Vacuum-purified Hawking radiation from evaporating black holes: Lessons from moving mirrors

TL;DR

The paper analyzes Hawking-like radiation in the moving-mirror model as an analogue for black-hole evaporation and asks whether late-time vacuum fluctuations can purify emitted quanta without incurring a large energy cost. By separating information flux (partner modes) from energy flux and introducing an analytically tractable evaporating-black-hole–like trajectory, the authors demonstrate that purification can occur with negligible additional energy under certain late-time dynamics, while also clarifying the conditions under which an indirect energy cost might arise. They show that the partner modes’ localization and their energy content depend on the mirror’s trajectory, and that the purification mechanism does not require independent, energy-carrying partners; rather, purification can be tied to a global correlation structure in a Gaussian vacuum. The work highlights a fundamental decoupling between information purification and energy flow, embodied in a large late-time Bondi-frame boost, and discusses the potential relevance for real black holes, suggesting that purification without prohibitive energy costs remains a viable possibility deserving further exploration.

Abstract

This article investigates the possibility that Hawking-like quanta emitted by a moving mirror can be purified by late-time vacuum fluctuations, as proposed in Ref. [1]. Our motivation originates from recent discussions in Refs. [2,3] on whether vacuum purification necessarily entails a prohibitively large (indirect) energy cost, and our goal is to help clarify this issue. We identify the aspects of the mirror trajectory that determine the partners of Hawking quanta, as well as those that govern the energy carried to future null infinity. This allows us to highlight a fundamental disconnection within quantum field theory between the fluxes of quantum information (or purification) and energy. Throughout, we focus on quantities such as local correlation functions and energy fluxes, thereby avoiding reliance on a particle-based interpretation. Finally, we introduce an analytic mirror trajectory that produces Hawking radiation with an adiabatically varying temperature, mimicking the emission from an evaporating black hole. Our analysis identifies constraints on the mirror trajectory under which vacuum purification remains compatible with a prescribed energy budget, and we discuss the lessons that may be drawn from this model for realistic evaporating black holes.

Figures (3)

• Figure 1: Mirror trajectory beginning at $i^-$ and ending at $i^+$. The left panel depicts a mirror that initially emits thermal radiation to $\mathcal{I}^+_R$. At the point $T$, the trajectory changes and the emitted radiation ceases to be thermal. Strong correlations between widely separated points on $\mathcal{I}^+_R$ ensure that the total state of the field remains pure. The right panel shows the same mirror configuration as in the left panel, but focuses on an inertial particle arriving at $\mathcal{I}^+_R$ as a Hawking particle. The figure indicates the location of its progenitor mode on $\mathcal{I}^-_R$, as well as that of its partner mode. The location of the partner on $\mathcal{I}^-_R$ is identified by propagating the Hawking mode backwards in time and reflecting it across the asymptote of the mirror trajectory prior to point $T$. The shape of the partner mode upon reaching $\mathcal{I}^+_R$, as well as its energy content, is determined by the state of motion of the mirror in the neighborhood of the reflection event (point 2).
• Figure 2: This figure compares the exact trajectory defined by $p_{\rm evp}(u)$ (black line) with the exponential approximation $p_{\rm exp}(u) = v_\star^{(H)} - 4 M_\star \, \dot{p}_\star \, e^{-\frac{u - u_\star}{4 M_\star}}$ for two different values of $u_\star$, namely $-90$ and $-50$ (in units of $M_0$). The constants are fixed to the values $\alpha =1$, $u_0 = 0$, $p_0 = 1$, and $\dot{p}_0 = 1$. The figure shows that $p_{\rm exp}$ accurately describes the exact trajectory in an interval around $u_\star$. The dashed lines correspond to the would-be horizons associated with the two chosen values of $u_\star$. The values of such are, respectively, $v_\star^{(H)} = -430.6$ and $v_\star^{(H)} = -101.8$.
• Figure 3: Depiction of the purification of Hawking radiation by late-time vacuum fluctuations. The figure illustrates points 1, 2, and 3 described in the text above.