Why is the metric invertible?

TL;DR

This paper explores why the gravitational metric variable does not have zero vacuum, contrasting with other fields, and proposes modifications to the action to express the existence of its inverse through field equations.

Contribution

It formulates a novel approach to express the inverse of the metric via additional terms in the gravitational action without assuming it beforehand.

Findings

Proposes modifications to the gravitational action

Demonstrates the existence of the metric inverse through field equations

Highlights the unique nature of the gravitational metric in vacuum

Abstract

We raise, and provide an (unsatisfactory) answer to, the title's question: why, unlike all other fields, does the gravitational "metric" variable not have zero vacuum? After formulating, without begging it, we exhibit additions to the conventional action that express existence of the inverse through a field equation.