Exact Toda Black Holes of Rank-2 Lie Groups

TL;DR

This paper constructs exact spherically symmetric black hole solutions in Einstein-Maxwell-dilaton theory related to all rank-2 Lie groups, including new solutions for B2 and G2, and analyzes their thermodynamics.

Contribution

It introduces a method to derive exact black hole solutions associated with all rank-2 Lie groups using Toda equations, including novel B2 and G2 cases.

Findings

Derived explicit solutions for all rank-2 Lie group Toda black holes.

Identified new B2 and G2 Toda black hole solutions.

Confirmed thermodynamic quantities can be obtained without solving the full solutions.

Abstract

We consider Einstein gravity coupled to two Maxwell fields and one dilatonic scalar, and construct spherically-symmetric and static black holes that are charged under both Maxwell fields in general $D$ dimensions. We find that for suitable dilaton couplings, the equations of motion can be cast into one-dimensional Toda equations of all rank-2 Lie groups. We devise a brute-force approach to obtain the most general but remarkably elegant solutions to the Toda equations. This allows us to construct exact black holes associated with all the rank-2 Lie groups. The $B_2$ and $G_2$ Toda black holes are new. We study their thermodynamics and verify explicitly an earlier claim in the literature that all these thermodynamic quantities can be derived without having to solve for these black hole solutions.