SIGNATURE OF THE SIMPLICIAL SUPERMETRIC

We investigate the signature of the Lund-Regge metric on spaces of sim plicial 3-geometries which are important in some formulations of quant um gravity. Tetrahedra can be joined together to make a three-dimensio nal piecewise linear manifold. A metric on this manifold is specified by assigning a flat metric to the interior of the tetrahedra and value s to their squared edge lengths. The subset of the space of squared ed ge lengths obeying triangle and analogous inequalities is simplicial c onfiguration space. We derive the Lund-Regge metric on simplicial conf iguration space and show how it provides the shortest distance between simplicial 3-geometries among all choices of gauge inside the simplic es for defining this metric (Regge gauge freedom). We show analyticall y that there is always at least one physical time-like direction in si mplicial configuration space and provide a lower bound on the number o f spacelike directions. We show that in the neighbourhood of points in this space corresponding to flat metrics there are space-like directi ons corresponding to gauge freedom in assigning the edge lengths. We e valuate the signature numerically for the simplicial configuration spa ces based on some simple triangulations of the 3-sphere (S-3) and 3-to rus (T-3). For the surface of a 4-simplex triangulation of S-3 we find one time-like direction and all the rest space-like over all of the s implicial configuration space. For the triangulation of T-3 around Bat space we find degeneracies in the simplicial supermetric as well as a few gauge modes corresponding to a positive eigenvalue. Moreover, we have determined that some of the negative eigenvalues are physical, i. e. the corresponding eigenvectors are not generators of diffeomorphism s. We compare our results with the known properties of continuum super space.