The effects of boundary conditions on Rindler's spectral anomaly

TL;DR

The paper analyzes how uniformly accelerated boundaries in Rindler space generate a spectral anomaly for Klein-Gordon and Maxwell fields due to an emergent $-1/x^2$ potential. By enforcing a moving Dirichlet boundary at $x_m>0$ and employing Hankel functions with imaginary index, the authors establish a self-adjoint spectral problem, obtain discrete bound states in the accelerated region, and connect these to Bogoliubov transitions for particle production. A parallel non-relativistic analysis with a moving wall shows analogous piston-like spectral compression, while a thorough Maxwell treatment across multiple field components reveals polarization-dependent behavior and a consistent reduction to KG-type equations for the transverse modes. The work culminates in a complete operator-theoretic framework, clarifying the role of Sobolev spaces, completeness relations, and overlap integrals that underpin observable Unruh-like effects in the presence of accelerated boundaries, and it highlights the practical implications for photon production and energy quantization in accelerated cavities.

Abstract

Rindler's metric is an interesting way to incorporate a set of uniformly accelerated observers into space-time coordinates; this is consistent with special and general relativity. It is known that such an acceleration gives rise to the famous Unruh effect. Interestingly, its Galilean limit already shows the appearance of quantized modes for particles in free space, given by Airy functions. This happens when a wall or boundary condition is moving in an accelerated trajectory in free space and in the presence of a field. Here we show that such a boundary, when viewed as a material obstacle in motion, gives rise to quantized modes for the Klein-Gordon and Maxwell fields, as long as the boundary does not touch the singularity at the Rindler wedge. This corresponds to a quantum-mechanical problem with an anomalous fall-to-the-origin potential $-1/x^2$ supplemented with a Dirichlet condition. We provide further mathematical analysis regarding the completeness of the solutions in terms of Hankel functions $H^{(1)}$ of imaginary index and argument, and clarify the nature of the corresponding Sobolev spaces when the boundary condition disappears for the accelerated observer. A detailed interpretation of the transition amplitudes is given in connection with particle production obtained from a Bogoliubov transformation.

Figures (9)

• Figure 1: Spacetime diagram for a Rindler observer. The red lines represent the Rindler wedge and the blue hyperbolas correspond to uniformly accelerated observers.
• Figure 2: Accelerated polarizer in the direction of the $\hat{x}$ axis with perfect reflectivity along $\hat{x}$.
• Figure 3: Probability density for the wave function, comparison between the exact result and the WKB approximation.
• Figure 4: a) Hankel functions for various values of $\omega/\alpha$ (green curves) and density plot for $|H^{(1)}_{iy}(ix)|^2$. b) Frequency quantization for the polarizer as a boundary condition.
• Figure 5: A comparison of the zeros obtained with the approximation (\ref{['aproximación de ceros']}), red dots, and the exact graph of the wave function $\psi(x)=\sqrt{x}H^{(1)}_\nu(ik_\perp x)$, blue curves, is shown.