Partial Derivatives on Causal Sets

TL;DR

This paper compares two methods for estimating partial derivatives and metric components on causal sets, finding that the Moore-Penrose inverse approach is more accurate based on numerical tests, with potential improvements at higher densities.

Contribution

It introduces and compares two novel approaches for estimating derivatives on causal sets, highlighting the superior accuracy of the Moore-Penrose inverse method.

Findings

Moore-Penrose inverse approach outperforms the $ox$ operator method in accuracy.

Both methods show potential for improved accuracy at higher densities.

Numerical tests conducted on a causal diamond in 2D Minkowski space.

Abstract

We will discuss two approaches to estimating partial derivatives and the metric components; one utilizing past work describing a causal set $\Box$ operator, and one using a construction from linear algebra called the Moore-Penrose inverse. After running numerical tests on a causal diamond in $\mathbb{M}^2$, we find that the approach using the Moore-Penrose inverse is significantly more accurate. Despite the large variances in the method using the $\Box$ operator, there is reason to believe both approaches should become more accurate at higher densities.