Consistent derivation of the Hawking effect for both nonextremal and extremal Kerr black holes

Abstract

It is believed that extremal black holes do not emit Hawking radiation as understood by taking extremal limits of non-extremal black holes. However, it is debated whether one can make such conclusion reliably starting from an extremal black hole, as the associated Bogoliubov coefficients which relate ingoing and outgoing field modes do not satisfy the required consistency condition. We address this issue in a canonical approach firstly by presenting an exact canonical derivation of the Hawking effect for non-extremal Kerr black holes. Subsequently, for extremal Kerr black holes we show that the required consistency condition is satisfied in the canonical derivation and it produces zero number density for Hawking particles. We also point out the reason behind the reported failure of Bogoliubov coefficients to satisfy the required condition.