  We investigate the signature of the Lund-Regge metric on spaces of simplicial
three-geometries which are important in some formulations of quantum gravity.
Tetrahedra can be joined together to make a three-dimensional piecewise linear
manifold. A metric on this manifold is specified by assigning a flat metric to
the interior of the tetrahedra and values to their squared edge-lengths. The
subset of the space of squared edge-lengths obeying triangle and analogous
inequalities is simplicial configuration space. We derive the Lund-Regge metric
on simplicial configuration space and show how it provides the shortest
distance between simplicial three-geometries among all choices of gauge inside
the simplices for defining this metric (Regge gauge freedom). We show
analytically that there is always at least one physical timelike direction in
simplicial configuration space and provide a lower bound on the number of
spacelike directions. We show that in the neighborhood of points in this space
corresponding to flat metrics there are spacelike directions corresponding to
gauge freedom in assigning the edge-lengths. We evaluate the signature
numerically for the simplicial configuration spaces based on some simple
triangulations of the three-sphere (S^3) and three-torus (T^3). For the surface
of a four-simplex triangulation of S^3 we find one timelike direction and all
the rest spacelike over all of the simplicial configuration space. For the
triangulation of T^3 around flat 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
diffeomorphisms. We compare our results with the known properties of continuum
superspace.
