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?