Quantum fluxes and $⟨{\hat{\Phi }}^2⟩$ for a non-minimally coupled scalar field: ringdown and tail on approaching the polar Kerr inner horizon

Abstract

We compute $⟨{\hat{\Phi }}^2{⟩}_{\text{ren}}$ as well as the energy fluxes $⟨{\hat{T}}_{uu}{⟩}_{\text{ren}}$ and $⟨{\hat{T}}_{vv}{⟩}_{\text{ren}}$ (where $u$ and $v$ are the standard Eddington-Finkelstein coordinates) associated with a quantum massless real scalar field $\hat{\Phi }$, with a general curvature coupling constant $\xi$, near the inner horizon (IH) of a Kerr black hole, along the axis of rotation. The quantum field is in the Unruh state, corresponding to an evaporating black hole. We renormalize these quantities by the state-subtraction method. We drop the assumption of minimal coupling to the curvature, thereby generalizing the results of arXiv:2203.08502 for the fluxes at the IH. This requires understanding the asymptotic behavior of $⟨{\hat{\Phi }}^2{⟩}_{\text{ren}}$ neat the IH. State subtraction allows us to push the computation of $⟨{\hat{\Phi }}^2{⟩}_{\text{ren}}$ along the axis of rotation in the Kerr interior in arXiv:2409.17464 deeper into the near-IH region, exposing their final asymptotic behavior on approaching the IH. For $⟨{\hat{\Phi }}^2{⟩}_{\text{ren}}$ (a $\xi$-independent quantity in the Kerr case), we find that the approach to its finite asymptotic IH value is given, per $\ell$-mode, by a ringdown phase (namely exponentially damped oscillations), followed by an inverse-power tail, both in the tortoise coordinate $r_{\ast }$ (which diverges at the IH). Interestingly, in the regime where the ringing dominates, the ringing's complex frequencies are (numerically) found to match twice the well-known classical quasinormal-mode frequencies in Kerr, and the inverse-power tails are found to be $r_{\ast }^{-2\ell -3}$ (resembling Price's law in the classical black hole exterior, upon replacement $t\to r_{\ast }$). [Abridged]