Charged accelerating AdS black hole of $f(R)$ gravity and the Joule-Thomson expansion

A. A. Masoudi; B. Pourhassan; J. Sadeghi; M. Rostami; S. Miraboutalebi

REVIEW 4 major objections 5 minor 77 references

Charged accelerating AdS black hole of $f(R)$ gravity and the Joule-Thomson expansion

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A modified-gravity black hole shows the van der Waals inversion ratio 1/2

desk verdict The paper applies standard black hole thermodynamics to a charged accelerating AdS black hole in f(R) gravity, but the central results rest on an unjustified constant-a assumption and fail internal algebraic consistency checks. read the letter →

arxiv 1908.08410 v2 pith:YFTBU5XL submitted 2019-08-21 gr-qc

M. Rostami , J. Sadeghi , S. Miraboutalebi , A. A. Masoudi , B. Pourhassan This is my paper · ORCID

classification gr-qc MSC 83C5780A10 PACS 04.70.Dy04.50.Kd

keywords blackholethermodynamicsf(R)gravitychargedacceleratingAdSP-VcriticalityvanderWaalsphasetransitionJoule-Thomsonexpansioninversiontemperatureextendedspace

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that a charged, accelerating anti-de Sitter (AdS) black hole in $f(R)$ modified gravity behaves like a van der Waals fluid in extended phase space, where the cosmological constant acts as pressure. It constructs the thermodynamics of this black hole, finds a critical point in its pressure-volume diagram, and studies the Joule-Thomson (constant-enthalpy) expansion in which the mass is held fixed. The central quantitative result is that the ratio of the minimum inversion temperature to the critical temperature is $T_i^{\min}/T_c = 1/2$ for a specific acceleration parameter value, $a=1.12$. If correct, this extends a known black-hole/gas analogy to accelerating solutions in modified gravity and fixes the cooling-to-heating switching point of the black hole.

What carries the argument

The load-bearing device is the constant-curvature $f(R)$ setup combined with the assumption that $A r_+=a$ is a fixed constant. In $f(R)$ gravity with $R=R_0$, the theory behaves like Einstein gravity with a rescaled coupling, and the paper parametrizes the modification by $b=1+f'(R_0)>0$. Because the Hawking temperature diverges as $r_+\to 1/A$, the paper imposes $A r_+=a$ to make the temperature finite, turning $a$ into a free parameter in the equation of state. That equation of state, together with the Joule-Thomson coefficient $\mu=(\partial T/\partial P)_M$, produces the critical quantities, the inversion curves, and the ratio identity $T_i^{\min}/T_c=1/2$ at $a=1.12$. The relation between the two sides of the isenthalpic expansion is carried by the constant-mass condition, since for AdS black holes the mass is interpreted as enthalpy.

What would settle it

Recompute the Hawking temperature and the isenthalpic curves directly from the horizon condition $f(r_+)=0$ without imposing $Ar_+=a$, and check whether $A r_+$ changes with pressure; if it does, the equation of state and the inversion ratio shift. A simpler check is to repeat the calculation for nearby values such as $a=1.0$ and $a=1.2$ and see whether $T_i^{\min}/T_c$ remains close to $1/2$ or moves rapidly with $a$.

Watch

Extended reading notes

Core claim

The central claim is that the charged accelerating AdS black hole in $f(R)$ gravity with constant Ricci scalar $R_0$ admits a complete extended thermodynamics in which the pressure is $P=-bR_0/(32\pi)$ with $b=1+f'(R_0)>0$, and the mass is treated as enthalpy. From the metric and the first law, the paper derives an equation of state $P(T,r_+)$ that has a van der Waals-like critical point, with critical values expressed in terms of the charge $q$, the $f(R)$ parameter $b$, and the dimensionless acceleration parameter $a$. The heat capacity is computed and its roots and divergences are identified with phase transitions and stability changes. For the Joule-Thomson expansion, the paper obtains the inversion temperature and inversion curves in the $T$--$P$ plane, shows where the Joule-Thomson coefficient vanishes, and determines the reverse point on isenthalpic curves at which cooling turns into heating. The derived ratio $T_i^{\min}/T_c$ is shown to equal $1/2$ at $a=1.12$, matching a value previously reported for charged AdS black holes.

Load-bearing premise

The whole calculation hinges on treating $A r_+=a$ as a constant that does not depend on the horizon radius, pressure, or temperature; if that product varies, the equation of state, critical point, and the $1/2$ inversion ratio would no longer describe the black hole.

Editorial extensions

If this is right

The $P$--$V$ isotherms have a critical point set by $q$, $b$, and $a$, so for $T\approx T_c$ the black hole shows van der Waals-like first-order phase transitions.
In constant-mass Joule-Thomson expansion, the inversion curve separates cooling from heating, and its intersection with an isenthalpic curve marks the reverse point where the sign of $\mu$ changes.
The ratio $T_i^{\min}/T_c=1/2$ at $a=1.12$ reproduces the value reported for charged AdS black holes without acceleration, suggesting a common feature of van der Waals-type black hole thermodynamics.
Larger values of the $f(R)$ parameter $b$ raise the inversion temperature and make the inversion curves more uniform, while larger charge raises the inversion temperature at low pressure.
The $f(R)$ modification changes the critical ratio $\rho_c=P_c v_c/T_c$ away from the usual $3/8$ unless a special combination of parameters is chosen.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extension: If $A r_+$ is actually pressure-dependent, the advertised $1/2$ ratio should be seen as a consequence of the constant-$a$ assumption rather than a robust prediction; a small perturbation $a(P)$ would reveal how sensitive the ratio is.
Extension: Applying the same construction to other modified-gravity or hairy black hole solutions could test whether the $1/2$ inversion ratio is a universal feature of van der Waals-type black holes or an artifact of this particular parametrization.
Extension: A direct numerical scan of $T_i^{\min}/T_c$ for values of $a$ slightly below and above $1.12$ would show how finely tuned the result is, since the paper does not report the ratio's dependence on $a$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. The paper runs the familiar P–V criticality and Joule-Thomson formalism over a charged accelerating AdS black hole at constant curvature in f(R) gravity, and the headline result is a claimed universal ratio T_i^min/T_c = 1/2. Neither claim survives contact with the paper's own equations.

The setup is reasonable. The charged accelerating AdS metric is known, the f(R) constant-curvature treatment just rescales the gravitational constant by b = 1 + f'(R0), and the extended-phase-space machinery is standard. The bibliography covers the relevant literature on accelerating black holes, f(R) thermodynamics, and Joule-Thomson expansion. The paper is readable, and the figures do show the expected qualitative van der Waals–like shapes.

The problems start with the constant-a assumption in Section 2. The paper sets Ar_+ = a 'for simplicity' to dodge the T→∞ singularity. But a is then treated as a fixed parameter in all later thermodynamics, even though A is the acceleration parameter and r_+ is the horizon radius. If a is constant, A varies when r_+ varies along an isenthalpic curve or a critical-point scan, and the first law (18) contains no A work term. So the equation of state, critical values, and inversion curves are not the thermodynamics of a fixed-A black hole. On top of that, the specific value a = 1.12 is chosen so that Eq. (38) becomes 1/2. That is tuning to match Ref. [75], not a derived prediction.

Even within the a-constant assumption, the algebra does not close. Substituting v = 2r_+ into Eq. (19) gives a q^2 term with sign opposite to Eq. (20). The critical volume in Eq. (22) fails the inflection conditions of Eq. (21) when plugged into the displayed equation of state. And Eq. (32) versus Eq. (33) shows a missing factor of 3 and opposite signs for the b r_+^2 and q^2 terms. These are not peripheral typos; they support the claimed criticality and the inversion ratio. The figures also use inconsistent values of a (1, 0.972, 1.12, 0.4) without fixing the solution space.

The underlying solution is legitimate, and applying known methods to a new background has some catalogue value, but the displayed formulas are internally contradictory and the headline ratio is an artifact. A serious reader should redo every step before citing anything from Sections 3 and 4. If this came to me, I would not send it to referees in its current form; the authors need to fix the constant-a issue and the algebraic errors first. After that, it might become a modest but acceptable paper.

Referee Report

4 major / 5 minor

Summary. The manuscript investigates extended-phase-space thermodynamics and Joule-Thomson expansion for a charged, accelerating AdS black hole in f(R) gravity. The authors take a constant-curvature f(R) solution, impose Ar_+=a as a constant to avoid a temperature singularity, derive a Hawking temperature, entropy, and equation of state, and then analyze P-V criticality, heat capacity, and inversion curves. The central quantitative claim is that the ratio of the minimum inversion temperature to the critical temperature is T_i^min/T_c=1/2 for a=1.12, in agreement with Ref. [75]. I find that the central line of reasoning is not internally consistent: the constant-a assumption is not a legitimate thermodynamic constraint, the equation of state contains sign and factor inconsistencies, the critical volume does not satisfy the stated inflection conditions, and the advertised ratio is obtained by tuning a free parameter in a regime where the critical and inversion temperatures are negative.

Significance. If the results were sound, the paper would provide a useful example of van der Waals-like criticality and Joule-Thomson behavior in a modified-gravity accelerating black hole, extending a popular extended-phase-space calculation to f(R) backgrounds. The manuscript does present the full calculation chain from the metric to the equation of state, critical quantities, heat capacity, and inversion curves, and it uses the Wald entropy rather than the area law. However, the claimed results do not follow from the displayed equations, and the single quantitative agreement with the literature is a parameter choice rather than a derived prediction. I therefore cannot recommend publication in the present form.

major comments (4)

[Sec. 2, before Eq. (13)] The assumption that Ar_+=a is a constant is not a valid thermodynamic constraint. In the metric (1), A is a fixed acceleration parameter, whereas r_+ is the horizon position and hence a thermodynamic variable. If a is held fixed during the P-V and Joule-Thomson processes considered in Secs. 3 and 4, A must vary as a/r_+, but the first law in Eq. (18) contains no work term conjugate to A, so these are not processes within a fixed solution space. Conversely, if A is held fixed, a varies and the subsequent derivatives that treat a as constant in Eqs. (19), (22), and (32)-(38) are not the thermodynamics of the black hole. The singularity-avoidance rationale does not justify promoting Ar_+ to a thermodynamic constant.
[Sec. 3.1, Eqs. (19)-(22)] The equation of state is internally inconsistent. Substituting nu=2r_+ into Eq. (19) gives a q^2 term -3D q^2/(8 pi r_+^4), whereas Eq. (20) contains +6D q^2/(pi nu^4), which at nu=2r_+ is +3D q^2/(8 pi r_+^4). In addition, the critical volume quoted in Eq. (22) does not solve the inflection conditions (21) for the displayed EOS (20); applying (21) to (20) gives nu_c=2 sqrt(6/b) q, not sqrt(6/b) q. Thus the critical-point calculation that underlies the claimed van der Waals behavior is not supported.
[Sec. 4, Eqs. (37)-(39)] The value a=1.12 used for the headline ratio is outside the physical regime of the preceding formulas. Since a^2=1.2544, Eq. (22) gives T_c proportional to (1-a^2), which is negative, and Eq. (37) gives T_i^min proportional to (3D-1), which is also negative because D=(1-a^2)^2/(3-a^2) is approximately 0.037. The ratio T_i^min/T_c in Eq. (38) is therefore a ratio of two negative temperatures and does not describe a physical inversion or critical point.
[Sec. 4, Eqs. (38)-(39), Figs. 5-7] The agreement with Ref. [75] is obtained by selecting a=1.12; Eq. (38) depends on the free parameter a, and no independent determination of a is given. This is a one-parameter fit to the target value 1/2, not a prediction. The figures further use inconsistent values: Figs. 5 and 6 take a=1.12 while Fig. 7 takes a=0.4, and no justification is given for the change.

minor comments (5)

[Sec. 3.1, after Eq. (23)] The condition 'when B -> 2(3-a^2)/(3(1-a^2)b)' is unexplained; B is not defined, and the statement that 'nu_c changes to the form (3-a^2)/(3(1-a^2)b)' is dimensionally incompatible with Eq. (22), which has dimensions of q times a dimensionless factor.
[Eq. (31)] The thermodynamic volume V in Eq. (31) appears without the K factor that appears in Eq. (16), so the relation between V and r_+ used in the Joule-Thomson section should be stated explicitly.
[Sec. 4] Fig. 7 uses a=0.4 while Figs. 5 and 6 use a=1.12; the captions should state all relevant parameter values so that the inversion and isenthalpic curves can be reproduced.
[Throughout] There are recurring typographical and nomenclature issues: 'Joule-Thompson' for 'Joule-Thomson' in the Section 4 heading and elsewhere, 'unites' for 'units' in the introduction, and 'inverse temperature' used where 'inversion temperature' is meant.
[Ref. [75]] Ref. [75] is cited as an arXiv preprint without indicating which result in that paper gives the ratio 1/2; the authors should identify the specific equation or result being compared.

Circularity Check

1 steps flagged · score 6.0 of 10

The advertised ratio T_i^min/T_c = 1/2 is not predicted: it is obtained by choosing a=1.12 in a formula that depends on the free parameter a, then citing agreement with Ref. [75].

fitted input called prediction [Sec. 2 before Eq. (13); Sec. 4, Eqs. (38)-(39)]
"To avoid this singularity, one can consider the value of Ar+ as a constant for simplicity Ar+ = a. ... T min i TC = 1 2 9 √ 2D(3D − 1) 2(3 − a2)2 , (38) In the case of a = 1.12 we have, T min i TC = 1 2. (39) The above relation has a perfect agreement with the result of [75]."

Eq. (38) expresses the ratio T_i^min/T_c as a function of the dimensionless parameter a introduced in Sec. 2 by the ad hoc assumption Ar_+ = a. This a is not fixed by the f(R) solution, the metric, or any independent thermodynamic input; it is a free parameter of the calculation. Eq. (39) simply evaluates Eq. (38) at a = 1.12 and obtains 1/2, and the next sentence declares 'perfect agreement with the result of [75].' Because a = 1.12 is evidently selected so that Eq. (38) evaluates to the advertised 1/2, the ratio is not predicted from the black-hole model. The agreement with Ref. [75] is therefore not independent confirmation but a restatement of the parameter choice. The claimed universal ratio reduces by construction to a fitted input.

full rationale

The central circularity is confined to the headline inversion-temperature ratio. The paper introduces a free dimensionless parameter a by setting Ar_+ = a 'for simplicity,' derives the ratio T_i^min/T_c in Eq. (38) as a function of this a, and then chooses a = 1.12 in Eq. (39) to obtain 1/2, citing 'perfect agreement with the result of [75].' Since no independent physical principle fixes a = 1.12, the agreement is manufactured by parameter selection rather than derived. The other parts of the derivation — the equation of state, inflection-point criticality, and inversion curve from the condition mu = 0 — follow algebraically from the assumed metric, entropy, and first law, so they are not circular, although they may have separate algebraic or physical inconsistencies (e.g., the treatment of A as constant versus r_+ varying). Self-citations in the paper are to prior f(R) and black-hole thermodynamics work and are not load-bearing for this central claim; Ref. [75] is external, not self-citational, but the comparison is vacuous because the parameter was tuned to match it. The score of 6 reflects one central 'prediction' that reduces to a fitted parameter, while the surrounding framework remains self-contained.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the cited solution, the constant-curvature reduction, the extended phase-space identifications, and the ad hoc constancy of a. No new particles or fields are introduced. The free parameters a and b are chosen by hand and scanned; a in particular is set to 1.12 to force the quoted ratio.

free parameters (2)

a = A r_+ = 1.12 in Eq. (39); 0.972 in Fig. 3; 0.4 in Fig. 7
Assumed constant to avoid the r_+ -> 1/A temperature singularity; the advertised ratio 1/2 is obtained by choosing a=1.12.
b = 1 + f'(R_0) = varied: 1.0, 1.6; range in Fig. 2
f(R) parameter with no independent measurement; controls all thermodynamic quantities and is scanned in figures.

assumptions (4)

domain assumption Constant scalar curvature R=R_0 in the f(R) action
Invoked in Section 2 after Eq. (10): 'we consider only the case of constant scalar curvature R=R_0'. This reduces f(R) to Einstein gravity with rescaled G and a cosmological constant.
domain assumption The charged accelerating AdS metric (1)-(11) is a valid f(R) solution with the thermodynamic mass, charge, and entropy given in Refs. [65,66,59]
The paper cites [59] for the solution instead of deriving it; all following results inherit this solution's validity.
domain assumption Extended phase-space identifications P=-b R_0/(32 pi), V=4 pi r_+^3(1-a^2)/(3K), and M as enthalpy
Eqs. (15)-(17); needed to define pressure, volume, and the isenthalpic Joule-Thomson process.
ad hoc to paper A r_+ = a remains constant during thermodynamic processes
Introduced in Section 2 before Eq. (13) 'for simplicity'; all critical and inversion formulas depend on a.

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Pith. "Pith review of Charged accelerating AdS black hole of $f(R)$ gravity and the Joule-Thomson expansion." pith.science (2026). https://pith.science/paper/YFTBU5XL

@misc{pith2026190808410,
  author       = {Pith},
  title        = {Pith review of: Charged accelerating AdS black hole of $f(R)$ gravity and the Joule-Thomson expansion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YFTBU5XL}},
  note         = {Machine review of arXiv:1908.08410}
}

abstract

In this paper, the thermodynamical properties and the phase transitions of the charged accelerating anti-de Sitter (AdS) black holes are investigated in the framework of the $f(R)$ gravity. By studying the conditions for the phase transitions, it has been shown that the $P-V$ criticality and the van der Waals like phase transitions can be achieved for $ T \approx T_{c} $. The Joule-Thomson expansion effects are also examined for the charged accelerating AdS black holes of the $f(R)$ gravity. Here, we derive the inversion temperatures as well as the inversion curves. Then, we determine the position of the reverse point for different values of mass $M$ and parameter $b$ for the corresponding black hole. At this point, the Joule-Thompson coefficient is zero. So, in such case, we can say that such point is very important for the finding of cooling - heating regions. Finally, we calculate the ratio of minimum inversion temperature and critical temperature for such black hole.

Figures

Figures reproduced from arXiv: 1908.08410 by the authors.

**Figure 1.** The function f(r) in terms of the horizon radius with different values of b, and A = 0.5, q = 0.4, R0 = −12, m = 0.6. The position of the black hole event horizon is determined by f(r) = 0 (which is illustrated by blue dashed line of the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Temperature of the charged accelerating AdS black hole in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** The pressure diagram is plotted respect to the volume with diffe [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 3.** Figure 3: Pressure of the charged accelerating AdS black hole with t [PITH_FULL_IMAGE:figures/full_fig_p010_3.png]

**Figure 4.** Figure 4: The heat capacity in terms of r+ for R = −12, A = 0.5, and m = 0.3. (a) q = 0.1 (b) q = 0. In [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: The inversion curves for charged accelerating AdS black h [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: The inversion curves for charged accelerating AdS black h [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: The isenthalpic curves for different values of black hole mas [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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