Test fields cannot destroy extremal de Sitter black holes

Abstract

We determine the timelike Killing vector field that gives the correct definition of energy for test fields propagating in a Kerr-Newman-de Sitter spacetime, and use this result to prove that test fields cannot destroy extremal Kerr-Newman-de Sitter black holes. IntroductionIn a famous paper [1], Wald tested the weak cosmic censorship conjecture [2, 3] by dropping charged and/or spinning test particles into the event horizon of an extremal Kerr-Newman black hole. Both him and subsequent authors [4,5] found that the particle would not go in for values of the conserved quantities (energy, angular momentum, charge and/or spin) which would overspin/overcharge the black hole. Similar conclusions were reached by analyzing scalar and electromagnetic test fields propagating in extremal Kerr-Newman black hole backgrounds [6,7,8,9]. In this case, the fluxes of energy, angular momentum and charge across the event horizon were found to be always insufficient to overspin/overcharge the black hole. Some of these results were extended to higher dimensions [10,11,12] and also to the case when there is a negative cosmological constant [13,14,15]. At the same time, it was noticed that Wald's thought experiment might produce naked singularities when applied to nearly extremal black holes [16,17,18,19]. However, in this case the perturbation could not be assumed to be infinitesimal, and so backreaction effects would have to be taken into account; when this was done, the validity of the cosmic censorship conjecture appeared to be restored [20,21,22,23,24].In [25], the authors (in collaboration with L. Queimada) gave a general argument showing that extremal Kerr-Newman and Kerr-Newman-anti-de Sitter black holes cannot be overspun/overcharged by any type of test matter satisfying the null energy condition at the event horizon. This argument was later extended by Sorce and Wald [26] to the case of quasi-extremal Kerr-Newman black holes by considering the second order variation of the black hole mass.In all these gedanken experiments, however, one must be very careful with what is meant by the energy of the test matter, and how it relates to the increase in the black hole mass. In fact, from a logical point of view, these are independent concepts: the energy of the test matter is computed with respect to a given timelike Killing vector field, whereas the black hole mass is a parameter in a black hole solution of the Einstein-Maxwell field equations. In the asymptotic flat case, the two can be related via the ADM mass: indeed, the ADM mass of a spacetime containing an isolated black hole is precisely the black hole mass, whereas the energy of test matter located in the asymptotically flat region (measured with respect to the unique timelike Killing vector field) simply adds to the ADM mass; since this energy is conserved as the test matter moves into the black hole spacetime, the black hole mass should increase by precisely that amount when the test matter is absorbed. In the non-asymptotically flat cases, however, there is no ADM mass, an...

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