The Relativistic Boltzmann Equation and Two Times.

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 . The observed time of Einstein and Maxwell, in the presence of interaction, is not necessarily a monotonic function of τ . If t ( τ ) increases with τ , the worldline may be associated with a normal particle, but if it is decreasing in τ , it is observed in the laboratory as an antiparticle.

Entropy Measures as Geometrical Tools in the Study of Cosmology.

Classical chaos is often characterized as exponential divergence of nearby trajectories. In many interesting cases these trajectories can be identified with geodesic curves. We define here the entropy by S = ln χ ( x ) with χ ( x ) being the distance between two nearby geodesics.