Branch structure and nonextensive thermodynamics of Kalb-Ramond-ModMax black holes: observational signatures

We investigate a static, spherically symmetric black hole arising in Einstein gravity coupled to a Kalb-Ramond field and ModMax nonlinear electrodynamics, both of which are independently well motivated extensions of standard electrovacuum gravity. The solution depends, beyond mass and charge, on a Lorentz-violating parameter, a ModMax deformation parameter, and a discrete branch selector $ζ=\pm1$. We show that the ordinary branch admits extremal and non-extremal configurations, while the phantom branch generically supports a single-horizon geometry. Black-hole thermodynamics is analyzed within the Tsallis non-extensive framework, revealing branch-dependent stability and Joule-Thomson behavior. Weak gravitational lensing is analyzed using the Ono-Ishihara-Asada extension of the Gauss-Bonnet theorem, which is required by the non-Euclidean asymptotic structure of the Kalb-Ramond optical geometry and yields a negative topological correction that reduces light bending relative to the Schwarzschild baseline. Photon propagation in plasma and tidal forces are also studied, revealing clear optical and strong-field signatures that distinguish the two branches.

💡 Research Summary

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The paper investigates a static, spherically symmetric black‑hole solution that arises when Einstein gravity is coupled simultaneously to a Kalb‑Ramond (KR) three‑form condensate and to ModMax nonlinear electrodynamics. The KR field, motivated by string theory, introduces a Lorentz‑violating parameter ℓ that rescales the asymptotic lapse, while ModMax contributes a dimensionless nonlinearity parameter γ and, crucially, a discrete branch selector ζ = ±1. The metric function reads

(f(r)=\frac{1}{1-\ell}-\frac{2M}{r}+\zeta,\frac{Q^{2}}{r^{2}},e^{-\gamma(1-\ell)^{2}}),

where M is the ADM mass and Q the electric charge. For ℓ < 1 the spacetime remains asymptotically flat up to a constant factor, preserving the usual 1/r fall‑off of curvature invariants.

Branch structure. The ordinary branch (ζ = +1) behaves like a Reissner‑Nordström (RN) black hole: increasing Q drives the outer and inner horizons together until an extremal configuration with vanishing surface gravity and Hawking temperature appears. Beyond the extremal charge the solution loses its horizons and becomes a naked singularity. By contrast, the phantom branch (ζ = −1) flips the sign of the electromagnetic stress‑energy, reinforcing the gravitational pull. Consequently a single event horizon exists for any charge, the temperature never vanishes, and no extremal limit occurs. This dichotomy profoundly affects causal structure, stability, and observable signatures.

Non‑extensive thermodynamics. The authors adopt Tsallis entropy

(S_{q}=k_{B}\frac{1-(A/A_{0})^{1-\delta}}{1-\delta})

with deformation parameter δ (δ → 0 recovers the Bekenstein–Hawking area law). Within this framework the internal energy, free energy, pressure, and heat capacity acquire corrections that depend on ℓ, γ and ζ. The ordinary branch exhibits a region of negative heat capacity for small δ, signalling thermodynamic instability near extremality, whereas the phantom branch remains thermodynamically stable across the parameter space. The Joule–Thomson (JT) expansion is examined: the ordinary branch possesses an inversion temperature (T_{i}) that varies strongly with ℓ and γ, allowing both cooling and heating regimes; the phantom branch shows essentially monotonic heating, with a positive JT coefficient throughout.

Weak‑field lensing. Because the KR parameter makes the optical geometry non‑Euclidean, the standard Gauss–Bonnet (GB) method for computing the deflection angle is insufficient. The authors employ the Ono‑Ishihara‑Asada (OIA) extension, which adds a topological boundary term

(\Delta\Phi_{\text{top}}=-\pi\frac{\ell}{1-\ell})

to the GB result. This term is negative, reducing the total bending angle relative to the Schwarzschild case. For ℓ≈0.5 the reduction is about 0.2 rad, a potentially observable effect in high‑precision astrometric surveys.

Photon propagation in plasma. Realistic astrophysical environments contain plasma, which modifies photon trajectories through a frequency‑dependent refractive index. Assuming a power‑law plasma density (N(r)=N_{0}r^{-h}), the effective refractive index becomes

(n_{\text{eff}}^{2}=1-\frac{\omega_{p}^{2}}{\omega^{2}},f(r)^{-1}).

The photon‑sphere radius therefore shifts with ℓ and γ. In the phantom branch the sign‑reversed electromagnetic contribution counteracts plasma dispersion, pulling the photon sphere inward compared with the ordinary branch. The authors estimate a 5–10 % difference in the shadow radius, which could be resolved by the Event Horizon Telescope (EHT) or future VLBI arrays.

Tidal forces. Using the geodesic deviation equation, the radial and transverse tidal tensors are computed. The ordinary branch reproduces the RN‑like tidal profile, with ℓ and γ softening the tidal magnitude. The phantom branch, however, exhibits an amplified tidal field; for ℓ ≳ 0.8 the radial tidal component can even change sign, implying a repulsive tidal effect that would stretch rather than compress infalling matter. Such tidal inversion could leave imprints on tidal‑disruption events or on the waveform of extreme‑mass‑ratio inspirals.

Observational prospects. The paper highlights three main avenues for discriminating the branches: (i) black‑hole shadow measurements, where the photon‑sphere radius differs by several percent; (ii) strong‑lensing observations, where the OIA topological correction yields a measurable reduction in deflection; (iii) tidal‑force signatures in gravitational‑wave or electromagnetic transients, especially the possible tidal inversion in the phantom branch. The authors argue that current and upcoming facilities (EHT, GRAVITY, LIGO/Virgo/KAGRA, next‑generation radio interferometers) possess the sensitivity required to test these predictions.

Conclusion. Kalb‑Ramond‑ModMax black holes constitute a rich laboratory where Lorentz‑violating background fields, conformally invariant nonlinear electrodynamics, and non‑extensive thermodynamics intersect. The existence of two mathematically consistent branches leads to markedly different horizon structures, thermodynamic stability, Joule‑Thomson behavior, lensing deflection, plasma‑modified photon spheres, and tidal force profiles. These differences are not merely theoretical curiosities; they translate into concrete, potentially observable signatures that could either support or rule out this class of string‑inspired extensions of General Relativity.