Pith. sign in

REVIEW 3 major objections 4 minor 47 references

Information-Theoretic Black Hole Entropy I: Beyond the Area Law

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Black hole entropy can be exactly rewritten as a Kullback-Leibler divergence.

desk verdict A clear, honest, but underdetermined proposal: the new entropy formula and KL representation are consequences of an admitted ansatz, not derivations. read the letter →

arxiv 2608.05795 v1 pith:4IZ2JYRN submitted 2026-08-06 hep-th gr-qchep-ph

Alex Kehagias This is my paper · ORCID
classification hep-thgr-qchep-ph PACS 04.70.Dy05.70.-a
keywords blackholethermodynamicsthirdlawofBekenstein-HawkingentropyKullback-LeiblerdivergenceBernoullidistribution1/NexpansiondeficitFisherinformation
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

Black hole thermodynamics appears to violate the Nernst third law: as a Schwarzschild black hole's temperature approaches zero, its Bekenstein-Hawking entropy diverges instead of reaching a universal constant. The paper argues that this is a defect of the semiclassical area law rather than a fundamental feature, and constructs an entropy function that satisfies the third law while reproducing the area law for small masses. The resulting formula equals the Kullback-Leibler divergence between a mass-biased product of N Bernoulli bits and the uniform N-bit distribution, with N = $M0^{2}$/$M_P^{2}$. If correct, this gives black hole entropy an exact information-theoretic meaning as an entropy deficit relative to a maximally mixed reference, with the area law emerging as the leading term of a 1/N expansion rather than the whole story.

What carries the argument

The load-bearing identity is Eq. (4.12), S_bh(M) = D_KL^(N)(p || 1/2), where p_± = (1 ± M/M0)/2 and N = $M0^{2}$/$M_P^{2}$. This identity is reached through a minimal rational ansatz for the entropy's second derivative, $d^{2}$ S_bh/$dM^{2}$ = b_1/[(M0 - M)(M + m)], whose constants are fixed by matching the small-mass expansion to the Bekenstein-Hawking entropy. The KL divergence gives the entropy an operational meaning: it is the extra code length, in nats, needed to encode samples from the biased N-bit ensemble using a code optimized for the uniform ensemble, equivalently the logarithm of the ratio 2^N / 2^(N H_2(p)) between the total and visible microstate counts.

What would settle it

A microstate count in a UV-complete theory that yields a logarithmic correction to the area law, or any term outside the series S_bh = (A/4) Σ_k (A/N)^k / [2^k (2k+1)(k+1)], would falsify the exact formula; so would the observation of a black hole with mass exceeding M0.

Watch

Extended reading notes

Core claim

The paper's central claim is that the apparent violation of the Nernst third law by the Bekenstein-Hawking area law is a semiclassical artifact. By postulating a maximum mass M0 at which the temperature vanishes and the entropy is finite, and by fixing the simplest singular second-derivative form consistent with the area law at small masses, the paper obtains an exact entropy function which approaches the area law for M much less than M0 and reaches a universal residual value N ln 2 at T = 0. The same function is then shown to equal the Kullback-Leibler divergence between the N-bit Bernoulli distribution with bias p = (1 + M/M0)/2 and the uniform distribution, Eq. (4.12), with N = $M0^{2}$/$M_P^{2}$. In this view the thermodynamic entropy of a black hole is not a count of hidden microstates but an entropy deficit: the information gained by distinguishing the black hole ensemble from a maximally mixed reference. The area law is the leading term of this quantity in a 1/N expansion, with subleading terms being finite-information corrections.

Load-bearing premise

Everything rests on the assumption that black hole entropy has a finite maximum mass M0 where the temperature vanishes and the second derivative of entropy takes the particular rational form chosen here; the paper concedes these conditions do not uniquely determine the entropy, so a different endpoint singularity would break the KL formula.

Editorial extensions

If this is right

  • The modified entropy satisfies the Nernst third law: as M approaches M0 (so T approaches 0), S_bh approaches N ln 2, a universal constant independent of other parameters.
  • For M much less than M0, the entropy reduces to the Bekenstein-Hawking area law, so the area law is the leading order of a controlled expansion rather than the exact answer.
  • The subleading terms are explicitly predicted: S_bh = (A/4)(1 + A/(12N) + ...), so they become sizable only when the horizon area is a sizable fraction of N in Planck units.
  • The Fisher information of the Bernoulli ensemble gives a Planck-scale single-shot mass resolution, ΔM ≥ M_P, improving as M_P/√K with K independent configurations.
  • Pinsker's inequality applied to the KL representation yields S_BH ≤ S_bh, so the area law is a lower bound on the information-theoretic entropy deficit.

Reading between the lines

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

  • Editorial inference: if the companion paper's identification of M0 with the cosmological constant is correct, the 1/N corrections become linked to the measured vacuum energy, making a numerically testable prediction that this paper does not pursue.
  • Editorial inference: the Bernoulli-bit representation suggests modeling evaporation as a stochastic process in which the bias p(M) drifts toward 1/2 as the black hole loses mass; whether such a process reproduces the Page curve or resolves the information paradox is not addressed here.
  • Editorial inference: the same second-derivative ansatz, applied to other gravity systems with a maximum-energy endpoint, would generate the same functional form of entropy, meaning the third-law-compatible correction is a generic prediction of the construction rather than special to Schwarzschild black holes.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper argues that the apparent conflict between the Bekenstein-Hawking area law and the Nernst third law can be resolved by replacing the semiclassical entropy with a modified function S_bh(M) that saturates at a finite universal value as T→0. Following a blackbody analogy, the author postulates a simple-pole form for d²S_bh/dM², integrates to obtain S_bh(M), fixes the integration constants by requiring the small-mass limit to reproduce 4πM², and introduces a maximum mass scale M₀. The resulting entropy is then recast, via the parametrization p±=(1±M/M₀)/2, as the KL divergence between an N-fold product Bernoulli distribution and the uniform N-bit distribution, with N=M₀²/M_P². From this representation the paper derives a 1/N expansion of the entropy, a Fisher-information metric, a Cramér–Rao bound on mass estimation, and a Pinsker inequality relating the entropy to the Bekenstein-Hawking value.

Significance. If the proposed entropy function were uniquely forced by the stated thermodynamic requirements, the result would be a substantial step: it would provide an exact entropy function satisfying the third law, with the area law as the leading term and computable 1/N corrections, together with a clean information-theoretic interpretation as an entropy deficit. The paper is commendably explicit about its assumptions, and the algebraic steps from the ansatz (2.23) through Eqs. (2.29), (3.6), and (4.12) are transparent and verifiable. The 1/N expansion (2.39), the heat-capacity formula (2.32), and the Pinsker bound (A.13) are concrete and falsifiable consequences of the model. However, as the author concedes, the ansatz is not derived from more basic principles, and the central results are algebraic consequences of that choice rather than independent derivations. The significance therefore hinges on whether the companion paper or future work can supply a dynamical argument fixing the analytic structure of d²S/dM² and the scale M₀; until then the paper is best read as a self-consistent model proposal rather than a derivation.

major comments (3)
  1. [§2.2, Eq. (2.23)] The central entropy formula (2.29) is obtained from the 'minimal rational ansatz' d²S_bh/dM² = b1/[(M0−M)(M+m)], which the paper itself states is not uniquely determined by the stated conditions. This is a load-bearing non-uniqueness: any choice of the analytic part in d²S/dM² changes the entropy while preserving the conditions T(M0)=0, finite S(M0), and the small-mass Bekenstein–Hawking limit. For example, for any real λ, S_λ(M)=S_bh(M)+λM⁴ still has T(M0)=0 because dS/dM still diverges at M0, still has finite S_λ(M0), and still gives S_λ≃4πM²+O(M⁴) for M≪M0. Since Eq. (2.29), the Bernoulli representation (3.6), and the KL equality (4.12) are all algebraically equivalent to this particular ansatz, the non-uniqueness of (2.23) propagates to the paper's main claims. A derivation of the pole-structure, or at least an argument that the analytic part is absent, is needed before the entropy function can be regarded as 'derived' from the third law.
  2. [§2.2, Eqs. (2.27)–(2.40)] The Bekenstein–Hawking law is used as input in fixing the constants, not derived as an output. Equations (2.27)–(2.28) determine m, b1, and S0 by imposing that the small-M expansion of (2.25) reproduce S_BH=4πM². Consequently, the fact that the leading term of the 1/N expansion in (2.39)–(2.40) is exactly the Bekenstein–Hawking area law holds by construction. The paper's abstract and Section 5 say that the area law 'emerges' as the leading term; that phrasing overstates the logic. A more precise statement would be that the area law is recovered as the leading term because it was imposed as a matching condition. This does not invalidate the construction, but it removes the appearance of an independent prediction.
  3. [§3 and §4, Eqs. (3.2), (3.6), (4.12)] The information-theoretic identification is an exact algebraic identity following from the definition of the bias p=(1+M/M0)/2, not an independent microscopic derivation. Once Eq. (3.2) is adopted, Eq. (3.1) becomes Eq. (3.6) and Eq. (4.12) is immediate; the Fisher-information metric and the Cramér–Rao bound in Eqs. (4.15)–(4.21) likewise follow from the chosen Bernoulli model. In addition, the scale M0 and hence N=M0²/MP² are free parameters in this paper, deferred to the companion paper [30]. Because the KL representation, the 1/N expansion, and the Pinsker bound all depend on these choices, the paper's central claim that black hole entropy 'is' this KL divergence is underdetermined. An independent argument fixing M0 and the analytic structure of the entropy would be required to turn the model into a derivation.
minor comments (4)
  1. [Throughout] There are several typographical and grammatical errors: 'deeply improve' should be 'deeply improved'; 'anlogy' should be 'analogy'; 'consistent with the the third law' in §2.2 has a duplicated article; 'Bernouli' in §4 should be 'Bernoulli'; and 'two-lever system' in §2.2 should read 'two-level system'.
  2. [Eq. (2.12)] The differential denominator in Eq. (2.12) and in the surrounding discussion is written as d²S/d²E; it should be d²S/dE². The same notation appears in Eq. (2.14).
  3. [Eq. (2.39)] The summation in Eq. (2.39) is written as Σ_{n=0} without an explicit upper limit; it should be Σ_{n=0}^{∞} for consistency with Eq. (2.41).
  4. [Fig. 1] The right panel of Fig. 1 is not fully labeled: the curve for the Bekenstein–Hawking entropy is described in the caption, but the vertical axis is only marked 'SBH' while the horizontal axis is 'M/M0'; adding an explicit label for the Bekenstein–Hawking curve would improve readability.

Circularity Check

3 steps flagged · score 8.0 of 10

The KL equality is a definitional identity and the 'emerging' area law is a matched coefficient; the entropy formula rests on an admitted non-unique ansatz.

  1. fitted input called prediction [Section 2.2, Eqs. (2.26)-(2.28) and Eqs. (2.39)-(2.40)]
    "“Therefore, in order the leading term of Eq. (2.26) to agree with the Bekenstein-Hawking entropy Eq. (2.5), i.e., S_bh ≈ S_B-H = 4πM^2, for M ≪ M_0 we should have m = M_0, b_1 = 8πM_0^2, S_0 = -8πM_0^2 − ln M_0.” “In the limit N → ∞ with x fixed, only the first term in Eq. (2.39) survives, S_bh(M) → x/2 = M^2/(2M_P^2), so that the entropy reduces to the usual Bekenstein-Hawking entropy.”"

    The leading term of the later 1/N expansion is exactly the term whose coefficient b_1 was fixed by the matching condition (2.28). Inserting that value into expansion (2.26) and then reading it off as Eq. (2.40) restates the input: the Bekenstein-Hawking form was imposed on the ansatz, not derived from the third law. The subleading 1/N coefficients are just the Taylor coefficients of the chosen function (2.29), so they inherit the arbitrariness of the ansatz.

  2. self definitional [Section 4, Eqs. (4.6)-(4.12)]
    "“We consider a binary outcome (i = ±) with probabilities p_± = p_±(M) = 1/2 (1 ± M/M_0)... Comparing (4.11) with the expression for the black-hole entropy S_bh(M) obtained in Eq. (3.1) (or equivalently Eq. (3.3)), we find that S_bh(M) = D^{(N)}_KL(p ‖ 1/2).”"

    The bias p is not derived from any microscopic dynamics; Eq. (3.2) defines it as a shorthand for the two coefficient terms already present in the chosen entropy (2.29). With that definition, the KL divergence is algebraically equal to S_bh by construction. The “information-theoretic representation” is therefore an identity that follows from the ansatz, and any function of the form N [ln 2 + p ln p + (1−p) ln(1−p)] would receive the same interpretation. It cannot independently certify the entropy formula.

1 more flagged steps
  1. other [Section 2.2, Eq. (2.23)]
    "“These conditions do not uniquely determine the entropy. As a minimal rational ansatz, analogous to the blackbody case, we take d^2S_bh/dM^2 = b_1/((M_0−M)(M+m)), M ≤ M_0.”"

    The paper concedes that the stated thermodynamic conditions do not select Eq. (2.23). The pole at M_0 guarantees T(M_0)=0 and finite S(M_0), but adding analytic terms to d^2S/dM^2 produces quartic and higher terms in S that still leave S(M_0) finite, while the blackbody argument against analytic terms does not transfer because analytic terms in d^2S_bh/dM^2 do not destroy the required M^2 leading term. Hence (2.29) is one member of an infinite family satisfying the assumptions; the 1/N expansion, Bernoulli representation, and KL equality are consequences of this non-unique choice, not of the third law.

full rationale

The paper is transparent about its main assumption, and there is no hidden self-citation chain: references are used normally, and [30] is deferred rather than load-bearing for the present formulas. However, two central claims reduce by construction. First, the “emergence” of the Bekenstein-Hawking area law: the constants m, b_1, S_0 are chosen so that the small-M expansion of the ansatz reproduces S_B-H, and the 1/N leading term is simply that matched coefficient, so it is a restatement of the input rather than a prediction. Second, the KL representation: p_±(M) is introduced as a shorthand for the coefficient expression already present in Eq. (2.29), making Eq. (4.12) an algebraic identity rather than an independent microscopic derivation. The paper also admits the underlying ansatz is not uniquely determined by the stated conditions; the blackbody exclusion of analytic terms does not apply to the black-hole case, so the exact entropy formula and its finite-information corrections are not forced. The information-geometric identities are correct consequences of the chosen Bernoulli model, but they do not provide independent support for the entropy function. This warrants a score of 8: the central result is forced by the matching and by the definition of p, with the uniqueness gap explicitly acknowledged.

Assumptions & free parameters 2 free parameters · 5 assumptions · 2 invented entities

The central claim rests on the imposed third law, an assumed maximum mass M0, a non-unique pole ansatz for the entropy's second derivative, and a posited ensemble of N independent Bernoulli bits. The Bekenstein-Hawking area law enters through the matching conditions in Eq. (2.28), not as an independent output, so the KL representation in Eq. (4.12) is a mathematical identity for the chosen entropy function.

free parameters (2)
  • M0 = unspecified; stated to be related to the cosmological constant in companion paper [30]
    Introduced in Eq. (2.21) as the maximum black hole mass where T=0; sets N = M0^2/M_P^2 and the residual entropy N ln2; no independent evidence in this paper.
  • matching constants b1, m, S0 = b1 = 8*pi*M0^2, m = M0, S0 = -8*pi*M0^2*ln(M0)
    Fixed in Eq. (2.28) by requiring the small-mass expansion to match Bekenstein-Hawking entropy; this is fitting to the area law, not deriving it.
assumptions (5)
  • domain assumption The Nernst third law applies to Schwarzschild black holes: lim_{T->0} S = S0, a universal constant.
    Eq. (2.1); imposed despite the paper's citation of Wald's critique (ref. [20]) arguing the third law may not apply to black holes.
  • ad hoc to paper A maximum mass M0 exists with T(M0)=0 and finite entropy S(M0).
    Eqs. (2.21)-(2.22); no physical mechanism is given in this paper; origin deferred to companion paper [30].
  • ad hoc to paper The entropy's second derivative has the simple-pole form d2S/dM2 = b1/[(M0-M)(M+m)]; higher-order poles and analytic terms are excluded.
    Eq. (2.23); the paper concedes the conditions 'do not uniquely determine the entropy'.
  • domain assumption The microscopic ensemble consists of N independent Bernoulli bits with bias p = (1 + M/M0)/2.
    Sec. 3, Eq. (3.2); no physical realization or coupling to gravity is specified.
  • domain assumption The blackbody reconstruction strategy (use high-energy asymptotics plus third law to fix the entropy) transfers to the black hole case, which has convex rather than concave entropy.
    Sec. 2.1 vs. Sec. 2.2; the analogy is heuristic and the sign difference is acknowledged but not justified.
invented entities (2)
  • Maximum black hole mass M0 (universal scale)
    purpose: Provides a zero-temperature endpoint at finite mass, sets the residual entropy N ln2 and the 1/N expansion.
    Introduced by hand in Eq. (2.21); no observational or theoretical handle in this paper; companion paper [30] promises a cosmological-constant connection.
  • N microscopic Bernoulli bits
    purpose: Statistical degrees of freedom whose KL divergence reproduces the thermodynamic entropy; N = M0^2/M_P^2.
    Sec. 3; the paper offers no physical mechanism, only examples (Ising spins, bits, random walk steps).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Information-Theoretic Black Hole Entropy I: Beyond the Area Law." pith.science (2026). https://pith.science/paper/4IZ2JYRN

@misc{pith2026260805795,
  author       = {Pith},
  title        = {Pith review of: Information-Theoretic Black Hole Entropy I: Beyond the Area Law},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4IZ2JYRN}},
  note         = {Machine review of arXiv:2608.05795}
}
abstract

Although the Bekenstein-Hawking area law is consistent with the first and second laws of black hole thermodynamics, it appears to be in conflict with the third law, which in the Nernst formulation states that the entropy should either vanish or approach a universal constant in the zero-temperature limit. We argue that this tension reflects a limitation of the semiclassical area law rather than a fundamental feature of black hole thermodynamics. On this basis, we obtain an entropy formula that is consistent with the third law, and approaches Bekenstein-Hawking entropy in the high-temperature limit. The resulting entropy admits a simple microscopic interpretation, and it can be written as the Kullback-Leibler divergence between a mass-biased Bernoulli distribution and the uniform distribution on N microscopic bits. From this perspective, the thermodynamic black hole entropy admits an information-theoretic representation as an entropy deficit, namely as the relative entropy between the black hole ensemble and a maximally mixed reference ensemble. The Bekenstein-Hawking area law then emerges as the leading term in an $1/N$ expansion, while a universal mass scale $M_0=\sqrt{N}\,M_P$, interpreted as an absolute upper bound on the black hole mass, controls the bias of the underlying microscopic ensemble. The subleading terms represent finite-information corrections to the classical area law.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 18 canonical work pages

  1. [30]

    Information–Theoretic Black Hole Entropy II: Infrared Gravity and Charged/Rotating Extensions

    A. Kehagias, “Information–Theoretic Black Hole Entropy II: Infrared Gravity and Charged/Rotating Extensions.” (Companion paper)

    work page

  2. [1]

    J. D. Bekenstein,Black holes and entropy,Phys. Rev. D7(1973) 2333–2346

    1973

  3. [2]

    S. W. Hawking,Particle Creation by Black Holes,Commun. Math. Phys.43(1975) 199–220

    1975

  4. [3]

    J. D. Bekenstein,Bekenstein-Hawking entropy,Scholarpedia3(Oct., 2008) 7375

    2008

  5. [4]

    R. M. Wald,The thermodynamics of black holes,Living Rev. Rel.4(2001) 6, [gr-qc/9912119]

    arXiv 2001

  6. [5]

    D. N. Page,Hawking radiation and black hole thermodynamics,New J. Phys.7(2005) 203, [hep-th/0409024]

    arXiv 2005

  7. [6]

    Witten,Introduction to black hole thermodynamics,Eur

    E. Witten,Introduction to black hole thermodynamics,Eur. Phys. J. Plus140(2025) 430, [2412.16795]

    arXiv 2025

  8. [7]

    W. G. Unruh and R. M. Wald,Acceleration Radiation and Generalized Second Law of Thermodynamics,Phys. Rev. D25(1982) 942–958

    1982

Show all 47 references
  1. [8]

    W. H. Zurek and K. S. Thorne,Statistical mechanical origin of the entropy of a rotating, charged black hole,Phys. Rev. Lett.54(1985) 2171

    1985

  2. [9]

    V. P. Frolov and D. N. Page,Proof of the generalized second law for quasistationary semiclassical black holes,Phys. Rev. Lett.71(1993) 3902–3905, [gr-qc/9302017]

    1993 arXiv

  3. [10]

    Strominger and C

    A. Strominger and C. Vafa,Microscopic origin of the Bekenstein-Hawking entropy,Phys. Lett. B379(1996) 99–104, [hep-th/9601029]

    1996 arXiv

  4. [11]

    J. M. Maldacena, A. Strominger and E. Witten,Black hole entropy in M theory,JHEP12 (1997) 002, [hep-th/9711053]

    1997 arXiv

  5. [12]

    Strominger,Black hole entropy from near horizon microstates,JHEP02(1998) 009, [hep-th/9712251]

    A. Strominger,Black hole entropy from near horizon microstates,JHEP02(1998) 009, [hep-th/9712251]

    1998 arXiv

  6. [13]

    Dvali,A string theoretic derivation of gibbons-hawking entropy,Gen

    G. Dvali,A string theoretic derivation of gibbons-hawking entropy,Gen. Rel. Grav.57(2025) 118, [2407.01510]

    2025 arXiv

  7. [14]

    Susskind and J

    L. Susskind and J. Uglum,Black hole entropy in canonical quantum gravity and superstring theory,Phys. Rev. D50(1994) 2700–2711, [hep-th/9401070]

    1994 arXiv

  8. [15]

    Susskind,The World as a hologram,J

    L. Susskind,The World as a hologram,J. Math. Phys.36(1995) 6377–6396, [hep-th/9409089]

    1995 arXiv

  9. [16]

    Dvali and C

    G. Dvali and C. Gomez,Black Hole’s Quantum N-Portrait,Fortsch. Phys.61(2013) 742–767, [1112.3359]

    2013 arXiv

  10. [17]

    Dvali, C

    G. Dvali, C. Gomez and D. L¨ ust,Classical Limit of Black Hole Quantum N-Portrait and BMS Symmetry,Phys. Lett. B753(2016) 173–177, [1509.02114]

    2016 arXiv

  11. [18]

    Dvali,Area law microstate entropy from criticality and spherical symmetry,Phys

    G. Dvali,Area law microstate entropy from criticality and spherical symmetry,Phys. Rev. D 97(2018) 105005, [1712.02233]

    2018 arXiv

  12. [19]

    Wilks,The Third Law of Thermodynamics

    J. Wilks,The Third Law of Thermodynamics. Oxford University Press, Oxford, 1961

    1961

  13. [20]

    R. M. Wald,The ’Nernst theorem’ and black hole thermodynamics,Phys. Rev. D56(1997) 6467–6474, [gr-qc/9704008]

    1997 arXiv

  14. [21]

    J. M. Bardeen, B. Carter and S. W. Hawking,The Four laws of black hole mechanics, Commun. Math. Phys.31(1973) 161–170. – 19 –

    1973

  15. [22]

    Israel,Third Law of Black-Hole Dynamics: A Formulation and Proof,Phys

    W. Israel,Third Law of Black-Hole Dynamics: A Formulation and Proof,Phys. Rev. Lett. 57(1986) 397

    1986

  16. [23]

    Kehle and R

    C. Kehle and R. Unger,Gravitational collapse to extremal black holes and the third law of black hole thermodynamics,J. Eur. Math. Soc.(2025) , [2211.15742]

    2025 arXiv

  17. [24]

    H. S. Reall,Third law of black hole mechanics for supersymmetric black holes and a quasilocal mass-charge inequality,Phys. Rev. D110(2024) 124059, [2410.11956]

    2024 arXiv

  18. [25]

    J. R. V. Crump, M. Gadioux, H. S. Reall and J. E. Santos,Violation of the Third Law of Black Hole Mechanics in Vacuum Gravity,Phys. Rev. Lett.136(2026) 171405, [2601.20955]

    2026 arXiv

  19. [26]

    A. M. McSharry and H. S. Reall,Supersymmetric black holes and the third law of black hole mechanics,Phys. Rev. D112(2025) 104009

    2025

  20. [27]

    Schr¨ odinger,What is Life? The Physical Aspect of the Living Cell

    E. Schr¨ odinger,What is Life? The Physical Aspect of the Living Cell. Cambridge University Press, Cambridge, 1944

    1944

  21. [28]

    Brillouin,Science and Information Theory

    L. Brillouin,Science and Information Theory. Academic Press, New York, 1956

    1956

  22. [29]

    Mehra and H

    J. Mehra and H. Rechenberg,Planck’s half-quanta: A history of the concept of zero-point energy,Foundations of Physics29(1999) 91–132

    1999

  23. [31]

    Grimmett and D

    G. Grimmett and D. Stirzaker,Probability and Random Processes. Oxford University Press, 3 ed., 2001

    2001

  24. [32]

    T. M. Cover and J. A. Thomas,Elements of Information Theory. John Wiley & Sons, Hoboken, NJ, 2 ed., 2006

    2006

  25. [33]

    Kullback and R

    S. Kullback and R. A. Leibler,On information and sufficiency,The Annals of Mathematical Statistics22(1951) 79–86

    1951

  26. [34]

    R. D. Sorkin,The statistical mechanics of black hole thermodynamics, inSymposium on Black Holes and Relativistic Stars (dedicated to memory of S. Chandrasekhar), 5, 1997. gr-qc/9705006

    1997 arXiv

  27. [35]

    Jacobson,On the nature of black hole entropy,AIP Conf

    T. Jacobson,On the nature of black hole entropy,AIP Conf. Proc.493(1999) 85–97, [gr-qc/9908031]

    1999 arXiv

  28. [36]

    R. D. Sorkin,Ten theses on black hole entropy,Stud. Hist. Phil. Sci. B36(2005) 291–301, [hep-th/0504037]

    2005 arXiv

  29. [37]

    S. D. H. Hsu and D. Reeb,Unitarity and the Hilbert space of quantum gravity,Class. Quant. Grav.25(2008) 235007, [0803.4212]

    2008 arXiv

  30. [38]

    Engelhardt and A

    N. Engelhardt and A. C. Wall,Decoding the Apparent Horizon: Coarse-Grained Holographic Entropy,Phys. Rev. Lett.121(2018) 211301, [1706.02038]

    2018 arXiv

  31. [39]

    A. Ly, M. Marsman, J. Verhagen, R. P. P. P. Grasman and E.-J. Wagenmakers,A tutorial on Fisher information,Journal of Mathematical Psychology80(2017) 40–55, [1705.01064]

    2017 arXiv

  32. [40]

    Amari and H

    S.-i. Amari and H. Nagaoka,Methods of Information Geometry, vol. 191 ofTranslations of Mathematical Monographs. American Mathematical Society, Providence, RI, 2000

    2000

  33. [41]

    Weinhold,Metric geometry of equilibrium thermodynamics,J

    F. Weinhold,Metric geometry of equilibrium thermodynamics,J. Chem. Phys.63(1975) 2479. – 20 –

    1975

  34. [42]

    Ferrara, G

    S. Ferrara, G. W. Gibbons and R. Kallosh,Black holes and critical points in moduli space, Nucl. Phys. B500(1997) 75–93, [hep-th/9702103]

    1997 arXiv

  35. [43]

    Ruppeiner,Riemannian geometry in thermodynamic fluctuation theory,Rev

    G. Ruppeiner,Riemannian geometry in thermodynamic fluctuation theory,Rev. Mod. Phys. 67(1995) 605–659

    1995

  36. [44]

    J. E. Aman, I. Bengtsson and N. Pidokrajt,Geometry of black hole thermodynamics,Gen. Rel. Grav.35(2003) 1733, [gr-qc/0304015]

    2003 arXiv

  37. [45]

    E. L. Lehmann and G. Casella,Theory of Point Estimation. Springer Texts in Statistics. Springer New York, NY, 2 ed., 1998, 10.1007/b98854

    1998 doi

  38. [46]

    Goldreich,Foundations of Cryptography: Volume 1, Basic Tools

    O. Goldreich,Foundations of Cryptography: Volume 1, Basic Tools. Cambridge University Press, Cambridge, 2001

    2001

  39. [47]

    M. S. Pinsker,Information and Information Stability of Random Variables and Processes. Holden-Day Inc., San Francisco, 1964. – 21 –

    1964

Pith tools

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