Gravitational Waves on Kerr Black Holes

A suite of Mathematica notebooks containing analytic expressions for the metric perturbations of a Kerr black hole.

Notebook Descriptions

MetricPerturbations.nb

This notebook provides analytic expressions for the metric perturbations of a Kerr black hole. The metric components are given in Boyer-Lindquist, ingoing, and outgoing coordinates and in two gauges: ingoing radiation gauge (IRG) and outgoing radiation gauge (ORG). These explicit expressions are saved in the directory metric-perturbations/.

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EinsteinFieldEquations.nb

This notebook:

1. Computes the linearized Einstein tensor associated with an arbitrary perturbation of the Kerr background and saves its components in the directory linearized-Einstein-tensor/. This step takes a few minutes of runtime on a laptop.
2. Imports the components of the metric perturbations from MetricPerturbations.nb and verifies that they solve the linearized Einstein vacuum equations.

This check is carried out for both radiation gauges (IRG and ORG) but only in Boyer-Lindquist coordinates. A simple coordinate transformation is then sufficient to establish correctness of the components in ingoing and outgoing coordinates.

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ExampleUsage.nb

This notebook showcases how the explicit expressions from MetricPerturbations.nb can be used for symbolic or numeric computations. As an example, we display plots of metric perturbation components.

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R. Berens, T. Gravely, and A. Lupsasca, “Gravitational Waves on Kerr Black Holes I: Reconstruction of Linearized Metric Perturbations,” 2024. arXiv:2403.20311

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