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Laps and shift function

  1. May 26, 2010 #1
    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 [tex] t^a \nabla_a t [/tex], where t is "time function".
    What is the difference between [tex]g^{ab}\nabla_b t [/tex]
    and t^a. Thanks for answer.
  2. jcsd
  3. May 27, 2010 #2

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

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

    Last edited: May 27, 2010
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