[Physics] What does the Ricci tensor represent

Question

I'm new to this site so I am sorry if I get the format wrong. However, I'm having some trouble understanding the Ricci tensor. I know it is found by contracting the Riemann tensor and that this is done so that its indicies match with the stress energy tensor in the Einstein field equations. I've tried many online resources, but none give a clear explanation of what the Ricci tensor represents, what I mean by this is for example the Weyl tensor, that is to do with stretching and squeezing of spacetime, however it's just the Ricci tensor I'm having issues with and what it's physical meaning.

Answer 1

It is important to consider the Ricci scalar first. I put here a diagram of a two dimensional sphere with radius $r$. From the pole a vector is transported to the equator and back so that the angle at $A$ is $\pi/2$ Now divide the angle by the surface area of the region enclosed by the parallel transport. This is $1/8^{th}$ the area of the sphere $4\pi r^2$ The result is the Ricci curvature for the region $R~=~1/r^2$ which is the Ricci scalar curvature of the sphere. In general for a parallel transport of a vector around a loop the deviation in the angle of the vectors defines the Ricci curvature as $$ R~=~\frac{\theta}{\cal A}. $$

In general we may think of the Ricci tensor as due to deviation from flatness of a metric so that $$ g_{\mu\nu}~=~\eta_{\mu\nu}~-~\frac{1}{3}R_{\mu\alpha\nu\beta}x^\alpha x^\beta~+~O(x^3), $$ where $\eta_{\mu\nu}$ is the metric for flat spacetime. The metric volume element is $\sqrt{det(g)}$ or often written as $\sqrt{-g}$ and this is then $$ \sqrt{-g}~=~\left(1~-~\frac{1}{6}R_{\alpha\beta}x^\alpha x^\beta\right)\sqrt{-\eta}. $$ This means that the Ricci tensor is associated with changing the volume of a region of space. This is compared to the Weyl tensor that defines a volume preserving diffeomorphism. The Ricci tensor defines then a Ricci flow of the metric $$ \frac{dg_{ij}}{dt}~=~-2R_{ij} $$ In four dimensions we may think of this as the flow of a spatial metric with respect to time. This also has connections to conformal structure.