arXiv subjects

William Ballik

Publications and source records attributed to William Ballik.

3 recordsLinked to original sources

arXiv2026

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 $H^I_\chi$ associated with Killing vector $\chi$. $E$ and $F$ are given by $H^I_\xi$ and $H^I_\beta$ respectively where $\xi$ is asymptotically hypersurface-orthogonal and $\beta$ is proportional to the divergence of the Principal Conformal Killing--Yano tensor $\boldsymbol{h}$. I show that the first law will be satisfied by $H^I_\chi$ 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 $\boldsymbol{h}$.

gr-qc↗

arXiv2013

The Vector Volume and Black Holes

By examining the rate of growth of an invariant volume $\mathcal V$ of some spacetime region along a divergence-free vector field $v^α$, we introduce the concept of a "vector volume" $\mathcal{V}_v$. This volume can be defined in various equivalent ways. For example, it can be given as $\mathrm d \mathcal V(μ) / \mathrm d μ$, where $v^α\partial_α= \mathrm d / \mathrm d μ$, and $μ$ is a parameter distance along the integral curve of $v$. Equivalently, it can be defined as $\int v^α\mathrm d Σ_α$, where $\mathrm d Σ_α$ is the directed surface element. We find that this volume is especially useful for the description of black holes, but it can be used in other contexts as well. Moreover, this volume has several properties of interest. Among these is the fact that the vector volume is linear with respect to the the choice of vector $v^α$. As a result, for example, in stationary axially symmetric spacetimes with timelike Killing vectors $t^α$ and axial symmetric Killing vectors $ϕ^α$, the vector volume of an axially symmetric region with respect to the vector $t^α+ Ωϕ^α$ is equal for any value of $Ω$, a consequence of the additional result that $ϕ^α$ does not contribute to $\mathcal{V}_v$. Perhaps of most interest is the fact that in Kerr-Schild spacetimes the volume element for the full spacetime is equal to that of the background spacetime. We discuss different ways of using the vector volume to define volumes for black holes. Finally, we relate our work to the recent wide-spread thermodynamically motivated study of the "volumes" of black holes associated with non-zero values of the cosmological constant $Λ$.

gr-qc↗

arXiv2010

The volume of stationary black holes and the meaning of the surface gravity

The invariant four-volume $\mathcal{V}$ of a complete black hole (the volume of the spacetime at and interior to the horizon) diverges. However, if one considers the black hole set up by the gravitational collapse of an object, and integrates only a finite time to the future of the collapse, the resultant volume is well defined and finite. In this paper we examine non-degenerate stationary black holes (and cosmological horizons) and find that $\mathcal{V}_{s} \varpropto \ln(λ)$ where $s$ is any shell that terminates on the horizon, $λ$ is the affine generator of the horizon and the constant of proportionality is the Parikh volume of $s$ divided by the surface gravity. This provides an alternative local and invariant definition of the surface gravity of a stationary black hole.

gr-qc↗