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Black Holes Thermodynamics and Generalised Non-Extensive Entropy

T0 review · 4 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read For a Schwarzschild black hole, the Hawking temperature and the Bekenstein-Hawking entropy are the unique thermodynamic temperature and entropy, so non-extensive generalized entropies cannot describe it unless the black hole has hair.

desk verdict A solid, conditional no-go for non-extensive entropies in Schwarzschild thermodynamics, with useful reviews and a few new consistency checks; needs fixing of dimensional slips and clearer caveats, but worth a serious referee. read the letter →

arxiv 2502.05801 v1 pith:NKBZNDWM submitted 2025-02-09 gr-qc hep-th

Emilio Elizalde , Shin'ichi Nojiri , Sergei D. Odintsov This is my paper · ORCID
classification gr-qchep-th PACS 04.70.Dy
keywords blackholethermodynamicsBekenstein-HawkingentropyHawkingtemperaturegeneralizedentropiesnon-extensiveRényiTsallishairyholes
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 argues that for a Schwarzschild black hole the Hawking temperature and the Bekenstein-Hawking entropy are not merely the standard choices but the only thermodynamically consistent ones. Because Hawking radiation has a thermal distribution fixed by the horizon geometry, any true temperature must be $T_H = P(r_H)/(4\pi)$; because a collapsing spherical dust shell conserves energy via Birkhoff's theorem, the thermodynamical energy must be the ADM mass $M$. The first law $dE = T dS$ then integrates to $S = A/(4G)$, which rules out Rényi, Tsallis, and multi-parameter generalized entropies for the Schwarzschild case. The paper then shows how generalized entropies could still survive for hairy black holes, where hair outside the horizon contributes to the ADM mass, and it reviews microscopic origins of generalized entropies in microcanonical, canonical, and grand canonical ensembles.

What carries the argument

The argument runs on three linked steps. First, demanding regularity of the Euclidean near-horizon geometry fixes the periodicity $4\pi/P(r_H)$ and hence the Hawking temperature $T_H = P(r_H)/(4\pi)$, independent of the gravity theory. Second, the thought experiment of a collapsing spherical dust shell identifies the thermodynamical energy with the ADM mass $M$ via Birkhoff's theorem and energy conservation. Third, the first law $dE = T dS$ integrates to $S = 4\pi G M^2 = A/(4G)$. For the hairy extension, the load-bearing device is the identity $\Theta \equiv h_3(r_H)\bigl(1 - m'(r_H)\bigr)^2 = 16G^2 \bigl[S'_{bh}(A)\bigr]^2$, which translates a candidate generalized entropy's derivative into the metric functions $h_3$ and mass function $m$ at the horizon, showing that Rényi, Tsallis, and the three- and six-parameter entropies correspond to specific metric choices.

What would settle it

Measure the temperature of the Hawking radiation from an isolated Schwarzschild black hole whose ADM mass is independently known and find $T \neq 1/(8\pi G M)$; that would break the claimed uniqueness of the temperature and of the energy identification.

Watch

Extended reading notes

Core claim

For a Schwarzschild black hole, the thermodynamical temperature and entropy are uniquely the Hawking temperature $T_H = 1/(8\pi G M)$ and the Bekenstein-Hawking entropy $S = A/(4G)$. Any attempt to use a generalized non-extensive entropy — Rényi, Tsallis, four-parameter, or five-parameter — as the thermodynamic entropy while keeping either $T = T_H$ or $E = M$ produces an inconsistent partner: the corresponding 'Rényi temperature' or 'Tsallis temperature' differs from what a Hawking-radiation detector measures, and forcing $T = T_H$ instead yields an energy $E \neq M$ that conflicts with energy conservation. The loophole is a black hole with hair: when fields outside the horizon contribute to the ADM mass, the simple first law acquires extra work terms, and in two-scalar-field models a generalized entropy can describe the thermodynamics of the black hole part, with the horizon temperature deviating from $T_H$ by a factor fixed by the metric.

Load-bearing premise

The load-bearing premise is that the thermodynamical energy of the black hole equals the ADM mass $M$, justified by the spherical dust-shell collapse via Birkhoff's theorem and energy conservation; if the energy should instead be a quasilocal mass, or if hair outside the horizon contributes, the uniqueness conclusion for temperature and entropy fails.

Editorial extensions

If this is right

  • No non-extensive entropy can describe a Schwarzschild black hole: each candidate either produces a temperature different from $T_H$ or an energy different from the ADM mass $M$.
  • Thermodynamic analyses of black holes that use Rényi or Tsallis entropies without altering the geometry are describing a different temperature or a different energy, not the Hawking radiation seen by observers.
  • For Reissner-Nordström black holes, a generalized entropy satisfying $T_H dS_g = dM$ is impossible in general; it exists only along a one-dimensional curve $Q = Q(M)$, such as $Q = q_0 M$, giving $S_g \propto M^2$.
  • In the two-scalar-field model, generalized entropies like Rényi and Tsallis arise naturally when the horizon temperature deviates from the naive Hawking temperature by the factor $C(r_H)/\sqrt{h_3(r_H)}$.
  • Current M87* and Sgr A* shadow observations constrain the parameter $\eta = r_H/(2GM(r_H))$ to be within about 0.73–1.33, consistent with Schwarzschild geometry and standard black hole thermodynamics.

Reading between the lines

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

  • If future shadow or lensing observations find $\eta$ significantly different from 1, that would indicate a non-Schwarzschild geometry and reopen the door for generalized entropies, although the paper does not establish a direct link between shadow size and entropy content.
  • The uniqueness argument should extend to the Kerr black hole in vacuum, where the ADM mass identification still holds; a natural check would be whether any generalized entropy can reproduce both $T_H$ and the ADM energy in the rotating case.
  • The phase-space measure construction points to a concrete quantum-gravity signature: if generalized entropies are realized by modified commutation relations, tiny deformations of uncertainty relations at the Planck scale should accompany them, potentially testable in future experiments.
  • The McLaughlin-expansion method gives a systematic dictionary: any entropy that is an analytic function of $S$ near $S=0$ can be mapped onto a canonical or grand-canonical ensemble, so the same technique could classify proposed entropies by their microscopic content.
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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

4 major / 6 minor

Summary. The paper argues that for a Schwarzschild black hole, the Hawking temperature and the Bekenstein-Hawking entropy are the unique thermodynamical temperature and entropy, provided the thermodynamical energy is identified with the ADM mass (E = M) and the temperature with the Hawking temperature (T = T_H). The derivation proceeds by integrating the first law dE = T dS. The paper then checks whether Rényi, Tsallis, four-parameter, and five-parameter generalized entropies can simultaneously reproduce the ADM mass and the Hawking temperature, concluding that they cannot without reducing to the Bekenstein-Hawking form. It then considers hairy black holes, particularly Reissner-Nordström and Einstein gravity coupled to two scalar fields, and shows that generalized entropies can be realized by appropriately choosing the mass function or the metric function entering the first law. The paper also computes photon-sphere and shadow radii for the scalar-field models, finding current shadow observations consistent with Schwarzschild geometry. Finally, it proposes microscopic interpretations of generalized entropies via microcanonical, canonical, and grand canonical ensembles, using a Taylor (McLaurin) expansion in the Gibbs entropy.

Significance. If the central conditional theorem is accepted, it provides a clean argument against the use of non-extensive entropies as the fundamental entropy of an isolated Schwarzschild black hole, clarifying the status of proposals in the recent literature. The explicit constructions for hairy black holes are useful as existence proofs that generalized entropies can arise in modified geometries, and the shadow constraints are a concrete phenomenological application. The paper also assembles a broad review of the authors' own prior work on generalized entropies. However, the novelty of the central uniqueness argument is limited (it is the standard first-law integration), and the constructive parts are significantly reverse-engineered, so the physical significance depends on the interpretation of the chosen functions M(σ) and h3(r).

major comments (4)
  1. [§IV.C.2, Eqs. (42)-(43)] The quantities T4 and T5 are defined as dS4/dM and dS5/dM, respectively. These are inverse temperatures, not temperatures, because for E = M the thermodynamic temperature is T = dE/dS = 1/(dS/dM). Comparing T4 and T5 directly to the Hawking temperature T_H = 1/(8πGM) is therefore dimensionally inconsistent and does not test whether the physical temperature equals T_H. The correct comparison would be between 1/T4 (and 1/T5) and T_H. The conclusion that S4 and S5 do not yield the Hawking temperature except in the Bekenstein-Hawking limit may survive this correction, but the argument as written is flawed and must be repaired.
  2. [§V.B.3, Eqs. (87)-(88)] The construction that realizes Rényi entropy and arbitrary generalized entropies is tautological: Eq. (88) defines the mass function M(σ) as an integral of the desired entropy S_g(r_H), so the relation T_H dS_g = dM holds by construction. Similarly, Eq. (87) is obtained by differentiating the Rényi entropy. This shows that one can always choose the free function M(σ) in the metric ansatz (83) to reproduce any prescribed entropy, but it does not demonstrate that the two-scalar-field model naturally produces or predicts such entropies. The paper should clearly label this as a reverse-engineering existence statement and discuss whether the resulting M(σ) is physically reasonable for large r_H (e.g., whether it is non-negative and monotonic).
  3. [§VIII.A, Eqs. (165)-(167)] The microscopic interpretation in Eqs. (165)-(167) is definitional rather than derivational. The coefficients f_g^{(n)} are taken from the McLaurin expansion of the desired generalized entropy S_g(S), and S0^n are defined in Eq. (162). Substituting these into Eq. (165) yields by construction the Taylor series of S_g evaluated at S0, provided the series converges. Thus Scan = S_g(S0) is an identity built into the definition, not a physical prediction. To present this as a microscopic origin, the paper must argue independently that the moments S0^n of the canonical or grand canonical distribution correspond to a natural physical ensemble, and that the specific combination in Eq. (165) is preferred over other possible resummations.
  4. [§Abstract and §III] The abstract states that the Hawking temperature and the Bekenstein-Hawking entropy are 'the only possible' thermodynamical temperature and entropy of the Schwarzschild black hole. This overstates the scope of the argument: the uniqueness conclusion depends on the interpretive assumption E = M, justified in §III by a dust-shell thought experiment, and on T = T_H. The paper itself acknowledges in §V that for non-Schwarzschild black holes, hair outside the horizon contributes to the ADM mass and the first law becomes dM = T_H dS + dM_hair, so the no-go does not extend to hairy cases. The abstract and the concluding summary should be rephrased to make the conditional nature of the theorem explicit.
minor comments (6)
  1. [Throughout] The expansion is consistently called the 'McLaughlin expansion'; the correct term is 'McLaurin expansion'. This appears in the abstract, in Section VIII, and in the summary.
  2. [§V.B.3, Eq. (87)] The expression M(σ) = (1/2)√(παG) d Arctan(...) is missing the factor 1/(2G) that appears in Eq. (86); as written it is not consistent with M'(r_H) = 1/(2G(1+απr_H^2/G)).
  3. [§V.A, text after Eq. (52)] The sentence 'the integrability condition requires ∂TH/∂Q because ∂Sg/∂M should not depend on Q' is incomplete and partly incorrect; the exactness of dM/TH imposes ∂(1/TH)/∂Q = 0 (or equivalently TH independent of Q in the relevant variables). The argument should be restated.
  4. [§IV.B.3, Eq. (26)] The formula contains a typo: 'S_T(E2)' should read 'S(E2)'.
  5. [§V.A, Eq. (47)] A parenthesis is missing: 'Eq. (47 shows' should read 'Eq. (47) shows'.
  6. [§II.A, Eq. (2)] The notation P(r) and the subsequent derivation of T_H would benefit from a statement about the dimensions of P(r) and the normalization of the time coordinate; the current presentation is somewhat implicit.

Circularity Check

3 steps flagged · score 6.0 of 10

Central uniqueness theorem is independent, but the generalized-entropy constructions are reverse-engineered and the Section VIII 'microscopic interpretation' is a Taylor-series identity.

  1. self definitional [Section V.B.3, Eq. (88)]
    "Similarly, for the generalised entropy Sg = Sg (rH), if we choose M(σ) in (83) by M(σ) = ∫ σ drH S′g(rH)/(4πrH), a model whose entropy is Sg can be constructed."

    The mass function M(σ), which sets the ADM mass and hence the exterior geometry, is defined as the integral of the derivative of the target entropy S_g divided by 4πrH. Because the Hawking temperature is T_H = 1/(4πrH), this choice makes dM = T_H dS_g identically. Thus the entropy S_g is not derived from the model; it is inserted into the model through the definition of M. Any generalized entropy can be realized this way, so the construction demonstrates consistency only by tautology.

  2. self definitional [Section V.B.4, Eqs. (102)-(106)]
    "Eq. (102) gives, h3(r = rH; ci(rH)) (1 − ∂m(r;ci(rH))/∂r |_{r=rH})^2 = 16G^2 [S′bh(A)]^2. Therefore for certain expressions of the general entropies, we find the corresponding form of Θ ... For example, in the case of the Rényi entropy (12), we obtain Θ = 1/(1+παrH^2/G)^2."

    The entropy S_bh was itself defined in Eq. (100) by integrating d m_eff / T. The formula then solves for the geometric factor Θ in terms of S′_bh(A). Substituting the derivative of a chosen entropy function into this relation manufactures a geometry whose integration gives back that same entropy. The 'generation' of Rényi, Tsallis, S3, and S6 entropies is therefore reverse-engineered: the target entropy is the input that fixes the geometry, not an output of independent thermodynamic reasoning.

1 more flagged steps
  1. self definitional [Section VIII.A, Eqs. (153), (165)-(167)]
    "Sg ≈ Σ∞ n=0 f(n)g/n! S^n. Here f(n)g is defined by f(n)g ≡ ∂nSg/∂Sn |_{S=0} ... Scan = Σ∞ n=0 f(n)g/n! S0^n ... Especially in the cases of the three-parameter entropy S3 in (29) and the four-parameter one S4 in (27), we obtain Scan3 = ... = 1/γ[(1+α/δ S0)^δ −1], Scan4 = ... = 1/γ[(1+α+/δ S0)^δ − (1+α−/δ S0)^−δ]."

    The coefficients f_g^{(n)} are defined as the Taylor coefficients of the very entropy S_g being interpreted. Substituting these coefficients into Scan is exactly writing the Taylor series of S_g and evaluating it at S0, so the final equalities reproduce S_g identically by construction. The 'microscopic interpretation' therefore contains no independent content: any analytic generalized entropy can be recovered from its own derivatives, making the claimed derivation a definitional identity rather than a physical explanation.

full rationale

The paper's central no-go statement, Sections II-III, is not circular. The Hawking temperature is obtained from Euclidean regularity, the ADM mass is identified with thermodynamic energy through the dust-shell/Birkhoff argument, and dS = dE/T then integrates to A/4G. This is a conditional theorem whose content depends on the E=M identification, and the paper itself acknowledges the limitation in Section V by introducing hair contributions. No step in that chain hides the conclusion in its premises. The circularity is confined to the constructive parts. Equation (88) defines the mass function from the derivative of a chosen entropy, so any S_g is realized by construction; Eq. (103) similarly solves for the geometry from S′_bh; and Section VIII defines Scan from the Taylor coefficients of S_g, making the recovered entropy an identity. These are self-definitional constructions presented as 'generating' or 'interpreting' generalized entropies, but they carry no independent predictive content. Because the main uniqueness claim remains independent, the overall circularity is partial rather than total, hence score 6.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central no-go argument relies on standard thermodynamics and the E=M identification; the hairy constructions rely on the two-scalar-field model and on freely chosen functions; no new physical entities are introduced.

free parameters (4)
  • α (Rényi)
    Deformation parameter in SR = (1/α) ln(1+αS), Eq. (12); chosen ad hoc in prior work, no data fit.
  • δ, A0 (Tsallis)
    Non-extensivity parameter and area scale in Eq. (18); A0 has dimension of area, chosen ad hoc.
  • α±, δ, γ (S4) and α±, δ, γ, ε (S5)
    Parameters of four- and five-parameter generalized entropies in Eqs. (27), (30); arbitrary deformation parameters from Refs. [4,5,49].
  • M(σ) and h3(φ,σ) functions = chosen by hand
    In Section V B 3 and V B 4, the mass function M(σ) and metric function h3 are chosen to reproduce a pre-selected generalized entropy (Eqs. (87), (88), (104)); these are free functions, not derived.
assumptions (5)
  • domain assumption The first law dE = T dS holds for black holes as a thermodynamical relation.
    Used throughout to derive entropy from E and T (e.g., Eq. (9)); assumes a well-defined equilibrium thermodynamics for black holes.
  • domain assumption Birkhoff's theorem ensures the exterior of a collapsing spherical dust shell is Schwarzschild and energy is conserved, so the ADM mass equals the thermodynamic energy.
    The dust-shell thought experiment in Section III is the basis for E=M; if gravitational wave emission or non-spherical effects occur, E=M may fail.
  • domain assumption The Bekenstein bound 2πRE > S is a valid criterion for physical entropy.
    Eq. (1) is used to reject Tsallis entropy with δ>1 (Section IV B); the bound is a conjecture, not proven for arbitrary generalized entropies.
  • ad hoc to paper Any spherically symmetric spacetime can be realized by Einstein gravity coupled to two scalar fields with Lagrange multiplier constraints, without propagating ghosts.
    Section V B relies on this construction from Refs. [45,61] to build hairy black hole models; it is an unproved (though cited) modeling tool specific to this paper's framework.
  • ad hoc to paper The modified phase-space measure e^{-M(q,p)} can be freely chosen to reproduce generalized entropies.
    In Section VII B, the measure is chosen ad hoc (e.g., Eq. (137)) to yield the desired entropy; this is an input, not a derived property.

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Pith. "Pith review of Black Holes Thermodynamics and Generalised Non-Extensive Entropy." pith.science (2026). https://pith.science/paper/NKBZNDWM

@misc{pith2026250205801,
  author       = {Pith},
  title        = {Pith review of: Black Holes Thermodynamics and Generalised Non-Extensive Entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NKBZNDWM}},
  note         = {Machine review of arXiv:2502.05801}
}
read the original abstract

The first part of this work provides a review of recent research on generalised entropies and their origin, as well as its application to black hole thermodynamics. To start, it is shown that the Hawking temperature and the Bekenstein-Hawking entropy are, respectively, the only possible thermodynamical temperature and entropy of the Schwarzschild black hole. Moreover, it is investigated if the other known generalised entropies, which include R\'enyi's entropy, the Tsallis one, and the four- and five-parameter generalised entropies, could correctly yield the Hawking temperature and the ADM mass. The possibility that generalised entropies could describe hairy black hole thermodynamics is also considered, both for the Reissner-Nordstr\"{o}m black hole and for Einstein's gravity coupled with two scalar fields. Two possibilities are investigated, namely, the case when the ADM mass does not yield the Bekenstein-Hawking entropy, and the case in which the effective mass expressing the energy inside the horizon does not yield the Hawking temperature. For the model with two scalar fields, the radii of the photon sphere and of the black hole shadow are calculated, which gives constraints on the BH parameters. These constraints are seen to be consistent, provided the black hole is of Schwarzschild type. Subsequently, the origin of the generalised entropies is investigated, by using their microscopic particle descriptions in the frameworks of a microcanonical and of a canonical ensemble, respectively. To finish, the McLaughlin expansion for the generalised entropies is used to derive, in each case, the microscopic interpretation of the generalised entropies, via the canonical and the grand canonical ensembles.

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Forward citations

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Reference graph

Works this paper leans on

100 extracted references · 32 canonical work pages · cited by 5 Pith papers

  1. [1]

    Can Magrans s/n, 08193 Barcelona, Spain

    Institute of Space Sciences (ICE, CSIC) C. Can Magrans s/n, 08193 Barcelona, Spain

  2. [2]

    Theory Center, High Energy Accelerator Research Organizat ion (KEK), Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan

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    Kobayashi-Maskawa Institute for the Origin of Particles an d the Universe, Nagoya University, Nagoya 464-8602, Japan

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    ICREA, Passeig Lluis Companys, 23, 08010 Barcelona, Spain 1 Abstract The first part of this work provides a review of recent researc h on generalised entropies and their origin, as well as its application to black hole thermo dynamics. To start, it is shown that the Hawking temperature and the Bekenstein-Hawking entrop y are, respectively, the only possibl...

  5. [5]

    Can the thermodynamical energy E be identified with the black hole mass M (i.e., E = M)?

  6. [6]

    black hole part

    Is the temperature of the black hole given by the Hawking temper ature, T = TH? For the first point, we should be careful in the following situation, th at is if BH is not the Schwarzschild one nor isolated one, there is no Arnowitt-Deser-Mis ner mass. Then the mass M may be the quasilocal mass contained in the horizon sphere or given b y the “black hole p...

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    The Birkhoff theorem [48] tells that the spac etime outside the shell is the Schwarzschild one (8)

    We assume an infalling spherically symmetric shell of dust with mass M and the initial radius sufficiently large. The Birkhoff theorem [48] tells that the spac etime outside the shell is the Schwarzschild one (8). The mass M is nothing but the mass of the shell. Inside the shell, the spacetime is empty and flat

  8. [8]

    A black hole is formed when the shell crosses the Schwarzschild radius rH = 2M in (8)

    By the collapse of the shell, the radius becomes smaller and smaller. A black hole is formed when the shell crosses the Schwarzschild radius rH = 2M in (8)

Show all 100 references
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    That 7 is, the geometry outside of the shell does not change during the co llapse

    The resulting geometry is always asymptotically flat and the shell m ass M appearing in the horizon radius is surely the energy E of the system, E = M because the energy should be conserved during the collapse of the shell due to the Birkh off theorem. That 7 is, the geometry ou...

  2. [10]

    R´ enyi temperature

    Assumption M = E If the mass M coincides with the energy E of the system due to the energy conservation [50–53], the consistency of the system with the thermodynamical e quation dS = dE/T requires to define the “R´ enyi temperature” TR by 1 TR ≡ dSR dM = 8πGM 1 + 4παGM 2 , (14)...

  3. [11]

    T = TH? Instead of assuming that the thermodynamical energy E is identical with BH mass M, we now assume that the thermodynamical temperature T coincides with the Hawking tem- perature TH. 9 By using the thermodynamical relation dE = T dS, the assumptions T = TH and S = SR sho...

  4. [12]

    Tsallis temperature

    M = E? Again by assuming that the thermodynamical energy E is given by BH mass M, we obtain A = 4π (2GM)2 = 16πG2E2 and the expression in (18) has the following form, ST = A0 4G (16πG2E2 A0 )δ , (19) 10 which may allow us to define “Tsallis temperature” as follows, TT ≡ dE dST ...

  5. [13]

    Tsallis energy

    T = TH? Instead of identifying the black hole mass M with the thermodynamical energy E, we now assume that the BH temperature is the Hawking temperature. Because we have A = 4π (4πTH)−2 = 1 4πTH2 , we find ST = A0 1−δ 4G ( 4πTH 2)δ , (21) which may lead to the “Tsallis energy” ...

  6. [14]

    Bekenstein-Hawking Entropy as Tsallis Entropy The standard thermodynamics is related to the extensive system. In the system, if we separate the system with thermodynamical energy E into two systems with E1 and E2 with E = E1 + E2, the standard entropy Sstandard(E) is extensive...

  7. [15]

    They obey the third law of thermodynamics, i.e., they vanish in the lim it of S → 0

  8. [16]

    They are monotonically increasing functions of the variable S

  9. [17]

    For the last point, when we consider the cosmology, A is given by the area of the apparent horizon, A = 4π H 2

    They diverge in the limit S → ∞ . For the last point, when we consider the cosmology, A is given by the area of the apparent horizon, A = 4π H 2 . Here H is the Hubble rate. Therefore S3, S4, and S6 diverge when the Hubble rate vanishes, H = 0 because the Bekenstein-Hawking en...

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    (31) Here E4 is the energy defined by the first relation dE4 = THdS4

    E = M ? The case of four-parameter generalised entropy S4 By substituting S = 4 πGM 2 in (10) with S0 = 0 to the four-parameter generalised entropy in (27), we find dE4 = THdS4 = 1 γ [ ( 1 + 4πα+ δ GM 2 )δ−1 α+ + ( 1 + 4πα− δ GM 2 )−δ−1 α− ] dM . (31) Here E4 is the energy defin...

  11. [19]

    (42) 15 Eq

    T = TH? In the case of the four-parameter generalised entropy in (27), if we identify the thermo- dynamical energy E with the mass M, the corresponding temperature T4 is given by T4 ≡ dS4 dM = 8πGM γ [ ( 1 + 4πα+ δ GM 2 )δ−1 α+ + ( 1 + 4πα− δ GM 2 )−δ−1 α− ] . (42) 15 Eq. (42)...

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    (63) Asymptotic flatness corresponds to lim r→+∞ λ(r) = 0 and we normalise the time coordinate t, to lim r→+∞ ν(r) = 0

    Black Hole with Scalar Hair We now consider the time-independent geometry, that is, static, s pherical, and asymp- totically flat spacetimes, ds2 = gµνdxµdxν = −e2ν(r)dt2 + e2λ(r)dr2 + r2dΩ 2 (2) . (63) Asymptotic flatness corresponds to lim r→+∞ λ(r) = 0 and we normalise the ti...

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    We do not include matter besides the two scalar fields φ and χ

    Examples In order to consider the examples, we now assume [58], e2ν = e −2λ = 1 h2(r) ( 1 − rH r ) , (79) 22 with a constant radius of the horizon rH. We do not include matter besides the two scalar fields φ and χ. Then the expressions (62) give, A(φ) = 1 κ2h2(φ) ( 1 − rH φ ){ ...

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    (83) Then when r is large, Eq

    Thermodynamics As an example, we consider the case h2 = 1 + 2GM (rH) − rH r = 1 + 2GM(σ) − σ φ . (83) Then when r is large, Eq. (79) tells e2ν = e −2λ ∼ 1 − 2GM (rH) r . (84) Therefore M (rH) is the ADM mass. In order to consider the possibility of (75), as an example, we cons...

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    dot” or “˙

    Thermodynamics Based on meff Here based on [58], we consider the thermodynamics by using meff in (78). Instead of (79), we assume, e−2λ(r) = e 2ν(r)h3(r) . (89) 24 Here h3(r) is a positive function of r. As in (82), the geometry (89) is realised by using (62) with the Lagrangian...

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