Black Holes and Entropy

So far, we have considered the laws of black hole mechanics and briefly discussed their correspondence to the laws of thermodynamics. We now wish to consider the laws of black hole mechanics as thermodynamical statements. This means that we will define a temperature and an entropy for our black holes. In thermodynamics, we may consider e.g. the grand canonical ensemble, where the system is characterized by quantities \(\mu , V, T\), nammely chemical potential, volume, and temperature, respectively. The first law of thermodynamics is then $$ dE = TdS - pdV + \mu dN~, $$