A notion encountered in field theory on non-globally hyperbolic manifolds is the notion of a spacetime being causally benign with respect to some field $\phi$, which is defined thusly :

  • A neighbourhood $U$ is causally regular if $\bar U$ has a neighbourhood $U_1$ such that for every solution $D(\phi') = 0$ on $U_1$, there is a smooth extension $\phi$ satisfying $D(\phi) = 0$ on $M$ and $\phi|_{U} = \phi'_{U}$.
  • A point $p$ is causally regular if every neighbourhood $U_p$ contains a causally regular neighbourhood which contains $p$.
  • The set of all causally regular points is denoted by $C$
  • We call a spacetime causally benign with respect to $D(\phi)$ if $C = M$.

Two of the results that bother me, much for the same reason, are the fact that $1)$ every globally hyperbolic spacetime is causally benign and $2)$ the Minkowski torus (Minkowski spacetime identified by $1$ timelike translation and $(n-1)$ spacelike translations) is also causally benign for the massless wave equation.

(for more details on all this cf for instance [1] or [2])

The reason why I find both of those strange is that in both case, consider the wave equation $\Box \phi = 0$. A priori, for any manifold locally isometric to Minkowski space we can consider that any neighbourhood (homeomorphic to a disk) can have imposed on it a solution of Minkowski space, by the Fourier decomposition

$$\phi(x,t) = \int d^2p\ [a(p) e^{i(px - E_pt)} + a^*(p) e^{-i(px + E_pt)}]$$

with $E^2 = p^2$. But on the other hand, if we consider the case of the torus, we get the solution (defined by Fourier series)

$$\phi(x,t) = \sum_{n} [a_n e^{i(p_nx - E_nt)} + a_n^* e^{-i(p_nx + E_nt)}]$$

(the same mostly applies to the cylinder $\mathbb R \times S$ which is globally hyperbolic)

In other words, not all modes of the field are allowed due to the boundary conditions. A solution on a local region with momentum plane-waves unrelated to the size of the torus won't extend to the entire manifold.

For a more concrete example, take some simply connected subset $U$ of the torus. Since it's just some simply connected open set with the Minkowski metric, it also admits as an extension Minkowski space itself, meaning that any solution of the metric equation, such as $\phi = \rho(x - t)$ for $\rho$ some compact support function, will also be a solution on $U$ via $\phi|_U$. But such a solution would not be a consistent one unless $\rho(x - t) = \rho(x + a_x - t) = \rho(x - t - a_t)$, which is not guaranteed for all such functions.

Does this conflict with the notion of the existence of causally regular neighbourhoods, or am I misunderstanding the notion? Or is the solution for the local neighbourhood not actually the Fourier transform presented? In particular, it is apparently untrue that it is causally benign if the mass isn't $0$, how does this relate to the problem?

Edit : I thought that maybe the problem was that I wasn't using the fact that $\phi_1$ was defined on $U_1$, and not $U$, but in this case, why can't I pick $U_1 = M$ for every neighbourhood $U$, in which case almost every spacetime will be benign except for the case where there are no solutions at all to the wave equation? (in the case of the wave equation, $\phi = 0$ is always a solution anyway)


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