Radiation of the Inner Horizon of the Reissner-Nordstr\"om Black Hole

Despite of over thirty years of research of the black hole thermodynamics our understanding of the possible role played by the inner horizons of Reissner-Nordstr\"om and Kerr-Newman black holes in black hole thermodynamics is still somewhat incomplete: There are derivations which imply that the temperature of the inner horizon is negative and it is not quite clear what this means. Motivated by this problem we perform a detailed analysis of the radiation emitted by the inner horizon of the Reissner-Nordstr\"om black hole. As a result we find that in a maximally extended Reissner-Nordstr\"om spacetime virtual particle-antiparticle pairs are created at the inner horizon of the Reissner-Nordstr\"om black hole such that real particles with positive energy and temperature are emitted towards the singularity from the inner horizon and, as a consequence, antiparticles with negative energy are radiated away from the singularity through the inner horizon. We show that these antiparticles will come out from the white hole horizon in the maximally extended Reissner-Nordstr\"om spacetime, at least when the hole is near extremality. The energy spectrum of the antiparticles leads to a positive temperature for the white hole horizon. In other words, our analysis predicts that in addition to the radiation effects of black hole horizons, also the white hole horizon radiates. The black hole radiation is caused by the quantum effects at the outer horizon, whereas the white hole radiation is caused by the quantum effects at the inner horizon of the Reissner-Nordstr\"om black hole.