Einstein tensor

The Einstein tensor G_{μν} = R_{μν} − (1/2) R g_{μν} is the symmetric (0,2) tensor built from the Ricci tensor and the metric that occupies the geometric side of Einstein's field equations. Its defining property — the property that picks it out from among all linear combinations of R_{μν} and R g_{μν} — is that it is divergence-free: ∇^μ G_{μν} = 0. This identity follows from contracting the second Bianchi identity ∇_λ R^ρ_{σμν} + (cyclic) = 0 twice, and it is what allows G_{μν} to couple consistently to the stress-energy tensor T_{μν}, which is itself conserved (∇^μ T_{μν} = 0) by the requirement of energy-momentum conservation. The Einstein tensor is therefore not a free choice; it is essentially uniquely fixed by the demand for a divergence-free, symmetric (0,2) tensor depending only on the metric and its first two derivatives.

In four spacetime dimensions the Einstein tensor has ten independent components, matching the ten independent components of the symmetric stress-energy tensor it equals on the right side of the field equations. In vacuum (T_{μν} = 0), the field equations reduce to G_{μν} = 0, which because of the trace structure is equivalent to R_{μν} = 0 — the vacuum Einstein equations. The Schwarzschild and Kerr black-hole solutions, gravitational waves in vacuum, and the de Sitter/anti-de Sitter spacetimes are all solutions of R_{μν} = 0 (with or without a cosmological-constant term Λ g_{μν}). Einstein introduced the tensor in his November 1915 Feldgleichungen der Gravitation paper.