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In Maggiore's book on Gravitational Waves, he derives the energy-momentum tensor $t_{\mu\nu}$ associated with a gravitational wave. From this, he gives in equation (1.161) that the momentum (in the $k$-direction) of gravitational waves inside a spherical shell $V$ at large distances away from the source is $$P_V^k = \frac1c\int_Vd^3x\ t^{0k}.$$ Differentiating both sides with respect to $x^0$ gives equation (1.162): $$\partial_0P_V^k = \frac1c\int_Vd^3x\ \partial_0t^{0k}.$$ However, in the next line Maggiore claims that this equals $$-\frac1c\int_SdA\ t^{0k}$$ where $S$ is the boundary of the volume.

How does this last step follow? The only tools I assume were used are the divergence theorem and the fact that $\partial^{\mu}t_{\mu\nu}=0$ in equation (1.141) for gravitational waves far from the source in approximately flat spacetime.

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  • $\begingroup$ Isn't there a conserved current involved here? So the time deriv is related to the divergence, and so the integral of the time deriv over the volume is related to the integral over the surface of the volume. $\endgroup$
    – puppetsock
    Commented Nov 1, 2019 at 17:35
  • $\begingroup$ Well, $\partial^{\mu}t_{\mu\nu}=0$ implies $\partial^{0}t_{0\nu}=-\partial^{i}t_{i\nu}$. Is that what you mean? $\endgroup$
    – J-J
    Commented Nov 1, 2019 at 17:37
  • $\begingroup$ Yes, exactly. You should be able to do the integral. The original author could have helped you by saying how it was done. en.wikipedia.org/wiki/Stokes%27_theorem#Divergence_theorem $\endgroup$
    – puppetsock
    Commented Nov 1, 2019 at 18:55
  • $\begingroup$ Indeed, I have used the divergence theorem to do just that. The issue is that this doesn't give me the same thing as the author's result. $\endgroup$
    – J-J
    Commented Nov 1, 2019 at 19:10
  • $\begingroup$ What do you get? $\endgroup$
    – puppetsock
    Commented Nov 1, 2019 at 19:14

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