# Application of Stokes' theorem to Minkowski spacetime and its asymptotic boundary

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Let $M$ be a $d$-dimensional smooth manifold with boundary $\partial M$. If $\omega$ is a $(d-1)$ form then Stokes' theorem reads: $$\int_M d\omega=\int_{\partial M}\omega\tag{1}$$

Now, Minkowski spacetime has one improper asymptotic boundary at infinity. Formally it is: $$\partial M = i^-\cup {\cal I}^-\cup i^0\cup {\cal I}^+\cup i^+\tag{2}$$

My question is on how Stokes' theorem (1) works for Minkowski spacetime and its improper boundary (2). I think this is relevant, for instance, when considering boundary terms of actions. Here are the issues that led me to ask this:

1. Minkowski spacetime isn't really a manifold with boundary. It is in fact conformally embedded into the interior of a manifold with boundary which is the unphysical spacetime introduced in the conformal compactification procedure. So I think (1) should be applied in the unphysical spacetime.

2. The Penrose conformal compactification gives ${\cal I}^\pm \simeq \mathbb{R}\times S^2$ as smooth null surfaces, but $i^\pm$ and $i^0$ become points. In standard integration theory since $\{i^+\},\{i^-\},\{i^0\}$ have measure zero the integral over $\partial M$ should have no contributions from these and be just $$\int_{\partial M}\omega=\int_{\mathcal{I}^+}\omega+\int_{\cal I^-}\omega\tag{3}$$ This is awkward because surely spatial infinity and timelike infinity should contribute depending on the form $\omega$, right? For instance, massive fields do not die off at $i^\pm$.

3. Moreover, the fact that in Penrose conformal compactification $i^\pm$ and $i^0$ are points leads me to think that $\partial M$ isn't really smooth. In fact, $i^\pm$ and $i^0$ seem like singular points. For example, $i^0$ seems like the apex of a double cone which I think would not be smooth as required by Stokes' theorem. In that setting, I think that instead of (3) we should have some contribution out of $i^0$ and $i^\pm$ coming from a sort of regularization of these singular points.

Summing it all up, my question is: what is the mathematically correct way to use Stokes' theorem to relate the integration on the bulk of Minkowski spacetime and integration on its asymptotic boundary (2)? How $i^\pm$ and $i^0$ contribute if they have measure zero and how does Stokes' theorem gets used since these points seem to spoil the smoothness of the boundary? Is there some sort of regularization that is required?

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