pith. sign in

arxiv: 2604.26586 · v1 · submitted 2026-04-29 · 🌀 gr-qc

Thermodynamic properties of the Kerr Black-hole in non-linear electrodynamics with cosmological constant

Vinayak S. Pawar , Siba Prasad Das This is my paper

Pith reviewed 2026-05-07 12:41 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Kerr black holenonlinear electrodynamicscosmological constantblack hole thermodynamicsmass profilehorizon functionslowly rotatingmagnetic charge
0
0 comments X

The pith

Nonlinear electrodynamics with a cosmological constant modifies the mass profile, horizon structure, and thermodynamic quantities of slowly rotating Kerr black holes.

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

The paper derives the charged density from a nonlinear electrodynamics Lagrangian and uses it to build a mass profile M(r) for a magnetically charged, slowly rotating Kerr black hole that also includes a cosmological constant. This mass profile levels off near the cosmological length scale L, creating a horizon function whose roots mark an inner Cauchy horizon, an outer event horizon, and a large cosmological horizon. The authors map the allowed regions of rotation parameter a versus mass M and then compute numerical values for temperature, horizon angular velocity, and entropy at each horizon surface. These results differ from the standard Kerr case because the nonlinear term and the cosmological constant reshape both the interior mass distribution and the surface properties. A reader would care if such changes affect predictions for black hole evaporation or the thermodynamics of charged rotating objects in an expanding universe.

Core claim

The mass profile M(r) of the magnetically charged slowly rotating Kerr black hole in nonlinear electrodynamics with cosmological constant reaches a plateau for r near the cosmological length L, where L squared equals plus or minus Lambda over 3. The horizon function Delta(r) is constructed as a quadratic polynomial in r whose zeros locate the inner, outer, and cosmological horizons; these locations depend on the rotation parameter a, the length L, and the modified mass profile. Temperature T, angular velocity Omega sub h, and entropy S are then evaluated numerically at the horizon surfaces for chosen values of magnetic charge q sub m, nonlinearity parameter beta, a, and L.

What carries the argument

The mass profile M(r) obtained by integrating the nonlinear electromagnetic charged density rho sub NLED(r), which then determines the quadratic horizon function Delta(r) used to locate the surfaces and compute the thermodynamic quantities.

If this is right

  • The mass profile M(r) reaches a plateau near the cosmological length L for any combination of magnetic charge and nonlinearity parameter.
  • Sharkfin diagrams in the a-M plane mark the allowed regions where the horizon function remains positive for different values of L.
  • The roots of Delta(r) give explicit inner, outer, and cosmological horizon radii that shift with a, L, and the modified M(r).
  • Tables of numerical values for T, Omega sub h, and S at each horizon incorporate the combined effects of nonlinear electrodynamics and the cosmological term.

Where Pith is reading between the lines

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

  • The plateau in M(r) may imply that the spacetime approaches a modified asymptotic behavior at large distances, similar to other models that include nonlinear electromagnetic terms.
  • Relaxing the slow-rotation limit while keeping the same Lagrangian could test whether the horizon modifications remain qualitatively the same at higher angular momentum.
  • If the tabulated thermodynamic values deviate from area-law expectations, they could be checked against independent calculations of surface gravity for the same metric.

Load-bearing premise

The black hole rotates slowly enough that the rotation parameter a is at most about 0.10 and the nonlinear electrodynamics Lagrangian is chosen in the specific form that produces the given charged density and mass profile.

What would settle it

Calculate the temperature at the outer horizon for fixed values of q sub m, beta, a, and L using both the paper's mass profile and the standard Kerr mass, then check whether the two temperatures differ by more than numerical precision.

Figures

Figures reproduced from arXiv: 2604.26586 by Siba Prasad Das, Vinayak S. Pawar.

Figure 1
Figure 1. Figure 1: The mass distribution of the Kerr-BH for different values of view at source ↗
Figure 2
Figure 2. Figure 2: The mass distribution of the Kerr-BH for different values of view at source ↗
Figure 3
Figure 3. Figure 3: The sharkfin diagram for spin parameter and the mass of the Kerr-BH for different view at source ↗
Figure 4
Figure 4. Figure 4: The variation of the discriminant ∆(r) term with radial distance (r) for different values of de-Sitter lengths and spin parameter of the Kerr-BH(a = 0.00 to 0.10). In Fig.4 we have shown the horizon function ∆(r) profile as a function of r for different values of rotation parameter a and dS length LdS. The horizon radii are the places where the values of ∆(r) = 0. We have understood that ∆(r) = 0 occurs wi… view at source ↗
Figure 5
Figure 5. Figure 5: The sharkfin diagram for spin parameter and the mass of the Kerr-BH for different view at source ↗
Figure 6
Figure 6. Figure 6: The variation of the discriminant ∆(r) term with radial distance (r) for different values of LAdS=1, 3, 5 and 10 and spin parameter of the Kerr-BH with a = 0.00, 0.05 and 0.10. a L r− r+ rC Tr+ TrC Ωr+ ΩrC Sr+ SrC 0.00 5 0 1.81 3.85 0.0268 0.0160 0.0 0.0 10.25 46.46 0.00 10 0 1.61 9.09 0.0301 0.0998 0.0 0.0 8.177 259.91 0.05 5 0.0016 1.80 3.85 0.0268 0.0160 0.0153 0.0039 10.24 46.47 0.05 10 0.0016 1.61 9.0… view at source ↗
read the original abstract

We study thermodynamic properties, in particular the Temperature~(T), Angular velocity~($\Omega_h$) and Entropy~(S) of the of magnetically charged slowly rotating (with rotation parameter $a \lsim 0.10$) Kerr black-hole(BH) with the inclusion of cosmological constant ($\Lambda$) in the background of nonlinear electrodynamics (NLED). At first we calculated the nonlinear electromagnetic magnetic charged density $\rho_{NLED}$$(r)$ which is needed to calculate the magnetic mass of the slowly rotating Kerr-BH. We showed the mass profile $M(r)$ of the BH for different combinations of magnetic charges~($q_m$) and non-linear parameter ($\beta$) presence in the Lagrangian density. We found that $M(r)$ attains a plateau for values of $r$ close to the the cosmological length~($L$), where $L^2$= $\pm \frac{\Lambda}{3}$, irrespective of the combinations of $q_m$ and $\beta$. The $\pm$ sign corresponds to the de-Sitter(dS) and Anti-de-Sitter(AdS) respectively. Afterwards we showed the allowed parameter spaces in $a$-$M$ plane using sharkfin diagram for different values of $L$ with positive values of the horizon function, $\Delta(r)$, and explain the extremal criterion and asymptotic limit. We showed the values of $r$ where the horizon function of the quadratic polynomials becomes zero and called them as the inner(Cauchy), outer(Event) and large cosmological horizons with different values of $a$. We showed that the horizon structure depends on $a$, $L$ and the mass profile $M(r)$. Finally, we tabulated the numerical values of three thermodynamic parameter, i.e., $T$, $\Omega_h$ and $S$ at those horizons surfaces. Our results demonstrate that NLED with cosmological constant significantly modifies both the internal structure and thermodynamic properties of Kerr-BH.

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

3 major / 2 minor

Summary. The paper studies thermodynamic properties (T, Ω_h, S) of slowly rotating (a ≲ 0.10) magnetically charged Kerr black holes in nonlinear electrodynamics (NLED) with cosmological constant Λ. It derives ρ_NLED(r) and the mass profile M(r) for various q_m and β, reports that M(r) reaches a plateau near the cosmological length L independent of q_m and β, analyzes horizon structure via sharkfin diagrams in the a-M plane for positive Δ(r), identifies inner, outer, and cosmological horizons, and tabulates numerical values of T, Ω_h, and S at those surfaces, concluding that NLED with Λ significantly modifies the internal structure and thermodynamics of Kerr BHs.

Significance. If the derivations prove rigorous and the claimed modifications are shown to exceed those from Λ alone, the work would extend black-hole thermodynamics to NLED backgrounds. However, the reported independence of the M(r) plateau from q_m and β, combined with the absence of explicit comparisons to the Maxwell limit, suggests the dominant effect is cosmological rather than from nonlinearity, which would limit the result's novelty and impact unless quantified deviations are demonstrated.

major comments (3)
  1. [Abstract] Abstract: The central claim that 'NLED with cosmological constant significantly modifies both the internal structure and thermodynamic properties' is not supported by the reported results. M(r) is stated to attain a plateau 'irrespective of the combinations of q_m and β', implying the plateau is driven by Λ rather than NLED. Explicit numerical or analytic comparison of T, Ω_h, and S to the β → ∞ (Maxwell) and q_m = 0 limits is required to establish the size of any NLED-induced deviations.
  2. [Abstract] Derivation of ρ_NLED(r) and M(r): The abstract outlines the steps to obtain ρ_NLED(r) and M(r) but supplies no explicit functional forms, integration details, or verification that the zeros of the horizon function Δ(r) follow directly without post-hoc parameter adjustments. Since the thermodynamic quantities are computed from these profiles, the absence of these derivations makes it impossible to confirm that the tabulated values are free of circularity or approximation artifacts under the a ≲ 0.10 restriction.
  3. [Results on horizons and thermodynamics] Horizon analysis and thermodynamics: The sharkfin diagrams and tabulated T, Ω_h, S values rest on the specific NLED Lagrangian and the slow-rotation assumption. Without error estimates on the a ≲ 0.10 approximation or a demonstration that the thermodynamic relations remain valid when the mass profile is inserted into the standard GR identities, the numerical results cannot be assessed as load-bearing evidence for the modification claim.
minor comments (2)
  1. [Abstract] Abstract contains a typographical error: 'the of the of magnetically charged'.
  2. [Abstract] The notation L^2 = ± Λ/3 should be clarified with the sign convention for dS versus AdS made explicit in the text.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment point by point below, providing clarifications and indicating revisions where they strengthen the presentation without altering the core results.

read point-by-point responses

  1. Referee: [Abstract] The central claim that 'NLED with cosmological constant significantly modifies both the internal structure and thermodynamic properties' is not supported by the reported results. M(r) is stated to attain a plateau 'irrespective of the combinations of q_m and β', implying the plateau is driven by Λ rather than NLED. Explicit numerical or analytic comparison of T, Ω_h, and S to the β → ∞ (Maxwell) and q_m = 0 limits is required to establish the size of any NLED-induced deviations.

    Authors: We agree that the reported independence of the asymptotic M(r) plateau from q_m and β correctly indicates that the large-r behavior is dominated by Λ. However, the NLED Lagrangian modifies the radial mass distribution at finite r, which in turn shifts the locations of the inner, outer, and cosmological horizons relative to the pure Kerr-Λ case. These shifts produce the tabulated differences in T, Ω_h, and S. To quantify the NLED contribution beyond Λ alone, we will add a new table in the revised manuscript that directly compares the thermodynamic quantities for the same a, L, and M values in three cases: the present NLED model, the Maxwell limit (β → ∞), and the uncharged limit (q_m = 0). This will make the magnitude of the deviations explicit. revision: yes

  2. Referee: [Abstract] Derivation of ρ_NLED(r) and M(r): The abstract outlines the steps to obtain ρ_NLED(r) and M(r) but supplies no explicit functional forms, integration details, or verification that the zeros of the horizon function Δ(r) follow directly without post-hoc parameter adjustments. Since the thermodynamic quantities are computed from these profiles, the absence of these derivations makes it impossible to confirm that the tabulated values are free of circularity or approximation artifacts under the a ≲ 0.10 restriction.

    Authors: The abstract is necessarily concise and therefore omits the explicit expressions. In Section 2 of the manuscript we start from the NLED Lagrangian, obtain the nonlinear magnetic charge density ρ_NLED(r) by the standard variational procedure, integrate to construct the position-dependent mass function M(r), and insert it into the horizon function Δ(r) = r² + a² - 2M(r)r - (Λ/3)r⁴. The roots are located by solving the resulting quartic equation numerically for each fixed set of parameters; no post-hoc tuning is performed. The a ≲ 0.10 restriction is an input to the slow-rotation metric ansatz, not an output of the root search. To improve accessibility we will insert the explicit functional form of ρ_NLED(r) and the integrated M(r) as a displayed equation early in the revised introduction, together with a short paragraph describing the numerical root-finding algorithm. revision: partial

  3. Referee: [Results on horizons and thermodynamics] Horizon analysis and thermodynamics: The sharkfin diagrams and tabulated T, Ω_h, S values rest on the specific NLED Lagrangian and the slow-rotation assumption. Without error estimates on the a ≲ 0.10 approximation or a demonstration that the thermodynamic relations remain valid when the mass profile is inserted into the standard GR identities, the numerical results cannot be assessed as load-bearing evidence for the modification claim.

    Authors: The thermodynamic quantities are evaluated with the standard Kerr expressions T = κ/2π, Ω_h = a/(r_h² + a²), S = π(r_h² + a²) once the horizon radii r_h are determined from Δ(r_h) = 0 with the NLED-derived M(r). These relations follow from the Killing horizon properties and the first law for stationary, axisymmetric spacetimes and remain valid when M is replaced by a radially dependent function, provided the metric retains the Kerr form (which it does by construction). The a ≲ 0.10 cutoff is chosen to keep higher-order rotation corrections below a few percent, consistent with the literature on slowly rotating black holes. In the revision we will add a dedicated paragraph citing the relevant GR identities, a brief estimate of the truncation error in the slow-rotation expansion, and a consistency check that the first law dM = T dS + Ω_h dJ holds numerically for the tabulated points. revision: yes

Circularity Check

0 steps flagged

No circularity: standard GR identities applied to independently derived M(r) from NLED Lagrangian.

full rationale

The derivation begins with a chosen NLED Lagrangian to compute ρ_NLED(r), integrates to obtain M(r) as a function of r (showing plateau near L independent of q_m, β), then applies the standard Kerr-(A)dS horizon function Δ(r) and thermodynamic relations T = κ/2π, S = A/4, Ω_h from the metric. No step defines a thermodynamic quantity in terms of itself or renames a fit as a prediction; no self-citation chain supports a uniqueness claim; the a ≲ 0.10 and specific Lagrangian are explicit assumptions, not hidden circular inputs. The central claim of modification rests on explicit numerical tabulation rather than tautological re-expression of inputs.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard Einstein equations with NLED stress-energy, the slow-rotation approximation for the metric, and the usual first-law thermodynamic relations applied to the horizons; no new entities are postulated.

free parameters (4)
  • beta
    Nonlinear parameter appearing in the NLED Lagrangian density; values are chosen to plot M(r) and horizons.
  • q_m
    Magnetic charge parameter used to generate rho_NLED(r) and M(r).
  • a
    Rotation parameter restricted to a ≲ 0.10 for the slow-rotation regime.
  • L
    Cosmological length scale set by Lambda; used to define asymptotic behavior.
axioms (2)
  • domain assumption The spacetime is described by a slowly rotating Kerr metric sourced by the NLED stress-energy tensor with cosmological constant.
    Invoked to obtain the metric functions and horizon locations.
  • domain assumption Black-hole thermodynamics (T, S, Omega_h) follows from the standard surface-gravity and area formulas applied to the horizons.
    Used to compute the tabulated thermodynamic quantities.

pith-pipeline@v0.9.0 · 5665 in / 1484 out tokens · 53741 ms · 2026-05-07T12:41:59.615448+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

52 extracted references · 52 canonical work pages

  1. [1]

    Jacobson,Lect

    T. Jacobson,Lect. Notes Phys.870, (2013) 1

    work page 2013

  2. [2]

    Curiel,Nature Astron.3, (2019) 27

    E. Curiel,Nature Astron.3, (2019) 27

    work page 2019

  3. [3]

    M. C. Miller and J. M. Miller,Phys. Rept.548, (2014) 1

    work page 2014

  4. [4]

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

    work page 1973

  5. [5]

    Jacobson, Thermodynamics of space-time: The Einstein equation of state, Phys

    T. Jacobson, Thermodynamics of space-time: The Einstein equation of state, Phys. Rev. Lett.75(1995), 1260-1263

    work page 1995

  6. [6]

    Padmanabhan, Thermodynamical Aspects of Gravity: New insights, Rept

    T. Padmanabhan, Thermodynamical Aspects of Gravity: New insights, Rept. Prog. Phys. 73(2010), 046901

    work page 2010

  7. [7]

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

    work page 1973

  8. [8]

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

    work page 1975

  9. [9]

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

    work page 1983

  10. [10]

    Chamblin, R

    A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, Phys. Rev. D60(1999), 064018

    work page 1999

  11. [11]

    Kubizˇ n´ ak and R

    D. Kubizˇ n´ ak and R. B. Mann, JHEP07(2012), 033

    work page 2012

  12. [12]

    J. M. Maldacena, Adv. Theor. Math. Phys.2(1998), 231

    work page 1998

  13. [13]

    Witten, Adv

    E. Witten, Adv. Theor. Math. Phys.2(1998), 253

    work page 1998

  14. [14]

    S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B428(1998), 105

    work page 1998

  15. [15]

    Casalderrey-Solana et al., Cambridge Univ

    J. Casalderrey-Solana et al., Cambridge Univ. Press (2014)

    work page 2014

  16. [16]

    S. A. Hartnoll, Class. Quant. Grav.26(2009), 224002

    work page 2009

  17. [17]

    Sarkar and P

    P. Sarkar and P. K. Das, New Astron.101, 102003 (2023)

    work page 2023

  18. [18]

    C. A. Escobar, R. Linares and B. Tlatelpa-Mascote, Int. J. Mod. Phys. A37, no.03, 2250011 (2022)

    work page 2022

  19. [19]

    Mignemi, Int

    S. Mignemi, Int. J. Mod. Phys. A37, no.11n12, 2250065 (2022). 14

    work page 2022

  20. [20]

    M. A. A. de Paula, L. C. S. Leite and L. C. B. Crispino, Eur. Phys. J. Plus137, no.7, 785 (2022)

    work page 2022

  21. [21]

    S. Giri, H. Nandan, L. K. Joshi and S. D. Maharaj, Mod. Phys. Lett. A36, no.31, 2150220 (2021)

    work page 2021

  22. [22]

    Garc´ ıa-Salcedo and N

    R. Garc´ ıa-Salcedo and N. Breton,Int. J. Mod. Phys. A15, (2000) 4341

    work page 2000

  23. [23]

    C. S. Camara, M. R. de Garcia Maia, J. C. Carvalho and J. A. S. Lima,Phys. Rev. D69 (2004) 123504

    work page 2004

  24. [24]

    Elizalde, J

    E. Elizalde, J. E. Lidsey, S. Nojiri and S. D. Odintsov,Phys. Lett. B574(2003) 1

    work page 2003

  25. [25]

    Novello, S

    M. Novello, S. E. Perez Bergliaffa and J. M. Salim,Phys. Rev. D69(2004) 127301

    work page 2004

  26. [26]

    Novello, E

    M. Novello, E. Goulart, J. M. Salim and S. E. Perez Bergliaffa,Class. Quant. Grav.24, (2007) 3021

    work page 2007

  27. [27]

    D. N. Vollick,Phys. Rev. D78, (2008) 063524

    work page 2008

  28. [28]

    S. I. Kruglov,Phys. Rev. D92, (2015) 123523

    work page 2015

  29. [29]

    S. I. Kruglov,Int. J. Mod. Phys. A31, (2016) 1650058

    work page 2016

  30. [30]

    S. I. Kruglov,Int. J.Mod. Phys. D25, (2016) 1640002

    work page 2016

  31. [31]

    S. I. Kruglov, Ann. Phys.441(2022)168449

    work page 2022

  32. [32]

    S. I. Kruglov, Ann. Phys. (Berlin)529(2017), 1700073

    work page 2017

  33. [33]

    S. I. Kruglov, Ann. Phys.428(2021), 168449

    work page 2021

  34. [34]

    K. A. Bronnikov,Phys. Rev. D63, (2001) 044005

    work page 2001

  35. [35]

    Born and L

    M. Born and L. Infeld,Proc. Royal Soc. (London) A144, (1934) 425

    work page 1934

  36. [36]

    Heisenberg and H

    W. Heisenberg and H. Euler,Z. Physik98, (1936) 714

    work page 1936

  37. [37]

    Schwinger,Phys

    J. Schwinger,Phys. Rev.82, (1951) 664

    work page 1951

  38. [38]

    S. L. Adler,Ann. Phys. (N.Y.)67, (1971) 599

    work page 1971

  39. [39]

    J. M. Bardeen,Proc. Int. Conf. GR5, Tbilisi(1968) 174

    work page 1968

  40. [40]

    Dymnikova,Gen

    I. Dymnikova,Gen. Rev. Grav.24, (1992) 235

    work page 1992

  41. [41]

    Ay´ on-Beato, A

    E. Ay´ on-Beato, A. Gar´ cia,Phys. Rev. Lett.80, (1998) 5056

    work page 1998

  42. [42]

    Breton,Phys

    N. Breton,Phys. Rev. D67, (2003) 124004

    work page 2003

  43. [43]

    S. A. Hayward,Phys. Rev. Lett.96, (2006) 31103

    work page 2006

  44. [44]

    J. P. S. Lemos and V. T. Zanchin,Phys. Rev. D83, (2011) 124005. 15

    work page 2011

  45. [45]

    Flachi and J

    A. Flachi and J. P.S. Lemos,Phys. Rev. D87, (2013) 024034

    work page 2013

  46. [46]

    S. H. Hendi,Ann. Phys.333, (2013) 282

    work page 2013

  47. [47]

    Balart and E

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

    work page 2014

  48. [48]

    S. I. Kruglov,Phys. Rev. D94, (2016)044026

    work page 2016

  49. [49]

    S. I. Kruglov,Europhys. Lett.115, (2016) 60006

    work page 2016

  50. [50]

    S. I. Kruglov,Ann. Phys. (Berlin)528, (2016) 588

    work page 2016

  51. [51]

    K. G. Managave, H. A. Redekar, R. B. Kumbhar, S. P. Das and K. Y. Rajpure, Int. J. Mod. Phys. A38, no.28, 2350152 (2023); H. A. Redekar, R. B. Kumbhar, S. P. Das and K. Y. Rajpure, [arXiv:2308.12639 [gr-qc]]

    work page arXiv 2023

  52. [52]

    Ravuri and T

    D. Ravuri and T. McMaken, Phys. Rev. D111, no.4, 044005 (2025). 16

    work page 2025