Complete Evaporation of Black Holes and Page Curves

Abstract

The problem of complete evaporation of a Schwarzschild black hole, the simplest spherically symmetric vacuum solution of the Einstein field equation, posed by Hawking, is that when the black hole mass M disappears, an explosion of temperature T=1/8πM takes place. We consider the Reissner–Nordstrom black hole, a static spherically symmetric solution to the Einstein–Maxwell field equations, and show that if mass M and charge Q<M satisfy the bound Q>M−CM3, C>0 for small M, then the complete evaporation of black holes without blow-up of temperature is possible. We describe curves on the surface of state equations such that the motion along them provides complete evaporation without temperature explosion. In this case, the radiation entropy follows the Page curve and vanishes at the end of evaporation. Similar results for rotating Kerr, Schwarzschild–de Sitter and Reissner–Nordstrom-(Anti)-de Sitter black holes are discussed.