Gravitation and irreversibility

The internal time operatorM is an alternative to the usual dynamic time, an independent parameter of motion. Even when the dynamical entity is the three-geometry⁽³⁾G and we are concerned with its evolution in superspace (the problem of cosmological evolution), dynamical time remains an independent parameter associated with a choice of lapse and shift functions. The quantityM is, on the contrary, an ensemble-dependent parameter related to the age of a process: the entirety of the ensemble's evolution. With this different view of time as age, we seek a geometrical counterpart toM for the⁽³⁾G as an ensemble. For a closed negatively curved universe, a Lyapounov function can be established which allows anM to be defined for the Robertson-Walker universe. The time component of superspace momentum_uv is introduced, and we identify its conjugate energyS/_uv with dissipation due to the evolving universe. A geometrical counterpart ofM is introduced by a conformal invariant . This quantity simultaneously expresses (i) the topological feature of orientation-preserving transformations, and (ii) the Hamiltonian treatment of dissipative systems. This dual character of , which links topological change to dissipative systems, suggests a geometrical basis forM. In this sense irreversibility is incorporated into the geometric structure of space-time, along with gravitation.