# Topology change and Canonical Formulation?

In the ADM formulation of general relativity it is assumed that the spacetime topology is $$\Bbb{R}\times \Sigma$$. Suppose I wanted to consider spacetimes that undergo topology change with foliation time, would it be possible to describe such dynamics within some canonical formulation? I have been searching for an extension of the ADM formalism that lifts the topological restriction, but I can't seem to find anything on the subject (there are a few papers describing topology change in classical GR in the covariant formalism but that is all I could find).

• You can look up the more general formalisms of the De Donder-Weyl/polysymplectic/multisymplectic formalism, which does not assume anything about the underlying space. Dec 8, 2022 at 21:24

In the ADM formulation the topology is not assumed but deduced. Thanks to Geroch  theorem, if spacetime is globally hyperbolic then topologically it is $$\mathbb{R} \times \Sigma$$ where the topology of $$\Sigma$$ do not change in time. The idea of the proof is as follows: consider two leaves of the foliation, $$\lbrace t_{0} \rbrace \times \Sigma_{0}$$ and $$\lbrace t_{1} \rbrace \times \Sigma_{1}$$. If spacetime admits everywhere a Lorentzian metric (and is orientable and time orientable) then there exist a non-vanishing vector field all over spacetime. Then we can use the integral curve of this vector field to map $$\Sigma_{0}$$ into $$\Sigma_{1}$$ showing they must have the same topology.