The Role of the Volume in Black Hole Thermodynamics

Gibbons et al. [arXiv:hep-th/0408217] found the energy $E$ of Kerr–anti-de Sitter black holes by integrating the first law of black hole thermodynamics. They found that $E$ corresponds to the Ashtekar–Magnon–Das (AMD) energy associated with an asymptotically nonrotating frame, whereas the AMD energy'' which I will call $F$ associated with an asymptotically rotating frame does not satisfy the first law. In Cveti\v{c} et al. [arXiv:1012.2888], the first law was extended by interpreting $E$ as an enthalpy and $\Lambda$ as being proportional to a pressure. The term conjugate to the pressure was then interpreted as the thermodynamic volume’’ $V_{th}$. Associated with the first law (with varying pressure) is a Smarr relation for $E$. The Smarr relation for $F$ also exists, and the term conjugate to the pressure in that Smarr relation is the ``geometric volume’’ $V_{geo}$, shown in [arXiv:1310.1935] to be equal to the vector volume $V_C$ of the black hole. To address why it is necessary to use $E$ rather than $F$ to have a viable first law but $V_C$ appears naturally in the Smarr relation associated with $F$ rather than $E$, I adapt Barnich and Comp`ere [arXiv:gr-qc/0412029], by defining a conserved quantity $H_{\chi }^I$ associated with Killing vector $\chi$. $E$ and $F$ are given by $H_{\xi }^I$ and $H_{\beta }^I$ respectively where $\xi$ is asymptotically hypersurface-orthogonal and $\beta$ is proportional to the divergence of the Principal Conformal Killing–Yano tensor $\mathit{h}$. I show that the first law will be satisfied by $H_{\chi }^I$ if both ${\chi }^a$ and the background anti-de Sitter metric have unvarying components, which holds for ${\xi }^a$ but not ${\beta }^a$, explaining why the first law works for $E$ but not $F$. I show that $V_C$ appears in the $\beta$-associated Smarr relation due to simplifications related to $\mathit{h}$.