Testing cosmic censorship conjecture near extremal black holes with cosmological constants

It has been shown previously that an extremal Reissner-Nordstr\"om or an extremal Kerr black hole cannot be overcharged or overspun by a test particle if radiative and self-force effects are neglected. In this paper, we consider extremal charged and rotating black holes with cosmological constants. By studying the motion of test particles, we find the following results: An extremal Reissner-Nordstr\"om anti-de Sitter (RN-AdS) black hole can be overcharged by a test particle but an extremal Reissner-Nordstr\"om de Sitter (RN-dS) black hole cannot be overcharged. We also show that both extrmal Kerr-de-Sitter (Kerr-dS) and Kerr-anti-de-Sitter (Kerr-AdS) black holes can be overspun by a test particle, implying a possible breakdown of the cosmic censorship conjecture. For the Kerr-AdS case, the overspinning requires that the energy of the particle be negative, a reminiscent of the Penrose process. In contrast to the extremal RN and Kerr black holes, in which cases the cosmic censorship is upheld, our results suggest some subtle relations between the cosmological constants and the comic censorship. We also discuss the effect of radiation reaction for the Kerr-dS case and find that the magnitude of energy loss due to gravitational radiation may not be enough to prevent the violation of the cosmic censorship.