Tomimatsu-Sato geometries, holography and quantum gravity

TL;DR

This paper explores the near horizon geometry of the $ ext{Tomimatsu-Sato}$ spacetime within the Kerr/CFT framework, calculating the dual CFT parameters and confirming the entropy matches the Bekenstein-Hawking formula.

Contribution

It demonstrates that the $ ext{Tomimatsu-Sato}$ spacetime's near horizon geometry fits a Ricci-flat class with $SL(2, ext{R})\times U(1)$ symmetry and identifies the dual CFT parameters in terms of intrinsic horizon charges.

Findings

The near horizon geometry belongs to a class including extremal Kerr geometries.

The central charge and temperature of the dual CFT are computed.

The Cardy formula reproduces the Bekenstein-Hawking entropy.

Abstract

We analyze the $\delta=2$ Tomimatsu-Sato spacetime in the context of the proposed Kerr/CFT correspondence. This 4-dimensional vacuum spacetime is asymptotically flat and has a well-defined ADM mass and angular momentum, but also involves several exotic features including a naked ring singularity, and two disjoint Killing horizons separated by a region with closed timelike curves and a rod-like conical singularity. We demonstrate that the near horizon geometry belongs to a general class of Ricci-flat metrics with $SL(2,\mathbb{R})\times U(1)$ symmetry that includes both the extremal Kerr and extremal Kerr-bolt geometries. We calculate the central charge and temperature for the CFT dual to this spacetime and confirm the Cardy formula reproduces the Bekenstein-Hawking entropy. We find that all of the basic parameters of the dual CFT are most naturally expressed in terms of charges defined intrinsically on the horizon, which are distinct from the ADM charges in this geometry.