pith. machine review for the scientific record. sign in

arxiv: 2604.08170 · v1 · submitted 2026-04-09 · 🌀 gr-qc · hep-th

Recognition: unknown

Thermodynamics and orbital structure of anti-de Sitter black holes in Palatini-inspired nonlinear electrodynamics

Edilberto O. Silva , Jo\~ao A. A. S. Reis , Faizuddin Ahmed
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:13 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords anti-de Sitter black holesnonlinear electrodynamicsPalatini gravityextended thermodynamicsphoton sphereshadow radiusgeodesicsblack hole thermodynamics
0
0 comments X

The pith

A Palatini-inspired nonlinear electrodynamics model yields a consistent anti-de Sitter black hole whose electromagnetic solution keeps its original parametric form while the metric gains the standard cosmological term.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs the anti-de Sitter completion of a static spherically symmetric black hole sourced by the Y^n Palatini-inspired nonlinear electrodynamics model. Starting from the Einstein-Hilbert action with negative cosmological constant plus the first-order nonlinear sector, the authors derive the field equations and demonstrate that the electromagnetic solution is unchanged in its parameters. The metric lapse function simply incorporates the usual AdS term. They then compute the horizon locations, Hawking temperature, extended phase-space thermodynamics and equation of state, followed by null and timelike geodesics including effective potentials, photon spheres, finite-distance shadow radii, and innermost stable circular orbits. A sympathetic reader would care because the construction supplies an exact AdS black-hole background whose thermodynamic and orbital features can be studied directly without solving a fully coupled nonlinear system.

Core claim

Starting from the Einstein-Hilbert action with a negative cosmological constant and the first-order PINLED sector, the nonlinear electromagnetic solution preserves its original parametric structure, while the lapse function acquires the standard AdS contribution. This furnishes the exact AdS extension of the asymptotically flat PINLED black hole.

What carries the argument

The first-order Palatini-inspired nonlinear electrodynamics Y^n sector, solved independently of the metric and then inserted into the Einstein equations with negative cosmological constant.

If this is right

  • The lapse function is the flat-space PINLED lapse plus the standard -r^2/L^2 AdS term, so horizon radii are shifted but retain the same qualitative structure.
  • Hawking temperature and extended phase-space thermodynamics produce an equation of state whose critical points can be compared with the flat-space case.
  • Null geodesics admit a photon sphere whose radius and the associated shadow radius for a static observer at finite distance are computable from the effective potential.
  • Timelike geodesics yield an innermost stable circular orbit whose location follows from the second derivative of the effective potential.
  • The resulting geometry supplies an exact background for numerical studies of thermodynamic stability and optical properties.

Where Pith is reading between the lines

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

  • Because the electromagnetic parameters remain fixed, thermodynamic quantities such as heat capacity may display the same functional dependence on mass and charge as in the asymptotically flat case, only shifted by AdS corrections.
  • The photon-sphere and shadow calculations provide a clean benchmark for comparing nonlinear-electrodynamics effects against the Reissner-Nordström-AdS case in the same coordinate system.
  • The preserved parametric structure suggests that any holographic dual interpretation, if pursued, would involve the same boundary gauge theory data as the flat-space model but with a different bulk geometry.

Load-bearing premise

The nonlinear electromagnetic sector can be solved on its own and then placed into the metric ansatz without the cosmological constant term altering the electromagnetic field's parametric structure.

What would settle it

An explicit integration of the electromagnetic field equations in the presence of the negative cosmological constant that yields a different radial dependence for the field strength or potential than the flat-space PINLED solution.

Figures

Figures reproduced from arXiv: 2604.08170 by Edilberto O. Silva, Faizuddin Ahmed, Jo\~ao A. A. S. Reis.

Figure 1
Figure 1. Figure 1: displays fAdS(ρ) for n = 2, q = 1, P˜ = 0.01, and four representative values of the ADM mass M˜ . Three qualitatively distinct regimes are apparent: • Sub-extremal (M˜ = 0.35): the lapse function is strictly positive everywhere above a minimum ra￾dius, indicating the absence of a horizon. The ge￾ometry describes a naked electromagnetic source. • Near-extremal (M˜ = 0.50): fAdS develops a local minimum that… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Dimensionless Hawking temperature [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Dimensionless Hawking temperature [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Dimensionless temperature [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Equation-of-state isotherms [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Gibbs free energy [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: shows CP /(2πℓ2 ) as a function of T˜ for n = 2, q = 1, and four pressures. On the LBH branch (solid lines), CP is positive and increases with T˜, characteris￾tic of thermally stable black holes. On the SBH branch (dashed lines), CP is negative, signaling local thermody￾namic instability. Both branches diverge at T˜ = T˜min (indicated by vertical dotted lines), where the heat ca￾pacity changes sign and the… view at source ↗
Figure 11
Figure 11. Figure 11: shows V null eff /L2 for n = 2, Mf = 0.75, Pe = 0.01, and four values of the charge q. Each potential exhibits a single maximum whose height, the capture cross-section, decreases as q increases: a larger nonlinear charge low￾ers the potential barrier and reduces the capture cross￾section, analogous to the effect of charge in the Reissner– Nordström case but with a modified radial dependence. The photon-sp… view at source ↗
Figure 12
Figure 12. Figure 12: shows the same potential for fixed q = 1, Pe = 0.01, and four values of the ADM mass Mf. As Mf increases, the black hole grows and the photon-sphere shifts to larger ρph, while the height of the barrier in￾creases monotonically. This is qualitatively consistent 0.40 2.05 3.70 5.35 7.00 ρ -0.02 0.12 0.27 0.41 0.55 V null eff /L 2 n = 2, Mf = 0.75, Pe = 0.01 (circles: photon sphere) q = 0.5 q = 1.0 q = 1.5 … view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Timelike effective potential [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Photon-sphere radius [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Shadow radius [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Shadow silhouettes of the PINLED-AdS black hole as seen by a static observer located at [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. ISCO radius [PITH_FULL_IMAGE:figures/full_fig_p019_17.png] view at source ↗
read the original abstract

We construct a consistent anti-de Sitter completion of the static and spherically symmetric black-hole solution sourced by the Palatini-inspired nonlinear electrodynamics \(Y^n\) model. Starting from the Einstein--Hilbert action with a negative cosmological constant and the first-order PINLED sector, we derive the full set of field equations and show that the nonlinear electromagnetic solution preserves its original parametric structure, while the lapse function acquires the standard AdS contribution. We then analyze the horizon structure, Hawking temperature, extended phase-space thermodynamics, and the associated equation of state. In addition, we investigate null and timelike geodesics, with emphasis on the effective potentials, photon sphere, shadow radius for a static observer at finite distance, and innermost stable circular orbit. The resulting framework furnishes the exact AdS extension of the asymptotically flat PINLED black hole and provides a coherent basis for numerical and phenomenological studies of its thermodynamic, optical, and orbital properties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript constructs a consistent anti-de Sitter completion of the static spherically symmetric black-hole solution sourced by the Palatini-inspired nonlinear electrodynamics Y^n model. Starting from the Einstein-Hilbert action with negative cosmological constant and the first-order PINLED sector, the authors derive the full set of field equations and claim that the nonlinear electromagnetic solution preserves its original parametric structure (with the same dependence on charge and the parameter n), while the lapse function acquires the standard AdS contribution. The paper then analyzes the horizon structure, Hawking temperature, extended phase-space thermodynamics and equation of state, followed by null and timelike geodesics with emphasis on effective potentials, photon sphere, shadow radius for a static observer at finite distance, and the innermost stable circular orbit.

Significance. If the central claim of parametric preservation holds under the coupled first-order system, the work supplies an exact AdS extension of the known asymptotically flat PINLED black holes. This furnishes a coherent framework for thermodynamic studies in extended phase space and for orbital/optical properties (photon spheres, shadows, ISCOs) that are directly relevant to AdS/CFT applications and black-hole phenomenology. The explicit treatment of both thermodynamic and geodesic sectors strengthens the utility for numerical follow-up work.

major comments (1)
  1. [Abstract / field-equation derivation] Abstract and the derivation of the field equations: the load-bearing claim that 'the nonlinear electromagnetic solution preserves its original parametric structure' after inclusion of the negative cosmological constant requires explicit demonstration that the first-order Palatini variations of the PINLED sector (with respect to the independent connection and auxiliary fields) receive no metric-dependent corrections from the Einstein equations with Lambda. The stress-energy tensor of the NED sector enters the metric equations, so any implicit dependence of the constitutive relations or connection on the metric functions could alter the integrated form of the electric field or auxiliary variable before the AdS term is added to the lapse; this decoupling must be shown by direct solution of the coupled system rather than by superposition from the flat-space case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address the major comment as follows.

read point-by-point responses

  1. Referee: Abstract and the derivation of the field equations: the load-bearing claim that 'the nonlinear electromagnetic solution preserves its original parametric structure' after inclusion of the negative cosmological constant requires explicit demonstration that the first-order Palatini variations of the PINLED sector (with respect to the independent connection and auxiliary fields) receive no metric-dependent corrections from the Einstein equations with Lambda. The stress-energy tensor of the NED sector enters the metric equations, so any implicit dependence of the constitutive relations or connection on the metric functions could alter the integrated form of the electric field or auxiliary variable before the AdS term is added to the lapse; this decoupling must be shown by direct solution of the coupled system rather than by superposition from the flat-space case.

    Authors: We agree with the referee that an explicit demonstration of the decoupling is necessary to substantiate the claim. In the revised version of the manuscript, we have expanded the derivation section to include a direct solution of the full coupled system. Specifically, we vary the action with respect to the independent connection and auxiliary fields, obtaining relations that fix the auxiliary variable Y and the connection in terms of the metric and the electromagnetic invariant without influence from the cosmological constant term. The resulting stress-energy tensor is then used in the metric field equations, which integrate straightforwardly to yield the AdS-modified lapse function while preserving the parametric form of the electromagnetic solution. This step-by-step solution confirms that no metric-dependent corrections arise in the NED sector due to the inclusion of Lambda. We believe this addresses the concern and strengthens the presentation of the field equations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from the coupled action

full rationale

The paper explicitly states that it begins from the Einstein-Hilbert action with negative cosmological constant plus the first-order PINLED sector, derives the full set of field equations, and then shows that the nonlinear electromagnetic solution preserves its parametric structure while the lapse acquires the AdS term. This constitutes an independent derivation from the action rather than a superposition of a pre-existing flat-space solution or a self-definitional fit. No load-bearing self-citation, ansatz smuggling, or renaming of known results is required for the central claim in the abstract; the preservation result is presented as emerging from the derived equations. The framework is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The construction rests on the Einstein-Hilbert action with negative cosmological constant plus the first-order Palatini nonlinear EM sector; n is a free model parameter and the metric ansatz is assumed spherically symmetric and static.

free parameters (1)
  • n
    Exponent in the nonlinear electrodynamics Lagrangian term Y^n; defines the specific model and is not derived from the action.
axioms (3)
  • standard math Einstein-Hilbert action supplemented by negative cosmological constant
    Gravitational sector used to derive the field equations.
  • domain assumption First-order Palatini formulation of the nonlinear EM sector
    The connection is varied independently; invoked when stating the PINLED model.
  • domain assumption Static spherically symmetric metric ansatz
    Assumed form of the line element before solving the equations.

pith-pipeline@v0.9.0 · 5475 in / 1660 out tokens · 49943 ms · 2026-05-10T17:13:33.111158+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

49 extracted references · 32 canonical work pages · 5 internal anchors

  1. [1]

    The SBH is locally unsta- ble (CP <0) and thermodynamically subdominant at all temperatures

    Forn= 2, the electromagnetic correctionK(y)∼ y/4produces a softer ˜P-visotherm than RN-AdS, and the system undergoes only a Hawking-Page first-order transition between thermal AdS and a stable large black hole. The SBH is locally unsta- ble (CP <0) and thermodynamically subdominant at all temperatures

  2. [2]

    This can produce a swallowtail in the˜G- ˜Tdiagram and a genuine VdW-like SBH-LBH first-order phase transition, qualitatively analogous to the RN-AdS case [25, 30]

    Forn≥3at sufficiently large chargeq, the energy densityK(y)∼[(3n−2)/(4n)]yn grows faster withy, generating a stronger electromag- netic contribution at small horizon radii. This can produce a swallowtail in the˜G- ˜Tdiagram and a genuine VdW-like SBH-LBH first-order phase transition, qualitatively analogous to the RN-AdS case [25, 30]. Thus, the nonlinear...

  3. [3]

    Figure 11 showsVnull eff /L2 forn= 2, ˜M= 0.75, ˜P= 0.01, and four values of the chargeq

    Null effective potential Forξ= 0, V null eff (ρ) =fAdS(ρ)L2 ρ2.(68) The maximum ofV null eff defines the photon sphere, which separates captured from scattered photon trajectories. Figure 11 showsVnull eff /L2 forn= 2, ˜M= 0.75, ˜P= 0.01, and four values of the chargeq. Each potential exhibits a single maximum whose height, the capture cross-section, decr...

  4. [4]

    For smallL(e.g.L= 2), thepotentialismonotonicallyincreasingbeyondthehori- zon; no local minimum exists, and no stable circular or- bit is available

    Timelike effective potential Forξ= 1, V timelike eff (ρ) =fAdS(ρ) (L2 ρ2 + 1 ) .(69) Figure 13 showsV timelike eff forn= 2,q= 1, ˜M= 0.75, ˜P= 0.01, and five values ofL. For smallL(e.g.L= 2), thepotentialismonotonicallyincreasingbeyondthehori- zon; no local minimum exists, and no stable circular or- bit is available. AsLincreases, a local minimum de- velo...

    2022

  5. [5]

    acknowledges the Inter University Center for Astronomy and Astrophysics (IUCAA), Pune, In- dia, for granting visiting associateship

    F.A. acknowledges the Inter University Center for Astronomy and Astrophysics (IUCAA), Pune, In- dia, for granting visiting associateship. J.A.A.S.R ac- knowledges partial financial support from UESB through Grant AuxPPI (Edital No. 267/2024), as well as from FAPESB–CNPq/Produtividade under Grant No. 12243/2025 (TOB-BOL2798/2025). DATA AVAILABILITY STATEME...

    2024

  6. [6]

    Born and L

    M. Born and L. Infeld, Proc. R. Soc. Lond. A144, 425 (1934)

    1934

  7. [7]

    Euler and B

    H. Euler and B. Kockel, Naturwissenschaften23, 246 (1935)

    1935

  8. [8]

    Demiánski, Found

    M. Demiánski, Found. Phys.16, 187 (1986)

    1986

  9. [9]

    Regular Black Hole in General Relativity Coupled to Nonlinear Electrodynamics

    E. Ayón-Beato and A. García, Phys. Rev. Lett.80, 5056 (1998), arXiv:gr-qc/9911046

    work page Pith review arXiv 1998

  10. [10]

    Ayon-Beato and A

    E. Ayón-Beato and A. García, Phys. Lett. B464, 25 (1999), arXiv:hep-th/9911174

    work page arXiv 1999

  11. [11]

    The Bardeen Model as a Nonlinear Magnetic Monopole

    E. Ayón-Beato and A. García, Phys. Lett. B493, 149 (2000), arXiv:gr-qc/0009077

    work page Pith review arXiv 2000

  12. [12]

    K. A. Bronnikov, Phys. Rev. D63, 044005 (2001), arXiv:gr-qc/0006014

    work page Pith review arXiv 2001

  13. [13]

    Bretón, Phys

    N. Bretón, Phys. Rev. D67, 124004 (2003), arXiv:hep- th/0301254

    work page arXiv 2003

  14. [14]

    S. I. Kruglov, Ann. Phys. (Berlin)527, 397 (2015), arXiv:1504.03941

    work page arXiv 2015

  15. [15]

    S. I. Kruglov, Annalen der Physik529, 1700073 (2017)

    2017

  16. [16]

    Bardeen, inProceedings of the 5th International Con- ference on Gravitation and the Theory of Relativity, edited by H.-J

    J. Bardeen, inProceedings of the 5th International Con- ference on Gravitation and the Theory of Relativity, edited by H.-J. Treder (Tbilisi University Press, Tbilisi,

  17. [17]

    Regular black holes with a nonlinear electrodynamics source

    L. Balart and E. C. Vagenas, Phys. Rev. D90, 124045 (2014), arXiv:1408.0306

    work page Pith review arXiv 2014

  18. [18]

    Construction of Regular Black Holes in General Relativity

    Z.-Y. Fan and X. Wang, Phys. Rev. D94, 124027 (2016), arXiv:1610.02636

    work page Pith review arXiv 2016

  19. [19]

    Verbin, B

    Y. Verbin, B. Puliçe, A. Övgün, and H. Huang, Phys. Rev. D111, 084061 (2025)

    2025

  20. [20]

    İ. İ. Çimdiker, A. Övgün, and Y. Verbin, (2026), arXiv:2603.10097v1, arXiv:2603.10097 [gr-qc]

    work page arXiv 2026

  21. [21]

    J. M. Maldacena, Adv. Theor. Math. Phys.2, 231 (1998), arXiv:hep-th/9711200

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  22. [22]

    Anti De Sitter Space And Holography

    E. Witten, Adv. Theor. Math. Phys.2, 253 (1998), arXiv:hep-th/9802150

    work page internal anchor Pith review arXiv 1998

  23. [23]

    S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B428, 105 (1998), arXiv:hep-th/9802109. 21

    work page internal anchor Pith review arXiv 1998

  24. [24]

    S. W. Hawking and D. N. Page, Commun. Math. Phys. 87, 577 (1983)

    1983

  25. [25]

    Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories

    E. Witten, Adv. Theor. Math. Phys.2, 505 (1998), arXiv:hep-th/9803131

    work page Pith review arXiv 1998

  26. [26]

    Charged AdS Black Holes and Catastrophic Holography

    A. Chamblin, R. Emparan, H. S. Reall, and R. C. Myers, Phys. Rev. D60, 064018 (1999), arXiv:hep-th/9902170

    work page Pith review arXiv 1999

  27. [27]

    Enthalpy and the Mechanics of AdS Black Holes

    D. Kastor, S. Ray, and J. Traschen, Class. Quant. Grav. 26, 195011 (2009), arXiv:0904.2765

    work page Pith review arXiv 2009

  28. [28]

    B. P. Dolan, Class. Quant. Grav.28, 125020 (2011), arXiv:1008.5023

    work page arXiv 2011

  29. [29]

    B. P. Dolan, Class. Quant. Grav.28, 235017 (2011), arXiv:1106.6260

    work page arXiv 2011

  30. [30]

    P-V criticality of charged AdS black holes

    D. Kubiznák and R. B. Mann, JHEP2012(07), 033, arXiv:1205.0559

    work page Pith review arXiv

  31. [31]

    Kubiznak, R

    D. Kubiznák, R. B. Mann, and M. Teo, Class. Quant. Grav.34, 063001 (2017), arXiv:1608.06147

    work page arXiv 2017

  32. [32]

    Altamirano, D

    N. Altamirano, D. Kubiznák, and R. B. Mann, Phys. Rev. D88, 101502 (2013), arXiv:1306.5756

    work page arXiv 2013

  33. [33]

    Kerr-AdS analogue of triple point and solid/liquid/gas phase transition,

    N. Altamirano, D. Kubiznák, R. B. Mann, and Z. Sherkatghanad, Class. Quant. Grav.31, 042001 (2014), arXiv:1308.2672

    work page arXiv 2014

  34. [34]

    R. A. Hennigar, W. G. Brenna, and R. B. Mann, JHEP 2015(07), 077, arXiv:1505.05517

    work page arXiv 2015

  35. [35]

    Gunasekaran, R

    S. Gunasekaran, D. Kubiznák, and R. B. Mann, JHEP 2012(11), 110, arXiv:1208.6251

    work page arXiv 2012

  36. [36]

    S. H. Hendi and M. H. Vahidinia, Phys. Rev. D88, 084045 (2013), arXiv:1212.6128

    work page arXiv 2013

  37. [37]

    M. K. Zangeneh, A. Sheykhi, and M. H. Dehghani, Phys. Rev. D91, 044035 (2015), arXiv:1505.01861

    work page arXiv 2015

  38. [38]

    J. L. Synge, Mon. Not. R. Astron. Soc.131, 463 (1966)

    1966

  39. [39]

    Luminet, Astron

    J.-P. Luminet, Astron. Astrophys.75, 228 (1979)

    1979

  40. [40]

    J. M. Bardeen, inBlack Holes (Les Houches 1972),edited by C. DeWitt and B. S. DeWitt (Gordon and Breach, New York, 1973) pp. 215–239

    1972

  41. [41]

    Ferrari and B

    V. Ferrari and B. Mashhoon, Phys. Rev. D30, 295 (1984)

    1984

  42. [42]

    Geodesic stability, Lyapunov exponents and quasinormal modes,

    V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin, Phys. Rev. D79, 064016 (2009), arXiv:0812.1806

    work page arXiv 2009

  43. [43]

    Event Horizon Telescope Collaboration, Astrophys. J. Lett.875, L1 (2019), arXiv:1906.11238

    work page internal anchor Pith review arXiv 2019

  44. [44]

    Event Horizon Telescope Collaboration, Astrophys. J. Lett.875, L4 (2019), arXiv:1906.11241

    work page Pith review arXiv 2019

  45. [45]

    Event Horizon Telescope Collaboration, Astrophys. J. Lett.930, L12 (2022), arXiv:2311.08680

    work page internal anchor Pith review arXiv 2022

  46. [46]

    Perlick, O

    V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Phys. Rev. D92, 104031 (2015), arXiv:1507.04217

    work page arXiv 2015

  47. [47]

    R. C. Pantig and A. Övgün, Ann. Phys.436, 168722 (2022), arXiv:2111.03802

    work page arXiv 2022

  48. [48]

    Plebański,Lectures on Nonlinear Electrodynamics (Nordita, Copenhagen, 1970)

    J. Plebański,Lectures on Nonlinear Electrodynamics (Nordita, Copenhagen, 1970)

    1970

  49. [49]

    Holography, Thermodynamics and Fluctuations of Charged AdS Black Holes

    A. Chamblin, R. Emparan, H. S. Reall, and R. C. Myers, Phys. Rev. D60, 104026 (1999), arXiv:hep-th/9904197

    work page Pith review arXiv 1999