Numerical Methods for Finding Stationary Gravitational Solutions

TL;DR

This paper reviews numerical methods for solving Einstein's equations to find stationary gravitational solutions, illustrating with examples like black rings and black holes in higher dimensions and anti-de Sitter space.

Contribution

It provides a comprehensive overview of mathematical foundations and practical techniques for numerically solving gravitational boundary value problems in higher-dimensional gravity.

Findings

Resolved asymptotically flat black rings

Constructed lumpy black holes in AdS

Analyzed Gregory-Laflamme zero modes in AdS_5×S^5

Abstract

The wide applications of higher dimensional gravity and gauge/gravity duality have fuelled the search for new stationary solutions of the Einstein equation (possibly coupled to matter). In this topical review, we explain the mathematical foundations and give a practical guide for the numerical solution of gravitational boundary value problems. We present these methods by way of example: resolving asymptotically flat black rings, singly-spinning lumpy black holes in anti-de Sitter (AdS), and the Gregory-Laflamme zero modes of small rotating black holes in AdS$_5\times S^5$. We also include several tools and tricks that have been useful throughout the literature.