Misconceptions About General Relativity in Theoretical Black Hole Astrophysics

TL;DR

This paper clarifies common misconceptions about coordinate transformations in general relativity, especially regarding rotating black hole metrics, emphasizing their non-diagonal nature and the limitations of local transformations.

Contribution

It corrects widespread misunderstandings about the applicability of coordinate transformations to rotating black hole metrics in general relativity.

Findings

No coordinate transformation can convert a rotating black hole metric to a Schwarzschild metric.

Rotating spacetime metrics cannot be globally diagonalized through coordinate changes.

Local coordinate transformations do not imply global equivalence of spacetimes.

Abstract

The fundamental role played by black holes in our study of microquasars, gamma ray bursts, and the outflows from active galactic nuclei requires an appreciation for, and at times some in-depth analysis of, curved spacetime. We highlight misconceptions surrounding the notion of coordinate transformation in general relativity as applied to metrics for rotating black holes that are beginning to increasingly appear in the literature. We emphasize that there is no coordinate transformation that can turn the metric of a rotating spacetime into that for a Schwarzschild spacetime, or more generally, that no coordinate transformation exists that can diagonalize the metric for a rotating spacetime. We caution against the notion of "local" coordinate transformation, which is often incorrectly associated with a global analysis of the spacetime.