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

We outline a proof of the stability of a massless neutral scalar field $ψ$ 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 $Ψ$ (related to a possible unstable mode $ψ$ by a non-local transformation) which satisfies the massless wave equation with respect to the auxiliary metric. By contrast with the case for $ψ$, the associated energy density of $Ψ$ is not only conserved but is also non-negative.

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