Acceleration radiation from vibrating atoms in Schwarzschild spacetime

TL;DR

This work addresses how curvature modifies acceleration radiation from a vibrating Unruh–DeWitt detector outside a Schwarzschild black hole by employing a (1+1) radial reduction and first-order perturbation theory within the Boulware vacuum. The authors derive a closed-form Floquet transition rate, showing Floquet lines at frequencies $\omega_n=\sqrt{f(R_0)}\,(n\Omega-\omega_0)$ with weights given by $J_n\left(\frac{(n\Omega-\omega_0)A}{\sqrt{f(R_0)}}\right)$ and a universal period factor $\frac{2\pi}{\Omega}$, exhibiting redshift amplification via $f(R_0)$ and recovering Minkowski results as $R_0\to\infty$. They discuss consistency checks, the regime of validity near the horizon, and outline extensions to (3+1) dimensions including greybody factors and different vacua, thereby linking flat-space vibrating-atom proposals to black-hole spacetimes. The results provide an analytically tractable bridge between flat-space acceleration radiation and curved-spacetime horizon physics, with potential extensions to more realistic detectors and vacua. Near-horizon subtleties reflect the known pathologies of the Boulware state, highlighting the need for a more complete treatment in that regime.

Abstract

Motivated by the work of Scully \textit{et al.} [ \textcolor{blue}{Proc. Nat. Acad. Sci. 115, 8131 (2018)}] and Dolan \textit{et al.}[ \textcolor{blue}{New J. Phys. 22, 033026 (2020)}], we study the acceleration radiation from a two-level Unruh-DeWitt detector that undergoes small-amplitude radial oscillations at fixed mean radius $R_0$ outside a Schwarzschild black hole. The massless scalar field is quantized in the Boulware vacuum to isolate curvature-modulated acceleration effects without a thermal Hawking background. Working in a (1+1) radial reduction and using first-order time-dependent perturbation, we evaluate the period-averaged transition rate (or the Floquet transition rate). The resulting particle emission spectrum exhibits a thermal Bose-Einstein-type profile with periodic trajectory yielding a Floquet resonance condition $nΩ> ω_0$ and a closed-form expression for the Floquet transition rate $\overline{P}_n$, which reduces to the flat Minkowski spacetime result as $R_0\to\infty$, in agreement with Near the horizon, $f(R_0)<1$ enhances the effective Bessel argument by $1/\sqrt{f(R_0)}$, providing a simple analytic demonstration of curvature/redshift amplification of acceleration radiation. In particular, the spectrum weighted by the Bessel function becomes ill-defined near the black hole horizon as $R_{0}\rightarrow 2M$, possibly manifesting the well-known pathological behavior of the Boulware vacuum state. We discuss the regime of validity (small amplitude, $R_0$ away from the horizon) and outline the extensions to (3+1) dimensions, including density-of-states and greybody factors, and to alternative vacuum choices. Our results offer an analytically tractable link between flat-space vibrating atom proposals and black-hole spacetimes.

Figures (3)

• Figure 1: Total Floquet transition rate from $\overline{P}_{\text{out}}^{(1+1)}$ (Eq. \ref{['eq:totalFR']}) vs normalized mean radius $R_{0}/M$ for two-level vibrating detector in Schwarzschild spacetime. The prefactor $\propto f^{-1/2}$ due to gravitational redshift and the Bessel argument $(n\Omega-\omega_{0})A\,f^{-1/2}$ lead to enhanced particle emission and smooth approach to the Minkowski limit as $R_{0}\rightarrow \infty$. We sum over integer harmonics obeying $n\in [\omega_{0}/\Omega,25]$ for all plots consistent with the emission selection rule $n\Omega>\omega_{0}$.
• Figure 2: Total Floquet transition rate $\overline{P}^{(1+1)}_{\rm out}$ from Eq. \ref{['eq:totalFR']}: (a) $(R_{0},A)$ with $\omega_{0}=0.3, \Omega=0.4$ , and (b) $(R_{0},\Omega)$ with $\omega_{0}=0.3, A=0.05$. Colors show the cumulative contribution $\Sigma_{n\geq n_{\text{min}}}\overline {P}_n^{(1+1)}$ with $n_{\text{min}}=\omega_{0}/\Omega+1$. The results tend to Minkowskian limit as $R_{0}\rightarrow \infty$ via $f(R_{0})=1-R_{0}/2M$. The redshift enhanced rate is clearly manifested in the near-horizon regime with $R_{0}\rightarrow 2M$.
• Figure 3: $(3+1)$-dimensional total Floquet transition rate $\overline{P}_{\text{out}}^{(3+1)}$ as a function of normalized mean distance $R_{0}/M$ and drive frequency $\Omega$ for two-level vibrating atom in Schwarzschild spacetime (in units $G=\hbar=c=1$). Plots represent radiative dynamics for four values of (a) $\omega_{0}=0.2$, (b) $\omega_{0}=0.4$, (c) $\omega_{0}=0.6$, (d) and $\omega_{0}=0.8$ . The chosen parameters include $M=1, g=1, \text{and}\ A=0.05$. Once again the range for harmonics $n\in \left[\omega_{0}/\Omega,25\right]$. For all four cases, the channel opening threshold condition $\omega_{0}/n$ pushes $\Omega$ to higher values as $\omega_{0}$ increases, leading to enhanced inert zone for transition rates. Each harmonic $n$ entails its own specific $\Omega$ threshold.