The effects of boundary conditions on Rindler's spectral anomaly

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.