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:

$R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$

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 $g_{\mu\nu}$ . 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 $G_{\mu\nu}$ describes the curvature of spacetime in how it relates to the presence of mass and energy described in the stress-energy tensor $T_{\mu\nu}$ .

$G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$

The Einstein tensor itself is composed of the Ricci curvature tensor $R_{\mu\nu}$ and Ricci scalar $R$ and spacetime metric $g_{\mu\nu}$ .

$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu}$

The Ricci curvature tensor and Ricci scalar are both composed of Christoffel symbols $\Gamma^\beta_{\mu\nu}$ , which are derived from the spacetime metric:

$\Gamma^\beta_{\mu\nu} = \frac{1}{2} g^{\beta\alpha} (g_{\alpha\mu,\nu} + g_{\alpha\nu,\mu} - g_{\mu\nu,\alpha})$

The comma signifies a partial derivative: $\frac{\partial g_{\alpha\beta}}{\partial x^\mu} = g_{\alpha\beta,\mu}$

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

$\frac{d^2 x^\beta}{ds^2} + \Gamma^\beta_{\mu\nu} \frac{d x^\mu}{ds} \frac{d x^\nu}{ds}$

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

$R_{\alpha\beta} = \partial_\rho \Gamma^\rho_{\beta\alpha} - \partial_\beta \Gamma^\rho_{\rho\alpha} + \Gamma^\rho_{\rho\lambda} \Gamma^\lambda_{\beta\alpha} - \Gamma^\rho_{\beta\lambda} \Gamma^\lambda_{\rho\alpha}$

$R = g^{\alpha\beta}R_{\alpha\beta}$

Metric2Ricci

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Calculate Ricci Curvature Tensors from Spacetime Metrics

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