# About the existenece and uniqueness of time-like vector fields in a time orientable manifold

I'm reading Wald's book "General Relativity" and I'm having a bit of trouble understanding his proof on the following lemma:

Let $$(\mathcal{M}, g_{ab})$$ be time orientable. Then, there exists a (highly non-unique) smooth non-vanishing time-like vector field $$t^a$$ on $$\mathcal{M}$$.

I'm going to copy/paste his proof to point out the parts I'm not fully understanding.

His proof: Since $$\mathcal{M}$$ is paracompact, we can choose a smooth Riemannian metric $$k_{ab}$$ on $$\mathcal{M}$$. At each $$p \in \mathcal{M}$$ there will be a unique future directed time-like vector $$t^a$$ which minimizes the value of $$g_{ab} v^a v^b$$ for vectors $$v^a$$ subject to the condition $$k_{ab} v^a v^b = 1$$. This $$t^a$$ will vary smoothly over $$\mathcal{M}$$ and thus provide the desired vector field.

What I don't understand is why is $$t^a$$ unique and why does it have to be? Couldn't multiple vector fields minimize $$g_{ab} v^a v^b$$ (which are subject to the condition $$k_{ab} v^a v^b = 1$$)? As for the second part of my question, my guess is that if it isn't unique then there isn't a continuous map in which we may choose an orientability on our manifold, however I'm not sure if my guess is correct.

• Aug 23 at 0:46