GR timelike vector show properties/decomposition holds

[binbagsss]

Homework Statement

Homework Equations

above

The Attempt at a Solution

I'm totally stuck as to how to get to started , how to use the hint to decompose a vector given a preferred spacetime direction. All I know is that any tensor can be decomposed into a anti symmetric and a symmetric tensor. Any hint to get me started greatly appreciated.

thanks

 

[andrewkirk]

It's a strange question. The thing the student is 'required to show' is easy and only involves the application of the definitions given and basic algebra, without needing to know anything about tensors.

The hard bit is working out what sort of decomposition the questioner has in mind - which is more like mind-reading than physics.

It occurs to me that a preferred vector $\vec v$ at a point on the manifold decomposes the tangent space at that point into a direct sum of the subspace $S$ generated by that vector, and the subspace $S^\bot$ orthogonal to it. That corresponds to a decomposition of the dual tangent space at that point into the space $U$ generated by dual of $\vec v$ and its orthogonal complement $U^\bot$.

Consider a two-tensor with components $T_{ab}$ that is a map from $V^2$ to $\mathbb R$, where $V$ is the local tangent space. This is isomorphic to a map $T'$ from the tangent space to the cotangent space such that $T'(\vec u)$ is the dual vector that maps vector $\vec w$ to $T(\vec u,\vec w)$. It is common to identify the tensor with its isomorph, and we will do that.

We can then write $T_{ab}=T^{||}_{ab}+T^\bot_{ab}$ where the first component maps the dual of a vector to its projection on $U$ and the second maps it to its projection on $U^\bot$.

So that's a decomposition that uses the preferred vector. I don't know whether it's the one the questioner had in mind.