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.Open Acces
