Coupling Non-Gravitational Fields with Simplicial Spacetimes

TL;DR

This paper develops a geometric, coordinate-free method to couple non-gravitational fields such as scalar, vector, and tensor fields to simplicial spacetimes in Regge Calculus, enabling consistent discretization of matter actions.

Contribution

It introduces a discrete exterior calculus framework for coupling matter fields to Regge lattice, providing a universal and consistent discretization approach for various fields.

Findings

Successfully reconstructed lattice actions for scalar, Maxwell, and Dirac fields.

Demonstrated a universal method for coupling matter to simplicial spacetimes.

Provided a geometric interpretation of matter coupling in discrete gravity.

Abstract

The inclusion of source terms in discrete gravity is a long-standing problem. Providing a consistent coupling of source to the lattice in Regge Calculus (RC) yields a robust unstructured spacetime mesh applicable to both numerical relativity and quantum gravity. RC provides a particularly insightful approach to this problem with its purely geometric representation of spacetime. The simplicial building blocks of RC enable us to represent all matter and fields in a coordinate-free manner. We provide an interpretation of RC as a discrete exterior calculus framework into which non-gravitational fields naturally couple with the simplicial lattice. Using this approach we obtain a consistent mapping of the continuum action for non-gravitational fields to the Regge lattice. In this paper we apply this framework to scalar, vector and tensor fields. In particular we reconstruct the lattice action for (1) the scalar field, (2) Maxwell field tensor and (3) Dirac particles. The straightforward application of our discretization techniques to these three fields demonstrates a universal implementation of coupling source to the lattice in Regge calculus.