A New Approach to the Calculation of Particle Creation from Analog Black Holes

TL;DR

The paper addresses the challenge of computing Bogoliubov coefficients for realistic moving-mirror trajectories by introducing the Inertial Replacement Method (IRM), a hybrid analytic-numeric framework that isolates the finite accelerating segment and replaces the far-field inertial regions with analytic extensions. It derives rigorous error bounds for both perfect and imperfect mirrors, demonstrates rapid convergence on benchmarks (Logex, Grav) and a fully numerical Chen-Mourou/AnaBHEL trajectory, and shows that the radiation spectrum is predominantly determined by the finite accelerating region. The results establish IRM as a reliable, broadly applicable tool for modeling analog Hawking radiation in realistic setups, enabling accurate predictions for upcoming experiments and guiding future 1+3D extensions. Overall, IRM provides a principled path to bridge analytic control and numerical evaluation in analog gravity, with clear implications for interpreting particle creation and horizon-like phenomena in laboratory systems.

Abstract

Accurate prediction of particle creation from accelerating mirrors is crucial for interpreting forthcoming analog Hawking radiation experiments such as AnaBHEL. However, realistic experimental setups render the associated Bogoliubov integrals analytically intractable. To address this challenge, we introduce the Inertial Replacement Method (IRM), a hybrid analytic-numerical framework for computing Bogoliubov coefficients for general moving-mirror trajectories. The IRM replaces the asymptotically inertial portions of a trajectory with analytic inertial extensions, so that numerical evaluation is required only for the finite accelerating segment. We derive perturbative error bounds for both perfectly and imperfectly reflecting mirrors, providing controlled accuracy estimates and guiding the choice of segmentation thresholds. The method is validated against analytically solvable trajectories and then applied to a fully numerical, PIC-based Chen-Mourou plasma-mirror trajectory relevant to the planned AnaBHEL experiment. A key physical insight emerging from this analysis is that the radiation spectrum is determined almost entirely by the finite accelerating region, with negligible sensitivity to the far-past and far-future inertial motion. These results establish the IRM as a reliable and broadly applicable computational tool for modeling particle creation in realistic analog-gravity systems such as AnaBHEL.

Figures (15)

• Figure 1: Schematic illustration of the ray-tracing construction used to obtain $v_m = p(u)$ and $u_m = f(v)$. A constant-$u$ null line intersects the mirror trajectory at coordinate $v_m$, and similarly for a constant-$v$ null line.
• Figure 2: Logex trajectory for representative values of the parameter $\kappa$.
• Figure 3: Left: Decomposition of the Logex trajectory $z(t)$ into the three IRM segments. Region I corresponds to the past asymptotically inertial portion, Region II is the accelerating segment, and Region III is the future asymptotically inertial portion. The red shaded band marks the interval where $|a(t)| > a_{\text{thres}} = 10^{-1.0}$, which defines the boundaries $t_A$ and $t_B$. The blue dashed line represents the inertial replacement trajectory $z_i(t)$, while the gray curve denotes the asymptotes $z_{\rm asym}(t)$. Their difference, $\delta z(t) = z_{\rm asym}(t) - z(t)$, quantifies how close the mirror is to its asymptotes. Right: The corresponding velocity (top) and acceleration (bottom). The Logex trajectory (red) approaches the asymptotic curve as $t \to \pm\infty$, reflecting its asymptotically null behavior. The inertial replacement (blue) matches the true trajectory at the boundaries $t_A$ and $t_B$, and becomes increasingly accurate as the threshold $a_{\text{thres}}$ is lowered. The dashed horizontal line indicates the chosen threshold acceleration.
• Figure 4: Dependence of the deviation $\Delta \left |\beta^{IRM}_{\omega \omega’} \right|^2)$ on the region I and III approximation methods for different threshold accelerations $a_{\mathrm{thres}}$ and frequencies $\omega'$. Each panel shows the logarithmic magnitude $\log_{10}\left( \Delta \left |\beta^{IRM}_{\omega \omega’} \right|^2 \right )$ evaluated over all combinations of methods $(\#^{(1)}, \#^{(3)})$ in regions I and III. The white square in each subplot marks the method pair $(\#^{(1)}, \#^{(3)})_{\mathrm{best}}$ that yields the minimal deviation.
• Figure 5: IRM deviation $\Delta|\beta^{\mathrm{IRM}}_{\omega\omega'}|^{2}$ for all $8\times 8$ method combinations in Regions I and III, plotted as functions of $\omega'$ for three values of $1/a_{\mathrm{thres}}$. Colored curves represent Region I methods, while dashed/dotted curves represent Region III methods. The gray curve in each panel marks the optimal pair $(\#^{(1)},\#^{(3)})_{\mathrm{best}}$ with minimal deviation. As $1/a_{\mathrm{thres}}$ increases, the IRM error decreases systematically.