Signature of the Simplicial Supermetric
James B. Hartle (UCSB), Warner A. Miller (LANL), Ruth M. Williams, (DAMPT)

TL;DR
This paper analyzes the signature of the Lund-Regge supermetric on simplicial three-geometries, revealing the presence of physical and gauge directions, and providing insights relevant for quantum gravity models.
Contribution
It derives the Lund-Regge metric on simplicial configuration space and analyzes its signature, including gauge freedoms and physical directions, with numerical evaluations on specific triangulations.
Findings
At least one physical timelike direction exists in simplicial configuration space.
Numerical analysis shows a consistent signature with one timelike and multiple spacelike directions.
Some negative eigenvalues are physical, not gauge modes.
Abstract
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…
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