3+1 Formalism and Bases of Numerical Relativity

TL;DR

This paper provides a comprehensive introduction to the 3+1 formalism of general relativity, essential for numerical relativity, including detailed derivations, examples, and discussions on initial data and evolution schemes.

Contribution

It offers a self-contained, detailed exposition of the 3+1 formalism, integrating geometric, matter, and electromagnetic aspects with practical considerations for numerical implementation.

Findings

Detailed derivations of 3+1 equations

Examples illustrating the formalism

Discussion on initial data and evolution schemes

Abstract

These lecture notes provide some introduction to the 3+1 formalism of general relativity, which is the foundation of most modern numerical relativity. The text is rather self-contained, with detailed calculations and numerous examples. Contents: 1. Introduction, 2. Geometry of hypersurfaces, 3. Geometry of foliations, 4. 3+1 decomposition of Einstein equation, 5. 3+1 equations for matter and electromagnetic field, 6. Conformal decomposition, 7. Asymptotic flatness and global quantities, 8. The initial data problem, 9. Choice of foliation and spatial coordinates, 10. Evolution schemes.