A Tale of Two Actions: A Variational Principle for Two-Dimensional Causal Sets

TL;DR

This paper introduces a variational principle for two-dimensional causal sets, comparing two proposed actions and testing their adherence to a discrete Einstein equation across various spacetime embeddings.

Contribution

It develops a variational framework for 2D causal sets and evaluates two different actions against the discrete Einstein equation in multiple spacetime models.

Findings

Certain causal sets satisfy the discrete Einstein equation on average.

Comparison shows differences between manifoldlike and nonmanifoldlike causal sets.

The method helps identify causal sets consistent with continuum spacetime geometries.

Abstract

In this paper we will explore two different proposals for the action for causal sets: the Benincasa-Dowker action and a modified version of the chain action. We propose a variational principle for two-dimensional causal sets and use it for both actions to determine which causal sets at least on average satisfy a discrete version of the Einstein equation. Specifically, we test this method on causal sets embedded in 2d Minkowski, de Sitter, and anti-de Sitter spacetimes and compare these results to the most prominent nonmanifoldlike causal sets, Kleitman-Rothschild causal sets.