Riemann Normal Coordinate expansions using Cadabra

TL;DR

This paper presents a method for expanding Riemann normal coordinates of geometrical quantities to fifth order in curvature using the computer algebra system Cadabra.

Contribution

It provides explicit fifth-order expansions of the metric and related quantities in Riemann normal coordinates computed with Cadabra, enhancing computational tools for differential geometry.

Findings

Explicit fifth-order expansions of the metric and geometrical quantities.

Use of Cadabra for automated tensor calculations in differential geometry.

Facilitates higher-order geometric analysis in theoretical physics.

Abstract

Riemann normal coordinate expansions of the metric and other geometrical quantities, including the geodesic arc-length, will be presented. All of the results are given to fifth-order in the curvature and were obtained using the computer algebra package Cadabra.