Repeated elastic reflexions of a particle in an expanding sphere

Abstract

For the description, including a relativistic one, of the motion of a particle in an expanding sphere it is found useful to define, for a particular elastic reflexion, a rebound function as the ratio of relative normal to tangential velocity. This function remains unaltered for repeated reflexions when the squared radius of the sphere is a quadratic function of the time and then explicit formulae are obtained for the motion of the particle. In particular, the wavelength to radius proportionality, derived by Wien, for radiation and slow expansion, is shown to hold for a single photon when the sphere’s radius increases at a uniform finite rate; there is an analogous result for the momentum of a material particle, treated according to classical mechanics, when the surface area of the sphere increases uniformly. Further results, for a monatomic gas and for radiation (i. e. a photonic gas) may be interpreted thermodynamically, e. g. there are examples of finite rate isentropic processes. The motion of a single particle according to relativistic mechanics and a case when the rebound function is not constant are also considered. Fundamental reflexion relations are proved in an appendix to be relativistically invariant and applicable to the reflexion of a particle at any moving surface.