the Ricci tensor
/ REE-chee /
The full Riemann tensor holds every detail of how spacetime is curved, twenty numbers at each point, more than Einstein's equations need to see. The Ricci tensor is a distilled part of it, obtained by averaging over directions, and it answers a specific, physical question: if you release a small ball of dust to fall freely, does its volume start to shrink, and how fast? It is the piece of curvature that matter directly responds to.
It is a contraction of the Riemann tensor, R_ab = R^c_acb, summing over one upper and one lower index. It is symmetric, R_ab = R_ba, and in four dimensions it has ten independent components, exactly matching the ten of the metric and the ten of the stress-energy tensor. Geometrically, the Ricci tensor controls the rate at which the volume of a small cloud of freely-falling test particles begins to change, so it measures the convergence or divergence of nearby geodesics due to curvature.
The Ricci tensor is the curvature that sits on the left of Einstein's field equations, coupled directly to the matter and energy that source gravity. A subtle and important honesty is required here: in empty space Einstein's equations set R_ab = 0, yet this does NOT mean spacetime is flat. The Riemann tensor has a remaining trace-free part, the Weyl tensor, that can be nonzero in vacuum, which is exactly how a region far outside a star, or a passing gravitational wave, can carry real tidal curvature while the Ricci tensor vanishes.
A cloud of coffee grounds released just above a massive planet does not merely fall; it also gets squeezed in some directions and stretched in others. The overall change in its volume as it falls is governed by the Ricci tensor, and in vacuum outside the planet that volume change is zero even though the tidal distortion is not.
Ricci controls volume change; the shape-only distortion is the Weyl part.
Ricci-flat (R_ab = 0) is not the same as flat. The Schwarzschild spacetime around a black hole is Ricci-flat everywhere outside the mass, yet it is strongly curved, its Riemann tensor is large near the center. Vacuum means no local source, not no curvature.