## Final Report Summary - HIDGR (Higher dimensional general relativity: explicit solutions and the classification and stability of black holes.)

General Relativity (GR) is Einstein's theory of gravity. There are several motivations for considering GR with extra spatial dimensions. For example, string theory predicts the existence of extra spatial dimensions and, at low energy, gravity in these extra dimensions is described by GR. The celebrated "gauge/gravity" correspondence of string theory relates GR with extra dimensions to "gauge theories" - non-gravitational quantum theories similar to the standard model of particle physics. The gauge theories that emerge this way are "strongly coupled", i.e. the particles interact strongly with each other. Studying strongly coupled theories directly is very difficult. The gauge/gravity correspondence shows that higher-dimensional GR can be used as a tool for calculating in such theories. For example, it is known that higher-dimensional black hole solutions of GR correspond to finite temperature states in gauge theory. Studying properties of such black holes has led to an improved understanding of some aspects of the quark/gluon plasma produced at heavy ion colliders such as LHC and RHIC.

Motivated by these applications, this project investigated properties of higher-dimensional GR and its black hole solutions. Work of the PI has shown that black holes can be qualitatively different in higher dimensions e.g. there exist "black rings": donut-shaped rotating black holes, which are impossible in three dimensions. We want to know what other kinds of black holes can exist, what are their basic properties, and what applications do they have via the gauge/gravity correspondence.

Even in three spatial dimensions, solving the equations of GR is very difficult. It is even more difficult in higher dimensions. A major part of the project involves developing new techniques for solving these equations. This ranges from new analytical ("pencil and paper") methods to solving the equations on a computer. This has led to the discovery of many new black hole solutions in higher dimensions. Some of these have been used to understand properties of strongly coupled gauge theories such as "plasma ball" configurations analogous to a ball of quark/gluon plasma, and "droplet" configurations that provide information about the properties of strongly coupled gauge theories in black hole spacetimes. Perhaps the most important new solutions constructed are "localized black holes" which describe the behaviour of strongly coupled gauge theories in finite volume at low temperature. These solutions were conjectured to exist in 1998 and we have constructed them for the first time.

A major aspect of this project concerns the stability of black holes: if a black hole is perturbed then will the perturbation decay, or will it grow and destroy the black hole? Even in three dimensions this is a difficult topic. We have developed new techniques for the investigation of black hole stability. We have demonstrated for the first time that black ring solutions are always unstable and used numerical simulations to study how this instability evolves. We found that slowly rotating black rings collapsed to form a spherical black hole. However, rapidly rotating rings suffer a "pinching" effect, and we gave strong evidence that this leads to a "singularity" where the curvature of spacetime becoming infinite. This is very exciting because it represents a violation of Penrose's "cosmic censorship" conjecture, which asserts that singularities are always hidden inside black holes. Since new physics is required to understand singularities, our result provides a possible way of making this new physics visible rather than hidden inside a black hole.

We have developed new methods for relating the geometry of space-time to the behaviour of perturbations. Specifically, some space-times have the property that gravity is so strong that it confines light rays within a finite region of space. We have argued that this is likely to lead to instability, i.e. growth of perturbations. We have applied this argument to "ultracompact" stars: stars so compact that they are almost, but not quite, black holes. We have also applied it to so-called "microstate geometries", which string theorists have conjectured to describe individual quantum states of black holes.

The extra dimensions predicted by string theory must be "compactified", curled up very small, so that we observe just three dimensions. Different ways of compactifying lead to the existence of different types of particles in three dimensions. Some models predict the existence of particles with a small, but non-zero, mass. These are interesting because it is known that rotating black holes can exhibit an instability in the presence of such fields. Team members have explored the astrophysical consequences of this possibility. They have also performed supercomputer simulations to study the evolution of this instability.

Some of the most important black hole solutions are "extreme", which means that they have the greatest possible charge or angular momentum consistent with their mass. Many observed black holes are almost extreme and extreme black holes are the only ones for which a quantum-mechanical description in string theory is understood. We have demonstrated that all extreme black hole solutions suffer from a new kind of instability in which an initial perturbation decays very slowly outside the black hole but does not decay at all on the event horizon (surface) of the hole. We have shown that an observer falling into such a black hole at late time would experience infinite forces at the event horizon. This is surprising because the "standard lore" is that nothing odd happens at the event horizon.

In higher dimensions, one can modify GR to obtain so-called Lovelock theories of gravity. These are closely related to three-dimensional theories that cosmologists have attempted to use to explain "dark energy". These theories have the interesting property that gravity can propagate faster or slower than light, unlike GR.Very little attention has been given to the mathematical consistency of such theories. We have studied this and shown that these theories suffer from a number of problems, e.g. they can form shocks analogous to those occurring in a fluid (e.g. in the air near a supersonic aircraft).

Motivated by these applications, this project investigated properties of higher-dimensional GR and its black hole solutions. Work of the PI has shown that black holes can be qualitatively different in higher dimensions e.g. there exist "black rings": donut-shaped rotating black holes, which are impossible in three dimensions. We want to know what other kinds of black holes can exist, what are their basic properties, and what applications do they have via the gauge/gravity correspondence.

Even in three spatial dimensions, solving the equations of GR is very difficult. It is even more difficult in higher dimensions. A major part of the project involves developing new techniques for solving these equations. This ranges from new analytical ("pencil and paper") methods to solving the equations on a computer. This has led to the discovery of many new black hole solutions in higher dimensions. Some of these have been used to understand properties of strongly coupled gauge theories such as "plasma ball" configurations analogous to a ball of quark/gluon plasma, and "droplet" configurations that provide information about the properties of strongly coupled gauge theories in black hole spacetimes. Perhaps the most important new solutions constructed are "localized black holes" which describe the behaviour of strongly coupled gauge theories in finite volume at low temperature. These solutions were conjectured to exist in 1998 and we have constructed them for the first time.

A major aspect of this project concerns the stability of black holes: if a black hole is perturbed then will the perturbation decay, or will it grow and destroy the black hole? Even in three dimensions this is a difficult topic. We have developed new techniques for the investigation of black hole stability. We have demonstrated for the first time that black ring solutions are always unstable and used numerical simulations to study how this instability evolves. We found that slowly rotating black rings collapsed to form a spherical black hole. However, rapidly rotating rings suffer a "pinching" effect, and we gave strong evidence that this leads to a "singularity" where the curvature of spacetime becoming infinite. This is very exciting because it represents a violation of Penrose's "cosmic censorship" conjecture, which asserts that singularities are always hidden inside black holes. Since new physics is required to understand singularities, our result provides a possible way of making this new physics visible rather than hidden inside a black hole.

We have developed new methods for relating the geometry of space-time to the behaviour of perturbations. Specifically, some space-times have the property that gravity is so strong that it confines light rays within a finite region of space. We have argued that this is likely to lead to instability, i.e. growth of perturbations. We have applied this argument to "ultracompact" stars: stars so compact that they are almost, but not quite, black holes. We have also applied it to so-called "microstate geometries", which string theorists have conjectured to describe individual quantum states of black holes.

The extra dimensions predicted by string theory must be "compactified", curled up very small, so that we observe just three dimensions. Different ways of compactifying lead to the existence of different types of particles in three dimensions. Some models predict the existence of particles with a small, but non-zero, mass. These are interesting because it is known that rotating black holes can exhibit an instability in the presence of such fields. Team members have explored the astrophysical consequences of this possibility. They have also performed supercomputer simulations to study the evolution of this instability.

Some of the most important black hole solutions are "extreme", which means that they have the greatest possible charge or angular momentum consistent with their mass. Many observed black holes are almost extreme and extreme black holes are the only ones for which a quantum-mechanical description in string theory is understood. We have demonstrated that all extreme black hole solutions suffer from a new kind of instability in which an initial perturbation decays very slowly outside the black hole but does not decay at all on the event horizon (surface) of the hole. We have shown that an observer falling into such a black hole at late time would experience infinite forces at the event horizon. This is surprising because the "standard lore" is that nothing odd happens at the event horizon.

In higher dimensions, one can modify GR to obtain so-called Lovelock theories of gravity. These are closely related to three-dimensional theories that cosmologists have attempted to use to explain "dark energy". These theories have the interesting property that gravity can propagate faster or slower than light, unlike GR.Very little attention has been given to the mathematical consistency of such theories. We have studied this and shown that these theories suffer from a number of problems, e.g. they can form shocks analogous to those occurring in a fluid (e.g. in the air near a supersonic aircraft).