Towards a discrete induced spatial curvature in causal sets

Abstract

Causal set theory posits that, in order to derive a quantum theory of gravity, the causal structure of a spacetime is the fundamental object to be quantized. The discrete objects so obtained, known as causal sets, are equipped solely with a partial ordering, to enforce causality, and a measure, to define volumes of subsets. Spacetimes are conjectured to be causal sets under the limit of a discreteness scale at the Planck scale, and therefore, the manifold structure of a spacetime must be recoverable under this limit. In this vein, we attempt to characterise inextendible antichains, structures which form the causal set-analogues of Cauchy hypersurfaces. Taking inspiration from discrete space curvatures in the literature, we use an induced distance function defined on these structures to define a new measure of induced spatial curvature for inextendible antichains, which we have explicitly derived in the continuum limit for a 3-dimensional globally hyperbolic spacetime. We then use numerical techniques to implement this curvature, and to explore its feasibility, in the simplest system: that of flat inextendible antichains in causal sets approximated by Minkowski spacetime. This preliminary investigation into the characterisation of inextendible antichain curvature would not only add to the growing list of evidence supporting the fundamental conjecture of causal set theory, but would also provide a valuable tool to characterise causal sets, and would bring us one step closer to a causal set theoretic understanding of the initial value formulation of general relativity.