Ricci scalar

The Ricci scalar R = g^{μν} R_{μν} is the trace of the Ricci tensor against the inverse metric and the simplest scalar invariant constructible from the curvature of a Riemannian or pseudo-Riemannian manifold. As a scalar field on the manifold, it is a coordinate-independent function R(x) whose value at each point characterises the local "average" curvature in a single number. On a 2-sphere of radius a, R = 2/a² is positive everywhere — the sphere is positively curved. On a hyperbolic 2-space of radius a, R = −2/a² is negative everywhere — the saddle-like geometry. On flat Minkowski spacetime, R = 0; on the Schwarzschild solution outside the horizon, R = 0 (vacuum solution), although the Riemann tensor is far from zero — the Ricci scalar is too coarse an invariant to detect tidal curvature.

The Ricci scalar plays a privileged role in the action principle for general relativity. The Einstein-Hilbert action S = (c⁴/16πG) ∫ R √(−g) d⁴x — varied with respect to the metric g^{μν} — gives Einstein's field equations in vacuum, and adding a matter Lagrangian recovers the full field equations with the stress-energy tensor on the right-hand side. The fact that R is the unique non-trivial scalar with the right dimensions and number of derivatives is the structural reason for the form of the field equations. Hilbert's November 1915 derivation by this variational method occurred almost simultaneously with Einstein's; the action is named for both.