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 both tangent and cotangent space at that point in a direct sum. This decomposition extends to all sort of their tensor products. Then we can split fiberwise the twice covariant tensor bundle which $g_{}$ is element of. This construction allows us to define the relevant projection $h_{ab} = g_{ab} + u_au_b$ of the tensor field $g_{ab}$.

 

[PeterDonis]

This construction allows us to define the relevant projection $h_{ab} = g_{ab} + u_au_b$ of the tensor field $g_{ab}$.

I don't see why you need "this construction" to do that. All you're doing is taking a linear combination of tensor fields in the tangent space. You already know you can do that for any tensor fields; you don't need any additional "construction" to make it possible.

 

[cianfa72]

You already know you can do that for any tensor fields; you don't need any additional "construction" to make it possible.

The point I was trying to make is technical. The set of smooth tensor fields of a given type on the smooth manifold M isn't a vector space, rather a module over the ring of $C^{\infty}(M)$ functions. So technically $h_{ab}\xi^a = 0$ holds fiberwise. Then one extends this to the tensor fields themselves.

 

[PeterDonis]

The set of smooth tensor fields of a given type on the smooth manifold M isn't a vector space

Again I'm not sure what you mean. The tangent space at each point of a smooth manifold is a vector space, and that is the vector space in which tensor equations like $h_{ab} = g_{ab} + u_a u_b$ and $h_{ab} \xi^a = 0$ are written. Note that the tangent space itself includes sufficient structure to define covariant derivatives, which are tensor operators. So every equation which is relevant in GR is a tensor equation in the tangent space.

I think you need to find an actual GR reference (textbook or peer-reviewed paper), or at least a differential geometry reference, that expounds whatever point you think you are trying to make.

 

[PeterDonis]

this projection operator I've discussed above is not $h_{ab}$, or even $h^a{}_b$.

In which case I don't see its relevance to what we're discussing in this thread.

I suppose $h^a{}_b$ lives in the tangent space

Of course it does. All tensors live in the tangent space.

it maps a vector in the tangent space at some point p to another vector that is orthogonal to the congruence at point p.

That's what $h^a{}_b$ does, yes. The only clarification I would make is that "the congruence at point p" is a particular vector $\xi^a$ in the tangent space, the one that is tangent at p to the worldline in the congruence that passes through p. The orthogonality property is then simply $h^a{}_b \xi^b = 0$.