The action of a casual set

Abstract

A causal set is a model for a discrete spacetime in which the “atoms of spacetime” carry a relation of ancestry. This order relation is mathematically given by a partial order, and is is taken to underly the macroscopic causal notions of before and after. The work presented in this thesis proposes a definition for the action of a causal set analogous to the continuum Einstein-Hilbert action. The path taken towards the definition of this action is somewhat indirect. We first construct a retarded wave operator on causal sets well-approximated by 4-dimensional spacetimes and prove, under certain assumptions, that this operator gives the usual continuum d’Alembertian and the scalar curvature of the approximating spacetime in the continuum limit. We use this result to define both the scalar curvature and the action of a causal set. This definition can be shown to work in any dimension, so that an explicit form of the action exists in all dimensions. We conjecture that, under certain conditions, the continuum limit of the action is given by the Einstein-Hilbert action up to boundary terms, whose explicit form we also conjecture. We provide evidence for this conjecture through analytic and numerical calculations of the expected action of various spacetime regions. The 2-dimensional action is shown to possess topological properties by calculating its expectation value for various regions of 2-dimensional spacetimes with different topologies. We find that the topological character of the 2d action breaks down for causally convex regions of the trousers spacetime that contain the singularity, and for non-causally convex rectangles. Finally, we propose a microscopic account of the entropy of causal horizons based on the action. It is a form of “spacetime mutual information” arising from the partition of spacetime by the horizon. Evidence for the proposal is provided by analytic results and numerical simulations in 2- dimensional examples. Further evidence is provided by numerical results for the Rindler and cosmic deSitter horizons in both 3 and 4-dimensions, and for a non-equilibrium horizon in a collapsing shell spacetime in 4-dimensions.