Self-Reflection in a Moving Mirror

Abstract

We present an analytic flat-spacetime accelerating boundary analog of Hawking-type emission that possesses infinite asymptotic acceleration (and radial acceleration in the black hole analog) but finite total radiated energy (and zero surface gravity in the black hole analog). We perform a unified study of its scattering symmetry, horizon formation, asymptotically extreme acceleration, finite total radiated energy, and the distinction between local energy flux and global particle production within a single closed-form model. The particle spectrum, energy spectrum, and equivalent spacetime metric are derived, revealing an interesting mix of normal and extremal black hole properties.

Figures (4)

• Figure 1: Space-time trajectory of the mirror, Eq. (\ref{['spacetimeofrho']}), in $(z,t)$ coordinates, with unit scale, plotted parametrically as a function of the rapidity magnitude $\rho>0$ running along the curve. The horizon has been set by convention to the advanced time $v=0$, corresponding to the left-moving null line through the origin. The mirror is taken to move to the left, giving the standard appearance of any ordinary thermal black-hole trajectory. In the asymptotic past, it is at rest at the spatial origin, $z=0$, and then accelerates toward the horizon. The dashed lines are the null rays $t=\pm z$. There is no visual hint in the causal structure that the energy emission would be finite.
• Figure 2: Energy density $E(\lambda)$ distributed over the frequency rapidity $\lambda=\frac{1}{2}\ln(\omega'/\omega)$, from Eq. (\ref{['E(lambda)']}). The curve is strongly localized, and note that the maximum occurs at $\lambda_{\rm max}\approx 0.967154$ (dashed line), showing that the radiated energy is dominated by logarithmic frequency shifts of order one. Integrating the distribution over all $\lambda$ yields the total energy, $\int_{-\infty}^{\infty} E(\lambda)\,d\lambda = 1/(12\pi)$, matching the stress-tensor energy.
• Figure 3: Rapidity-space energy densities $E_{\mathrm{flux}}(\rho)$ and $E_{\mathrm{part}}(\rho)$, defined in Eqs. (\ref{['flux']}) and (\ref{['part']}), respectively. Despite these different profiles of rapidity, the area under each curve is identical; both descriptions yield the same total energy, $E=1/(12\pi)$.
• Figure 4: Lapse function of the metric, comparing with the Schwarzschild geometry. The solid curve shows the standard Schwarzschild lapse $F(r)=1-2M/r$, while the dashed curve shows the lapse function of the new spacetime written in terms of areal radius $r$. The dashed curve is the shifted lapse function $F_{\rm new}(r)=\exp\!(2\,\operatorname{Ei}^{-1}(v_0-2r))$, where $v_0$ denotes the advanced-time location of the null shell; here $v_0=4M$ has been chosen so that the horizon is aligned at $r_H=2M$, matching Schwarzschild. The bottom panel zooms in near the horizon, showing that the slope of the lapse function (and hence the surface gravity) in our metric approaches zero.