Black hole enthalpy and an entropy inequality for the thermodynamic volume

In a theory where the cosmological constant $\ensuremath{\Lambda}$ or the gauge coupling constant $g$ arises as the vacuum expectation value, its variation should be included in the first law of thermodynamics for black holes. This becomes $dE=TdS+{\ensuremath{\Omega}}_{i}d{J}_{i}+{\ensuremath{\Phi}}_{\ensuremath{\alpha}}d{Q}_{\ensuremath{\alpha}}+\ensuremath{\Theta}d\ensuremath{\Lambda}$, where $E$ is now the enthalpy of the spacetime, and $\ensuremath{\Theta}$, the thermodynamic conjugate of $\ensuremath{\Lambda}$, is proportional to an effective volume $V=\ensuremath{-}\frac{16\ensuremath{\pi}\ensuremath{\Theta}}{D\ensuremath{-}2}$ ``inside the event horizon.'' Here we calculate $\ensuremath{\Theta}$ and…

• UD
  U.S. Department of Energy
• SA
  Science and Technology Facilities Council

  Awards: ST/F002998/1, ST/I002006/1