Numerical Methods for Finding Stationary Gravitational Solutions

TL;DR

This topical review surveys robust numerical strategies for stationary gravitational solutions, centering on the Einstein–DeTurck formulation to cast Einstein equations as well-posed elliptic boundary-value problems. It integrates linear perturbation theory, zero-mode analysis, and practical algorithms (Newton–Raphson and Ricci flow) with boundary-condition and gauge considerations, producing concrete implementations for black rings and AdS ultraspinning lumpy black holes. The work emphasizes seed selection, domain patching, and high-precision checks (first law, Smarr, DeTurck vector) to reliably map solution branches and phase structures in higher-dimensional gravity and holography. The resulting framework provides a versatile, physically informed toolkit for constructing and validating novel stationary gravitational configurations in diverse geometries, including AdS/CFT contexts.

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.

Figures (11)

• Figure 1: Graphical representation of both the exact and numerical solutions of (\ref{['eqs:polyeig']}). The red dots represent the numerical result and the solid black line the exact result (\ref{['eq:exactpoly']}).
• Figure 2: Onset of the rotating Gregory-Laflamme instability. The dotted line below $\Omega_H L = 1$ is the Hawking-Page transition and the dotted line above $\Omega_H L = 1$ is extremality. The left (right) vertical line is the onset of the $\ell = 1$ ($\ell = 2$) mode. We expect that the left side of each of these curves to be the unstable region. For $\Omega_H L = 0$, our results reproduce those in Hubeny:2002xnDias:2015pda.
• Figure 3: Warped rectangle in physical space and logic space.
• Figure 4: Patches and coordinates for black rings.
• Figure 5: Dimensionless angular momentum $j$ (in mass units) as a function of the angular frequency $\omega$ for $d=6$ (left) and $d=7$ (right). In both cases, $\omega$ reaches some maximum value, and must be decreased to continue the family of solutions.