Feeds:
Posts

## Dealing with noncompactness of S4 tangent bundle

The phase space of a 4-sphere is noncompact, the cotangent bundle. Essentially because the light speed is bounded, the noncompactness should not matter but there are many ways to address this. We are in the lucky situation that we can consider the complex sphere which is diffeomorphic to this space but which can be written as the solution of $z_1^2+\cdots+z_5^2 = const$ which implies that we can compactify it as a subvariety of $\mathbf{CP}^5$. This 4 complex dimensional subvariety is compact and is a compactification of the phase space. We thus have here the phase space of the actual universe correspond to a compact complex manifold, so we can apply both theorems about compact symplectic manifolds (useful in the context of results on periodic solutions to Hamiltonian flows) and we have a complex structure so theorems of algebraic geometry also apply. I am grateful to Peter Morrison for the idea of considering the complex sphere.