Duality and Four-Dimensional Black Holes

TL;DR

This paper studies how abelian duality transformations affect four-dimensional black hole solutions in string theory, revealing a symmetry that interchanges physical charges and can relate regular black holes to singular solutions.

Contribution

It demonstrates that duality maps a family of static, spherically symmetric solutions onto themselves, interchanging key physical parameters, and explores the implications for black hole properties and equivalence.

Findings

Duality interchanges mass and dilaton charge.

Duality swaps axion charge and Taub-NUT parameter.

Dual solutions can be singular and not physically equivalent.

Abstract

We consider the effects of abelian duality transformations on static, spherically-symmetric, asymptotically flat string spacetimes in four dimensions, where the dilaton, axion, metric, and gauge fields are allowed to be nonzero. Independent of the alpha' expansion, there is a six-parameter family of such configurations, labelled by the charges characterizing the asymptotic behaviour of the various fields: ie their mass, dilaton charge, axion charge, electric charge, magnetic charge, and Taub-NUT parameter. We show that duality, based on time-translation invariance, maps these solutions amongst themselves, with the effect of interchanging two pairs of these six labels, namely: (1) the mass and dilaton charge, and (2) the axion charge and the Taub-NUT parameter. We consider in detail the special case of the purely Schwarzschild black hole, for which the mass of the dual configuration vanishes to leading order in alpha'. Working to next-to-leading order in alpha' for the bosonic and heterotic strings, we find that duality takes a black hole of mass M to a (singular) solution having mass - 1/(alpha' M). Finally, we argue that two solutions which are related by duality based on a noncompact symmetry are {\it not} always physically equivalent.