Einstein tensor

In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci curvature) 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 that is consistent with conservation of energy and momentum.

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Definition The Einstein tensor $\backslash boldsymbol\{G\}$ is a tensor of order 2 defined over pseudo-Riemannian manifolds. In index-free notation it is defined as $$\backslash boldsymbol\{G\}=\backslash boldsymbol\{R\}-\backslash frac\{1\}\{2\}\backslash boldsymbol\{g\}R,$$ where $\backslash boldsymbol\{R\}$ is the Ricci tensor, $\backslash boldsymbol\{g\}$ is the metric tensor and $R$ is the scalar curvature, which is computed as the trace of the Ricci tensor $R\_\{\backslash mu\; \backslash nu\}$ by . In component form, the previous equation reads as $$G\_\{\backslash mu\backslash nu\}\; =\; R\_\{\backslash mu\backslash nu\}\; -\; \{1\backslash over2\}\; g\_\{\backslash mu\backslash nu\}R\; .$$

The Einstein tensor is symmetric $$G\_\{\backslash mu\backslash nu\}\; =\; G\_\{\backslash nu\backslash mu\}$$ and, like the on-shell stress–energy tensor, has zero divergence: $$\backslash nabla\_\backslash mu\; G^\{\backslash mu\backslash nu\}\; =\; 0\backslash ,.$$

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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: $$\backslash begin\{align\}$$

 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} \\
   &= \left(\delta^\gamma_\alpha \delta^\zeta_\beta - \frac{1}{2} g_{\alpha\beta}g^{\gamma\zeta}\right) R_{\gamma\zeta} \\
   &= \left(\delta^\gamma_\alpha \delta^\zeta_\beta - \frac{1}{2} g_{\alpha\beta}g^{\gamma\zeta}\right)\left(\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}\right), \\[2pt]
 G^{\alpha\beta}
   &= \left(g^{\alpha\gamma} g^{\beta\zeta} - \frac{1}{2} g^{\alpha\beta}g^{\gamma\zeta}\right)\left(\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}\right),
     

\end{align} where $\backslash delta^\backslash alpha\_\backslash beta$ is the Kronecker tensor and the Christoffel symbol $\backslash Gamma^\backslash alpha\{\}\_\{\backslash beta\backslash gamma\}$ is defined as $$\backslash Gamma^\backslash alpha\{\}\_\{\backslash beta\backslash gamma\}\; =\; \backslash frac\{1\}\{2\}\; g^\{\backslash alpha\backslash epsilon\}\backslash left(g\_\{\backslash beta\backslash epsilon,\backslash gamma\}\; +\; g\_\{\backslash gamma\backslash epsilon,\backslash beta\}\; -\; g\_\{\backslash beta\backslash gamma,\backslash epsilon\}\backslash right).$$ and terms of the form $\backslash Gamma\; ^\backslash alpha\; \_\{\backslash beta\; \backslash gamma,\; \backslash mu\}$ or $g\_\{\backslash beta\backslash gamma,\backslash mu\}$ represent partial derivatives in the μ-direction, e.g.: $$\backslash Gamma^\backslash alpha\{\}\_\{\backslash beta\backslash gamma,\; \backslash mu\}\; =\; \backslash partial\; \_\backslash mu\; \backslash Gamma^\backslash alpha\{\}\_\{\backslash beta\backslash gamma\}\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; x^\backslash mu\}\; \backslash Gamma^\backslash alpha\{\}\_\{\backslash beta\backslash gamma\}$$

Before cancellations, this formula results in $2\; \backslash times\; (6\; +\; 6\; +\; 9\; +\; 9)\; =\; 60$ 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: $$\backslash begin\{align\}$$

 G_{\alpha\beta}
   & = g^{\gamma\mu}\left[ g_{\gamma[\beta,\mu]\alpha} + g_{\alpha[\mu,\beta]\gamma} - \frac{1}{2} g_{\alpha\beta} g^{\epsilon\sigma} \left(g_{\epsilon[\mu,\sigma]\gamma} + g_{\gamma[\sigma,\mu]\epsilon}\right)\right] \\
   & = g^{\gamma\mu} \left(\delta^\epsilon_\alpha \delta^\sigma_\beta - \frac{1}{2} g^{\epsilon\sigma}g_{\alpha\beta}\right)\left(g_{\epsilon[\mu,\sigma]\gamma} + g_{\gamma[\sigma,\mu]\epsilon}\right),
     

\end{align} where square brackets conventionally denote antisymmetrization over bracketed indices, i.e. $$g\_\{\backslash alpha\backslash beta,\backslash gamma\backslash epsilon\}\; \backslash ,\; =\; \backslash frac\{1\}\{2\}\; \backslash left(g\_\{\backslash alpha\backslash beta,\backslash gamma\backslash epsilon\}\; -\; g\_\{\backslash alpha\backslash gamma,\backslash beta\backslash epsilon\}\backslash right).$$

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Trace The trace of the Einstein tensor can be computed by contracting the equation in the definition with the metric tensor . In $n$ dimensions (of arbitrary signature):

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