Consider the following Lemma from Wald (Lemma 8.1.1): Let ##(M,g_{ab})## be a time-orientable space-time. Then there exists a (highly non-unique) smooth nonvanishing timelike vector field ##t^{a}## on ##M##.(adsbygoogle = window.adsbygoogle || []).push({});

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

Now he doesn't justify any of his claims so I'm left here trying to see why any of what he says is true (everything other than the existence of the Riemannian metric on ##M## which is a trivial consequence of paracompactness and partitions of unity).

He never rigorously defined what it means for ##(M,g_{ab})## to be time-orientable in the discussion preceding this Lemma; all he says in fact is that in a time-orientable space-time there exists a continuous designation of the future and past half of a lightcone in ##T_p M## as one varies ##p##. I'm assuming this means that there exists a continuous time-like vector field ##X^{a}## on ##M## in analogy with the usual definitions of orientability.

I can figure out the first "half" of his proof. Let ##S\subseteq T_p M = \{v^{a}\in T_p M: k_{ab}v^{a}v^{b} = 1\}##. ##S## is compact and ##g_p:S\rightarrow \mathbb{R},v^{a} \mapsto g_{ab}v^{a}v^{b}## is continuous because any bilinear map on a finite dimensional normed space is continuous. Hence there exists some ##t_p^{a}\in S## such that ##g_{ab}t_p^{a}t_p^{b}## is a a minimum; note that if ##v^{a}\in T_p M## is any time-like vector (meaning ##g_{ab}v^{a}v^{b} < 0##) then ##u^{a} = \frac{v^{a}}{(k_{ab}v^{a}v^{b})^{1/2}}## is also time-like by bilinearity and ##u^a\in S## by construction so we know there necessarily exists a time-like vector in ##S## hence the minimizing vector ##t_p^{a}## must be time-like. Hopefully these are the justifcations Wald had in mind when he wrote that "proof".

What I don't get is why this minimizing vector must necessarily be future directed, where in the world the time-orientability of ##(M,g_{ab})## is even used, and why the vector field ##t^{a}## defined as ##t^{a}(p) = t_p^{a}## at each ##p \in M## is necessarily smooth. Could anyone kindly explain those parts. Thanks in advance.

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# Wald Lemma 8.1.1-help with proof

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