Duality between Electric and Magnetic Black Holes

TL;DR

This paper explores the duality between electric and magnetic black holes, demonstrating their quantum property equivalence through semi-classical analysis and discussing implications for black hole entropy in cosmology.

Contribution

It provides a semi-classical proof that electric and magnetic black holes have equivalent quantum properties, extending the concept of duality to gravitational solitons.

Findings

Chemical potential and charge projection compensate for action differences

Electric black hole pair production is not a runaway process

Entropy definition requires caution in cosmological contexts

Abstract

A number of attempts have recently been made to extend the conjectured $S$ duality of Yang Mills theory to gravity. Central to these speculations has been the belief that electrically and magnetically charged black holes, the solitons of quantum gravity, have identical quantum properties. This is not obvious, because although duality is a symmetry of the classical equations of motion, it changes the sign of the Maxwell action. Nevertheless, we show that the chemical potential and charge projection that one has to introduce for electric but not magnetic black holes exactly compensate for the difference in action in the semi-classical approximation. In particular, we show that the pair production of electric black holes is not a runaway process, as one might think if one just went by the action of the relevant instanton. We also comment on the definition of the entropy in cosmological situations, and show that we need to be more careful when defining the entropy than we are in an asymptotically-flat case.