Cauchy-horizon flux coefficients in the reduced Polyakov model

Abstract

We derive the leading Cauchy-horizon flux coefficient in the stationary reduced Polyakov sector of spherically symmetric charged black holes. For a nonextremal inner horizon with affine coordinate \(V_-=-e^{-κ_-v}\), a finite late-time Eddington--Finkelstein flux \(F_-^{(\infty)}=\lim_{v\to+\infty}\langle T_{vv}\rangle\) is amplified as \(\langle T_{V_-V_-}\rangle\sim F_-^{(\infty)}/(κ_-^2V_-^2)\). In the stationary reduced Polyakov model, \(F_-^{(\infty)}=t_v-Nκ_-^2/(48π)\). Thus the leading pure \(V_-^{-2}\) Polyakov coefficient is absent precisely on the inner-horizon cancellation surface \(t_v=Nκ_-^2/(48π)\). The future event horizon determines the distinct outgoing condition \(t_u=Nκ_+^2/(48π)\), so the two horizons select different loci in the stationary \((t_u,t_v)\) state space. Standard outer prescriptions, such as the asymptotically flat Unruh prescription and the outer-horizon thermal/KMS prescription, generically lie away from the inner-horizon cancellation surface and generate nonzero inner-horizon coefficients. We then analyze the total flux hierarchy \(T_{vv}^{\rm tot}=F_0+Av^{-p}+o(v^{-p})\): cancellation of the pure quadratic coefficient is the constant-level condition \(F_0=0\), while nonzero Price-tail terms give logarithmically weakened divergences. This state-space formulation gives an exact characterization of Cauchy-horizon flux amplification in the anomaly-induced radial sector and shows that, when the total coefficient is nonzero, the corresponding radial null curvature diverges.