Black-Hole Polarization and Cosmic Censorship

The destruction of the black-hole event horizon is ruled out by both cosmic censorship and the generalized second law of thermodynamics. We test the consistency of this prediction in a (more) ‘dangerous’ version of the gedanken experiment suggested by Bekenstein and Rosenzweig. A U(1)-charged particle is lowered slowly into a near extremal black hole which is not endowed with a U(1) gauge field. The energy delivered to the black hole can be red-shifted by letting the assimilation point approach the black-hole horizon. At first sight, therefore, the particle is not hindered from entering the black hole and removing its horizon. However, we show that this dangerous situation is excluded by a combination of two factors not considered in former gedanken experiments: the effect of the spacetime curvature on the electrostatic selfinteraction of the charged system (the black-hole polarization), and the finite size of the charged body. The cosmic censorship hypothesis, introduced by Penrose [1] thirty years ago, is one of the corner stones of general relativity. Moreover, it is being envisaged as a basic principle of nature. The validity of this conjecture is, however, still an open question in classical general relativity. The destruction of a black hole (i.e., its event horizon) is ruled out by this principle because that would expose naked singularities to distant observers. Moreover, the horizon area A of a black hole is associated with entropy Sbh = A/4 (we use gravitational units in 1 which G = c = 1). Therefore, without any obvious physical mechanism to compensate for the loss of (black-hole) entropy, the destruction of the black-hole event horizon is expected to violate the generalized second law of thermodynamics [2]. For these two reasons, any process which seems, at first sight, to have a chance of removing the black-hole horizon is expected to be unphysical. For the advocates of the cosmic censorship principle the task remains to find out how such candidate processes eventually fail to remove the horizon. In this paper we inquire into the physical mechanism which protects the black-hole horizon from being eliminated by the capture of a charged particle which “supersaturate” the extremality condition for black holes. As is well known, the Kerr-Newman metric with M < Q + a (where M,Q and a are the mass, charge, and angular momentum per unit mass of the configuration) does not contain an event horizon, and therefore describes a naked singularity. We begin with the type of gedanken experiment suggested by Bekenstein and Rosenzweig [3] (our version of the gedanken experiment is, however, more ‘dangerous’ than the one considered in [3]): suppose there exist two different types of local charge, namely typeq ∈ U(1) and type k ∈ U (1), e.g., electric and magnetic charge. A Reissner-Nordström spacetime with two different charges Q ∈ U(1) and K ∈ U (1) displays an event horizon only if Q +K ≤ M. The black hole is assumed to be a near extremal Reissner-Nordström black hole, possessing a U (1) charge K, but no U(1) charge. Thus, the black hole is not endowed with a U(1) gauge field, and an infalling charge q [where q ∈ U(1)] seems to encounter no repulsive electrostatic potential barrier. Therefore, at first sight, the particle is not forbidden from crossing (and removing) the black-hole horizon. An assimilation of a charged body with proper energy (energy-at-infinity) E, and charge q, results with a change dM = E in the black-hole mass and a change dQ = q in its charge; Thus, a necessary condition for removal of the horizon after the assimilation of the body is E ≤ (q +K −M)/2M ≤ q/2M . Bekenstein and Rosenzweig [3] considered the infall of the charged particle from spatial infinity. For this case E ≥ μ, where μ is the particle’s rest mass, and a (necessary) condition