The universality of the Laplace-Runge-Lenz symmetry in all rotationally symmetric systems is discussed. The independence of the symmetry on the type of interaction is proven using only the most generic properties of the Poisson brackets. General-
A Lorentz-covariant formalism for a gas which interacts randomly with radiation is developed and used to prove, in an Einstein model, that in thermal equilibrium the relativistic Boltzmann distribution for the gas and Planck's distribution for the radiation are consistent, in contrast to a recent claim by Boyer. It is shown that in a Lorentzcovariant model energy and momentum must be conserved in elementary processes (in contrast to conservation only in the average required in nonrelativistic models), and that this requirement is viblated in Boyer's model. Relativistic Fokker-Planck equations and a relativistic fluctuation-dissipation theorem are found as by-products.
Relativistic rigid motion suggests a new version for the so-called 'twin paradox', comparing the ages of two astronauts on a very long spaceship. Although there is always an instantaneous inertial frame in which the whole spaceship, being rigid, is simultaneously at rest, the twins' ages, measured as the proper-times along their individual world lines, are different when they are located at remote parts of the spaceship. The age, or proper-time, difference depends on the distance at rest between the astronauts and the rapidity difference between start to end. The relation of the age difference with the relative Doppler shift of light signals transmitted between the astronauts, and implications for the possibility to assign common age (proper-time) to complex, spatially extended, relativistic systems, are also discussed. The condition for simultaneous arrival of light signals emitted simultaneously from the opposite ends of a rigidly accelerating spaceship is resolved. ‡ PACS numbers: 03.30.+p Keywords : twin paradox, relativistic rigid motion, proper-time, relativistic age, extended relativistic systems, rapidity § It should be pointed out that trying to estimate proper-time relations just by looking at the diagrams may be misleading, because we are used to see Euclidean relations, while Minkowski space-time is pseudo-Euclidean. Figure 2, for instance, demonstrates equal proper-time intervals that look to us larger and larger with growing velocity. On the other hand, world-line intervals that seem to be of equal lengths may correspond to different proper-time lapses, the higher the speed the shorter the proper-time lapse.
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