Kinematical conditions in the construction of spacetime

Abstract

We adopt the point of view that a solution of Einstein's equations is an evolution of given initial Cauchy data. Implementing the evolution equations necessarily requires a determination, not directly dictated by the field equations, of the kinematics of the observers in terms of which the evolution is represented. In this paper we study the observers' kinematics (velocities and accelerations) in terms of the geometry of their congruences of world lines relative to families of time slicings of spacetime, which contrasts with the more usual approach of imposing particular "gauge" or "coordinate conditions." The types of conditions we suggest are adapted to the exact Einstein equations for general strong-field, dynamic spacetimes that have to be calculated numerically. Typically, the equations are three-dimensionally covariant, elliptic, and linear in the kinematical functions (the lapse function and shift vector) that they determine. The gravitational field enters in nonlinear form through the presence of curvature in the equations. We present a flat-space model of such elliptic equations (e.g. for maximal slicing) which suggests that this curvature leads to an exponential decrease in the proper time between time slices at late times. We show how the use of maximal slicing with minimal-distortion observers generalizes the notion of a stationary rest frame to dynamical asymptotically flat spacetimes. In cosmological spacetimes the use of minimum-distortion observers is shown to differentiate between those universes which contain only kinematic time dependence (e.g. open Kasner universe) and those in which dynamical degrees of freedom are present (e.g. mixmaster universe). We examine many examples and construct new coordinate systems in both asymptotically flat and cosmological solutions to illustrate these properties.