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

Abstract

We study a massive scalar field on the rotating Kalb–Ramond black hole of Kumar et al. [7], near extremality in

. The Klein–Gordon–Fock equation separates, with

$$\Lambda _\textrm{rad}=\Lambda _\textrm{ang}+a^{2}\omega ^{2}-2am\omega $$ Λ rad = Λ ang + a 2 ω 2 - 2 a m ω

. At extremality

$$\Delta $$ Δ

and K ( r ) share a zero, keeping

$$K^{2}/\Delta $$ K 2 / Δ

regular and reducing the radial problem to Whittaker form on the Kerr line

$$\gamma =0$$ γ = 0

and the Kerr–Newman line

$$\lambda =1$$ λ = 1

. Stationary resonances saturate

$$\omega =m\,\Omega _{H}$$ ω = m Ω H

with

$$\mu _\textrm{cloud}>m\,\Omega _{H}$$ μ cloud > m Ω H

, overtones accumulating from above. In the genuine Kalb–Ramond interior

$$0<\lambda <1$$ 0 < λ < 1

no such reduction exists; we solve the exact radial equation numerically, validated against the closed forms to eleven digits. Off extremality,

$$a=a_\textrm{ext}(1-\delta ^{2})$$ a = a ext ( 1 - δ 2 )

, the clouds join the co-rotating zero-damped branch

$$\omega _{n}=m\,\Omega _{H}-i\,\kappa \,(n+\tfrac{1}{2}+\beta )$$ ω n = m Ω H - i κ ( n + 1 2 + β )

, governed by the near-horizon exponent

$$\beta $$ β

and distinct from the photon-sphere family. Greybody factors and superradiance follow from direct integration, recovering the classical

$$\ell =m=1$$ ℓ = m = 1

Kerr result. Strong-field lensing follows in the Bozza–Tsukamoto limit, the shadow from Carter separation. Intersecting the Event Horizon Telescope window of Zubair et al. [8] with our cloud-existence region bounds

$$(\gamma ,\lambda )$$ ( γ , λ )

more tightly than either alone.