Abstract
Motivated by the study of conserved Aretakis charges for a scalar field on the horizon of an extremal black hole, we construct the metrics for certain classes of four-dimensional and five-dimensional extremal rotating black holes in Gaussian null coordinates. We obtain these as expansions in powers of the radial coordinate, up to sufficient order to be able to compute the Aretakis charges. The metrics we consider are for 4-charge black holes in four-dimensional STU supergravity (including the Kerr-Newman black hole in the equal-charge case) and the general 3-charge black holes in five-dimensional STU supergravity. We also investigate the circumstances under which the Aretakis charges of an extremal black hole can be mapped by conformal inversion of the metric into Newman-Penrose charges at null infinity. We show that while this works for four-dimensional static black holes, a simple radial inversion fails in rotating cases because a necessary conformal symmetry of the massless scalar equation breaks down. We also discuss that a massless scalar field in dimensions higher than four does not have any conserved Newman-Penrose charge, even in a static asymptotically flat spacetime.
Highlights
In the last few years there have been many studies that have revealed that the horizon of an extremal black hole is unstable to small perturbations
The instabilities stem from the existence of conserved charges on the future horizon of the extremal black hole, which imply that physical perturbations do not decay at large values of the advanced time v
As we show later, in the weakened falloff of the weakly asymptotically flat (WAF) metrics in the stationary case, the Ricci scalar does not fall off fast enough at infinity, and this provides an obstruction to being able to relate the Aretakis charges to the Newman-Penrose charges of the conformally inverted WAF metric, at least if we assume a simple inversion of the radial coordinate
Summary
In the last few years there have been many studies that have revealed that the horizon of an extremal black hole is unstable to small perturbations. This is straightforward for a simple static example such as the extremal Reissner-Nordström solution, but it is rather less simple for a stationary metric such as the extremal Kerr solution In such a case one cannot construct an exact expression for the metric in Gaussian null form, but it is sufficient to determine just the first few orders in a GNC expansion of the metric in powers of the radial coordinate measuring distance away from the horizon. This metric approaches Minkowski spacetime at infinity, but with rather weaker falloff conditions than those of an asymptotically flat spacetime written in Bondi-Sachs coordinates It was shown in [12] that Newman-Penrose charges can be computed in the WAF spacetime obtained by conformal inversion of the original extremal black hole, and in various static examples the mapping between Aretakis and Newman-Penrose charges was exhibited. This happens because there is a term in the large-r expansion for the scalar equation written in Bondi-Sachs coordinates that presents an obstruction to the existence of conserved charges, and this term occurs with a dimension-dependent coefficient ðn − 2Þðn − 4Þ that is absent in n 1⁄4 4 dimensions but not when n ≥ 5
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