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 × S 5 .We also include several tools and tricks that have been useful throughout the literature. * Electronic address: ojcd1r13@soton.ac.uk † Electronic address: jss55@cam.ac.uk ‡ Electronic address: bw356@cam.ac.uk 1 arXiv:1510.02804v1 [hep-th] 9 Oct 2015 Contents

The gravitational dual of an insulator/superconductor transition driven by increasing the chemical potential has recently been constructed. However, the system was studied in a probe limit and only a part of the phase diagram was obtained. We include the backreaction and construct the complete phase diagram for this system. For fixed chemical potential there are typically two phase transitions as the temperature is lowered. Surprisingly, for a certain range of parameters, the system first becomes a superconductor and then becomes an insulator as the temperature approaches zero. As a byproduct of our analysis, we also construct the gravitational dual of a Bose-Einstein condensate of glueballs in a confining gauge theory.

In Lovelock theories, gravity can travel faster or slower than light. The causal structure is determined by the characteristic hypersurfaces. We generalise a recent result of Izumi to prove that any Killing horizon is a characteristic hypersurface for all gravitational degrees of freedom of a Lovelock theory. Hence gravitational signals cannot escape from the region inside such a horizon. We investigate the hyperbolicity of Lovelock theories by determining the characteristic hypersurfaces for various backgrounds. First we consider Ricci flat type N spacetimes. We show that characteristic hypersurfaces are generically all non-null and that Lovelock theories are hyperbolic in any such spacetime. Next we consider static, maximally symmetric black hole solutions of Lovelock theories. Again, characteristic surfaces are generically non-null. For some small black holes, hyperbolicity is violated near the horizon. This implies that the stability of such black holes is not a well-posed problem.

Sufficiently small Schwarzschild black holes in global AdS 5 ×S 5 are GregoryLaflamme unstable. We construct new families of black hole solutions that bifurcate from the onset of this instability and break the full SO(6) symmetry group of the S 5 down to SO(5). These new "lumpy" solutions are labelled by the harmonics . We find evidence that the = 1 branch never dominates the microcanonical/canonical ensembles and connects through a topology-changing merger to a localised black hole solution with S 8 topology. We argue that these S 8 black holes should become the dominant phase in the microcanonical ensemble for small enough energies, and that the transition to Schwarzschild black holes is first order. Furthermore, we find two branches of solutions with = 2. We expect one of these branches to connect to a solution containing two localised black holes, while the other branch connects to a black hole solution with horizon topology S 4 × S 4 which we call a "black belt".

Abstract:We numerically construct asymptotically anti-de Sitter (AdS) black holes in four dimensions that contain only a single Killing vector field. These solutions, which we coin black resonators, link the superradiant instability of Kerr-AdS to the nonlinear weakly turbulent instability of AdS by connecting the onset of the superradiance instability to smooth, horizonless geometries called geons. Furthermore, they demonstrate nonuniqueness of Kerr-AdS by sharing asymptotic charges. Where black resonators coexist with Kerr-AdS, we find that the black resonators have higher entropy. Nevertheless, we show that black resonators are unstable and comment on the implications for the endpoint of the superradiant instability.Keywords: Black Holes, AdS-CFT Correspondence, Classical Theories of Gravity JHEP12 (2015)171 Introduction. As the simplest of gravitating objects, black holes (BHs) play a fundamental role in our understanding of general relativity. Indeed, four-dimensional, asymptotically flat BHs are stable and uniquely specified by their asymptotic charges [1]. However, there are circumstances where stability and uniqueness can be violated, such as those in higher dimensions [2][3][4][5][6][7][8][9][10][11]. We will argue that this can also be accomplished in four dimensions with asymptotically anti-de Sitter (AdS) BHs. Unlike Minkowski or de Sitter space, AdS contains a timelike boundary at conformal infinity where reflecting (energy and angular momentum conserving) boundary conditions are typically imposed to render the initial value problem well posed [12]. The presence of this boundary has drastic consequences for the stability of solutions in AdS. For example, rotating BHs may contain an ergoregion from which energy can be extracted by the Penrose process [13]. For waves, this phenomenon is called superradiance [14][15][16] (see [17] for a review). In AdS, these waves return after scattering from the boundary and extract more energy. The process continues until the waves contain enough energy to backreact on the geometry, causing the so-called superradiant instability [18][19][20].The reflecting boundary also has implications for the stability of AdS itself. A nonlinear instability may occur if an excitation with arbitrarily small, but finite energy around AdS continues to reflect off the boundary and eventually forms a BH. There is numerical evidence in support of this instability with a spherically symmetric scalar field [21][22][23]. There is additionally a proposed perturbative explanation for this instability [21] which applies to pure gravity and beyond spherical symmetry [24]. At linear order in perturbation theory, AdS contains an infinite tower of evenly-spaced normal modes. At higher orders, resonances between modes cause higher modes to be excited that grow linearly in time. In the generic case, this leads to a breakdown of perturbation theory, and is interpreted as the beginnings of a nonlinear instability. This instability is called weakly turbulent due to this energy shift from longer to sho...