# On the discrete Christoffel symbols

## TL;DR

This paper introduces a discrete analogue of Christoffel symbols within piecewise flat spacetime, connecting it to a finite difference form of the Einstein-Hilbert action, advancing discrete gravity formulations.

## Contribution

It defines a discrete Christoffel symbol in piecewise flat spacetime and relates it to a finite difference form of the Einstein-Hilbert action, providing a new approach to discrete gravity.

## Key findings

- Discrete Christoffel symbols are constructed from edge lengths and vertex coordinates.
- The discrete action reduces to a finite difference form of the Einstein-Hilbert action.
- The approach offers a second order form of the Regge action for specific structures.

## Abstract

The piecewise flat spacetime is equipped with a set of edge lengths and vertex coordinates. This defines a piecewise affine coordinate system and a piecewise affine metric in it, the discrete analogue of the unique torsion-free metric-compatible affine connection or of the Levi-Civita connection (or of the standard expression of the Christoffel symbols in terms of metric) mentioned in the literature, and, substituting this into the affine-connection form of the Regge action of our previous work, we get a second order form of the action. This can be expanded over metric variations from simplex to simplex. For a particular periodic simplicial structure and coordinates of the vertices, the leading order over metric variations is found to coincide with a certain finite difference form of the Hilbert-Einstein action.