Positive Energy Functional for Massless Scalars in Rotating Black Hole Backgrounds of Maximal Ungauged Supergravity

TL;DR

This paper proves the stability of massless scalar fields in complex rotating black hole backgrounds within supergravity by transforming the problem into an auxiliary metric where energy positivity is established.

Contribution

It introduces a novel method to demonstrate scalar field stability in rotating supergravity black holes using an auxiliary metric approach to handle super-radiance.

Findings

Scalar field stability is established in a broad class of supergravity black hole backgrounds.

The auxiliary metric approach ensures non-negative energy density despite super-radiance.

The method circumvents difficulties posed by ergo-regions in stability analysis.

Abstract

We outline a proof of the stability of a massless neutral scalar field $\psi$ in the background of a wide class of four dimensional asymptotically flat rotating and ``electrically charged'' solutions of supergravity, and the low energy limit of string theory, known as STU metrics. Despite their complexity, we find it possible to circumvent the difficulties presented by the existence of ergo-regions and the related phenomenon of super-radiance in the original metrics by following a strategy due to Whiting, and passing to an auxiliary metric admitting an everywhere lightlike Killing field and constructing a scalar field $\Psi$ (related to a possible unstable mode $\psi$ by a non-local transformation) which satisfies the massless wave equation with respect to the auxiliary metric. By contrast with the case for $\psi$, the associated energy density of $\Psi$ is not only conserved but is also non-negative.