The Geometry of Tangent Spaces on Causal Sets

TL;DR

This paper develops a framework for defining tangent spaces, connections, and curvature on causal sets, with numerical results indicating convergence to expected flat spacetime values as density increases.

Contribution

It introduces a novel method to construct tangent spaces and geometric quantities on causal sets, extending previous estimators and demonstrating their convergence.

Findings

Numerical results approach flat spacetime values with increasing density.

Defined tangent spaces, connections, and curvature on causal sets.

Validated methods through numerical experiments.

Abstract

In this paper, we expand on previous work describing partial derivatives and metric component estimators to define tangent spaces on causal sets. Partial derivative operators are the basis vectors of the tangent space, and the metric defines the inner product. First, we use partial derivatives of the metric components to define the connection and partial derivatives of the connection to define the curvature. Numerical results show that both of these approach the expected values for a flat spacetime as density increases. Then we used the connection to define parallel transport and geodesics.