The Generalized Second Law for Fermionic and Bosonic Fields

Abstract

We evaluate the validity of the generalized second law for Kerr black holes perturbed by fermionic and bosonic fields. We derive that the critical frequency for a test field below which the area of a Kerr black hole would decrease, coincides with the superradiance limit which pertains to bosonic fields. The fact that the absorption of fermionic fields with arbitrarily low frequencies is allowed, leads to a generic violation of the generalized second law as both the black hole and the environment lose entropy. The result does not contradict the proof of the area theorem which pre-assumes the validity of the null energy condition. We also construct a thought experiment involving bosonic fields to check whether the minimum increase in the area can compensate for the decrease in the entropy of the environment. We minimize the entropy increase by considering a black hole at the extremal limit, perturbed by a bosonic field at the superradiance limit. We show that the generalized second law remains valid for bosonic fields that satisfy the null energy condition. The result does not require the employment of entropy bounds when one assigns von Neumann entropy to test fields.