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Reading Tao's book: Nonlinear Dispersive Equations I came upon an identity (the energy flux identity for the wave equation, page 90) for which the proof uses the Stokes theorem. In this case he uses the Stokes theorem on a truncated backward lightcone:

$\{(t,x):0\leq t \leq t_1, |x| \leq T_* -t\}$

The problem here is that, when integrating along the boundary of the cone; the curved part (i.e., the mantle):

$\{(t,x) : 0 < t < t_1, |x| = T_* -t\}$

is a null hypersurface with respect to the usual Minkowski metric, which he uses. When you restrict the Minkowski metric to a null hypersurface you get a degenerate metric. He explains that we can fix this apparent burden in a footnote:

Strictly speaking, $\Sigma_1$ [which is this boundary] is not quite spacelike, whic causes dS [the induced area form] to degenrate to zero and $n_\beta$ [the normal] to elongate to infinity. But the area from $n_\beta dS$ remains well defined in the limit; we omit the standard details.

I am aware that Stokes theorem is oblivious of the metric (that is, it works on any manifold, with or without metric). Furthermore, in this particular case the volume form in the lightcone coincides with the volume form given by the usual Euclidean metric, which we can use, and thus interpret the integral as an integral in Euclidean space; and then the induced metric is perfectly valid and I obtain the identity verbatim. It, however, worries me that this procedure probably doesn't extend to general Lorentz manifolds. Also, in the explanation given in the footnote he describes what seems to be another way to fix the problem, which apparently is standard but for which I haven't been able to find any reference.

I guess I could integrate in a stretched out "lightcone" with a non-lightlike boundary and take limits as the boundary becomes lightlike; and this will probably give me the same answer, it however doesn't seem to be what he's describing (right?).

Is there anything where I could learn how to do this more generally?

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The canonical reference for this would be Friedlander's The Wave Equation on a Curved Space-time, in particular look for the term "Leray form". In practice, note that in an energy estimate you are actually using Stoke's theorem in the following form: let $J$ be the energy current (a vector field) and let $\omega$ be the space-time volume form, if $\Omega$ is a domain with piece-wise smooth boundaries, consider Stokes theorem applied to the form $\iota_J\omega$ where $\iota$ is the interior derivative. We have then

$$ \int_{\partial\Omega} \iota_J\omega = \int_\Omega \mathrm{d}\iota_J\omega = \int_\Omega (\mathrm{div} J)\omega $$

The key part that may be confusing is how to convert this to a parametrised integral on a null boundary piece. But this part is just done by choosing local coordinates! (In fact, most of the content of the Leray form formalism is down to choosing local coordinates in a way that is "compatible".)


One way to approach Leray forms is to consider the following: let $u$ be a defining function of the null boundary. We can find a form $\eta$ such that $\omega = \mathrm{d}u \wedge \eta$. This form gives a volume form on the null boundary, but depends on the choice of $u$: if $u' = fu$ where $f$ is a nonvanishing smooth function, then $\mathrm{d}u' = u\mathrm{d}f + f\mathrm{d}u$, on the null hypersurface $u$ vanishes since it is a defining function, so you see that along the surface the corresponding $\eta'$ has to be $f^{-1}\eta$. The point is that now you can interpret

$$ \int_{\partial\Omega} \iota_J\omega = \int_{\partial\Omega} \iota_J (\mathrm{d}u\wedge \eta) = \int_{\partial\Omega} J(u) \eta $$

If you fix $u$ and $\eta$ this allows you to compare the integrals for different $J$s.


One other way of approach is to start with a null vector field $\ell$ along the null hypersurface, and foliate the null hypersurface with space-like sections. On each of the space-like sections you get the canonical induced volume form $\mathrm{d}S$. We can take $\bar{\ell}$ to be a null vector field such that $g(\ell,\bar{\ell}) \neq 0$, $\iota_{\bar{\ell}} \mathrm{d}S = 0$. Then the metric lowering $\ell^\flat \wedge \bar{\ell}^\flat \wedge \mathrm{d}S$ is proportional to the volume form. So by a scalar renormalisation of $\ell$ and/or $\bar{\ell}$ we can assume that the above form is in fact the volume form. Then you can use $\bar{\ell}^\flat \wedge \mathrm{d}S$ as the volume form on the null hypersurface and consider $\int_{\partial\Omega} \iota_J\omega = \int_{\partial\Omega} g(\ell,J) \bar{\ell}^\flat \wedge \mathrm{d}S$. This is probably closer to what Terry sketched in his book. (Note that this paragraph and the previous one actually describe isomorphic procedures, just phrased slightly differently.)

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