On the Action, Topology and Geometric Invariants in Quantum Gravity

TL;DR

This paper discusses how the definition of the action in general relativity and quantum gravity depends on the topology and boundary conditions of spacetime, impacting the formulation of quantum theories of gravity.

Contribution

It highlights the importance of topology, boundary conditions, and geometric invariants in defining the action for quantum gravity approaches.

Findings

The action in GR requires fixed topology for well-definition.

Global topology influences boundary conditions in quantum gravity.

Geometric invariants are affected by topology and boundary choices.

Abstract

The action in general relativity (GR), which is an integral over the manifold plus an integral over the boundary, is a global object and is only well defined when the topology is fixed. Therefore, to use the action in GR and in most approaches to quantum gravity (QG) based on a covariant Lorentzian action, there needs to exist a prefered (global) timelike vector, and hence a global topology $R \times S^3$, for it to make sense. This is especially true in the Hamiltonian formulation of QG. Therefore, in order to do canonical quantization, we need to know the topology, appropriate boundary conditions and (in an open manifold) the conditions at infinity, which affects the fundamental geometrical scalar invariants of the spacetime (and especially those which may occur in the QG action).