A Relativistic Zeno Effect

A Zenonian supertask involving an infinite number of identical colliding balls is generalized to include balls with different masses. Under the restriction that the total mass of all the balls is finite, classical mechanics leads to velocities that have no upper limit. Relativistic mechanics results in velocities bounded by that of light, but energy and momentum are not conserved, implying indeterminism. By entertaining the possibility that the missing energy and momentum are carried away by a photon, however, one can restore the conservation laws and determinism. 1 Classical Zeno process It is a commonplace that, in the mechanics of a system composed of massive, elastically colliding particles, the conservation laws of momentum and energy may be applied, either individually to each pair of particles as they collide, or to the whole system. In the former case, momentum and energy are conserved at each collision, and in the latter the total momentum (the vector sum of all the individual momenta) and the total energy (the scalar sum of all the energies) are constant in time. However, consider the infinite set of points on a straight line, xn = 2−n, where n = 0, 1, 2,..., which we shall call ‘Zeno points’. Suppose that there are equal point masses, ‘Zeno balls’, at each of these Zeno points. If all the balls are at rest except the zeroth one, which has a negative velocity, it will collide after a finite time with the first ball, transferring all its momentum to that ball (the collision is presumed to be perfectly elastic). This first ball then collides with its neighbour to the left, which will subsequently collide with its left neighbour, and so on ad infinitum until, after a finite time has elapsed, all the balls have been briefly in motion, but all have been brought to rest. Since there is no longer any motion, the momentum and kinetic energy of the original ball have been lost. Momentum and energy conservation have been violated! Such is the scenario described by Pérez Laraudogoitia (1995); and, invoking the time-reversal invariance of Newtonian mechanics, this author proclaimed the demise of classical determinism. At any time after the balls have been brought to rest, the time reversal transformation does not change the configuration, and Newton’s laws of motion yield as a viable evolution the simple possibility that al