Long-Range Forces Between Non-Identical Black Holes With Non-BPS Extremal Limits

TL;DR

This paper investigates the long-range forces between non-identical extremal black holes in theories with multiple gauge fields, revealing that such black holes generally repel each other unless they are identical, and exploring conditions for zero-force interactions.

Contribution

It extends the analysis of black hole interactions to non-identical and non-BPS extremal black holes in arbitrary dimensions, using solutions governed by $SL(N+1,R)$ Toda equations.

Findings

Non-identical extremal black holes repel each other.

Pairs of non-extremal black holes can exhibit zero-force conditions.

Indications of similar behavior in higher $SL(N+1,R)$ models.

Abstract

Motivated by recent studies of long-range forces beween identical black holes, we extend these considerations by investigating the forces between two non-identical black holes. We focus on classes of theories where charged black holes can have extremal limits that are not BPS. These theories, which live in arbitrary spacetime dimension, comprise gravity coupled to $N$ 2-form field strengths and $(N-1)$ scalar fields. In the solutions we consider, each field strength carries an electric charge. The black hole solutions are governed by the $SL(N+1,R)$ Toda equations. In four dimensions the black-hole solutions in the $SL(3,R)$ example are equivalent to the "Kaluza-Klein dyons." We find that any pair of such extremal black holes that are not identical (up to overall scaling) repel one another. We also show that there can exist pairs of non-extremal, non-identical black holes which obey a zero-force condition. Finally, we find indications of similar results in the higher examples, such as $SL(4,R)$.