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. The notion that both determinism and the conservation laws might be salvaged via photon creation is shown to be flawed.
A Beautiful Supertask
Throughout this article I will consider elastic collisions between point particles which move in one and the same unidimensional space, the axis X. As we will see, this restriction is not essential; we could have taken particles of a finite size submitted to the restriction of moving in a straight line. Let us take a reference frame in which the particle A, of mass m, moves rightward and approaches the point 0 of the axis A'at a velocity v while particle B, also of mass m, moves to the left and also approaches 0 at velocity v. Let them collide at 0. Since the collision is elastic, the total kinetic energy of the particles will be conserved and, by symmetry, A will move away to the left from 0 at velocity v and B will move away to the right from 0 also at velocity v. Let us describe this collision in a frame of reference which moves to the right at velocity v with respect to the former. In it the initial situation corresponds to a particle A at rest which is approached by B (whose mass is identical to A) at velocity 2v and moving towards the left! After colliding, B remains at rest and A starts to move towards the left at the velocity of B before the collision, that is 2v. This is a general well-known result in the classical mechanics of elastic collisions (valid even in relativistic mechanics) which we will make use of later. These preceding comments are intended to make this result more intuitively plausible. Assume a reference frame in which there is an infinite number of particles P,, each one of mass m, at rest in the points X,= j , (i e {1, 2, 3, 4, ... }) while at the point X1, a particle Po of mass m and constant velocity v is moving towards P,. After the collision, Po will remain at rest in Xt = 2 , while P, will approach and collide with particle P2 at velocity v, P2 at rest in X7 \ . After this second collision P, will remain at rest in X2 while P2 will approach P3 at velocity v, and so on. In general, for any i € {1,2 ,3 ,4 , . . . }, P, will move during a certain interval of time at velocity v from X, = jj to Xi+I y+r , but it will remain at rest in XM = yn. The space covered by P, is X-XHX = j.-^rn T+i and the time taken to fi cover it is obviously t, = -fin . This is the time that transpires between the two collisions to which P, is subjected (/ e {1, 2, 3, 4, ... }). Consequently, once Po and P, collide, all successive collisions will have taken place after the period of time given by the sum of the series
Time evolution of large classical systems
We begin with some very general and elementary remarks about nonequilibrium statistical mechanics. We then establish our notation for discussing finite systems of classical point particles, construct the microcanonical ensemble, and sketch some of the relations between statistical mechanics and ergodic theory.