicit form of the Einstein tensor, the Einstein tensor is anonlinear function of the metric tensor, but is linear in the secondpartial derivatives of the metric. As a symmetric order-2 tensor, the Einstein tensor has 10 independent components in a 4-dimensional space. It follows that the Einstein field equations are a set of 10quasilinear second-order partial differential equations for the metric tensor.

Thecontracted Bianchi identities can also be easily expressed with the aid of the Einstein tensor:$${\displaystyle {\mathrm{\nabla }}_{\mu }{G}^{\mu \nu }=0.}$$

The (contracted) Bianchi identities automatically ensure the covariant conservation of thestress–energy tensor in curved spacetimes:$${\displaystyle {\mathrm{\nabla }}_{\mu }{T}^{\mu \nu }=0.}$$

The physical significance of the Einstein tensor is highlighted by this identity. In terms of the densitized stress tensor contracted on aKilling vector⁠${\displaystyle {\xi }^{\mu }}$⁠, an ordinary conservation law holds:$${\displaystyle {\mathrm{\partial }}_{\mu }\left(\sqrt{-g}{T}^{\mu }{}_{\nu }{\xi }^{\nu }\right)=0.}$$

David Lovelock has shown that, in a four-dimensionaldifferentiable manifold, the Einstein tensor is the onlytensorial anddivergence-free function of the${\displaystyle g_{\mu \nu }}$ and at most their first and second partial derivatives.^{[1][2][3][4][5]}

However, theEinstein field equation is not the only equation which satisfies the three conditions:^[6]

1. Resemble but generalizeNewton–Poisson gravitational equation
2. Apply to all coordinate systems, and
3. Guarantee local covariant conservation of energy–momentum for any metric tensor.

Many alternative theories have been proposed, such as theEinstein–Cartan theory, that also satisfy the above conditions.

• Contracted Bianchi identities
• Vermeil's theorem
• Mathematics of general relativity
• General relativity resources

• Ohanian, Hans C.; Remo Ruffini (1994).Gravitation and Spacetime (Second ed.).W. W. Norton & Company.ISBN 978-0-393-96501-8.
• Martin, John Legat (1995).General Relativity: A First Course for Physicists. Prentice Hall International Series in Physics and Applied Physics (Revised ed.).Prentice Hall.ISBN 978-0-13-291196-2.

• Tensors in general relativity
• Albert Einstein

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