Towards a Definition of Locality in a Manifoldlike Causal Set

TL;DR

This paper proposes a new way to define local regions in causal sets that resemble spacetime, using order-intervals, and demonstrates its effectiveness through analytic expressions and simulations.

Contribution

It introduces a novel definition of locality in manifoldlike causal sets based on order-intervals and provides analytic and simulation evidence supporting its validity.

Findings

Analytic expressions for order-interval functions in Minkowski spacetime

A new continuum dimension estimator based on local regions

Simulations support the proposed locality definition

Abstract

It is a common misconception that spacetime discreteness necessarily implies a violation of local Lorentz invariance. In fact, in the causal set approach to quantum gravity, Lorentz invariance follows from the specific implementation of the discreteness hypothesis. However, this comes at the cost of locality. In particular, it is difficult to define a "local" region in a manifoldlike causal set, i.e., one that corresponds to an approximately flat spacetime region. Following up on suggestions from previous work, we bridge this lacuna by proposing a definition of locality based on the abundance of m-element order-intervals as a function of m in a causal set. We obtain analytic expressions for the expectation value of this function for an ensemble of causal set that faithfully embeds into an Alexandrov interval in d-dimensional Minkowski spacetime and use it to define local regions in a manifoldlike causal set. We use this to argue that evidence of local regions is a necessary condition for manifoldlikeness in a causal set. This in addition provides a new continuum dimension estimator. We perform extensive simulations which support our claims.