[Physics] The Weyl tensor and gravitational waves

Question

How exactly is the Weyl tensor is connected with information about gravitational waves? And what are physical reasons for that?

Answer 1

The Weyl tensor is the trace-free part of the Riemann tensor. The latter describes the curvature of spacetime. In the absence of sources, the trace part of the Riemann tensor will vanish due to the Einstein equations, but the Weyl tensor can still be non-zero. This is the case for gravitational waves propagating in vacuum.

The physical reason is that even in the absence of sources, there can be curvature of space time. Gravitational waves represent small variations of curvature, i.e. non-zero curvature. An analogous situation would be electromagnetic waves propagating in vacuum. One can describe them by the source-free Maxwell equations without specifying anything about charges or currents that might have produced them.