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