Causal structure and light-conical singularities in Lorentzian simplicial quantum gravity
The definition of a Lorentzian gravitational path integral requires specifying which causal struc- tures are admitted off shell. Of particular importance are light-conical singularities, codimension- two irregularities of the light cone structure that render the gravitational action complex and can therefore lead to exponential enhancement or suppression of configurations. We investigate the prevalence and structure of such singularities in Lorentzian Regge gravity. We develop an efficient algorithm for randomly generating realizable Regge geometries satisfying the generalized triangle inequalities, and use it to sample the configuration space of four-dimensional triangulations. We find that light-conical singularities are generic: for bones shared by many four-simplices, configurations with irregular light cone structure overwhelmingly dominate, with trouser-type singularities gen- erally entropically favored. Restrictions on the causal character of subsimplices, such as requiring all tetrahedra to be spacelike, can substantially alter these statistics. We further show that both yarmulke- and, in four dimensions, trouser-type singularities can occur on unbounded regions of configuration space associated with simple refinement moves. Beyond light-conical singularities, we analyze the causal structure of Lorentzian triangulations more generally, for example by introducing the notion of pairwise embeddable chrono-topologies and a set of discrete causality data. Our re- sults demonstrate that causal irregularities are an intrinsic and abundant feature of the unrestricted Lorentzian Regge configuration space, sharpening the question of which causality conditions should be imposed in Regge gravity, spin-foam models, and ultimately the continuum gravitational path integral.