# The Tomimatsu-Sato metric reloaded

## TL;DR

This paper provides exact formulas for the ergoregion boundaries in the Tomimatsu-Sato metric, corrects previous errors, and supplies numerical data for various parameters, enhancing understanding of this spacetime solution.

## Contribution

We derived precise analytic expressions for ergoregion boundaries in the Tomimatsu-Sato metric and corrected earlier inaccuracies in the literature.

## Key findings

- Exact formulas for ergoregion surfaces derived
- Identification of an event horizon inside the singularity
- Numerical tables for ergoregion boundaries provided

## Abstract

In this work, we derive exact analytic formulae for the inner and outer surfaces representing the boundary of the ergoregion appearing in the Tomimatsu-Sato (TS) metric. Exact expressions for the radii of the ergoregion in prolate spheroidal coordinates and in Boyer-Lindquist coordinates are obtained. We also found that in addition to the ring-shaped naked singularity there is an event horizon placed in the inner region inside the aforementioned curvature singularity. In comparing our results with previous studies, we also uncovered and corrected several errors in the literature. Finally, we provide tables of numerical values for the inner and outer boundaries of the ergoregion for different values of the rotational parameter. We hope this study will be a useful resource for all researchers interested in the Tomimatsu-Sato metric.