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.


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