Particle Dynamics and Thermal Properties in Kalb-Ramond ModMax Black Holes: Theoretical Predictions for Observational Tests of Exotic Physics

Ahmad Al-Badawi; Faizuddin Ahmed; \.Izzet Sakall{\i}

REVIEW 5 major objections 5 minor 3 cited by

Particle Dynamics and Thermal Properties in Kalb-Ramond ModMax Black Holes: Theoretical Predictions for Observational Tests of Exotic Physics

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper argues that ordinary (ζ=+1) and phantom (ζ=−1) branches of the Kalb-Ramond ModMax black hole can be told apart observationally: phantom branches are hotter, thermally unstable, and bend light with electromagnetic corrections of…

desk verdict A competent parameter scan of what is effectively Reissner-Nordström with rescaled parameters; the thermal section has real sign and algebra errors, and the phantom-branch claims overstate what survives correction. read the letter →

arxiv 2508.03226 v1 pith:R7PFAGBG submitted 2025-08-05 gr-qc hep-th

Faizuddin Ahmed , Ahmad Al-Badawi , İzzet Sakall{i} This is my paper · ORCID

classification gr-qchep-th MSC 83C5783C1083C2283D05 PACS 04.70.-s04.20.-q11.30.Cp

keywords Kalb-RamondgravityLorentzsymmetrybreakingModMaxelectrodynamicsblackholethermodynamicsshadowgravitationallensingGauss-Bonnetmethodinnermoststablecircularorbit

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

This paper aims to establish that a black hole built from Kalb-Ramond (Lorentz-symmetry-breaking) gravity plus ModMax (duality-invariant nonlinear) electrodynamics carries observational signatures controlled by a discrete sign ζ. In the ordinary branch (ζ=+1) the spacetime behaves like a deformed Reissner-Nordström hole, while in the phantom branch (ζ=−1) the electric term has the opposite sign. The paper argues the two branches differ in the innermost stable circular orbit (phantom ISCOs are 5–10 times larger), in Hawking temperature (phantom is always hotter), in thermal stability (phantom has negative specific heat; ordinary shows a second-order phase transition), and most cleanly in the electromagnetic correction to the gravitational deflection angle, which has opposite signs for the two branches. If these predictions hold, lensing, shadow, or thermal observations could test Lorentz symmetry breaking and ModMax nonlinearity against ordinary general relativity.

What carries the argument

The load-bearing object is the metric function in Eq. (2), $F(r)=1/(1-\ell)-2M/r+\zeta e^{-\gamma}Q^2/((1-\ell)^2 r^2)$, together with the two continuous parameters $\ell$ (Lorentz-symmetry-breaking strength) and $\gamma$ (ModMax nonlinearity) and the discrete branch sign $\zeta$. From it the paper constructs the neutral-particle effective potential $V_{\rm eff}(r)=(1+L^2/r^2)F(r)$, the charged-particle potential $U_\pm(r)=qA_t \pm \sqrt{F(r)(1+L^2/r^2)}$, the Hawking temperature $T_H=F'(r_+)/(4\pi)$, the specific heat $C_+=dM/dT_H$, the photon-sphere condition $r_{\rm ph}F'(r_{\rm ph})-2F(r_{\rm ph})=0$, and the optical metric used in the Gauss-Bonnet theorem method (a curvature-integral way to compute light deflection instead of solving null geodesics). The sign $\zeta$ controls every qualitative difference the paper reports; $\ell$ sets the size of Lorentz-breaking corrections and $\gamma$ screens the electromagnetic contributions through $e^{-\gamma}$.

What would settle it

Measure the light-deflection angle of a charged black-hole candidate with known mass, charge, and impact parameter: Eq. (53) predicts the electromagnetic contribution to the bending angle has sign $-\zeta$, so a measured charge-dependent shift of the wrong sign, or a magnitude that cannot be fit by $e^{-\gamma}Q^2$ for any allowed $\gamma$ and $\ell$, would refute the central claim. A second decisive check is the shadow radius: Eq. (42) predicts, for example, $R_{\rm sh}\approx 9.45$ for $\ell=-0.5,\gamma=0.2,Q=0.5$ but about $5.01$ for the same parameters with $\ell=0$; a shadow measurement outside the model's parameter range would rule it out.

Watch

Extended reading notes

Core claim

The central claim is that the metric $F(r)=1/(1-\ell)-2M/r+\zeta e^{-\gamma}Q^2/((1-\ell)^2r^2)$ defines two physically distinct black-hole families. In the geodesic sector, the effective potential moves the ISCO inward for ordinary branches and outward by factors of 5–10 for phantom branches, and charged-particle motion can become chaotic. In the thermal sector, the Hawking temperature $T_H=F'(r_+)/(4\pi)$ is higher for phantom than for ordinary branches by $Q^2 e^{-\gamma}/(2\pi(1-\ell)^2 r_+^3)$; the ordinary branch has divergent specific heat at a critical horizon radius, while the phantom branch has everywhere-negative specific heat, and its free energy changes sign, indicating Hawking-Page-type transitions. In the optical sector, the photon-sphere and shadow radii shift strongly with $\ell$, and the Gauss-Bonnet deflection angle of Eq. (53) contains an electromagnetic term proportional to $-\zeta e^{-\gamma}Q^2/r_0^3$, so the two branches produce opposite-sign corrections to the bending of light.

Load-bearing premise

The thermal predictions assume the Hawking temperature is simply $F'(r_+)/(4\pi)$, even though the metric's time coordinate is not normalized to unit rate at infinity; correcting that normalization multiplies all temperatures by $\sqrt{1-\ell}$ and shifts the phase-transition locations.

Editorial extensions

If this is right

Ordinary-branch KR ModMax holes allow stable neutral orbits closer to the horizon than Schwarzschild, while phantom-branch ISCOs sit 5–10 times farther out, so the inner edge of an accretion disk could indicate which branch is realized.
Phantom branches are hotter and always thermally unstable (negative specific heat); ordinary branches show a second-order phase transition at a critical horizon radius, giving distinct temperature-mass evolution.
The deflection angle's electromagnetic term flips sign with $\zeta$, so precision lensing of a charged black-hole candidate can separate ordinary from phantom branches.
Shadow radii in this model vary by up to roughly a factor of two with $\ell$, putting the predicted differences within reach of horizon-scale interferometry.
The free-energy sign change in the phantom branch implies Hawking-Page-type transitions, connecting the model to equilibrium thermodynamics between a black hole and thermal radiation.

Reading between the lines

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

Beyond the paper's calculations: the same branch-dependent curvature that reverses the deflection correction should also split the quasi-normal-mode ringdown spectrum; a perturbation analysis would give an independent test.
Beyond the paper's calculations: the factor-of-5–10 ISCO shifts suggest high-frequency quasi-periodic oscillations in accreting compact objects as a sharper probe than shadows; the paper does not develop this channel.
Beyond the paper's calculations: because phantom branches have everywhere-negative specific heat, an astrophysical phantom black hole should evaporate or migrate rapidly once perturbed; adding an evaporation-time calculation would turn this instability into a lifetime prediction.
Beyond the paper's calculations: Eq. (53) is a weak-field expansion, so strong-lensing observables such as relativistic images remain open; extending the Gauss-Bonnet calculation there could sharpen the branch discriminant.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

What you should know: this paper runs the standard black-hole phenomenology pipeline — effective potentials, ISCO, charged-particle orbits, Hawking temperature, shadows, GBT lensing — on the Kalb-Ramond-ModMax metric of Eq. (2). The machinery is applied competently and the paper is self-contained, but the metric is Reissner-Nordström with rescaled parameters: (1−ℓ)F(r) = 1 − 2M(1−ℓ)/r + ζe^{−γ}Q²/((1−ℓ)r²). So for ζ=+1 the orbit, shadow, and lensing results are RN results with M_eff = M(1−ℓ) and Q_eff² = e^{−γ}Q²/(1−ℓ), and the "5–10×" phantom ISCO shifts are just phantom-RN with a large effective charge. The paper's claim that this is genuinely novel physics overstates the case.

What it does well: the geodesic and shadow sections are standard and, as far as I checked, correct. The photon-sphere equation (40) and the ISCO polynomial (14) follow from the stated metric, and the charged-particle effective potential (27) is set up honestly. No circularity: the algebra is self-contained from the cited metric. The self-citations are to earlier applications of the same methods; heavy, but not circular.

Now the soft spots, in order of severity. Eq. (34) has the wrong sign. Differentiating Eq. (2) and using Eq. (31) gives F'(r_+) = ((1−ℓ)r_+² − ζe^{−γ}Q²)/((1−ℓ)²r_+³), so the displayed Hawking temperature flips the ζ term. The correct derivative still gives T(ζ=−1) − T(ζ=+1) > 0, so the abstract's "phantom is hotter" claim survives the fix. The stress-test note is right about the displayed formula but wrong to say the headline is undermined. Eq. (35) is not the difference of Eq. (34), and both need to be redone together.

Eq. (38) is wrong. Computing dM/dT from (31) and the corrected (34) gives C_+ = 2πr²[(1−ℓ)r² − ζe^{−γ}Q²]/(3ζe^{−γ}Q² − (1−ℓ)r²), which reduces to −2πr² for Schwarzschild; the paper's formula gives −4πr². The qualitative branches survive (phantom always negative, ordinary divergent), but the quantitative claims do not.

Eq. (4) is inconsistent with the horizon equation (3); the extremal mass and radius are off by powers of √(1−ℓ). The deflection angle (53) also worries me: the electromagnetic correction enters at 1/r₀³ rather than the 1/b² you expect for RN-type metrics, so I would re-derive (51)–(53) before trusting the opposite-sign phantom lensing claim. Add the F(∞)=1/(1−ℓ) issue: absolute temperatures are off by √(1−ℓ), a systematic factor, so relative comparisons survive. The charged-particle chaos claim is asserted rather than demonstrated, but that is a minor point.

Bottom line: a salvageable parameter study, not a breakthrough. The geodesic and shadow tables are useful reference for this specific metric; the thermodynamic and lensing sections should not be quoted until corrected. A serious referee should see it because the errors are concrete and checkable; my own verdict is reject in current form, with the expectation of a major revision.

Referee Report

5 major / 5 minor

Summary. The paper studies geodesic motion, thermodynamics, photon spheres, shadows, and gravitational lensing of Kalb-Ramond ModMax black holes, with the stated goal of identifying observational discriminants between ordinary (ζ=+1) and phantom (ζ=-1) branches. The analysis is purely algebraic, starting from the metric function in Eq. (2), and produces numerical tables and figures for ISCO radii, horizon structures, temperatures, free energies, specific heats, and deflection angles. The authors claim that phantom branches have higher Hawking temperatures, negative specific heat, and opposite-sign electromagnetic lensing corrections compared to ordinary branches, as summarized in the abstract and Sections 4-6.

Significance. If the results were correct, the paper would provide a systematic set of theoretical predictions for Lorentz-symmetry-breaking and ModMax effects, with potentially testable shadow sizes and lensing signatures. The work is self-contained, does not fit parameters to data, and presents its derivations explicitly, which are strengths. However, the central thermal and lensing claims rest on several algebraic errors and internal contradictions, including the sign error in Eq. (34), the incorrect specific-heat formula in Eq. (38), the inconsistent extremal condition in Eq. (4), and dimensionally inconsistent terms in the lensing computation. As written, the paper does not provide reliable predictions for the advertised observational discriminants.

major comments (5)

[Section 4, Eq. (34)] The displayed Hawking temperature has the wrong sign in the electromagnetic term. Differentiating Eq. (2) and using Eq. (31) gives F'(r_+) = ((1-ℓ) r_+^2 - ζ e^{-γ} Q^2) / ((1-ℓ)^2 r_+^3), so T_H = ((1-ℓ) r_+^2 - ζ e^{-γ} Q^2) / (4π(1-ℓ)^2 r_+^3). Eq. (34) has a '+' before ζ e^{-γ} Q^2. Consequently the difference of the displayed formula gives T_H(ζ=-1) - T_H(ζ=+1) < 0, contradicting Eq. (35). The text's temperature-positivity constraint for the phantom branch, (1-ℓ) r_+^2 > Q^2 e^{-γ}, is the condition appropriate to the erroneous sign; the corrected formula is automatically positive. The abstract's phantom-higher-temperature claim is thus inconsistent with the paper's own Eq. (34), although it would follow from the corrected derivative.
[Section 4, Eqs. (37)-(38)] The free energy in Eq. (37) is consistent with the corrected temperature carrying '-ζ', not with Eq. (34), indicating an internal inconsistency in the derivation chain. The specific heat C_+ = dM/dT_H computed from Eqs. (31) and the corrected T_H is C_+ = 2π r_+^2 [(1-ℓ) r_+^2 - ζ e^{-γ} Q^2] / [3ζ e^{-γ} Q^2 - (1-ℓ) r_+^2], which does not match Eq. (38): the numerical factor and r-dependence differ, and Eq. (38) misses the ζ term in the numerator. The claims of second-order phase transitions for ordinary branches and uniformly negative specific heat for phantom branches are therefore not established by the derivation as written.
[Section 2, Eq. (4)] The extremal condition contradicts the horizon equation (3). Setting the discriminant in Eq. (3) to zero gives M = e^{-γ/2}|Q|/(1-ℓ)^{3/2} and r_ext = e^{-γ/2}|Q|/√(1-ℓ) (for ζ=+1); the paper's Eq. (4) instead gives M = e^{-γ} Q^2/((1-ℓ)^2 r_ext) with r_ext = e^{-γ/2}|Q|/(1-ℓ), which does not solve F(r)=0. For Q>0 and ζ=1, substitution yields F(r_ext) = ℓ/(1-ℓ) ≠ 0. The horizon analysis that follows, including the extremal discussion, is therefore affected.
[Section 6, Eqs. (51) and (53)] The deflection angle in Eq. (53) contains the term -π ζ e^{-γ} Q^2/((1-ℓ) r_0^3), which has dimensions of inverse length and cannot appear in a dimensionless deflection angle; the known leading electromagnetic contribution for Reissner-Nordström is of order Q^2/b^2. The optical curvature in Eq. (51) similarly mixes terms of different mass dimension (M/r^3 versus Q^2/r^5 with the displayed prefactors). The claimed opposite-sign electromagnetic corrections between ordinary and phantom branches are therefore not supported by the presented Gauss-Bonnet computation.
[Section 4, normalization of T_H] Because F(∞) = 1/(1-ℓ) ≠ 1, the Killing vector ∂_t is not unit-normalized at infinity. The standard surface-gravity normalization rescales the temperature by a factor √(1-ℓ) relative to T_H = F'(r_+)/(4π). The paper never states this normalization; the absolute temperatures, free energies, and phase-transition locations in Section 4 shift under the standard normalization. A common rescaling preserves the sign of T_H(ζ=-1) - T_H(ζ=+1) but changes the quantitative predictions that the paper presents.

minor comments (5)

[Section 3.2, Eq. (30)] Eq. (30) is not a well-formed expression; the parentheses are unbalanced and the algebraic structure is garbled, making it impossible to verify the claimed stability condition.
[References] Reference [59] is incomplete: it lists only authors with no title, journal, or year, and cannot be located by the reader.
[Figure 3] Figure 3 caption (a) lists 'ζ = 0' even though ζ is defined as ±1, and panel (o) uses ℓ=1, which makes the metric function F(r) singular; these appear to be typographical errors that should be corrected.
[Section 3.2] The paper claims chaotic trajectories for charged particles but provides no quantitative diagnostics such as Lyapunov exponents or Poincaré sections; the claim remains qualitative.
[Section 3.2, Eq. (20) vs Eq. (29)] Eq. (20) uses a sign '∓' for the electromagnetic potential, but Eq. (29) uses a single sign without connecting it to ζ; the sign convention should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: thermal/geodesic/shadow/lensing results are algebra from the cited metric, not fits or self-citation chains.

full rationale

The paper's output quantities (ISCO radii, Hawking temperature, entropy, free energy, specific heat, photon sphere, shadow radius, and deflection angle) all follow by direct differentiation and substitution from the metric function F(r) in Eq. (2), which is attributed to Ref. [59] (Araújo Filho, Heidari, and Lobo), not to the present authors. None of the quoted 'predictions' is fitted to observational data or to the target values; each is a closed-form expression in M, Q, ℓ, γ, and ζ. The frequent self-citations are to the authors' previous applications of standard methods (effective potentials, Gauss–Bonnet theorem method, etc.) and serve as bibliographic context rather than load-bearing proof. No cited uniqueness theorem is invoked to forbid alternatives, and no ansatz is smuggled in via a self-citation. The reviewer-noted sign inconsistency between Eq. (34) and the derivative of Eq. (2), as well as the asymptotic-normalization issue F(∞)=1/(1−ℓ)≠1, are internal-correctness or physical-normalization concerns; they are not circularity, because the contested claims do not reduce to their own inputs by construction. The derivation chain is self-contained algebra from a stated input metric.

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

The central claim uses no fitted parameters and introduces no new entities. It depends on the KR ModMax solution from prior work and on standard but delicate black hole temperature and GBT formulas, both of which are handled incorrectly in places.

free parameters (4)

Kalb-Ramond parameter ℓ
Dimensionless LSB parameter in Eq (2); scanned over values -0.75 to 0.75 in figures and 0 to 0.5 in tables. Not fitted; it is an input of the model.
ModMax parameter γ
Nonlinearity parameter in Eq (2); scanned over values 0 to 1. Input of the model, not fitted.
black hole charge Q
Charge in Eq (2); scanned over values 0.5 to 2. Input of the model, not fitted.
branch parameter ζ
Discrete sign ±1 in Eq (2) selecting ordinary vs phantom branch; scanned over both.

assumptions (4)

domain assumption The metric in Eq (2) is a valid black hole solution of KR gravity coupled to ModMax electrodynamics.
Taken from Ref [59]; the paper does not derive the solution and the validity of the underlying theory is assumed.
domain assumption The Hawking temperature is T_H = F'(r_+)/(4π) with the Killing time ∂_t treated as unit-normalized at infinity.
Used in Section 4 Eq (34); F(∞)=1/(1-ℓ), so a lapse normalization factor is needed; the paper does not justify this convention.
standard math Entropy satisfies the area law S = π r_+^2.
Stated in Eq (36); standard if the first law holds, but the paper does not verify the first law under the modified geometry.
standard math Gauss-Bonnet theorem applies to the optical metric with the standard straight-line approximation and background subtraction.
Used in Section 6; the calculation of the optical curvature and boundary terms contains algebraic errors.

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Cite this review

Pith. "Pith review of Particle Dynamics and Thermal Properties in Kalb-Ramond ModMax Black Holes: Theoretical Predictions for Observational Tests of Exotic Physics." pith.science (2026). https://pith.science/paper/R7PFAGBG

@misc{pith2026250803226,
  author       = {Pith},
  title        = {Pith review of: Particle Dynamics and Thermal Properties in Kalb-Ramond ModMax Black Holes: Theoretical Predictions for Observational Tests of Exotic Physics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R7PFAGBG}},
  note         = {Machine review of arXiv:2508.03226}
}

abstract

We present a comprehensive theoretical study of geodesic motion and thermodynamic behavior in Kalb--Ramond (KR) black hole (BH) spacetimes sourced by ModMax electrodynamics. Both neutral and charged test particle dynamics are investigated, highlighting how the Lorentz symmetry breaking (LSB) parameter $\ell$, the ModMax nonlinearity parameter $\gamma$, and the discrete branch parameter $\zeta$ significantly modify orbital structures compared to classical Schwarzschild and Reissner--Nordstr\"{o}m (RN) solutions. Effective potential analysis reveals notable shifts in the innermost stable circular orbit (ISCO): ordinary branches allow stable orbits closer to the horizon, while phantom branches shift them outward by factors of 5--10. For charged particles, the combined influence of modified gravity and nonlinear electromagnetic fields may induce chaotic trajectories in certain regimes. On the thermodynamic side, we derive full expressions for Hawking temperature, entropy, and Helmholtz free energy. Ordinary branches exhibit divergent specific heat indicating second-order phase transitions, whereas phantom branches yield consistently negative specific heat, implying thermal instability. Phantom BHs are found to possess higher Hawking temperatures and show distinct thermodynamic phase structures, including Hawking--Page-type transitions. Observational features such as BH shadows and gravitational lensing are explored, revealing parameter-dependent changes in photon sphere radii and deflection angles. Notably, the deflection angle analysis via the Gauss--Bonnet theorem method (GBTm) shows opposite-sign electromagnetic corrections for phantom versus ordinary branches, suggesting potential observational discriminants.

Figures

Figures reproduced from arXiv: 2508.03226 by the authors.

**Figure 1.** Plot of the metric function F(r) versus the parameters ℓ (left) and γ (right). Here, M = 1 and q = 0.5. 0 2 4 6 8 -2 -1 0 1 2 r F(r) Phantom branch ℓ -0.4 -0.2 0 0.2 0.4 0 1 2 3 -1.0 -0.5 0 0.5 1.0 r F(r) Phantom branch γ 0 0.2 0.4 0.6 0.8 1.0 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Plot of the metric function F(r) versus the parameters ℓ (left) and γ (right). Here, M = 1 and q = 0.5. Figures 1 and 2 provide comprehensive visualization of how the metric function F(r) responds to variations in the fundamental parameters ℓ and γ for both ordinary and phantom BH branches. The ordinary branch ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Embedding diagrams of the KR ModMax BH for various parameter values of ℓ, ζ, and γ. The mass and coupling parameters are set to M = 1 and Q = 1. The horizons (red rings) are governed by the horizons served in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: Behavior of the effective potential Veff given in Eq. (8) for different values of LSB parameter ℓ and ModMax parameter γ. Here, we set M = 1 = L. RN-BH, ℓ=0=γ RN-BH in KR-gravity, ℓ=0.2, γ=0 Current BH, ℓ=0.2, γ=0.2 0.5 1.0 1.5 2.0 2.5 3.0 0 2 4 6 8 10 r Veff RN-BH, ℓ=…

**Figure 5.** Figure 5: Comparison of the effective potential Veff given in Eq. (8) for different BH scenarios. Here, we set M = 1 = L. These conditions translate into a constraint on the metric function: F F′′ + 3 F F′ r − 2 F ′2 = 0, (13) leading to the polynomial equation for the ISCO radi…

**Figure 6.** Figure 6: Variation of the rISCO for KR ModMax BH with various values of the parameters ℓ (left), γ (middle) and Q (right). Here, M = 1 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: Illustration of neutral particle trajectories around an ordinary BH (ζ = 1) for different values of KR-field parameter ℓ. Here, we set M = 1, L = 1, Q = 1, and γ = 0.3. -10 -5 0 5 10 -10 -5 0 5 10 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 (d) γ = 0.2 (e) γ = 0.6 …

**Figure 8.** Figure 8: Illustration of neutral particle trajectories around an ordinary BH (ζ = 1) for different values of ModMax parameter γ. Here, we set M = 1, L = 1, Q = 1 and ℓ = 0.3. The trajectory visualizations in Figures 7, 8, and 9 reveal the intricate orbital patterns that emerge …

**Figure 9.** Figure 9: Illustration of neutral particle trajectories around an ordinary BH (ζ = 1) for combined values of ModMax parameter γ and KR-field parameter ℓ. Here, we set M = 1, L = 1, Q = 1. observational consequences for stellar orbits around supermassive BHs and could potentially…

**Figure 10.** Figure 10: The behavior of the Hawking temperature for KR ModMax BH with various values of the parameters ℓ (left) and γ (right). Here, M = 1 and Q = 0.5. 0 10 20 30 40 0 0.02 0.04 0.06 0.08 r TH Phantom branch ℓ -0.75 -0.50 -0.25 0 0.25 0.50 0.75 0 1 2 3 4 5 0 0.1 0.2 0.3 r TH …

**Figure 11.** Figure 11: The behavior of the Hawking temperature for KR ModMax BH with various values of the parameters ℓ (left) and γ (right). Here, M = 1 and Q = 0.5. the LSB parameter ℓ systematically raises the temperature across all horizon radii, reflecting how violations of Lorentz sym…

**Figure 12.** Figure 12: The behavior of The Helmholtz energy of the KR ModMax BH with various values of the parameters ℓ (left) and γ (right). Here, M = 1 and Q = 0.5. 0 10 20 30 40 0 10 20 30 r+ G Phantom branch ℓ -0.75 -0.50 -0.25 0 0.25 0.50 0.75 0 1 2 3 4 5 -2 -1 0 1 r+ G Phantom branch …

**Figure 13.** Figure 13: The behavior of the Helmholtz energy of the KR ModMax BH with various values of the parameters ℓ (left) and γ (right). Here, M = 1 and Q = 0.5. The Helmholtz free energy analysis presented in Figures 12 and 13 reveals fundamental differences in global thermodynamic st…

**Figure 14.** Figure 14: Variation of the specific heat capacity C+ as a function of horizon radius rh for KR ModMax BH with various values of the parameters ℓ (left) and γ (right). Here, M = 1 and Q = 0.5. 0 10 20 30 40 -30 000 -20 000 -10 000 0 r+ C+ Phantom branch ℓ -0.75 -0.50 -0.25 0 0.2…

**Figure 15.** Figure 15: Variation of the specific heat capacity C+ as a function of horizon radius rh for KR ModMax BH with various values of the parameters ℓ (left) and γ (right). Here, M = 1 and Q = 0.5. characteristic divergences in the heat capacity at critical horizon radii, marking sec…

**Figure 16.** Figure 16: Variation of the photon sphere rph for KR ModMax BH with various values of the parameters ℓ (left) and γ (right). Here, M = 1 and Q = 0.5 [PITH_FULL_IMAGE:figures/full_fig_p017_16.png]

**Figure 17.** Figure 17: Variation of the shadow radius Rsh for KR ModMax BH with various values of the parameters ℓ (left) and γ (right). Here, M = 1 and Q = 0.5. branches, though the magnitude of these effects differs significantly between the two configurations. This parameter dependence s…

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Forward citations

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