Ricci tensor

The Ricci tensor R_{μν} is the contraction of the Riemann curvature tensor over its first and third indices: R_{μν} = R^λ_{μλν} = g^{λρ} R_{ρμλν}. It is a symmetric (0,2) tensor field on the manifold; in four spacetime dimensions it has ten algebraically independent components, fewer than the Riemann tensor's 20 because the trace operation collapses the algebraic structure. Geometrically, R_{μν} measures how the volume of a small geodesic ball deviates from the corresponding volume in flat space along the directions selected by μ and ν — the trace part of curvature, the part that controls volume contraction (or expansion) under geodesic flow, as distinct from the trace-free Weyl part of Riemann that controls tidal shearing.

The physical importance of the Ricci tensor is that it appears directly in the geometric side of Einstein's field equations: G_{μν} = R_{μν} − (1/2) R g_{μν} = (8πG/c⁴) T_{μν}. Among all tensors derivable from the metric and its first two derivatives, the combination on the left-hand side is essentially uniquely picked out by the requirement of being symmetric, having the right number of derivatives, and being divergence-free (so that it can couple consistently to a conserved stress-energy tensor T_{μν}). The Ricci tensor is named after Gregorio Ricci-Curbastro, Levi-Civita's collaborator on the 1900 Méthodes de calcul différentiel absolu. The trace of the Ricci tensor with the inverse metric gives the Ricci scalar R, the simplest scalar invariant of curvature.