The setting is a three-parameter family of rotating black holes derived by Kumar, Ghosh and Wang by applying the Newman–Janis algorithm to the static Kalb–Ramond solution of Lessa, Silva, Maluf and Almeida. The Kalb–Ramond field is an antisymmetric two-form that arises as a closed-string excitation in the heterotic string; when it acquires a constant nonzero vacuum expectation value, the local Lorentz group is broken spontaneously while general covariance survives. The parameters are the spin a, the Kalb–Ramond coupling γ tied to the field’s vacuum expectation value, and the exponent λ encoding the nonminimal coupling. Two limits serve as anchors: setting γ to zero recovers Kerr, while λ = 1 turns the horizon polynomial into its Kerr–Newman form with γ playing the role of a squared electric charge. In between lies 0 < λ < 1, the genuinely Kalb–Ramond interior that is the central novelty of the new work.

The first task was to map extremality itself. Extremality occurs when the horizon polynomial Δ(r) acquires a double root, and the authors derived closed-form expressions for the extremal mass and spin as functions of γ and λ. These formulas reduce correctly to the Kerr endpoint a = M and to the Kerr–Newman relation a² + γ = M². On the extremal surface the Hawking temperature vanishes identically while the horizon angular velocity and entropy stay finite. Numerically, at fixed γ the Kalb–Ramond exponent inflates the horizon faster than it raises the spin: the horizon angular velocity falls from 0.50 at the Kerr endpoint to 0.293 at (γ, λ) = (0.80, 0.70), and the entropy reaches 2.49270 in units of πM² at (γ, λ) = (0.80, 0.30), overshooting the Kerr value of 2. Two independent computational pipelines, one symbolic at thirty-digit precision and one numerical, reproduced every tabulated value to five significant figures.

Onto this geometry the authors placed a massive scalar field obeying the Klein–Gordon–Fock equation. Because the rotating Kalb–Ramond metric retains the Carter structure of Kerr, the equation separates exactly, with the radial and angular separation constants related by Λ_rad = Λ_ang + a²ω² − 2amω. At extremality the horizon polynomial and the radial function K(r) share a zero, which keeps K²/Δ regular and reduces the radial problem to Whittaker form on the Kerr and Kerr–Newman lines, where the resonance condition ½ + β − σ = −n fixes the cloud mass spectrum analytically. In the genuine Kalb–Ramond interior no such reduction exists, so the team solved the exact radial equation numerically as a two-point boundary value problem, validated against the closed forms to eleven digits.

The clouds that emerge are stationary bound states saturating the superradiant threshold, ω = mΩ_H, carrying zero net energy flux through the horizon. A genuine bound state requires μ > mΩ_H, a constraint the authors emphasize is not optional: below it the asymptotic tail becomes oscillatory and radiative rather than exponentially decaying. The spectrum behaves like a gravitational atom, with overtones accumulating at the synchronization threshold from above and their separation from the threshold shrinking monotonically. The effective height of each cloud, defined as the peak of the radial probability density with no fitted parameter, grows with the Kalb–Ramond coupling — from 2.374 at the Kerr endpoint to 8.145 at (γ, λ) = (0.60, 0.70) for the dipole fundamental. The authors also report a negative result: heavy quadrupole clouds on the Kerr and Kerr–Newman lines sit much closer to the horizon, reaching a height of 0.064, which does not violate Hod’s no-short-hair theorem because that bound’s numerical form carries no force on an extremal rotating background.

Just off extremality, with the spin reduced by a factor (1 − δ²), the clouds acquire small imaginary parts and become quasinormal modes. The study identifies them as the co-rotating zero-damped branch, ω_n = mΩ_H − iκ(n + ½ + β), governed by the same near-horizon exponent β that shaped the clouds — so clouds and ringdown are two limits of a single calculation. As δ vanishes the real part tends to mΩ_H and the damping to zero, recovering the clouds exactly. The Kalb–Ramond signature in this branch lives almost entirely in the real part: the horizon angular velocity drops by 26 percent between the Kerr endpoint and (γ, λ) = (0.60, 0.70), an order of magnitude more than the few-percent shift of the photon-sphere family, while the damping varies by only about 5 percent because the Kalb–Ramond interior lowers the temperature and raises β by compensating amounts. Notably, the derived β satisfies the known Kerr zero-damped relation of Yang and collaborators to machine precision, and the WKB barrier-top method, which fails for near-extremal co-rotating modes, was deliberately avoided.

The scattering calculation recovered a classical benchmark. For a massless scalar wave in the ℓ = m = 1 channel on Kerr at a = 0.99, direct integration of the radial equation yields a peak superradiant amplification of 0.366 percent at ωM = 0.379, consistent with the roughly 0.3 percent value established by Starobinsky and by Press and Teukolsky. In the Kalb–Ramond interior, however, the saturation frequency migrates downward with Ω_H and the peak amplification is strongly suppressed, falling to 0.034 percent at (γ, λ) = (0.60, 0.70) — a factor of eleven weaker than Kerr. The modified horizon polynomial raises the centrifugal barrier the wave must tunnel through, so the channel through which scalar clouds would deplete a black hole’s rotation is markedly less efficient, which the authors note weakens the reach of LIGO–Virgo ultralight-boson bounds on this background.

Lensing and shadow calculations supply the observational side. Using the corrected rotating photon-orbit condition rather than a commonly quoted but erroneous non-rotating expression, and working in the Bozza–Tsukamoto strong-deflection limit, the team found that near-extremal spin splits the prograde and retrograde photon orbits by a factor of three in radius, with critical impact parameters of 2.252M and −6.983M at the Kerr endpoint. The prograde logarithmic deflection coefficient is large, roughly 6.2 to 7.5 across the parameter space and sensitive to the Kalb–Ramond parameters, because the redshifted Lyapunov exponent of the near-horizon orbit grows without bound toward extremality — meaning relativistic images are, in principle, more widely separated and easier to resolve. Shadow contours constructed from Carter separation at an inclination of 60 degrees and near-extremality δ = 0.10 all fall within the 1σ Event Horizon Telescope window for M87*, whose measured angular diameter is 42 ± 3 microarcseconds, corresponding to an average shadow radius between 4.10 and 5.00 in units of the mass. At the tabulated points the current shadow data alone therefore cannot distinguish the Kalb–Ramond interior from Kerr.

The unifying message is that three independent probes — cloud existence, ringdown frequencies and shadow morphology — can be crossed in a single parameter plane. Cloud existence requires the horizon angular velocity to exceed the field’s mass-to-charge ratio, excluding the deepest part of the Kalb–Ramond interior, and intersecting that region with the telescope-allowed window bounds the pair (γ, λ) more tightly than either constraint alone. The authors flag limits honestly: the analysis is theoretical and perturbative, confined to near-extremal spins and massless or massive scalar probes, and the current Event Horizon Telescope data leave the parameter window wide open at 1σ confidence. As next-generation shadow observations of M87* and Sgr A* combine with LIGO–Virgo ringdown measurements and possible S2-orbit constraints on hypothetical clouds around Sgr A*, the corridor of allowed Kalb–Ramond parameters could close from both sides. Extensions to fully nonlinear hairy solutions and charged Proca clouds are left to future work.

Subject of Research: Stationary scalar clouds, quasinormal modes and observational constraints on near-extremal rotating Kalb–Ramond black holes

Article Title: Stationary scalar clouds, quasinormal ringing, and observational signatures of (near-)extremal rotating Kalb–Ramond black holes: from the superradiant threshold to EHT bounds

Article References: Aydemir, G., & Sakallı, İ. (2026). Stationary scalar clouds, quasinormal ringing, and observational signatures of (near-)extremal rotating Kalb–Ramond black holes: from the superradiant threshold to EHT bounds. The European Physical Journal C, 86(10), Article 1222. https://doi.org/10.1140/epjc/s10052-026-16352-1

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16352-1

Keywords: Kalb–Ramond field, black holes, scalar clouds, superradiance, quasinormal modes, greybody factors, strong-field lensing, black hole shadow, Event Horizon Telescope, Lorentz symmetry violation, string theory, near-extremal rotation