Gravitational wave propagation in GR - follow up

[cianfa72]

The subspace of the tangent space that Wald refers to is a vector space in its own right.

Yes. The point is that we have a tensor field (the metric tensor field $g_{ab}$) and a timelike congruence $\xi^a$. The latter, evaluated at each point, allows us to split tangent and cotangent space at that point in a direct sum. This decomposition extends to their tensor products. Then we can split fiberwise the twice covariant tensor bundle. This construction allows us to define the relevant projection $h_{ab} = g_{ab} + u_au_b$ of the tensor field $g_{ab}$.