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.
تحميل البحث