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X. Curvature
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):
\[2{D_\mu }{R^\mu }_\nu - {D_\nu }R =
0\]
where ${R_{\mu \nu }}$ is the Ricci (10.17) and $R$ the curvature scalar
(10.18).
By
defining the particular combination, known as the Einstein tensor
\[{G_{\mu \nu }}: = {R_{\mu \nu }} -
\frac{1}{2}R{g_{\mu \nu }} \qquad \operatorname{G} ({{\mathbf{g}}_\mu }): =
\operatorname{Ric} ({{\mathbf{g}}_\mu }) - \frac{1}{2}R{{\mathbf{g}}_\mu }\]
it is simply seen that the contracted Bianchi identity (10.22) is equivalent
to
\[{D_\mu }{G^\mu }_\nu = {D_\mu }{G^{\mu
\nu }} = 0 \qquad {D_\mu }{\text{G}}({{\mathbf{g}}^\mu }) = 0\]
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:
- identically zero when spacetime is flat;
- symmetric due to the symmetry of the Ricci tensor and the metric;
- linearly related to the Riemann curvature tensor;
- constructed from the Riemann and the metric;
- determined by first and second derivatives of the metric
only;
- covariantly conserved.