3+1 Regge calculus with conserved momentum and Hamiltonian constraints

The Einstein action is evaluated for space-times whose three-metrics on a family of spacelike hypersurfaces are piecewise flat. The 3+1 action of Lund and Regge [F. Lund and T. Regge (private communication)], recently generalized by Piran and Williams [T. Piran and R. M. Williams, Phys. Rev. D 33, 1622 (1986)], is recovered in this way. A natural interpretation of the momentum constraint is obtained for simplicial initial data sets; and, by incorporating a nonzero shift vector and a nonconstant lapse, one finds a formalism in which the constraints are preserved by the time evolution. (In contrast to the continuum case, the constraints are not conserved if the lapse and shift are chosen a priori.) A consistent Hamiltonian formalism is readily obtained by the standard (Bergmann–Dirac) procedure or, alternatively, by algebraically solving the constraint equations for the lapse and shift on each three-simplex. Explicit solutions to the classical equations are found for spaces built from congruent simplices. In this special case, the action is that of a free relativistic particle moving in a curved space-time with indefinite metric and a conformal timelike Killing vector. For general space-times, if one a priori sets the shift vector to zero, the action has the form of a sum of such free-particle actions, but one for which the different particles interact by having coordinates in common.