Einstein tensor

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 that is consistent with conservation of energy and momentum.

Definition

The Einstein tensor <math>\boldsymbol{G}</math> is a tensor of order 2 defined over pseudo-Riemannian manifolds. In index-free notation it is defined as

<math display="block">\boldsymbol{G}=\boldsymbol{R}-\frac{1}{2}\boldsymbol{g}R,</math>

where <math>\boldsymbol{R}</math> is the Ricci tensor, <math>\boldsymbol{g}</math> is the metric tensor and <math>R</math> is the scalar curvature, which is computed as the trace of the Ricci tensor <math>R_{\mu \nu}</math> by . In component form, the previous equation reads as

The Einstein tensor is symmetric

and, like the on-shell stress–energy tensor, has zero divergence:

<math display="block">\nabla_\mu G^{\mu\nu} = 0\,.</math>

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:

<math display="block">\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}</math>

where <math>\delta^\alpha_\beta</math> is the Kronecker tensor and the Christoffel symbol <math>\Gamma^\alpha{}_{\beta\gamma}</math> is defined as

<math display="block">\Gamma^\alpha{}_{\beta\gamma} = \frac{1}{2} g^{\alpha\epsilon}\left(g_{\beta\epsilon,\gamma} + g_{\gamma\epsilon,\beta} - g_{\beta\gamma,\epsilon}\right).</math>

and terms of the form <math>\Gamma ^\alpha _{\beta \gamma, \mu}</math> or <math>g_{\beta\gamma,\mu}</math> represent partial derivatives in the μ-direction, e.g.:

<math display="block">\Gamma^\alpha{}_{\beta\gamma, \mu} = \partial _\mu \Gamma^\alpha{}_{\beta\gamma} =

\frac{\partial}{\partial x^\mu}

\Gamma^\alpha{}_{\beta\gamma}</math>

Before cancellations, this formula results in <math>2 \times (6 + 6 + 9 + 9) = 60</math> 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:

<math display="block">\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}</math>

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

<math display="block">g_{\alpha[\beta,\gamma]\epsilon} \, = \frac{1}{2} \left(g_{\alpha\beta,\gamma\epsilon} - g_{\alpha\gamma,\beta\epsilon}\right).</math>

Trace

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

<math display="block">\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) =