"Cosmic flashing" in four dimensions

Quantum field theory allows violation of the weak energy condition in the form of locally negative energy densities and fluxes. If the laws of physics place no restrictions on the extent of energy condition breakdown, then dramatic violations of the second law of thermodynamics, causality, and cosmic censorship might become possible. In this paper, we explore the possibility that manipulation of negative energy fluxes could lead to the production of naked singularities. This might be accomplished by injecting a negative energy flux into an extreme ($Q=M$) charged black hole. However, quantum field theory requires that an initial negative energy flux due to quantum coherence effects must be followed by a more than compensating positive flux. Thus any singularity resulting from this process would be only "temporarily" naked. In an earlier publication, we dubbed the occurrence of a naked singularity with limited duration "cosmic flashing." There, in a two-dimensional analysis where the fluxes were produced by moving mirrors, we showed that quantum field theory imposed limits on the magnitude and duration of the negative energy flux in the form of an uncertainty-principle-type inequality. If $\left|\Delta M\right|$ is the magnitude of the change in the mass of the black hole due to the absorption of negative energy, and $\Delta T$ is the effective lifetime of the naked singularity thus produced, then we showed that $\left|\Delta M\right|\Delta T<\mathrm{}1\mathrm{}$, in units where $c=\hslash =1\mathrm{}$. The current paper analyzes the behavior of a minimally coupled, quantized, massless scalar field propagating in a four-dimensional extreme Reissner-Nordström black-hole background. In this case a similar inequality is shown to hold for a general negative energy flux, irrespective of how the flux is produced. A numerical analysis shows that the angular-momentum-dependent potential barrier around the black hole screens out the contributions to the flux from the higher $l$ modes. We estimate the metric perturbations produced by the negative energy flux. In an order of magnitude estimate, we show that these are smaller than the metric fluctuations expected from quantum gravity. Therefore we conclude that quantum field theory prevents an unambiguous violation of cosmic censorship.