Pith. sign in

REVIEW 2 major objections 5 minor 1 cited by

The Smarr formula within the Geroch-Held-Penrose formalism

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read The paper argues that the Penrose–Rindler K-curvature identity, evaluated on a horizon, is exactly the Smarr relation: Gaussian curvature is internal energy, a null-expansion derivative is temperature, and a Ricci term is pressure.

desk verdict Solid GHP derivation of the Smarr formula for static constant-curvature horizons, with a factor-of-2 typo in the RN check and an honest ad hoc Taub-NUT step. read the letter →

arxiv 2412.09682 v2 pith:OS342TGY submitted 2024-12-12 gr-qc

Alberto Guilabert , Pedro Bargueño This is my paper · ORCID
classification gr-qc MSC 83C5780A10
keywords blackholethermodynamicsSmarrformulaGeroch-Held-PenroseformalismtrappinggravityPenrose-RindlerK-curvatureReissner-Nordström-AdSf(R)horizontopology
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 tries to show that black hole thermodynamics is not an analogy grafted onto gravity but a reading of a purely geometric identity. Working in the GHP spin-coefficient formalism, the authors evaluate the field equations on the horizon's spatial cross-sections and find that the Penrose–Rindler K-curvature identity, $k_g=K+\bar{K}$, integrates to the Smarr relation $U = 2TS - 3PV$. Along the way they propose a revised definition of trapping gravity, $\kappa = \sqrt{A/A_0}\,\text{þ}'\rho$, that reproduces surface gravity for static spherically symmetric horizons, and they derive the Smarr formula for Reissner–Nordström–AdS black holes as $M = 2TS + Q\Phi - 2P_\Lambda V$. If right, the result would put the Smarr formula on a quasi-local geometric footing, extend it to toroidal and hyperbolic horizon topologies, and connect it to extended gravity theories such as $f(R)$.

What carries the argument

The load-bearing object is the Penrose–Rindler K-curvature, a spin-coefficient quantity whose sum with its complex conjugate gives the Gaussian curvature $k_g$ of the 2-surfaces orthogonal to the two null directions. The GHP field equations connect $k_g$ to the derivative of the null expansion, $\text{þ}'\rho$, and to the Ricci scalars; evaluating these equations where $\rho=0$ and integrating over the horizon produces the Penrose–Rindler Smarr formula. A second mechanism is the revised trapping gravity $\kappa = \sqrt{A/A_0}\,\text{þ}'\rho$, which supplies the temperature in the formula and repairs the failure of earlier quasi-local surface-gravity definitions in static spherically symmetric cases.

What would settle it

Compute $\text{þ}'\rho$ at two different polar angles on a Kerr horizon: the proposed temperature $T\propto \sqrt{A/A_0}\,\text{þ}'\rho$ would vary between the two points, whereas the temperature of a stationary black hole in equilibrium must be constant. The authors themselves note that $\text{þ}'\rho$ is not constant over the Kerr horizon, so this is a concrete check of the definition's domain of validity.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Penrose–Rindler identity $k_g = K + \bar{K}$, which expresses the Gaussian curvature of a horizon's spatial sections in terms of spin coefficients, is not just a kinematic identity but the Smarr formula itself once the GHP field equations are imposed. Evaluated on a future outer marginal surface (the 2-surface where the null expansion vanishes, foliating a trapping horizon) and integrated, it gives $U = 2TS - 3PV$, with the explicit identifications $U = \frac{\chi(H)}{4}\sqrt{A/A_0}$, $T = \frac{\hbar}{2\pi}\sqrt{A/A_0}\,\text{þ}'\rho$, and $P = -\frac{1}{4\pi}(\Phi^{ph}_{11}+3\Lambda^{ph})$. The authors show this reproduces the Smarr formula for the Reissner–Nordström–AdS black hole, $M = 2TS + Q\Phi - 2P_\Lambda V$, after identifying the left-hand combination with the black hole mass, and they extend it to $f(R)$ gravity by multiplying the geometric identity by $F(R)$. They also show that the same curvature identity carries the Smarr formula for charged Taub–NUT–AdS spacetimes, with the caveat that the length factor $\sqrt{A/A_0}$ must be replaced by an ad hoc factor.

Load-bearing premise

The argument assumes a static spacetime whose horizon cross-sections are compact surfaces of constant Gaussian curvature, with the two null directions geodesic, shear-free, and non-rotating; remove that assumption and the term-by-term identification between geometry and thermodynamics breaks down.

Editorial extensions

If this is right

  • The Smarr formula becomes a geometric statement about the curvature of horizon cross-sections, so it can be derived for any static spacetime with a constant-curvature horizon foliation without solving the full global spacetime.
  • The revised trapping gravity recovers the standard surface gravity for static spherically symmetric black holes, repairing a known failure of earlier quasi-local definitions.
  • The same formula imposes a topology law: in General Relativity with the dominant energy condition, only spherical horizons with non-zero temperature can exist.
  • For $f(R)$ gravity the entropy is $S = A F(R)/4\hbar$ and the Smarr relation follows by multiplying the geometric identity by $F(R)$.
  • The proposed temperature does not apply to stationary axisymmetric horizons, since the relevant expansion derivative is not constant over the Kerr horizon; extending the construction to that case is left open.

Reading between the lines

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

  • A quasi-local first law could be built for horizons that are not global event horizons, giving a route from local horizon geometry to gravitational energy without relying on asymptotic flatness.
  • If the temperature formula is taken seriously, the Kerr-horizon failure suggests the transverse expansion derivative must be replaced by an averaged or frame-corrected quantity in stationary spacetimes; finding that quantity is a concrete next step.
  • Because the $f(R)$ extension rescales energy and entropy by $F(R)$ while leaving the geometric temperature unchanged, comparing the resulting Smarr formula with independent first-law derivations would test the revised definitions.
  • The ad hoc Taub–NUT length factor indicates that the $\sqrt{A/A_0}$ normalization is tied to constant-curvature foliations; deriving that factor geometrically for NUT-type horizons would either broaden or delimit the claimed universality.
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

2 major / 5 minor

Summary. The manuscript derives a 'Penrose-Rindler Smarr formula' from the Geroch-Held-Penrose (GHP) form of the Einstein field equations. For static spacetimes foliated by compact orientable 2-surfaces of constant Gaussian curvature and metric ansatz ds^2 = f(r)dt^2 - f(r)^{-1}dr^2 - r^2 dOmega^2, the authors reduce the GHP field equations to Eq. (28), multiply by a geometrical length factor, and identify the Gaussian-curvature term with internal energy, the þ'rho term with 2TS, and the Ricci-scalar term with -3PV. This yields U = 2TS - 3PV (Eq. (41)) and, after the enthalpy identification, the standard Smarr relation for Reissner-Nordström-AdS (Eq. (47)). The paper also proposes a revised trapping gravity (Eq. (35)) and temperature (Eq. (34)), applies the construction to f(R) gravity by multiplication by F(R), and sketches the charged Taub-NUT-AdS case. The main limitations are stated by the authors: the temperature definition is not valid for stationary axisymmetric horizons, and the Taub-NUT analysis requires an ad hoc length factor.

Significance. If the derivation is accepted, the paper provides a compact geometric packaging of black-hole thermodynamics: the K-curvature identity in Eqs. (22)-(23) is a genuine field-equation identity, and the reduction to Eq. (28) is transparent and checkable. The proposed revised trapping gravity (35) is a concrete, testable improvement over Hayward's definition in the static spherically symmetric limit, and the recovery of the RN-AdS Smarr relation is a nontrivial consistency check. The paper is also commendably explicit about its assumptions and about the limits of the temperature proposal. On the other hand, the thermodynamic dictionary is constructed so that T and V reproduce the known Smarr terms, so the central result is a reformulation rather than an independent derivation of black-hole thermodynamics. The factor-1/2 error in Eq. (44) currently makes the key example non-reproducible. With that error corrected and the dictionary's status clarified, the paper would be a useful contribution to quasi-local black-hole thermodynamics.

major comments (2)
  1. [Sec. IV, Eq. (44)] Equation (44) states Phi_11^ph = Q^2/r^4 for the Reissner-Nordström solution, but the standard GHP value is Phi_11^ph = Q^2/(2r^4), as follows from Eq. (26) with the electromagnetic G^r_r = -Q^2/r^4. Because P in Eq. (36) enters -3PV, the printed value makes the charge term in -3PV equal to QPhi rather than (1/2)QPhi; consequently Eq. (45) cannot be obtained from Eq. (43), and the temperature that would be inferred from Eq. (32) would disagree with the Hawking temperature by a factor of 2 in the charge term. The final Smarr relation (47) is recovered only after the correction Phi_11^ph -> Q^2/(2r^4). This is a load-bearing error in the manuscript's central validation example and must be corrected, with Eqs. (43)-(47) rechecked.
  2. [Sec. IV, Eqs. (32)-(41)] The 'Penrose-Rindler Smarr formula' is a rearrangement of Eq. (28), not a consequence of the field equations alone. The temperature T is defined in Eq. (34) precisely so that 2TS matches the þ'rho term in Eq. (32), and the thermodynamic volume V is defined in Eq. (39) precisely so that -3PV matches the (Phi_11^ph + 3Lambda^ph) term. Thus Eq. (41) holds by construction. To support the paper's claim that the Smarr formula is encoded in the GHP equations, the authors should specify independent physical constraints on the dictionary (for example, demanding agreement with Hawking temperature and with the known RN-AdS thermodynamic volume) and show that Eqs. (34) and (39) are the unique or natural choices satisfying those constraints. Without such a specification, the correspondence (62) is an identity of definitions rather than a derivation.
minor comments (5)
  1. [Eqs. (19) and (22)] In Eqs. (19) and (22), the term rho rho' (or 2 Re(rho rho')) is written twice; one occurrence should presumably be sigma sigma'. The later horizon evaluation is unaffected because rho = sigma = 0, but the displayed general identity should be corrected.
  2. [Eqs. (18), (32), (33)] The symbol chi is used both for the Euler characteristic chi(H) in Eqs. (32)-(33) and (49) and for the GHP gauge normalization in Eq. (18); please use different symbols to avoid confusion.
  3. [Sec. IV, Eq. (31)] The text calls A0 a dimensionless constant, but Eqs. (31) and (34) treat sqrt(A/A0) as a length; please clarify the units and conventions so that the dimensions of U, T, and kappa are explicit.
  4. [Sec. V, Eq. (57)] The replacement sqrt(A/A0) -> (r_+^2 + n^2)/r_+ in Eq. (57) is introduced without a derivation; since the authors state that its geometric meaning is beyond the scope of the work, the section should be labeled more explicitly as an ansatz rather than a derivation.
  5. [Sec. VI] The final paragraph already concedes that Eq. (34) fails for stationary axisymmetric horizons; this limitation is central enough that it should also appear in the abstract or introduction so that readers are not misled about the scope of the revised temperature.

Circularity Check

2 steps flagged · score 4.0 of 10

The GHP identity is independent, but the T/P/V dictionary and the Taub-NUT length factor are chosen such that Eq. (32) becomes the known Smarr relation; partial circularity.

  1. fitted input called prediction [Sec. IV, Eqs. (32)-(41), especially Eqs. (34), (36), (39) and (41)]
    "Moreover, the key feature of this Penrose-Rindler Smarr formula is that it motivates a revised definition of black hole temperature given by T = ℏ/2π sqrt(A/A0) þ′ρ ... after identifying the Bekenstein-Hawking entropy S = A/4ℏ in the GR case. This motivates the definition of the thermodynamic volume of the black hole as V = sqrt(A/A0)(A/3) ... Considering the aforementioned volume, we are able to write the Penrose-Rindler Smarr formula in the same terms as Padmanabhan's result. That is, U = 2TS − 3PV."

    With S = A/4ℏ, the definitions (34) and (39) make 2TS exactly equal to the second term of Eq. (32) and 3PV exactly equal to the third term after using the pressure definition (36). Therefore Eq. (41) is an algebraic rewriting of the geometric identity (32), not an independent prediction of the Smarr relation. The factor 1/3 in the volume definition is imported so that the pressure term acquires the standard Smarr coefficient 3; it is not derived from the GHP equations. The non-circular content is the external check that the temperature so defined coincides with the surface gravity in the spherically symmetric case, which is asserted but not computed in the paper.

  2. fitted input called prediction [Sec. V, around Eqs. (56)-(61)]
    "the Smarr formula can be recovered by multiplying Eq. (56) by 1/8π (r_+^2+n^2)/r_+ (that is, we are replacing sqrt(A/A0) → (r_+^2+n^2)/r_+), leading to ... The geometric motivation and interpretation of the factor ... is beyond the scope of this manuscript and it motivates an interesting path for linking Eq. (22) with the Smarr formula in even more generic configurations."

    The replacement sqrt(A/A0) → (r_+^2+n^2)/r_+ is not derived from the GHP equations; it is an unexplained length factor inserted so that the first term of Eq. (57) becomes the known Taub-NUT mass U = (r_+^2+n^2)/(2r_+) and so that the full expression matches Eq. (61), the known Smarr formula. The paper concedes that the factor's geometric motivation is beyond the scope of the manuscript, so the claimed recovery of the Smarr relation is by construction rather than by a first-principles derivation.

full rationale

Equation (28) is an independent GHP identity obtained from the field equations, and the identification of the integrated Gaussian curvature with internal energy has a non-trivial geometric anchor. The proposed temperature also has external support: for static spherically symmetric horizons it is asserted to reproduce the standard surface gravity, although the printed RN calculation contains a factor-of-two error in Φ_11 that would spoil it as written. The circular element lies in the dictionary: T and V are defined so that Eq. (32) becomes U = 2TS − 3PV, making the Smarr form a rewriting of the geometric identity. The Taub-NUT extension is more openly circular, since the length factor needed to reach Eq. (61) is chosen ad hoc and its motivation is declared beyond scope. The self-citation [41] used for the mass identification in the RN-AdS example is load-bearing, but that relation is standard and externally checkable, so it does not by itself add to the circularity score. Overall, the central GHP content is independent, but the Smarr derivations are to a significant degree constructed to match the target formula.

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

The central derivation depends on a specific static foliation ansatz, the assumption that field equations take the form -R_mu_nu = Delta t_mu_nu, and the choice of length scales A0 and the Taub-NUT factor. These are not fitted to data in an empirical sense, but they are chosen by hand to make the thermodynamic identification work, which raises the circularity burden.

free parameters (2)
  • Unit horizon area A0 = 4*pi*|g-1| for non-flat horizons; Lx*Ly for flat torus
    Chosen so that sqrt(A/A0) supplies the length scale converting the geometric identity into thermodynamic quantities; enters U (Eq. 31), T (Eq. 34), and V (Eq. 39). Not derived from first principles.
  • Taub-NUT length factor = (r+^2 + n^2)/r+
    Introduced in Sec. V to replace sqrt(A/A0) because the temperature definition Eq. (34) fails for charged Taub-NUT-AdS; the paper states its geometric motivation is beyond the scope. This factor is needed to recover the known Smarr formula.
assumptions (4)
  • domain assumption Field equations of the theories considered can be written as -R_mu_nu = Delta t_mu_nu with an effective stress-energy tensor independent of the Ricci tensor (Eq. 8).
    The paper gives examples (GR, f(R), scalar-tensor) but explicitly says no comprehensive analysis exists of which Lagrangians allow this form (Sec. III).
  • domain assumption Spacetime is static and foliated by compact orientable 2-surfaces of constant Gaussian curvature, with metric ds^2 = f dt^2 - g dr^2 - r^2 dOmega^2 and g = f^{-1} for the core derivation.
    This ansatz (Sec. IV, App. A) ensures the Ricci scalars are real, Phi_01 = Phi_02 = 0, and the principal null directions are geodesic, shear-free, and non-rotating; it excludes Kerr and generic Taub-NUT.
  • ad hoc to paper Gauge choice chi = 1 in Hayward's trapping gravity, with the claim that any other normalization leads to the same results.
    Stated in Sec. III around Eq. (18); no proof is given for the gauge-independence claim.
  • domain assumption Bekenstein-Hawking entropy S = A/(4 hbar) holds in GR and is used to identify temperature from the geometric term.
    Takes S = A/(4 hbar) as input (Sec. IV) to convert þ'rho into temperature; this is standard black hole thermodynamics.
invented entities (2)
  • Revised trapping gravity kappa = sqrt(A/A0) þ'rho
    purpose: Replaces Hayward's definition so that the temperature T = hbar/(2 pi) sqrt(A/A0) þ'rho reproduces surface gravity in static spherical symmetry and makes Eq. (32) read as 2TS.
    A new definition, not a new physical object; anchored to the Hawking temperature in the spherical limit but no independent prediction for exotic topologies.
  • Generalized internal energy U = (chi(H)/4) sqrt(A/A0)
    purpose: Provides a quasi-local energy that matches Padmanabhan's a/2 for spheres and gives zero for tori and negative values for hyperbolic horizons.
    A proposed definition; the sign and topology behavior matches prior work [41], but it is not a measured quantity.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Smarr formula within the Geroch-Held-Penrose formalism." pith.science (2026). https://pith.science/paper/OS342TGY

@misc{pith2026241209682,
  author       = {Pith},
  title        = {Pith review of: The Smarr formula within the Geroch-Held-Penrose formalism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OS342TGY}},
  note         = {Machine review of arXiv:2412.09682}
}
abstract

The connection between classical thermodynamics and black hole horizons is a fundamental topic in gravitational physics, offering a potential pathway to understanding quantum aspects of gravity. However, while black hole mechanics exhibits well-known thermodynamic parallels, a rigorous geometric interpretation of thermodynamic variables directly from the field equations warrants further research. In this manuscript, we present a thermodynamic formulation of the field equations through the decomposition of the Riemann tensor, employing the Geroch-Held-Penrose (GHP) formalism, to clarify a strong correspondence between black hole thermodynamic variables and geometrical quantities derived from horizon geometry. Our analysis reveals an intrinsic connection between the Penrose and Rindler $K$-curvature and the Smarr relation, motivating a revised definition of both trapping gravity and black hole internal energy. Additionally, we derive through this GHP formalism the Smarr formula for the Reissner-N\"ordstrom black hole cointained in an AdS spacetime and we explore the implications of this relationship for black holes with exotic topologies and in the context of extended theories, exemplified by $f(R)$ gravity. These findings suggest a deeper geometrical basis for black hole thermodynamics, potentially advancing our understanding of gravitational energy, horizon entropy, and their significance within quantum gravity frameworks.

Figures

Figures reproduced from arXiv: 2412.09682 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic diagram showing the relation between the fi [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Penrose-Rindler equation and horizon thermodynamics of stationary black holes

    gr-qc 2026-02 conditional novelty 6.0 of 10

    For static and Kerr-like rotating black holes the horizon condition is equivalent to the Penrose–Rindler equation, which, read as a pressure equilibrium, yields a quasi-local Smarr formula with a newly defined 'Smarr volume.'

Reference graph

Works this paper leans on

55 extracted references · 50 canonical work pages · cited by 1 Pith paper

  1. [1]

    Hawking, Black holes in general relativity , Communications in Mathematical Physics 25 (1972) 152–166

    S.W. Hawking, Black holes in general relativity , Communications in Mathematical Physics 25 (1972) 152–166

    work page 1972

  2. [2]

    Woolgar, Bounded area theorems for higher-genus black holes , Classical and Quantum Gravity 16 (1999) 3005–3012

    E. Woolgar, Bounded area theorems for higher-genus black holes , Classical and Quantum Gravity 16 (1999) 3005–3012 . 15

    work page 1999

  3. [3]

    Kontou and V

    E.-A. Kontou and V. Sacchi, A generalization of the hawking black hole area theorem , General Relativity and Gravitation 56 (2024)

    work page 2024

  4. [4]

    and ( 23). To illustrate these results, we consider two specific examples: (i) wit hin GR, we employ the GHP formalism to reobtain the Smarr formula for the Reissner-Nordstr¨ om-AdS bla ck hole and (ii) we adapt the Penrose-Rindler Smarr formula to f (R) gravity. Reissner-N¨ ordstrom black hole in AdS background: a GHP der ivation The Reissner–Nordstr¨ om ...

    work page

  5. [5]

    Christodoulou and R

    D. Christodoulou and R. Ruffini, Reversible transformations of a charged black hole , Physical Review D 4 (1971) 3552–3555

    work page 1971

  6. [6]

    Harvie, A penrose-type inequality for static spacetimes , (2024), [ gr-qc/2410.12203]

    B. Harvie, A penrose-type inequality for static spacetimes , (2024), [ gr-qc/2410.12203]

    work page arXiv 2024

  7. [7]

    Bekenstein, Black holes and entropy , Physical Review D 7 (1973) 2333–2346

    J.D. Bekenstein, Black holes and entropy , Physical Review D 7 (1973) 2333–2346

    work page 1973

  8. [8]

    Wald, Black hole entropy is the noether charge , Physical Review D 48 (1993) 3427–3431

    R.M. Wald, Black hole entropy is the noether charge , Physical Review D 48 (1993) 3427–3431

    work page 1993

Show all 55 references
  1. [9]

    Christodoulou, Reversible and irreversible transformations in black-hol e physics , Physical Review Letters 25 (1970) 1596–1597

    D. Christodoulou, Reversible and irreversible transformations in black-hol e physics , Physical Review Letters 25 (1970) 1596–1597

    1970

  2. [10]

    Padmanabhan, Thermodynamical aspects of gravity: new insights , Reports on Progress in Physics 73 (2010) 046901

    T. Padmanabhan, Thermodynamical aspects of gravity: new insights , Reports on Progress in Physics 73 (2010) 046901

    2010

  3. [11]

    Bamba and C.-Q

    K. Bamba and C.-Q. Geng, Thermodynamics in f(r) gravity in the palatini formalism , Journal of Cosmology and Astroparticle Physics 2010 (2010) 014–014

    2010

  4. [12]

    Peralta and S

    C. Peralta and S. Jor´ as, Thermodynamics of f(r) theories of gravity , Journal of Cosmology and Astroparticle Physics 2020 (2020) 053–053

    2020

  5. [13]

    Susskind, Some speculations about black hole entropy in string theory , (1993), [ hep-th/9309145]

    L. Susskind, Some speculations about black hole entropy in string theory , (1993), [ hep-th/9309145]

    1993 arXiv

  6. [14]

    Faraoni, Black hole entropy in scalar-tensor and ƒ(r) gravity: An overview , Entropy 12 (2010) 1246–1263

    V. Faraoni, Black hole entropy in scalar-tensor and ƒ(r) gravity: An overview , Entropy 12 (2010) 1246–1263

    2010

  7. [15]

    Lee, Are black holes black bodies? , Nuclear Physics B 264 (1986) 437–486

    T. Lee, Are black holes black bodies? , Nuclear Physics B 264 (1986) 437–486

    1986

  8. [16]

    Dalui, B.R

    S. Dalui, B.R. Majhi and T. Padmanabhan, Thermal nature of a generic null surface , Physical Review D 104 (2021)

    2021

  9. [17]

    Bardeen, B

    J.M. Bardeen, B. Carter and S.W. Hawking, The four laws of black hole mechanics , Communications in Mathematical Physics 31 (1973) 161–170

    1973

  10. [18]

    Appels, R

    M. Appels, R. Gregory and D. Kubizˇ n´ ak,Thermodynamics of accelerating black holes , Physical Review Letters 117 (2016)

    2016

  11. [19]

    Hawking, Particle creation by black holes , Communications In Mathematical Physics 43 (1975) 199–220

    S.W. Hawking, Particle creation by black holes , Communications In Mathematical Physics 43 (1975) 199–220

    1975

  12. [20]

    Hayward, General laws of black-hole dynamics , Physical Review D 49 (1994) 6467–6474

    S.A. Hayward, General laws of black-hole dynamics , Physical Review D 49 (1994) 6467–6474

    1994

  13. [21]

    Hayward, Spin coefficient form of the new laws of black hole dynamics , Classical and Quantum Gravity 11 (1994) 3025–3035

    S.A. Hayward, Spin coefficient form of the new laws of black hole dynamics , Classical and Quantum Gravity 11 (1994) 3025–3035

    1994

  14. [22]

    In this particular case, the spacetime is of type D and the congruen ces associated to the principal null directions are geodesic, shear-free and non-rotating

    and ( 23) as a thermodynamic relation. In this particular case, the spacetime is of type D and the congruen ces associated to the principal null directions are geodesic, shear-free and non-rotating. Then κ =σ = 0 and ω = Im(ρ) = 0, as is shown in App. A. This implies that ρ is...

  15. [23]

    Nielsen and J.H

    A.B. Nielsen and J.H. Yoon, Dynamical surface gravity , Classical and Quantum Gravity 25 (2008) 085010

    2008

  16. [24]

    Gibbons and S.W

    G.W. Gibbons and S.W. Hawking, Cosmological event horizons, thermodynamics, and particl e creation, Physical Review D 15 (1977) 2738–2751

    1977

  17. [25]

    Eling, R

    C. Eling, R. Guedens and T. Jacobson, Nonequilibrium thermodynamics of spacetime , Physical Review Letters 96 (2006)

    2006

  18. [26]

    Padmanabhan, Classical and quantum thermodynamics of horizons in spheri cally symmetric space-times , Class

    T. Padmanabhan, Classical and quantum thermodynamics of horizons in spheri cally symmetric space-times , Class. Quant. Grav. 19 (2002) 5387–5408 [gr-qc/0204019]

    2002 arXiv

  19. [27]

    Fodor, K

    G. Fodor, K. Nakamura, Y. Oshiro and A. Tomimatsu, Surface gravity in dynamical spherically symmetric spacet imes, Physical Review D 54 (1996) 3882–3891

    1996

  20. [28]

    This coincides with the energy term obtained by Padmanabhan, E = a/2, for the static and spherically symmetric case [10]

    is strongly related with the Hayward definition of quasi-local energ y [ 21] U = 1 4π √ A 16π ∫ H ⋆kg, (29) whereA denotes the area of the black hole horizon and ⋆ denotes the Hodge operator defined on the horizon H. This coincides with the energy term obtained by Padmanabhan, E...

  21. [29]

    Jacobson, Thermodynamics of spacetime: The einstein equation of stat e, Physical Review Letters 75 (1995) 1260–1263

    T. Jacobson, Thermodynamics of spacetime: The einstein equation of stat e, Physical Review Letters 75 (1995) 1260–1263

    1995

  22. [30]

    Cvetiˇ c, G.W

    M. Cvetiˇ c, G.W. Gibbons, D. Kubizˇ n´ ak and C.N. Pope,Black hole enthalpy and an entropy inequality for the thermodynamic volume , Physical Review D 84 (2011)

    2011

  23. [31]

    ( 28), this can be rewritten as χ(H) 4 √ A A0 − 1 4π √ A A0 þ′ρA = 1 4π √ A A0 (Φ ph 11 + 3Λ ph)A, (32) which will be referred as the Penrose-Rindler Smarr formula

    in Eq. ( 28), this can be rewritten as χ(H) 4 √ A A0 − 1 4π √ A A0 þ′ρA = 1 4π √ A A0 (Φ ph 11 + 3Λ ph)A, (32) which will be referred as the Penrose-Rindler Smarr formula. Here, we can identify the first term on the previous relation as the total internal energy of the black ho...

  24. [32]

    Kastor, S

    D. Kastor, S. Ray and J. Traschen, Enthalpy and the mechanics of ads black holes , Classical and Quantum Gravity 26 (2009) 195011

    2009

  25. [33]

    Smarr, Mass formula for kerr black holes , Physical Review Letters 30 (1973) 71–73

    L. Smarr, Mass formula for kerr black holes , Physical Review Letters 30 (1973) 71–73

    1973

  26. [34]

    Kubizˇ n´ ak, R.B

    D. Kubizˇ n´ ak, R.B. Mann and M. Teo,Black hole chemistry: thermodynamics with lambda , Classical and Quantum Gravity 34 (2017) 063001

    2017

  27. [35]

    Newman and R

    E. Newman and R. Penrose, An Approach to Gravitational Radiation by a Method of Spin Co efficients , Journal of Mathematical Physics 3 (1962) 566–578

    1962

  28. [36]

    Kothawala and T

    D. Kothawala and T. Padmanabhan, Thermodynamic structure of lanczos-lovelock field equatio ns from near-horizon symmetries, Physical Review D 79 (2009)

    2009

  29. [37]

    Kothawala, S

    D. Kothawala, S. Sarkar and T. Padmanabhan, Einstein ’s equations as a thermodynamic identity: The case s of stationary axisymmetric horizons and evolving sphericall y symmetric horizons , Physics Letters B 652 (2007) 338–342

    2007

  30. [38]

    Penrose and W

    R. Penrose and W. Rindler, Spinors and Space-Time , Cambridge University Press (Oct., 1984), 10.1017/cbo9780511564048

    1984 doi

  31. [39]

    Quiros, Selected topics in scalar–tensor theories and beyond , International Journal of Modern Physics D 28 (2019) 1930012

    I. Quiros, Selected topics in scalar–tensor theories and beyond , International Journal of Modern Physics D 28 (2019) 1930012

    2019

  32. [40]

    Banerjee and B.R

    R. Banerjee and B.R. Majhi, Statistical origin of gravity , Physical Review D 81 (2010)

    2010

  33. [41]

    Bargue˜ no and E

    P. Bargue˜ no and E. Contreras, The Geroch-Held-Penrose Calculus: Fundamentals and Appli cations, Springer Nature Switzerland (2023), 10.1007/978-3-031-42948-4

    2023 doi

  34. [42]

    Geroch, A

    R. Geroch, A. Held and R. Penrose, A space-time calculus based on pairs of null directions , Journal of Mathematical Physics 14 (1973) 874–881

    1973

  35. [43]

    Medved, D

    A.J.M. Medved, D. Martin and M. Visser, Dirty black holes: spacetime geometry and near-horizon sym metries, Classical and Quantum Gravity 21 (2004) 3111–3125

    2004

  36. [44]

    Hayward, Quasilocal gravitational energy , Physical Review D 49 (1994) 831–839

    S.A. Hayward, Quasilocal gravitational energy , Physical Review D 49 (1994) 831–839

    1994

  37. [45]

    Taub, Empty space-times admitting a three parameter group of moti ons, The Annals of Mathematics 53 (1951) 472

    A.H. Taub, Empty space-times admitting a three parameter group of moti ons, The Annals of Mathematics 53 (1951) 472

    1951

  38. [46]

    Villalba, P

    F. Villalba, P. Bargue˜ no, A. Vargas and E. Contreras, Newman–penrose scalars and black hole equations of state , Physics of the Dark Universe 30 (2020) 100614

    2020

  39. [47]

    Faraoni and S

    V. Faraoni and S. Capozziello, Beyond Einstein Gravity: A Survey of Gravitational Theorie s for Cosmology and Astrophysics, Springer, Dordrecht (2011), 10.1007/978-94-007-0165-6. 16

    2011 doi

  40. [48]

    Sharif and M

    M. Sharif and M. Zubair, Thermodynamics in f(r,t) theory of gravity , Journal of Cosmology and Astroparticle Physics 2012 (2012) 028–028

    2012

  41. [49]

    Hawking and G.F.R

    S.W. Hawking and G.F.R. Ellis, The large-scale structure of space-time. , Cambridge Monographs on Mathematical Physics, Cambridge University Press (1973)

    1973

  42. [50]

    Durkee, V

    M. Durkee, V. Pravda, A. Pravdov´ a and H.S. Reall, Generalization of the geroch–held–penrose formalism to hi gher dimensions, Classical and Quantum Gravity 27 (2010) 215010

    2010

  43. [51]

    Newman, L

    E. Newman, L. Tamburino and T. Unti, Empty-space generalization of the schwarzschild metric , Journal of Mathematical Physics 4 (1963) 915–923

    1963

  44. [52]

    Liu and H.-S

    J.-F. Liu and H.-S. Liu, Thermodynamics of taub-nut-ads spacetimes , The European Physical Journal C 84 (2024)

    2024

  45. [53]

    Bordo, D

    A.B. Bordo, D. Kubiznak and T.R. Perche, Taub-nut solutions in conformal electrodynamics , (2020), [hep-th/2011.13398]

    2020 arXiv

  46. [54]

    Hennigar, D

    R.A. Hennigar, D. Kubizˇ n´ ak and R.B. Mann,Thermodynamics of lorentzian taub-nut spacetimes , Physical Review D 100 (2019)

    2019

  47. [57]

    That is, we just showed that Eq

    is identically to M = 2(TS +ψN −PV ) +φeQe +φmQm, (61) where all the well-known thermodynamic variables are described in [ 48, 49]. That is, we just showed that Eq. ( 22) is again related with the Smarr formula even in more general ansatz , although finding the necessary physic...

    2022

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.