Volume dependent extension of Kerr-Newman black hole thermodynamics

Abstract

We show that the Hawking–Bekenstein entropy formula is modified by a factor of 8/3 if one also considers a work term in the 1st law of thermodynamics by a pressure stemming from the Hawking radiation. We give an intuitive definition for the corresponding thermodynamical volume by the implicit definition ϵ=Mc2/V, which is the average energy density of the Hawking radiation. This volume scales as V∼M5, agreeing with other suggestions. As a result the corresponding Smarr relation describes an extensive entropy and a stable effective equation of state S(E,V)∼E3/4V1/4. These results pertain for charged and rotating Kerr-Newman black holes. Keywords: Black holes, Volume, Entropy, Thermodynamics, Heat capacity, Thermal stability