Locally Metric Spacetimes

TL;DR

This paper investigates spacetimes characterized by connections that are locally but not globally Levi-Civita, offering a new perspective beyond the traditional global metric approach, with an example based on Robertson-Walker spacetimes.

Contribution

It introduces a method to construct spacetimes using connections that are locally Levi-Civita but not globally, expanding the framework of spacetime geometry.

Findings

Provides a general method for non-degenerate connections

Presents an example modifying Robertson-Walker spacetimes

Explores implications of locally metric connections

Abstract

Spacetimes have conventionally been described by a global Lorentzian metric on a differentiable four-manifold. Herein we explore the possibility of spacetimes defined by a connection, which is locally but not globally Levi-Civita. The general method of obtaining such connections is presented for the non-degenerate case followed by an example that modifies the Robertson-Walker spacetimes for flat spacelike hypersurfaces.