About Time

The concept of time is discussed in the context of the canonical formulation of the gravitational field. Using a hypersurface orthogonal foliation, the arbitrariness of the lapse function is eliminated and the shift vector vanishes, allowing a consistent definition of time. I. ON LAPSES AND SHIFTS As it well know, the space-time of Newtonian mechanics is foliated by globally defined 3-dimensional simultaneous sections which characterizes a hypersurface orthogonal propagation vector field. This is mainly a consequence of the Galilei symmetry and the absolute time may be described as foliation time, defined by the affine parameter of an integral curve of that vector field. Likewise, in special relativity a local foliation of Minkowski space-time may be introduced as a sequence of light cones, whose structure is invariant under the Poincarè group. Then the local time of special relativity may be also defined as a parameter (proportional to the arclength) of the integral curves of the propagation vector of that foliation. When we move to general relativity, the Poincarè group is replaced by the manifold mapping group and time is usually taken to be just one of the local coordinates. When this is combined with the principle of general covariance, it becomes very difficult to characterize a clear notion of time, specially when working with problems which demand an explicit time parameter, such as the canonical formulation of general relativity [1]. The purpose of this note is to reexamine concept of time in that context and to propose an appropriate foliation which may be used to define a coordinate independent notion of time. We start with a brief review of the standard Dirac-ADM canonical formulation of the gravitational field [2], [3]. Given a space-time V4, with metric components Gαβ in arbitrary coordinates, at each point it may be locally decomposed into a space-like 3-dimensional hypersurface Σ(t) with E-Mail: [email protected]