## Periodic Reporting for period 4 - NewNGR (New frontiers in numerical general relativity)

Reporting period: 2020-03-01 to 2021-02-28

The main aim of the project is to get a better understanding of gravity as described by Einstein's theory of general relativity (GR). In the past, applications to astrophysics and cosmology motivated by the prospect of detecting gravitational waves were the main driving force in the field. In recent years, GR has led to many developments in areas of theoretical and mathematical physics beyond its traditional playground of astrophysics and cosmology. The main reason for this is the so called the AdS/CFT correspondence, also known as the gauge/string duality. This is a conjectured equivalence between certain string theories (and hence quantum gravity) in (d+1)-dimensional asymptotically anti-de Sitter space (AdS) and certain d-dimensional gauge theories, similar to real world theories of particles. In the regime in which these string theories reduce to classical GR in AdS spaces, the dual gauge theory becomes strongly interacting. Therefore, by studying classical GR in higher dimensional asymptotically AdS spaces we can access, from first principles and in a controlled manner, the strongly interacting regime of certain gauge theories. This is particularly relevant and useful to study far-from-equilibrium processes in such theories because traditional methods in quantum field theory do not work. This has found many applications outside the boundaries of gravitational physics; in particular to high energy physics, in condensed matter physics and more recently in fluid mechanics.

It turns out that GR beyond the astrophysical setting is poorly understood. For instance, black holes can have new shapes and they may suffer from dynamical instabilities whose endpoints are not known. Moreover, gravity in AdS space exhibits new phenomena, which are not present in the astrophysical context, and which often arise in the full non-linear regime of the theory. Hence numerical methods are the only tool available to study them. The main aim of this project is to get a better understanding of gravity, as described by GR, in scenarios that naturally arise in the context of the AdS/CFT correspondence, namely higher dimensional and/or AdS spaces, using numerical methods. This project has two general objectives:

1) Develop new numerical codes to study time-dependent spaces.

2) Find new types of equilibrium black holes.

It turns out that GR beyond the astrophysical setting is poorly understood. For instance, black holes can have new shapes and they may suffer from dynamical instabilities whose endpoints are not known. Moreover, gravity in AdS space exhibits new phenomena, which are not present in the astrophysical context, and which often arise in the full non-linear regime of the theory. Hence numerical methods are the only tool available to study them. The main aim of this project is to get a better understanding of gravity, as described by GR, in scenarios that naturally arise in the context of the AdS/CFT correspondence, namely higher dimensional and/or AdS spaces, using numerical methods. This project has two general objectives:

1) Develop new numerical codes to study time-dependent spaces.

2) Find new types of equilibrium black holes.

Generically, newly relevant length scales are randomly generated during the evolution of certain black hole instabilities in higher dimensions and in AdS. To accurately describe such systems numerically, we developed a code which has fully flexible adaptive mesh refinement (AMR). This allows us to resolve multiple scales in a completely automatic way, whilst keeping the cost of the computation under control. The result is GRChombo. Our code is entirely written in C++14 using hybrid MPI/OpenMP parallelism and vector intrinsics to achieve good performance on the latest architectures.

We have successfully accomplished our goals in Objective 1). First, we have managed to evolve the various instabilities that afflict certain asymptotically flat black holes in 5, 6 and 7 dimensions and found compelling evidence for their endpoints. We have also considered collisions of stable spinning black holes in 6 and 7 dimensions. In all these cases, we showed that if the initial angular momentum is large enough, a naked singularity forms via the Gregory-Laflamme instability. This result implies that Penrose’s weak cosmic censorship conjecture is violated in higher dimensions and, consequently, general relativity generically loses its predictive power. However, this loss of predictivity is minimal, only involving microscopic amounts of mass. We have also developed a general method to solve the Einstein equations in AdS as an initial boundary value problem. This method allows us to carry out large scale simulations and use AMR. With this new code, we have investigated the non-linear evolution of generic perturbations of global AdS beyond spherical symmetry. We also studied the process of black hole formation in confining backgrounds and the equilibration of large perturbations of black branes. Our results allowed us to get a better understanding of the applicability of hydrodynamics in strongly coupled gauge theories. In terms of Objective 2), we have developed a general code to construct new types of black holes in 10 and 11 dimensions and study their properties near the region where they merge with new types of black holes with new shapes. This work has allowed us to extract new predictions for the AdS/CFT correspondence and understand how the field theory “detects” the changes of shape in black holes. In addition, we have developed a new method to find black holes in theories of gravity with multiple speeds of propagation and applied it to construct rotating black holes in Einstein-Aether theory of gravity.

Our work has been published in high impact journals and our GRChombo code is publicly available on our website: http://www.grchombo.org/.

We have successfully accomplished our goals in Objective 1). First, we have managed to evolve the various instabilities that afflict certain asymptotically flat black holes in 5, 6 and 7 dimensions and found compelling evidence for their endpoints. We have also considered collisions of stable spinning black holes in 6 and 7 dimensions. In all these cases, we showed that if the initial angular momentum is large enough, a naked singularity forms via the Gregory-Laflamme instability. This result implies that Penrose’s weak cosmic censorship conjecture is violated in higher dimensions and, consequently, general relativity generically loses its predictive power. However, this loss of predictivity is minimal, only involving microscopic amounts of mass. We have also developed a general method to solve the Einstein equations in AdS as an initial boundary value problem. This method allows us to carry out large scale simulations and use AMR. With this new code, we have investigated the non-linear evolution of generic perturbations of global AdS beyond spherical symmetry. We also studied the process of black hole formation in confining backgrounds and the equilibration of large perturbations of black branes. Our results allowed us to get a better understanding of the applicability of hydrodynamics in strongly coupled gauge theories. In terms of Objective 2), we have developed a general code to construct new types of black holes in 10 and 11 dimensions and study their properties near the region where they merge with new types of black holes with new shapes. This work has allowed us to extract new predictions for the AdS/CFT correspondence and understand how the field theory “detects” the changes of shape in black holes. In addition, we have developed a new method to find black holes in theories of gravity with multiple speeds of propagation and applied it to construct rotating black holes in Einstein-Aether theory of gravity.

Our work has been published in high impact journals and our GRChombo code is publicly available on our website: http://www.grchombo.org/.

The results obtained are mostly beyond the state of the art: Previous to our work, there had been only one study of the endpoint of the Gregory-Laflamme instability of black strings in 5 dimensional Kaluza-Klein space, which is effectively a (2+1)-dimensional problem. In our work, the study of the instabilities of asymptotically flat black in various dimensions is a (3+1)-dimensional problem and hence significantly more difficult and computationally expensive. In addition, we have also studied collisions of stable spinning black holes in 6 and 7 dimensions and shown that the violations of the weak cosmic censorship conjecture are truly generic. Therefore, our work shows that this conjecture is false in higher dimensions. On the technical side, in order to do the simulations that went into this project we had to develop a new code (GRChombo) and new numerical techniques that had never been used before in numerical general relativity in order to get stable evolutions. Most notoriously, the use of diffusion inside the black hole. Similarly, our work on the endpoint of the ultraspinning instability was novel and beyond the state of the art since we could determine the details of the approach to the singularity in asymptotically flat black hole spacetimes. In this work we encountered spherical apparent horizons described by surfaces that are not starred-shaped; finding such surfaces required a completely novel approach that had not been used before. In our studies of the dynamics of gravity in AdS, we have a developed a completely new approach to solve the Einstein equations in AdS as an initial boundary value problem. This has the advantage compared to other approaches used in the community that we can run large scale simulations on supercomputers and use AMR. With our new code we have studied the non-linear instability of AdS beyond spherical symmetry for the first time. We have also studied gravitational collapse and equilibration in AdS and confining backgrounds for the first time. This has allowed us to compare various theories of relativistic hydrodynamics with the microscopic predictions provided by holography. Our construction of critical black holes and the studies of topology changes are completely new. Furthermore, the method to construct black holes in theories with multiple speeds of propagation (and hence multiple horizons) is also new, as are the rotating black holes in Einstein-Aether.