Calculate Ricci Curvature Tensors from Spacetime Metrics

Files:

The Metric2Ricci.mlx code contains the function that does the computations using the MATLAB® Symbolic Math Toolbox™.

The ExamplesOfMetrics.mlx code describes and sets up some examples of metric tensors and spacetime metrics.

Purpose:

The primary purpose of this code is to assist academics in solving Einstein's field equations:

This can be a cumbersome task if done by hand. This code allows the user to input a spacetime metric or metric tensor of their own design and output the Christoffel symbols, Ricci tensor and Ricci scalar. They can then, for example, test if the metric is a valid solution to Einstein's field equations.

Matthew E Chasco

Mathworks

Details:

Einstein’s Theory of General Relativity describes how matter and energy curve spacetime, giving rise to what we observe as gravitational forces. The curvature of spacetime can be described in a matrix known as a metric tensor . If we want evaluate whether or not a given metric is a solution to the Einstein field equations, we compute the Einstein tensor. The Einstein tensor describes the curvature of spacetime in how it relates to the presence of mass and energy described in the stress-energy tensor .

The Einstein tensor itself is composed of the Ricci curvature tensor and Ricci scalar and spacetime metric .

The Ricci curvature tensor and Ricci scalar are both composed of Christoffel symbols , which are derived from the spacetime metric:

The comma signifies a partial derivative:

The Christoffel symbols play an explicit role in the geodesic equation:

The Ricci curvature tensor and scalar can then be described in terms of the Christoffel symbols and spacetime metric: