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 $$-1/x^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> supplemented with a Dirichlet condition. Our mathematical treatment enables explicit computations of radiation production that avoids divergences, in contrast with continuous spectra. We provide further mathematical analysis regarding the completeness of the solutions in terms of Hankel functions $$H^{(1)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> 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. Finally, our classical calculations provide the reaction force on a mirror subjected to radiation pressure, and our quantum mechanical transition coefficients allow us to compute photocurrents in an absorptive surface. Ultimately, the wave functions of this study are mathematically sound, and provide a simple concept to measure radiation reaction effects.