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Einstein Maxwell Scalar Black Hole: Thermodynamic Properties with Logarithmic Barrow Entropy

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read With logarithmically corrected Barrow entropy, the Einstein–Maxwell–Scalar black hole temperature falls to a finite nonzero plateau, so evaporation freezes and a stable remnant survives.

desk verdict The paper's central remnant plateau is an artifact of an entropy that is not the horizon entropy of the EMS metric, and the first law fails; the rest is routine black hole thermodynamics. read the letter →

arxiv 2505.17172 v2 pith:F2QIRMJH submitted 2025-05-22 gr-qc

Ritabrata Biswas , Satyajit Pal This is my paper · ORCID
classification gr-qc PACS 04.70.Dy04.70.-s05.70.Ce
keywords Einstein-Maxwell-ScalartheoryBarrowentropylogarithmiccorrectionstoblackholethermodynamicsphasetransitionsP-VcriticalityJoule-Thomsonexpansionremnants
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

This paper studies a static, spherically symmetric Einstein–Maxwell–Scalar (EMS) black hole after replacing the standard area entropy with Barrow's fractalised entropy plus a logarithmic correction. The central claim is that under this entropy model the black hole temperature decreases as entropy grows but approaches a finite nonzero constant at large entropy, so evaporation freezes at a minimal temperature rather than running away to zero mass. From that plateau the paper derives a stable black hole remnant, a four-phase specific-heat structure, multivalued free energy with first-order phase transitions, and van der Waals-like P–V criticality. The authors read this as supporting remnant-based resolutions of the information-loss paradox and as a possible route to dark-energy and dark-matter candidates from primordial remnants.

What carries the argument

The load-bearing object is the logarithmically corrected Barrow entropy $S_{BLC}=(\pi r_h)^{1+\delta/2}+\alpha[\log(2D+r_h)+(1+\delta/2)\log(\pi r_h)]+\beta$, built from the EMS horizon radius $r_h$ through equation (12), together with the mass function $M(r_h)$ obtained from the metric (5). This entropy replaces the standard area law everywhere: temperature, specific heat, free energy, P–V criticality, and inversion curves are all recomputed from $S_{BLC}$, and the log term is what prevents the entropy from vanishing at small areas, flattening the temperature onto a finite plateau. The central mechanism is therefore the substitution of $S_{BLC}$ for the area entropy rather than any new gravitational equation.

What would settle it

Compute $T_H$ from the surface gravity in equation (14) and independently from $T_H=dM/dS_{BLC}$ using the metric's true horizon area $4\pi r_h(r_h-q^2/M)$ in the entropy formula; if the two temperatures disagree for generic $q,\lambda$, the thermodynamic system is inconsistent, and a direct scan for whether the true-area temperature still tends to a nonzero constant would decide the remnant claim.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that combining fractalised Barrow entropy with its logarithmic correction changes the late-stage thermodynamics of an EMS black hole. Using the entropy $S_{BLC}$ defined in equation (12) and the horizon temperature $T_H$ from equation (14), the temperature is monotonically decreasing in entropy but tends to a positive constant $T_{\min}$ instead of diverging or vanishing. The specific heat $C_P$ then exhibits four phases—unstable, stable, unstable, stable—with a divergence at one entropy scale and smooth sign changes at two others, while the Gibbs free energy develops negative minima that mark globally stable remnant states and a multivalued $F(T_H)$ indicating coexistence and first-order transitions. The paper also derives P–V critical curves, Joule–Thomson inversion behavior, and geometrothermodynamic Ricci divergences as further consequences of the same entropy replacement.

Load-bearing premise

The whole result depends on taking equation (12) as the entropy of this black hole and on assuming $T_H=dM/dS_{BLC}$ holds; if the entropy should instead be built from the actual horizon area $r_h(r_h-q^2/M)$, or if the first law fails, the plateau, remnant, and phase structure need not be properties of the EMS black hole.

Editorial extensions

If this is right

  • Late-stage evaporation halts at a finite temperature $T_{\min}$, leaving a high-entropy, low-temperature Planck-scale remnant instead of a runaway endpoint.
  • The specific heat shows two smooth crossovers and one divergent transition, so small and large EMS black holes can be thermodynamically stable while intermediate sizes are unstable.
  • Negative free-energy minima make the remnant the globally favored configuration at fixed temperature, and the multivalued free energy implies coexistence of phases with a first-order transition.
  • The critical pressure falls with critical volume as $P_c\sim 1/V_c^2$, so the van der Waals-like critical structure survives the fractal and logarithmic corrections.
  • The Joule–Thomson inversion pressure diverges at a finite horizon radius, marking a size threshold below which adiabatic expansion cannot cool the black hole, consistent with a frozen remnant.

Reading between the lines

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

  • Independent of the paper's claims, the remnant plateau, if real, makes the remnant a natural dark-matter candidate because it is cold, dense, and stable; the paper leaves its cosmological abundance unquantified.
  • The same entropy substitution should, by the paper's own logic, yield a similar plateau in other static black holes such as Reissner–Nordström backgrounds, which is a testable extension the paper does not pursue.
  • The entropy formula in equation (12) uses the flat-space area $4\pi r_h^2$ rather than the metric's actual horizon area $r_h(r_h-q^2/M)$; a first-law check would show whether the plateau is genuine or an artifact of that choice.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The manuscript studies the extended-phase-space thermodynamics of the static, spherically symmetric Einstein–Maxwell–scalar (EMS) black hole of Gao and Zhang, replacing the Bekenstein–Hawking entropy with the logarithmically corrected Barrow entropy S_BLC of Eq. (12). The authors compute the Hawking temperature, specific heat, and Gibbs free energy as functions of S_BLC; scan the responses to the electric charge q, the coupling λ (read as pressure), the Barrow index δ, and the log-correction parameters α and β; and claim several novel features: a finite minimum-temperature plateau with a stable remnant that freezes evaporation, multi-stage C_P phase structure, free-energy branches indicating first-order transitions, P–V criticality with rectangular-hyperbola P_c–V_c curves, a Joule–Thomson inversion analysis with a divergent inversion pressure, and geometrothermodynamic Ricci-curvature divergences. The central advertised result is that quantum-gravitational (Barrow plus logarithmic) corrections halt evaporation at T_min, supporting remnant-based resolutions of the information-loss paradox.

Significance. If the central claims were sound, the paper would be a useful phenomenological contribution connecting quantum-gravity-inspired entropy corrections to remnants and phase structure in a scalarized black hole. Credit is due for assembling a large set of explicit thermodynamic diagnostics, for summarizing parameter dependencies in Table 1, and for the paper's own transparent admission in Section 4 that the two critical-point equations (21) and (22) are not simultaneously satisfiable. However, the advertised results hinge entirely on identifying S_BLC with the thermodynamic entropy of the EMS black hole and on T_H being its conjugate temperature; both legs fail (Major Comments 1–2). Once the entropy used is not the horizon entropy of metric (3), the plateau/remnant claim, the phase structure, the P–V criticality, and the Joule–Thomson analysis are not statements about the EMS black hole. The significance of the manuscript as it stands is therefore limited, and a correct version would require a complete rederivation of every result.

major comments (5)
  1. [§2, Eq. (12)] The entropy S_BLC in Eq. (12) is not the Barrow entropy of the metric (3). The horizon two-sphere has area A = 4πf(r_h) = 4πr_h(r_h − q²/M) = 4πr_h(2D + r_h) with D = −q²/(2M), so substituting A into the template (11) yields S = [πr_h(2D + r_h)]^{1+δ/2} + α(1+δ/2)[log(2D + r_h) + log(πr_h)] + β. Eq. (12) instead contains the power term (πr_h)^{1+δ/2}, which misses the factor (2D + r_h)^{1+δ/2} for all D and δ, and its log term multiplies log(2D + r_h) by α rather than α(1+δ/2) whenever δ ≠ 0. Thus Eq. (12) matches neither the true horizon area nor the stated template (11), and every quantity derived from it in Sections 3–5 is built on a misidentified entropy.
  2. [§3, Eqs. (13)–(16)] The pair (T_H, S_BLC) does not satisfy the first law. In the λ = 0 sector, Eqs. (13)–(14) give M = r_h/2 and T_H = 1/(4πr_h), while the δ = α = β = 0 truncation of Eq. (12) is S_BLC = πr_h; then dM/dS_BLC = 1/(2π), which equals T_H only at the isolated point r_h = 1/2. Since C_P = T_H(∂S_BLC/∂T_H)_P in Eq. (15) and F = M − T_H S_BLC in Eq. (16) presuppose the standard first law, the phase-transition claims of Table 1 and Figures 1–3, and the minimal-temperature plateau that motivates the remnant claim, are computed from a non-entropy with a non-conjugate temperature. Moreover, for λ = 0 Eq. (14) gives T_H → 0 as r_h → ∞ while S_BLC → ∞, so the claimed finite plateau at large S_BLC is not an asymptotic property of the paper's own formulas.
  3. [§4, Eqs. (19)–(22)] The paper states that Eqs. (21) and (22), obtained respectively from ∂P/∂V = 0 and ∂²P/∂V² = 0, do not satisfy one another simultaneously, and that thermal fluctuations must be included to determine the critical points; no thermal-fluctuation treatment appears anywhere in Sections 4–7. In addition, Eq. (21) gives V_C ∝ ((3Mr − q²)/(Mr³))^{3/2}, which decreases with horizon radius and has inverse-volume dimensions, contradicting the definition V = (4π/3)r³ in Eq. (17). The P–V criticality analysis and the P_c–V_c hyperbola of Figure 5 are therefore not established.
  4. [§5, Eqs. (23)–(28)] The Joule–Thomson coefficient (25), the inversion temperature (26), the inversion pressure (27), and the final expression for μ_JT (33) are quoted rather than derived. Eq. (25) is presented as following from differentiation of the Smarr relation, but no Smarr relation for this EMS–Barrow system is stated or proved, and the derivation cannot be sound given the first-law failure of Major Comment 2. Eq. (26) is asserted as T_i = V(∂T/∂V) without a computation from Eq. (14), and the electric potential U in Eq. (23) and Φ = q/r_h in Eq. (30) are not derived from the EMS solution with coupling K(φ) = e^{2φ}. The claimed divergences of the inversion pressure and μ_JT, and the 'thermodynamically frozen remnant' interpretation of Section 5, rest on these unverified formulas.
  5. [§2 and Figs. 1–3] The numerical scans treat M, D, q, λ, δ, α, and β as independent parameters, but for the solution (4)–(5) the scalar charge is fixed by D = −q²/(2M), as the text itself states with reference [33]. With M = 1 and D = 1, real electric charge would require q² = −2, so the ranges q = 0.1–2.5 used in Figures 1a, 2a, 3a, and 4 are inconsistent with the spacetime whose thermodynamics is under study. The claimed influence of scalar and electric charges on stability is consequently not a property of the EMS black hole (3)–(5).
minor comments (5)
  1. [§3] The sentence 'Temperature for very low q and high α cases, however, vanishes in spite of ending at a finite plateau' is self-contradictory and should be reworded.
  2. [Throughout] There are numerous typos and editing errors, including 'Meisser 87' for Messier 87, 'sapce-time' in Section 4, 'phrases' for 'phases' in Sections 4 and 7, and garbled superscript/subscript formatting in the Barrow-area discussion in Section 2.
  3. [References] Several references are mismatched to the claims they support: Refs. [44] and [45] (ATLAS Collaboration papers) are cited for properties of Tsallis/Rényi entropy and the generalized second law; Ref. [52] (a Hawkes-process paper) is cited for Gauss–Bonnet/Lovelock inversion curves; and Refs. [53] and [54] (a nuclear-physics paper and an f(Q) gravity paper) are cited for black-hole Joule–Thomson behavior. Reference [12] is also incomplete, with its title cut off.
  4. [§6] The Ricci-scalar plots are announced as 'figure 8a–8c' but appear as Figures 9a–9c; Eq. (35) contains unmatched brackets and is nearly illegible; and the caption of Figures 9a–9c uses 'd = 1' where the text uses D and reports '=. 1' for the value of q, which is presumably a typo.
  5. [Table 1] Several cells of Table 1 are empty or malformed (for example, the δ row contains an empty 'S_BLC < S3' entry), and the caption refers to parameter orderings without defining the entropy thresholds S1–S5 that organize the phase statements.

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed minimal-temperature plateau/remnant is a definitional artifact: S_BLC contains a free additive constant β while T_H does not, so 'large S_BLC at fixed T_H' is built into Eq. (12), not predicted by EMS gravity.

  1. self definitional [Eq. (12), Section 2; Section 3 (Hawking temperature and T_min plateau, Fig. 1e)]
    "S_BLC = (π rh)^{1+δ/2} + α[ log(2D+rh) + (1+δ/2) log(π rh)] + β . ... In general, in all the cases, temperature is followed to decrease with the increment of entropy but temperature approaches a finite nonzero constant at large S_BLC, i.e., evaporation stops or freezes out at a minimal temperature T_min."

    T_H in Eq. (14) contains only r_h, M, q, λ; it contains no β. S_BLC in Eq. (12) contains β as an additive constant, so S_BLC can be made arbitrarily large at fixed r_h by increasing β while T_H remains exactly constant. The paper's central qualitative claim—'temperature approaches a finite nonzero constant at large S_BLC'—is therefore a direct consequence of how S_BLC was defined, not a result of the EMS field equations. The 'minimal temperature plateau/remnant' is the free additive constant β renamed as a physical freezing of evaporation. The same definitional character is reinforced because Eq. (12) is explicitly obtained by substituting the Schwarzschild area into the Barrow template, so the EMS horizon area 4π r_h(2D+r_h) never enters the entropy that generates the plateau.

full rationale

The central 'prediction' of a minimal-temperature plateau and stable remnant is not a consequence of the EMS field equations; it is encoded in the definition of S_BLC. Equation (12) contains an additive free constant β, while the Hawking temperature (14) is independent of β, so the regime 'large S_BLC at fixed T_H' is reachable by construction. The paper's own Section 7 confirms the ordering: 'We expected, in this work, that the combo of fractalised entropy and logarithmic correction may give minimum temperature, stable remnants... This leads us to choose...' The remaining thermodynamic quantities (C_P, F, JT coefficient, GTD Ricci) are algebraic rearrangements of the same assumed S_BLC together with T_H, so they inherit the definitional plateau rather than adding independent content. There is also a serious internal-consistency problem that is not itself circularity: Eq. (12) is obtained by substituting the Schwarzschild area into the Barrow template (11), while the EMS horizon area from metric (3) is 4π r_h(2D+r_h); the paper never verifies T_H = dM/dS_BLC, so C_P and F computed from this S_BLC are not guaranteed to be thermodynamic quantities. No load-bearing self-citation appears: [38] and [40] are external sources for the entropy ansatz. On balance, one central claim reduces by construction to the assumed entropy, giving partial circularity (score 6).

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

The central calculation imports the EMS solution and the Barrow/log-corrected entropy as given, then derives temperature, specific heat, free energy, PV criticality, and JT expansion. The main free choices are the entropy model parameters delta, alpha, beta, and the treatment of D as independent. No new entities are introduced. The key burden is that S_BLC is not the area entropy of the metric in eq. (3).

free parameters (4)
  • delta (Barrow fractal index) = 0 <= delta <= 1, e.g., 0.5 in most plots
    Sets the power-law of the Barrow entropy term; chosen by hand, not derived.
  • alpha (logarithmic correction coefficient) = e.g., alpha=10 in Section 6 plots
    Controls the strength of the log correction; arbitrary.
  • beta (entropy constant shift) = e.g., beta=2 in Figure 4
    Constant offset in entropy; affects free energy and specific heat plots.
  • D (scalar charge in entropy) = D=1 in plots
    Appears in log(2D+r_h); physically D = -q^2/(2M) but treated as independent.
assumptions (4)
  • domain assumption The Gao-Zhang dilaton-de Sitter solution (eqs. 3-5) is the spacetime background for the thermodynamics.
    Imported from ref. [34]; not re-derived or validated.
  • domain assumption Barrow entropy with log correction, eq. (12), is the system's entropy.
    Postulated rather than derived; uses Schwarzschild area instead of the EMS horizon area 4*pi*f(r_h).
  • domain assumption Extended phase space with P = lambda/3 and V = 4*pi*r^3/3 applies to this solution.
    Standard for spherical AdS black holes with f(r)=r^2; questionable when f(r)=r(r-q^2/M).
  • domain assumption The first law dM = T dS + Phi dq + V dP holds with T from surface gravity and S from eq. (12).
    No check of T = dM/dS; given the entropy-area mismatch, consistency is doubtful.

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Cite this review

Pith. "Pith review of Einstein Maxwell Scalar Black Hole: Thermodynamic Properties with Logarithmic Barrow Entropy." pith.science (2026). https://pith.science/paper/F2QIRMJH

@misc{pith2026250517172,
  author       = {Pith},
  title        = {Pith review of: Einstein Maxwell Scalar Black Hole: Thermodynamic Properties with Logarithmic Barrow Entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F2QIRMJH}},
  note         = {Machine review of arXiv:2505.17172}
}
read the original abstract

The thermodynamics of black holes (BHs) within the Einstein Maxwell Scalar (EMS) framework, incorporating Barrow entropy and its logarithmic corrections to analyze quantum gravity effects is investigated here. A static, spherically symmetric BH solution is obtained by coupling the scalar field nonminimally to the electromagnetic field through a scalar dependent function. The thermodynamic properties including temperature, specific heat, and Gibbs free energy are derived and explored in the context of Barrow-modified entropy. We identify phase transitions and critical behavior by analyzing PV criticality and uncover the influence of scalar and electric charges on stability. Furthermore, the Joule Thomson expansion is examined to understand the inversion behavior and thermodynamic responses under adiabatic expansion. Our findings suggest the presence of thermodynamic instabilities, remnants, and nontrivial critical phenomena, providing new insights into BH thermodynamics in modified gravity scenarios with quantum corrections.

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Reviewed August 7, 2026 · model on record in the stance chip above.