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Tetrads in General Relativity

X. Curvature

Einstein Tensor

   An especially useful form of the second Bianchi identity comes from contracting twice on (10.14) to obtain the so-called contracted Bianchi identity (first derived by the mathematician Aural Voss, 1880):

10.22

\[2{D_\mu }{R^\mu }_\nu - {D_\nu }R = 0\]

where ${R_{\mu \nu }}$ is the Ricci (10.17) and $R$ the Ricci scalar (10.18). These  four geometric identities are valid for arbitrary metric. They derive from the general covariance of GRT under coordinate transformations; see Bianchi Identities.

   By defining the particular combination, known as the Einstein tensor

10.23

\[{G_{\mu \nu }}: = {R_{\mu \nu }} - \frac{1}{2}R{g_{\mu \nu }} \qquad \operatorname{G} ({{\mathbf{g}}_\mu }): = \operatorname{Ric} ({{\mathbf{g}}_\mu }) - \frac{1}{2}R{{\mathbf{g}}_\mu }\]

it is simply seen that the contracted Bianchi identity (10.22) is equivalent to

10.24

\[{D_\mu }{G^\mu }_\nu = {D_\mu }{G^{\mu \nu }} = 0 \qquad {D_\mu }{\text{G}}({{\mathbf{g}}^\mu }) = 0\]

That is, the Einstein tensor is covariantly conserved on account of the contracted Bianchi identity (10.22) that holds for any torsionless spacetime.

   Unique properties of the Einstein tensor are:

  1. identically zero when spacetime is flat;
  2. symmetric due to the symmetry of the Ricci tensor and the metric;
  3. linearly related to the Riemann curvature tensor;
  4. constructed from the Riemann and the metric;
  5. determined by first and second derivatives of the metric only;
  6. covariantly conserved.

In $d=4$ dimensions, the most general symmetric, divergence-free tensor of rank two constructed from the metric and its first two derivatives is $a G_{\mu\nu} + b g_{\mu\nu}$ with constants $a$ and $b$. [Lovelock's Theorem]