The Relativistic Boltzmann Equation and Two Times.

L P Horwitz

doi:10.3390/e22080804

Abstract

We discuss a covariant relativistic Boltzmann equation which describes the evolution of a system of particles in spacetime evolving with a universal invariant parameter . The observed time t of Einstein and Maxwell, in the presence of interaction, is not necessarily a monotonic function of . If increases with , the worldline may be associated with a normal particle, but if it is decreasing in , it is observed in the laboratory as an antiparticle. This paper discusses the implications for entropy evolution in this relativistic framework. It is shown that if an ensemble of particles and antiparticles, converge in a region of pair annihilation, the entropy of the antiparticle beam may decreaase in time.

Highlights

Covariant MechanicsStueckelberg, in 1941 [1], wrote that the worldline of a particle in spacetime can be thought of as generated by an event, a point in spacetime, moving according to dynamical laws and generating such a worldline
We discuss a covariant relativistic Boltzmann equation which describes the evolution of a system of particles in spacetime evolving with a universal invariant parameter τ
In the case of the flow of an ensemble of particles, such as a beam, if the particles encounter a region of interaction which can turn the flow back in time, and the entropy continues to increase in τ for the backward flowing beam, constituting a beam of antiparticles, one would observe the antiparticLe beam as decreasing in entropy, becoming less disordered as the beam approaches the annihilation region

Summary

Covariant Mechanics

Stueckelberg, in 1941 [1], wrote that the worldline of a particle in spacetime can be thought of as generated by an event, a point in spacetime, moving according to dynamical laws and generating such a worldline. In Stueckelberg’s original paper [1], he envisaged the possibility of the world line of a particle as starting as a free particle, straight and increasing in t, but under interaction, curving continuously to return in the negative direction of t From this he observed, as remarked above, that t is not single valued, and the introduction of a new parameter along the motion is necessary. In 1973, Horwitz and Piron [6] generalized the Stueckelberg theory to be applicable to many body systems by postulating that the parameter τ is universal, as for Newtonian time [2] They were able to solve the classical relativistic two-body central potential problem with a Hamiltonian of the form p1 μ p1 μ p μp μ. In this sense we can formulate the notion of probability distributions and, eventually, a Boltzmann equation [10]

Boltzmann Equation

Entropy Flow

Conclusions