Observables and cohomology classes for classical gravitational field

TL;DR

This paper explores classical gravitational observables as Diff(M)-gauge invariant cohomology classes, linking global spacetime topology to physical phenomena and suggesting implications for quantum gravity and spacetime classification.

Contribution

It defines a set of classical geometric observables for gravity as gauge-invariant cohomology classes, connecting spacetime topology with physical characteristics and potential Lagrangian construction.

Findings

Observables are linked to spacetime topology.

Manifolds in 4+ dimensions are hard to classify.

Different topologies may share local geometry.

Abstract

Some of the most outstanding questions in the field of gravitation and geometry remain unsolved as a result of our limited understanding of the global structure of the spacetime geometry and the role played by global spacetime diffeomorphism group in quantum gravity. Some insight into these important questions may be gained by looking at certain aspects of general covariance and observables in classical gravitational theory. In this paper I shall define as set of classical geometric observables of the gravitational field by which I mean Diff(M)-gauge invariant cohomology classes defined on a Lorentzian structure. They represent global characteristics of the physical gravitational phenomena, are linked to the topology of the spacetime, and can be used in constructing new Lagrangians. The problem of finding a complete set of data out of observables is related perhaps to the fact that at present moment, manifolds in dimension 4 and above cannot be effectively classified. One could interpret this result as a pointer to the possibility that there might be spacetimes with different topologies (i.e., different global characteristics) which have indistinguishable local spacetime geometry.