Minkowski metric signature as a statistical prediction of piecewise-flat quantum gravity

Abstract

We study an ab initio calculation of the signature of the spacetime metric, within the framework of piecewise-flat models of quantum gravity. Applying numerical techniques, we evaluate the probability that any 4-simplex in a triangulation of a spacetime manifold has a given metric signature. For this we use a toy-model of quantum gravity based on quantum Regge calculus, where we ignore any effects due to dynamics and focus only on the kinematics of the configuration space. Our results may hold even for other QG models, as long as they behave similarly in the same approximation. We obtain that for a suitably chosen spacelike domain of edge lengths and the UV cutoff, the probability of observing Minkowski signature reaches approximately 85%. In contrast, allowing both spacelike and timelike edge lengths leads to the split signature being dominant, with probability between 72% and 81%. The results are essentially independent of the choice of topology and triangulation of the spacetime manifold. Our findings demonstrate that the signature of the metric does not necessarily need to be postulated in a given model of quantum gravity, but can instead be predicted from it, and we estimate the relative size of the UV cutoff and the domain for edge lengths that a given model should incorporate in order to predict the Minkowski signature with high probability. There is a potential to further increase this probability by taking into account the dynamics of the theory.