Cohomology and MP Spacetimes

TL;DR

This paper develops a cohomological approach to gravitation, representing MP spacetimes and Schwarzschild black holes via a gravitational potential within a cochain complex framework.

Contribution

It introduces a novel cohomological formulation of gravity using a specific tensor complex, linking it to sheaf cohomology and enabling potential-based representations of spacetimes.

Findings

Cohomology of the complex is isomorphic to cech cohomology.

MP spacetimes can be represented by a gravitational potential.

Schwarzschild black hole admits a potential-based representation.

Abstract

This paper pursues a cohomological formulation for gravitation in which gravity might be expressed in terms of a gravitational potential, much in the spirit of electrodynamics. To this end we introduce a cochain complex consisting of $(n+2)$-tensors, symmetric in two indices and skew-symmetric in the remaining $n$ indices. The cohomology of the complex is shown to be isomorphic to the \v{C}ech cohomology of an appropriately defined sheaf of functions. Furthermore, it is demonstrated that in isotropic coordinates MP spacetimes as well as the Schwarzschild black hole may be thusly represented, that is, by means of the coboundary of a potential, as defined by the differential operator of the complex.