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In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field equations for gravitation that describe spacetime curvature in a manner consistent with energy.

Definition

The Einstein tensor $$\mathbf{G} is a rank 2 tensor defined over pseudo-Riemannian manifolds. In index-free notation it is defined as \( \mathbf{G}=\mathbf{R}-\frac{1}{2}\mathbf{g}R,$$

where $$\mathbf{R}$$ is the Ricci tensor, $$\mathbf{g}$$ is the metric tensor and R is the scalar curvature. In component form, the previous equation reads as

$$G_{\mu\nu} = R_{\mu\nu} - {1\over2} g_{\mu\nu}R.$$

The Einstein tensor is symmetric

$$G_{\mu\nu} = G_{\nu\mu}\,$$

and, like the stress–energy tensor, divergenceless

$$G^{\mu\nu}{}_{; \nu} = 0\,.$$

Explicit form

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:

\begin{align} G_{\alpha\beta} &= R_{\alpha\beta} - \frac{1}{2} g_{\alpha\beta} R \\ &= R_{\alpha\beta} - \frac{1}{2} g_{\alpha\beta} g^{\gamma\zeta} R_{\gamma\zeta} \\ &= (\delta^\gamma_\alpha \delta^\zeta_\beta - \frac{1}{2} g_{\alpha\beta}g^{\gamma\zeta}) R_{\gamma\zeta} \\ &= (\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}), \end{align}

where $$\delta^\alpha_\beta$$ is the Kronecker tensor and the Christoffel symbol $$\Gamma^\alpha_{\beta\gamma}$$ is defined as

$$\Gamma^\alpha_{\beta\gamma} = \frac{1}{2} g^{\alpha\epsilon}(g_{\beta\epsilon,\gamma} + g_{\gamma\epsilon,\beta} - g_{\beta\gamma,\epsilon}).$$

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

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:

\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}

where square brackets conventionally denote antisymmetrization over bracketed indices, i.e.

$$g_{\alpha[\beta,\gamma]\epsilon} \, = \frac{1}{2} (g_{\alpha\beta,\gamma\epsilon} - g_{\alpha\gamma,\beta\epsilon}).$$

Trace

The trace of the Einstein tensor can be computed by contracting the equation in the definition with the metric tensor $$g^{\mu\nu}$$ . In n dimensions (of arbitrary signature):

\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}

The special case of 4 dimensions in physics (3 space, 1 time) gives $$G\,,$$ the trace of the Einstein tensor, as the negative of $$R\,,$$ the Ricci tensor's trace. Thus another name for the Einstein tensor is the trace-reversed Ricci tensor.

Use in general relativity

The Einstein tensor allows the Einstein field equations (without a cosmological constant) to be written in the concise form:

$$G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}.$$

which becomes in geometrized units (i.e. c = G = 1),

$$G_{\mu\nu} = 8 \pi \, T_{\mu\nu}.$$

From the explicit form of the Einstein tensor, the Einstein tensor is a nonlinear function of the metric tensor, but is linear in the second partial derivatives 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 quasilinear second-order partial differential equations for the metric tensor.

The Bianchi identities can also be easily expressed with the aid of the Einstein tensor:

$$\nabla_{\mu} G^{\mu\nu} = 0.$$

The Bianchi identities automatically ensure the covariant conservation of the stress–energy tensor in curved spacetimes:

$$\nabla_{\mu} T^{\mu\nu} = 0.$$

The physical significance of the Einstein tensor is highlighted by this identity. In terms of the densitized stress tensor contracted on a Killing vector $$\xi^\mu$$ , an ordinary conservation law holds:

$$\partial_{\mu}(\sqrt{-g} T^\mu{}_\nu \xi^\nu) = 0.$$

Uniqueness

David Lovelock has shown that, in a four-dimensional differentiable manifold, the Einstein tensor is the only tensorial and divergence-free function of the g_{\mu\nu} and at most their first and second partial derivatives.   

However, the Einstein field equation is not the only equation which satisfies the three conditions:

Resemble but generalize Newton-Poisson gravitational equation
Apply to all coordinate systems, and
Guarantee local covariant conservation of energy–momentum for any metric tensor.

Many alternative theories have been proposed, like Einstein–Cartan theory, etc... that also satisfy the above conditions.
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References

Lovelock, D. (1971). "The Einstein Tensor and Its Generalizations". Journal of Mathematical Physics 12 (3): 498–502. Bibcode:1971JMP....12..498L. doi:10.1063/1.1665613.
Lovelock, D. (1969). "The uniqueness of the Einstein field equations in a four-dimensional space". Archive for Rational Mechanics and Analysis 33 (1): 54–70. Bibcode:1969ArRMA..33...54L. doi:10.1007/BF00248156.
Farhoudi, M. (2009). "Lovelock Tensor as Generalized Einstein Tensor". General Relativity and Gravitation 41 (1): 17–29.
Rindler, Wolfgang (2001). Relativity: Special, General, and Cosmological. Oxford University Press. p. 299. ISBN 0-19-850836-0.

Schutz, Bernard (May 31, 2009). A First Course in General Relativity (2 ed.). Cambridge University Press. p. 185. ISBN 0-521-88705-4.

Ohanian, Hans C.; Remo Ruffini (1994). Gravitation and Spacetime (Second edition ed.). W. W. Norton & Company. ISBN 0-393-96501-5.
Martin, John Legat (1995). General Relativity: A First Course for Physicists. Prentice Hall International Series in Physics and Applied Physics (Revised edition ed.). Prentice Hall. ISBN 0-13-291196-5.

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