# What is the difference between the Ricci tensor and the scalar curvature?

What is the difference of physics meaning (for beginner) between the Ricci tensor $$R_{\mu\nu}$$ and the scalar curvature $$R$$ terms ?

Wikipedia gives the same explanation for the two, as we could see below, so it does not help to understand the difference between the two:

Wikipedia, Ricci Curvature:

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a narrow conical piece of a small geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space

and

Wikipedia, Scalar Curvature:

Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space

• Perhaps you could erase any reference to Einstein field equation in your question since the is a general Differential geometry topic Mar 16, 2020 at 8:57
• all right, I do that Mar 16, 2020 at 16:43

The factor $$k_i$$ in arc length can be regarded as the deviation from Euclidean space. In a small enough region, the space becomes close Euclidean ($$k_i \approx 1$$). Curvature corresponds to the second derivative of $$k_i$$ as radius $$R_i$$ goes to zero. For a two dimensional space, two directions and magnitudes are necessary. In four dimensional spacetime four are needed, meaning a rank-4 tensor, Riemann. Contract with two indices and that gives you Ricci, and contract the remaining two indices to get scalar curvature.