Laps and Shift Function: Understanding the Role of t^a in Wald's GR Book (p.255)

[paweld]

Could anyone explain me what is the interpretation of t^a filed in
Wald's GR book (p.255). It's defined as any (?) field which
fulfills condition $$t^a \nabla_a t$$, where t is "time function".
What is the difference between $$g^{ab}\nabla_b t$$
and t^a. Thanks for answer.

 

[dodelson]

Hi,

Another way of stating $$t^a\nabla_a t=1$$ is that the Lie derivative of $$t$$ along $$t^a$$ equals 1. So Wald is just saying that the vector field $$t^a$$ is properly normalized so that the function $$t$$ changes at a constant rate of 1 along its integral curves. This normalization would be impossible to achieve if, for example, $$t^a$$ were parallel to the Cauchy surfaces, as $$t$$ would not change at all along its integral curves. The condition $$t^a\nabla_a t=1$$ makes sure that $$t^a$$ is properly normalized as to generate time flow.

$$g^{ab}\nabla_b t$$ is just equal to $$\nabla^at$$. This doesn't satisfy the above condition, since $$\nabla^at\nabla_a t\not=1$$ (not necessarily, at least).

Cheers,
Matthew