Geodesic instability and internal time in relativistic cosmology

The concept of "internal time" is applied to a cosmological model having spatial hypersurfaces of negative curvature. It is then possible to ascribe an irreversible evolution to the expanding universe without resorting to any "coarse graining" or "loss of information." The key observation which enables this description to be used is that geodesic flow on a four-manifold can be reduced to geodesic flow on a three-manifold when the Robertson-Walker metric is used. If the three-surface is compactified in such a way as not to change the metric, and if it has negative curvature, the geodesic system is a Bernoulli flow—a dynamical system which has the highest degree of instability. We draw various conclusions about mixing in the system pertinent to the microwave background, the observational consequences of negative curvature for objects moving with respect to the galaxies, and we show that the requirement of negative curvature always leads to a particle horizon, a conclusion which has some bearing on the physical spectrum of the internal time operator and on the possibility of removing the cosmological singularity to the infinite past.