Show that, for a vector field $v^a$ that vanishes on a hypersurface S enclosing a region V of an N-dimensional manifold:

$$\int_V(\nabla_av^a)\sqrt{-g}d^Nx=0$$ Where $g$ stands for the determinant of the metric.

This is one of the maths exercises on my physics book, and it puzzles me to the point that I don't even know where to start on, so even a hint will do!

  • 1
    $\begingroup$ You might want to use some theorem linking an integral over $V$ to an integral over $\partial V$, like the divergence theorem. Yet I'm not sure if $v^a$ necessarily vanishes on $\partial V$. It might just yield that it's tangential to it. $\endgroup$
    – dennis_s
    Mar 11, 2021 at 9:06
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    $\begingroup$ $\nabla_{a}v^{a} = \text{div } v$ . Since you are using $\sqrt{-g}d^{N}x$ for the volume form, I assume you are working on a Lorentzian manifold. The statement follows directly from the Divergence Theorem for Lorentzian manifolds. $\endgroup$
    – SYZ
    Mar 11, 2021 at 17:18
  • $\begingroup$ I Guess I could just take the Divergence Theorem and see that the term integrates over the boundary vanishes, but we never got to see that in the book yet, so this exercise is probably meant to be solved by doing some "brute" calculus that I don't even know how to start... $\endgroup$ Mar 12, 2021 at 11:22


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