Visualizing Ricci Tensor

Question

By definition Ricci Tensor is a Tensor formed by contracting two indices of Riemann Tensor. Riemann Tensor can be visualized in terms of a curve, a vector is moving and orientation of the initial and final vector. However, the operation of contraction of indices which give rise to Ricci Tensor doesn't give any visual sense.

I would like you guys to help me visualize Ricci Tensor.

This post imported from StackExchange Physics at 2014-03-22 16:50 (UCT), posted by SE-user Qubit

Answer 1

I've always liked John Baez' coffee-grounds visualization of the Ricci and Weyl tensors.

Basically you look at a small ball of free falling coffee grounds and use them to test how the geodesics are moving with respect to each other. Ricci curvature measures the change in volume of this ball. Weyl curvature measures the stretching and squashing.

This post imported from StackExchange Physics at 2014-03-22 16:50 (UCT), posted by SE-user twistor59

Answer 2

Direct geometric meaning.

This post imported from StackExchange Physics at 2014-03-22 16:50 (UCT), posted by SE-user NiftyKitty95