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arxiv: 2606.30507 · v1 · pith:3BGFVWYJnew · submitted 2026-06-29 · 🌀 gr-qc

The Role of the Volume in Black Hole Thermodynamics

William Ballik This is my paper

Pith reviewed 2026-06-30 04:48 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black hole thermodynamicsKerr-AdS black holesfirst lawSmarr relationconserved chargesthermodynamic volumeKilling vectorsAMD energy
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The pith

The first law of black hole thermodynamics holds for energy E but not F because only the associated Killing vector keeps unvarying components.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper adapts the Barnich-Compère definition to associate a conserved quantity H^I_χ with a Killing vector χ in Kerr-anti-de Sitter spacetimes. It demonstrates that this quantity satisfies the first law of black hole thermodynamics, including with varying pressure, precisely when both the Killing vector and the background metric have components that do not vary. This condition is met for the vector ξ defining E, the energy in an asymptotically nonrotating frame, but not for β defining F in the rotating frame. The same framework accounts for the geometric volume appearing in the Smarr relation tied to F through properties of the principal conformal Killing-Yano tensor.

Core claim

By defining the conserved quantity H^I_χ associated with a Killing vector χ, where E equals H^I_ξ and F equals H^I_β, the first law is satisfied if both χ^a and the background anti-de Sitter metric have unvarying components. This holds for ξ^a but not β^a, which explains why the first law works for E but not F. The vector volume V_C appears in the β-associated Smarr relation due to simplifications related to the principal conformal Killing-Yano tensor h.

What carries the argument

The conserved quantity H^I_χ defined by adapting the Barnich-Compère prescription to a Killing vector χ, which yields the AMD energies E and F for the specific choices of ξ and β.

Load-bearing premise

The adapted Barnich-Compère definition of the conserved quantity applies directly to the Kerr-AdS black holes with the given choices of Killing vectors and without further adjustments.

What would settle it

Compute the first law variation explicitly for H^I_β and check if it fails to hold when the components of β vary with the black hole parameters.

Figures

Figures reproduced from arXiv: 2606.30507 by William Ballik.

Figure 2.1
Figure 2.1. Figure 2.1: The collapse of a timelike boundary surface Σ that terminates at the central singularity simultaneously with the null cone δ and produces a black hole with horizon H. The null cone v is any null cone to the future of δ. The invariant four-volume V calculated here is bounded by δ and v and is to the interior of H. [Clarification: the details of this are in [9], but the idea is that for one of the calculat… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: The vertical lines represent the congruence of integral curves of v. The region R is shown in light grey and the specific region R(µ) shown in darker grey. The hypersurface Γ is labelled and the other surface represents a movement a parameter distance µ along the integral curves. The coordinates x i = const are shown. Since v α ∂α = d dµ , (2.4.3) 33 [PITH_FULL_IMAGE:figures/full_fig_p076_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: The situation where the region R is something like a closed ball, with boundary ∂R corresponding to a 3-sphere (if R is four-dimensional). Here a hypersurface Γ cuts R into two regions, R1 and R2, shown. The boundary ∂R1 is divided into one region which is a common boundary to both R and R1, identified as ∂R ∩ ∂R1, and the intersection of R with Γ, R ∩ Γ. Then we have, choosing suitable orientation, Z R1… view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: The shaded region is R, and the vertical lines represent the integral curves of vector v within it. Γ here is a hypersurface which in this case “cuts into the side of R,” and so does not intersect with each integral curve once (intersecting some twice and some zero times). The region Q is the region “between Γ and ∂R” and the boundaries of Q are the portion of ∂Q lying within ∂R, given by ∂Q∩∂R, and the … view at source ↗
read the original abstract

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 $\boldsymbol{h}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that the first law holds for the AMD energy E but not the rotating-frame quantity F in Kerr-AdS black holes because E = H^I_ξ and F = H^I_β, where H^I_χ is an adaptation of the Barnich-Compère conserved quantity. The first law is satisfied by δH^I_χ precisely when both the Killing vector χ^a and the fixed AdS background have unvarying components (true for the asymptotically hypersurface-orthogonal ξ but not for β proportional to div h). This distinction explains the thermodynamic viability of E (interpreted as enthalpy) versus F, while V_C enters the β-Smarr relation via simplifications from the PCKY tensor h.

Significance. If the adaptation of Barnich-Compère is valid without extraneous terms and the unvarying-components criterion follows directly, the work supplies a unified kinematic explanation for the choice of E over F in extended black-hole thermodynamics and for the distinct roles of V_th and V_geo = V_C. It connects the first-law and Smarr structures to the asymptotic properties of the Killing vectors and the background metric, potentially clarifying why the pressure term appears differently in the two cases.

major comments (2)
  1. The central claim rests on the adaptation of the Barnich-Compère definition of H^I_χ to the Kerr-AdS setting and the direct correspondence E = H^I_ξ, F = H^I_β. The manuscript must explicitly verify that this adaptation introduces no additional boundary or variation terms when χ = β (whose components vary), as this is the load-bearing step for why δH^I_β fails the first law while δH^I_ξ succeeds.
  2. The assertion that δH^I_χ obeys the first law exactly when both χ^a and the background AdS metric have unvarying components requires a step-by-step derivation from the definition of H^I_χ, including explicit checks against the concrete forms of ξ and β in the Kerr-AdS metric (and against the equations of the three cited papers). Without this derivation or the checks, the sufficiency of the unvarying-components condition remains unverified.
minor comments (2)
  1. Notation for the vectors ξ and β, and for the PCKY tensor h, should be introduced with explicit definitions or references to their components in the Kerr-AdS coordinates before the statements about unvarying components.
  2. The abstract refers to 'the section defining H^I_χ' implicitly; the manuscript should number the relevant equations for the definition and the variation δH^I_χ so that the first-law condition can be traced directly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the constructive feedback. We agree that the adaptation and the unvarying-components criterion require more explicit verification and will revise the manuscript to include the requested step-by-step derivations and checks.

read point-by-point responses

  1. Referee: The central claim rests on the adaptation of the Barnich-Compère definition of H^I_χ to the Kerr-AdS setting and the direct correspondence E = H^I_ξ, F = H^I_β. The manuscript must explicitly verify that this adaptation introduces no additional boundary or variation terms when χ = β (whose components vary), as this is the load-bearing step for why δH^I_β fails the first law while δH^I_ξ succeeds.

    Authors: We agree this verification is necessary. The revised manuscript will add an explicit section adapting the Barnich-Compère formula to Kerr-AdS, with direct computation showing that no extraneous boundary or variation terms appear for χ = β. This will confirm the distinction between δH^I_ξ and δH^I_β arises only from the component variation. revision: yes

  2. Referee: The assertion that δH^I_χ obeys the first law exactly when both χ^a and the background AdS metric have unvarying components requires a step-by-step derivation from the definition of H^I_χ, including explicit checks against the concrete forms of ξ and β in the Kerr-AdS metric (and against the equations of the three cited papers). Without this derivation or the checks, the sufficiency of the unvarying-components condition remains unverified.

    Authors: We will include a new derivation subsection in the revision. Starting from the definition of H^I_χ, we derive the first-law condition and verify it holds iff both χ^a and the AdS background have unvarying components. Explicit checks will use the concrete Kerr-AdS expressions for ξ and β, cross-referenced to the equations in the three cited papers. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation adapts external Barnich-Compère definition to Kerr-AdS and derives unvarying-components condition independently

full rationale

The paper's central steps consist of (1) adapting the Barnich-Compère conserved quantity H^I_χ from the cited external reference arXiv:gr-qc/0412029, (2) identifying E = H^I_ξ and F = H^I_β using the given vector definitions, and (3) showing that δH^I_χ obeys the first law precisely when both χ^a and the fixed AdS background have unvarying components. These steps rely on external citations (Gibbons et al., Cvetič et al., Barnich-Compère) whose results are not reproduced or fitted inside the present work; the unvarying-components criterion is presented as a derived property of the adapted definition rather than a self-fit or self-citation chain. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on standard GR assumptions about Killing vectors and conserved charges plus the specific definitions of E, F, and the principal conformal Killing-Yano tensor taken from the cited papers; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Existence of a conserved charge H^I_χ associated with any Killing vector χ via the Barnich-Compère construction
    Invoked when defining E = H^I_ξ and F = H^I_β
  • domain assumption The background AdS metric and the chosen Killing vector can be compared for component invariance
    Central to the condition that makes the first law hold

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discussion (0)

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