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Inside a black hole, entropy grows with time but trails the horizon

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A doctoral thesis compiles and reproduces prior semiclassical results claiming the interior scalar-mode entropy of black holes is proportional to horizon entropy with a coefficient less than one.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection A PhD thesis repackaging known CR-Zhang results, with a load-bearing proportionality claim that is not actually derived because it depends on an unfixed constant gamma; useful as a survey, not as a new research paper. the 4 major comments →

arxiv 2509.03042 v1 pith:2K3E64KP submitted 2025-09-03 gr-qc hep-phhep-thmath-phmath.MP

Aspects of the Black Hole Interior Volume and Entropy

Shad Ali This is my paper
classification gr-qc hep-phhep-thmath-phmath.MP MSC 83C5783C45 PACS 04.70.Dy04.70.-s
keywords black hole thermodynamicsHawking radiationblack hole interior volumeinterior entropyBekenstein-Hawking entropyblack hole evaporationinformation paradoxf(R) gravity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 thesis sets out to show that the interior of a black hole is not a small dead end but a growing region: the volume of the largest spacelike hypersurface inside the horizon increases linearly with advanced time, and the entropy of the massless scalar quantum modes occupying that volume rises with it. The central claim is that this interior entropy remains directly proportional to the Bekenstein-Hawking horizon entropy throughout quasi-static Hawking evaporation, with a proportionality constant less than one. If true, the horizon does not have to be the final storehouse of infallen information; most of the information is lost at the horizon, but the interior modes keep a proportional, growing record. The same linearity and proportionality are claimed for Schwarzschild, Reissner-Nordstrom, Kerr, d-dimensional charged, and f(R) black holes, with the coefficient shrinking as dimension or the f(R) parameter grows.

Core claim

For a black hole formed in collapse, the interior volume is defined as the maximum proper volume of a spacelike spherically symmetric 3-dimensional hypersurface bounded by a sphere on the horizon. At late advanced time the maximal surface sits at a constant radius, giving an interior volume proportional to advanced time; for Schwarzschild this is 3*sqrt(3)*pi*M^2*nu. Counting scalar-field modes in this volume with WKB and phase-space methods gives an entropy proportional to that volume, and setting the mode temperature to the Hawking temperature while letting the mass decrease through the Stefan-Boltzmann law yields the differential relation dS_int = -F(M,Q) dS_BH. The same pattern is claime

What carries the argument

The maximal-hypersurface interior volume: for a spherically symmetric black hole, the volume bounded by a horizon sphere is defined as the largest proper volume of a spacelike, spherically symmetric 3-surface. Solving the geodesic equation in the auxiliary metric r^4(-f(r)*nu_dot^2 + 2*nu_dot*r_dot) gives a constant-radius maximal surface and a volume linear in advanced time. The entropy computation combines this volume with quantum-statistical counting of modes, the free energy of a thermal scalar gas, and the Hawking-temperature equilibrium; the evaporation link is the Stefan-Boltzmann law. This construction does the work of turning a geometric volume into a thermodynamic entropy and then

Load-bearing premise

The interior scalar field is assumed to sit in equilibrium at the Hawking temperature while the hole evaporates quasi-statically as a blackbody; if the modes just inside the horizon are not at that temperature, the claimed proportionality between the two entropies fails.

What would settle it

Compute the effective temperature of the massless scalar modes on the constant-radius maximal hypersurface inside a Schwarzschild black hole using a fully dynamical vacuum state appropriate to collapse rather than assuming equilibrium at the Hawking temperature. If the occupation spectrum gives a temperature different from 1/(8*pi*M), then dS_int will not equal -F(M) dS_BH, and the central proportionality fails; the same check can be done by evaluating the one-loop free energy in the interior volume and comparing its mass dependence with F(M).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the interior volume grows linearly with advanced time, an evaporating black hole can keep absorbing quantum modes without saturating, giving a concrete storage place for information that Hawking radiation appears to lose.
  • The proportionality dS_int = -F dS_BH with F less than one means the interior entropy rises as the horizon shrinks, so the combined entropy need not decrease; this is the proposed route toward the information paradox.
  • Rotation and charge do not break the storage mechanism: the same linear growth is claimed for Kerr and Reissner-Nordstrom black holes, with the coefficient depending on mass, charge, and spin but staying below unity.
  • In higher dimensions and in f(R) gravity the coefficient shrinks as the dimension or the modified-gravity parameter grows, implying that such black holes radiate their horizon entropy away faster relative to the growth of interior entropy.
  • At late evaporation the interior volume and entropy remain nonzero down to Planck-scale masses, supporting the idea of a remnant carrying residual information.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The paper assumes the interior mode temperature equals the Hawking temperature; a direct calculation of the renormalized stress-energy tensor just inside the horizon would test whether the proportionality survives without that equilibrium.
  • Integrating dS_int = -F dS_BH from formation to Planck mass would give a Page-curve-like prediction for when interior entropy overtakes horizon entropy; the paper does not perform that integration.
  • The analysis fixes charge and angular momentum during evaporation; allowing Q and J to vary while the hole radiates would modify F(M,Q,J) and could break the simple proportionality.
  • The same counting method should apply to photon or graviton modes rather than a massless scalar; if the coefficient depends on spin, the proportionality may not be universal.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. This manuscript, a 2019 PhD thesis posted to arXiv in 2025, revisits the Christodoulou–Rovelli (CR) interior volume and Baocheng Zhang's interior entropy. It computes the CR volume for Schwarzschild, Reissner–Nordström, Kerr, and d-dimensional charged black holes, then uses quantum-statistical counting in the interior volume to obtain scalar-field entropy. The central claimed results are: (i) interior volume and entropy grow linearly with advanced time; (ii) interior entropy is proportional to Bekenstein–Hawking entropy with proportionality constant less than unity, both in the static case and during Hawking evaporation, dS_int = -F(M,Q)dS_BH; (iii) the same relation holds for f(R) black holes with a coefficient depending on the modified-gravity parameter b. A final chapter studies coexistence curves and configurational entropy of f(R)-AdS black holes. The derivations are sketched, with several steps deferred to partial appendices.

Significance. Should the proportionality result hold with a derived, parameter-free coefficient, it would provide a concrete bridge between the black-hole interior and horizon thermodynamics and would be relevant to the information-paradox literature. The paper has the merit of collecting and extending an existing line of work, and the maximal-hypersurface volume part follows the established CR construction. It also explicitly presents the WKB/statistical counting argument and compares two Hamiltonian approaches. However, as it stands the central numerical claim is not an output of the derivation: the coefficient depends on the unspecified constant γ, two displayed formulas disagree by a factor of two, and the key f(R) coefficient is withheld. The paper therefore does not currently deliver a falsifiable parameter-free prediction.

major comments (4)
  1. [§2.3.1, Eq. (2.44)–(2.45) and Eq. (2.2)] Eq. (2.45) gives S_int/S_BH = √3 γ/(15360π), while Eq. (2.2) gives √3 γ/(30720π); the two displayed formulas differ by a factor of 2. For a single massless scalar in four dimensions the blackbody constant is γ = 15360π in the normalization of Eq. (2.44), so Eq. (2.45) yields √3 > 1 and Eq. (2.2) yields √3/2. The claim that the proportionality constant is 'less than unity' is therefore not a derived result but an implicit constraint on γ. The statement in §2.3.4 that 'Its value doesn't affect our discussion' is incorrect; it changes the sign of the claimed inequality.
  2. [§2.3.3, Eq. (2.98)–(2.99)] The differential coefficient F(M,Q) is defined with γ in Eq. (2.98), but the definition after Eq. (2.99) drops γ. This makes the central relation dS_int = -F(M,Q)dS_BH ambiguous. Figure 2.5, which plots F(M,Q), cannot be used to verify the proportionality or its magnitude unless it is stated whether γ is included. The same ambiguity affects the Q=0 limit, where the evaporation-law coefficient would differ by the factor γ.
  3. [§4.3, Eq. (4.20)–(4.21)] The neglect of the second term in Eq. (4.20) is unjustified. The quasi-static assumption only gives dM/dv ≪ 1, but v itself is large; for the Schwarzschild-like scaling v ~ γM³ and dM/dv ~ -1/(γM²), the product v(dM/dv) is O(M), not negligible. No such bound is provided for the f(R) case. In addition, the function γ(M,Q;b) is not given; the text states 'the concrete expression is not given here'. Consequently Eq. (4.21) and Fig. 4.1 cannot be checked, and the claimed f(R) proportionality coefficient is unsupported.
  4. [§4.3 / §2.3.3] The central comparison assumes the interior massless scalar is in equilibrium at the Hawking temperature and that evaporation is a quasi-static blackbody process. These assumptions are asserted rather than justified. If the interior field is not at the horizon temperature, the proportionality between interior and Bekenstein–Hawking entropy fails. Because the paper's main claim is the existence of this proportionality, the conclusion is conditional on an unexamined physical assumption.
minor comments (6)
  1. [Throughout] 'Plank mass' should be 'Planck mass'; 'advance time' should be 'advanced time'.
  2. [Eq. (1.7)] The angular term is garbled ('θ sin² dϕ²'); it should be r²(dθ² + sin²θ dϕ²).
  3. [Appendices] The text repeatedly refers to 'See Appendix 6' (e.g., §2.3.1 and §2.3.2), but the appendices are labeled A–D and are only partial; the referenced derivations are not fully present.
  4. [Eqs. (2.44), (2.93), (4.17)] The constant γ in Eq. (2.44) is not tied to the Stefan–Boltzmann constant σ used in Eq. (2.93) and Eq. (4.17); the notation should be unified and the relation between γ and σ stated.
  5. [Chapter 5] The phase-transition/configurational-entropy chapter is disconnected from the abstract's central claim about interior volume and entropy; no link is made between the two parts of the thesis.
  6. [Fig. 4.1] The plot is for M=2, Q=1, R0=-12, but the formula for γ(M,Q;b) is missing, so the plot cannot be reproduced or independently checked.

Circularity Check

2 steps flagged

The '<1' interior-entropy claim is not a derived prediction: it is an implicit constraint on the free Stefan-Boltzmann constant γ, and the two displayed formulas for S_CR disagree by a factor 2.

specific steps
  1. fitted input called prediction [Eq. (2.44)-(2.45) and Eq. (2.98)-(2.99), Sec. 2.3.1]
    "the rate of mass loss from Schwarzschild black hole due to Hawking radiation can be written as dM/dν = −1/(γM²) ... Using ... β = 8πM, then the entropy equation becomes SCR = 3√3γM/(45×8^3) = 3√3γA/((45×8^4)π) ... the proportional relation ... dSCR = −F(M,Q)dSBH"

    The factor multiplying A in Eq. (2.45) (and its factor-2 variant in Eq. (2.2)) is proportional to γ, a constant introduced as unspecified in the mass-loss law. The headline statement that the proportionality constant is 'less than unity' is therefore not an output of the derivation; it is an implicit bound on γ. The subsequent evaporation result dS_int = −F(M,Q)dS_BH inherits the same γ. Fixing γ by the blackbody normalization invoked in the same section makes the inequality marginal or reversed depending on which displayed formula is used. Thus the quantitative central claim reduces to a constraint on an input parameter rather than being a prediction.

  2. other [Eq. (2.2) vs Eq. (2.45), Sec. 2.3.1; Eq. (6.2)]
    "SCR = 3√3γ/(90×8^4)π A (Eq. 2.2); SCR = 3√3γM/(45×8^3) = 3√3γA/((45×8^4)π) (Eq. 2.45)"

    The two displayed formulas are presented for the same interior entropy after substituting A = 16πM^2, but they differ by a factor of 2. A central quantitative claim such as 'proportionality constant less than unity' cannot be considered a derived result when the manuscript itself gives two mutually inconsistent normalizations. The claimed inequality is therefore not robust; it depends both on the free parameter γ and on which of the paper's own formulas is selected.

full rationale

The thesis's strongest claim is that the entropy of scalar modes in the black-hole interior is proportional to the Bekenstein-Hawking entropy with a constant less than unity, both statically and during Hawking evaporation. The CR volume result V_CR = 3√3πM^2ν and the statistical entropy S = π^2V/(45β^3) are derived inside the manuscript and have independent content. However, the step from these to the area-proportional 'less than unity' result passes through dM/dν = −1/(γM^2), where γ is an unspecified input. The displayed coefficient of A is therefore proportional to γ, and the '<1' statement is not derived: it is an implicit condition on γ. The same free parameter propagates into the evaporation-law relation dS_int = −F(M,Q)dS_BH. The manuscript also displays two factor-of-2 different expressions for S_CR (Eq. 2.2 and Eq. 2.45), so the claimed numerical proportionality is not even internally unique. The d-dimensional and f(R) chapters repeat the same structural calculation; the f(R) chapter retains a free σ in Eq. (4.21), and the coefficient γ(M,Q;b) is left as an unevaluated ratio. Chapter 5 is a standard phase-transition/configurational-entropy analysis and is not part of the circular chain. Overall, the central quantitative 'prediction' reduces by construction to the normalization of the evaporation input, while the purely structural proportionality remains an algebraic consequence of the assumptions. This is partial, not total, circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The derivation depends on the CR volume definition, the equilibrium statistical treatment of interior modes, the Hawking-temperature identification, and the unspecified gamma constant. No new entities are introduced; the central result is heavily reliant on prior published work, including the author's own papers.

free parameters (2)
  • gamma
    Unspecified positive constant in the Stefan-Boltzmann mass-loss rate dM/dv = -1/(gamma*M^2); it appears in the final entropy proportionality and is never determined.
  • b = varied; b=1 is the Einstein limit
    Modified gravity coefficient b=1+f'(R0) in the f(R) action; chapter 4 conclusions are functions of b and are scanned numerically without independent determination.
axioms (5)
  • domain assumption The CR definition of interior volume as the volume of the largest spherically symmetric spacelike hypersurface is a valid physical measure of black hole interior size.
    Invoked in Section 2.2 and used in all subsequent chapters; not derived.
  • domain assumption Equilibrium statistical mechanics applies on the maximal hypersurface even though the interior is time-dependent; the proper time between adjacent maximal hypersurfaces tends to zero.
    Stated in Section 2.3.1 and Section 4.3 to justify using a thermal free energy.
  • domain assumption Hawking radiation can be treated as blackbody radiation and the evaporation as quasi-static; the interior scalar field has the Hawking temperature.
    Announced in Section 2.3.1 and Section 4.3; load-bearing for the entropy relation.
  • domain assumption The f(R) black hole solutions in Eq. (4.5) and their thermodynamic quantities are valid.
    Used without independent derivation in chapter 4.
  • standard math WKB approximation and phase-space cell counting in (2*pi)^3 is valid in the curved interior.
    Used in Section 2.3.1, Eqs. (2.32)-(2.40).

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of Aspects of the Black Hole Interior Volume and Entropy." pith.science (2026). https://pith.science/paper/2K3E64KP

@misc{pith2026250903042,
  author       = {Pith},
  title        = {Pith review of: Aspects of the Black Hole Interior Volume and Entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2K3E64KP}},
  note         = {Machine review of arXiv:2509.03042}
}
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abstract

The basic and conceptual notion of this work starts from the recent investigations of Marios Christodoulou and Carlo Rovelli (CR) in their paper entitled ''How big is a black hole?''. This work is related to the black hole interior volume and the entropy by Baocheng Zhang. In CR work, a spherically symmetric Schwarzschild black hole is considered and defines the interior volume, as the volume inside a sphere $S$ is the maximal proper volume of a space-like spherically symmetric $3d$ hypersurface bounded by the sphere $S$. Using this definition, they found the interior volume of a black hole is proportional to the advanced time. This is the main characteristic of this formulation. Which means that a black hole could store a large amount of infalling information. Using this special property, one can prob the nature of Hawking radiation emitted by a black hole. They also extend their result to the charged static black hole and found a consistent result. Their numerical analysis showed that the special character of this formulation is that the volume linearly increases with advancing time. Later, the CR work is followed by Baocheng Zhang, who investigated the entropy of scalar quantum modes in the interior of the Schwarzschild Black hole. He found the entropy of scalar quantum modes in the interior of Black hole is proportional to the lack hole surface area. The proportionality constant is found to be less than unity, which means that the interior entropy is less as compared to the Bekenstein Hawking entropy (horizon entropy). Note that these analyses are only applicable for black holes with mass greater than the Planck mass. If massless than Planck's mass, then one needs to understand the uncertainty relation.

Figures

Figures reproduced from arXiv: 2509.03042 by Shad Ali.

Figure 1.1
Figure 1.1. Figure 1.1: Black hole singularity position physical laws without gravity, which is also a failure of this theory, because one can’t con￾sider the zero gravity at any orientation or can’t consider the relative motion of two bodies in different frames of reference. It also deals with the motion of bodies under the uniform speed relative to each other. The major problem for this theory is how to explain the bodies in … view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: In Schwarzschild coordinates the light cone appears to close up as r → 2M and for light cone opens when r > 2M As the photon moves inwards, the distance to black hole decreases with time, this time is termed as advanced time. (t, r) → (ν, r). So from the above Eq. (1.11), we can write as, ds2 =  1 − 2M r  dµdν − r 2 dΩ (1.14) From first part of Eq. (1.13), we have e r ∗−r 2M =  r 2M − 1  (1.15) r ∗ =… view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Conformal diagram of Kruskal coordinates • The space outside the black hole is in the right quadrant, which places equal r (radius) taking a form of hyperbolas going up and down. Here the event horizon looks like a straight diagonal line (r = 2M). This is the space in which our original coordinates are well defined. • Following the future directed null rays, we reach to the region 2. Space inside the bla… view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: Conformal diagram of maximally extended R-N black hole (a) Q < M and (b). Extremal case (M = Q • Case 2. The second case is Q = M as extremal limit, so from above Eq. (1.28), we can write as [r] = 1 − 2M r + Q2 r 2 =  1 − Q r 2 (1.31) and the metric becomes ds2 =  1 − Q r 2 dt2 +  1 − Q r −2 dr2 + r 2 dΩ 2 (1.32) In this case, we have to assign a single charge. Let we take the charge is positive th… view at source ↗
Figure 1.5
Figure 1.5. Figure 1.5: A sketch of Kerr black hole with different parts [PITH_FULL_IMAGE:figures/full_fig_p029_1_5.png] view at source ↗
Figure 1.6
Figure 1.6. Figure 1.6: Summarization of black hole types [PITH_FULL_IMAGE:figures/full_fig_p029_1_6.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: The Volume of Space- like hyper-surface bounded by Minkowski space-time with θ < 450 Where V ′ R is the volume with respect to some other space like surface also bounded by S. This means that if hyper-surface P is the largest one, then the change in the volume must be zero and if we compare this volume VR of the largest hypersurface to the volume V ′ R bounded by some other hyper-surface P′ , then it mus… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: t − r diagram, showing the hyper-surface of infinite length In case of non-eternal black hole formed under collapsing process, the maximal hyper￾surface don’t have infinite volume, because the maximal surface starting form horizon and approaching to the surface rν = 3 2M don’t extend to infinity. Actually, it is the surface that prevents the object to go into singularity. The reason is that the surface r… view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: The Penrose diagram, which shows the largest hyper-surface, which guarantees the interior volume of a static spherically symmetric black hole formed under collapsed process (blue line Part-2). Part-1 is the null part and part-3 is the collapsed part with finite volume. 2.2.1 Volume in the interior of Schwarzschild Black hole Consider the Schwarzschild geometry in Eddington Finkelstein coordinates (ν, r, … view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: The plot of Fmax r M  vs r M . The function is maximum at ( r M ) = 1.4039 and ( a M ) = 0.2 If we plot the function Fmax r M  vs r M , it gives This plot shows that the position of the hyper-surface is at rν = 1.4039 when a = 0.2. For the detail see [35]. Considering Kerr metric given in Eq. (2.26) the interior volume is calculated as in Eq. (2.30). The extrinsic curvature for a Kerr black hole is cal… view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: Plot between F(M, Q) and M for (Q = 0.5), which shows the proportional relation of the two entropies during the evaporation Using the Eq. (2.95) and (2.97), the proportional relation between the two type of entropy is calculated as dSCR = − γ q 27M4 − 8MQ2p 9M2 − 8Q2 − 36M2Q2 + 9M3 p 9M2 − 8Q2 + 8Q4 720√ 2π  M + p M2 − Q2 2 dSBH (2.98) Let us simply it can be written as dSCR = −F(M, Q)dSBH (2.99) where… view at source ↗
Figure 2.6
Figure 2.6. Figure 2.6: Plot of fmax( a M ) vs ( a M ) Now to calculate the proportional relation between the horizon entropy and the entropy of massless scalar field, we calculate the variation of Bekenstein Hawking entropy [18,52,53] defined as SBH = A 4 = π(r 2 + − a 2 ) Where A = 4π(r 2 + + a 2 ) the area of the event horizon for Kerr black hole. The differential form is S˙ BH = 2π(1 + √ M2 − a 2 ) 2 √ M2 − a 2 MM˙ (2.106) … view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Plots of the dimension versus F  Q M  , where  Q M  = 0.5. the variation of Bekenstein-Hawking entropy. This means that, Bekenstein-Hawking entropy decreases rapidly with the increase in dimensions of space-time as compare to the entropy of the scalar field. In other words, the Hawking radiation can getting faster with the increase in dimensions of space-time. Summarizing this work, we calculated the… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: The proportionality coefficient γ(M, Q; b) as a function of b as M = 2, Q = 1 and R0 = −12. value will approach to zero at b = 1.6. It means that the variation of the scalar field entropy is much more less than the variation of Bekenstein-Hawking entropy. In other words, the growth rate of the scalar field entropy in the interior volume of the black hole is much more less than the decreasing rate of Beke… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Plot of reduced pressure ˜p. vs reduced temperature ˜τ for d = 4 of f(R) AdS black hole. The coexistence curve (blue) and critical point (red small circle on the top of coexistence curve) is also shown for (˜p, τ˜)-plane We see that the difference in number densities as a function of reduced temperature. This relation between the difference in number densities n1−n2 nc and reduced temperature ˜τ is plott… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Plot of difference between number densities of small and large molecules n1−n2 nc vs reduced temperature ˜τ for d = 4 of an f(R) AdS black hole. −Scon = p 3 − √ p˜+ p 3 − 3 √ p˜  √ p˜ √ 2 log   p 3 − √ p˜+ p 3 − 3 √ p˜  2 p 3 − √ p˜   + p 3 − √ p˜− p 3 − 3 √ p˜  √ p˜ √ 2 log   p 3 − √ p˜− p 3 − 3 √ p˜  2 p 3 − √ p˜   or we can write as Scon = − s p˜ 2 q 3 − p p˜ + q 3 − 3 p p˜  log 1 + … view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Plot of configurational entropy vs reduced temperature for d = 4 [PITH_FULL_IMAGE:figures/full_fig_p091_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Plot of configurational entropy vs reduced temperature for d = 5. This shows that as the dimension d increases, the (Scon − τ˜) plot becomes more and more concave [PITH_FULL_IMAGE:figures/full_fig_p091_5_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: The plot of difference in number densities n1−n2 nc of small and large f(R) AdS black hole vs dimension d. This result show that change in number density of small and large black hole is not same. 5.5 Discussions and conclusions In this chapter, we consider a d-dimensional f(R) AdS black hole for accounting the effects of modified gravity parameter b on the investigation of coexistence curves, difference… view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 5, 2026.