## Periodic Reporting for period 3 - NewNGR (New frontiers in numerical general relativity)

Reporting period: 2018-09-01 to 2020-02-29

"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 all these developments, black holes played and continue to play fundamental role since they are fundamental objects in the theory. Black holes are regions of spacetime in which the gravitational force is so strong that nothing can escape, not even light and hence they appear completely black to external observers. Their importance in GR comes from the fact that they are extremely simple: they are built from the two most basic components of the theory, namely space and time. Moreover, black holes have a complete mathematical description, without any approximations, within the theory itself. Yet, in spite of their simplicity, they encode fundamental aspects of GR such as the non-trivial causal structure of the space-time and the presence of singularities where the theory breaks down. In fact, much of what we know about GR comes from the studies of black holes. This should be contrasted with other gravitational objects, such as neutron stars for which, apart from gravity, one has to incorporate other physics such as nuclear reactions, plasma physics, turbulence, etc., into their description. Therefore, almost inevitably, the description of such objects can only be approximate.

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 such as Quantum Chromodynamics (QCD). 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. Thanks to the correspondence, we can access the strongly interacting regime in those gauge theories by ""simply"" solving the classical equations of GR in AdS. 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 to model turbulence.

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 know, and a complete picture is lacking. Moreover, gravity in AdS space exhibits new phenomena, which are not present in the astrophysical context. These new gravitational phenomena often arise in the full non-linear regime of the theory and hence numerical methods are the only tool available to study them. Therefore, 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. I have two general objectives:

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

2) Find new types of equilibrium black holes.

Successfully achieving these two objectives will provide a deeper understanding of gravity as described by general relativity.

"

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 such as Quantum Chromodynamics (QCD). 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. Thanks to the correspondence, we can access the strongly interacting regime in those gauge theories by ""simply"" solving the classical equations of GR in AdS. 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 to model turbulence.

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 know, and a complete picture is lacking. Moreover, gravity in AdS space exhibits new phenomena, which are not present in the astrophysical context. These new gravitational phenomena often arise in the full non-linear regime of the theory and hence numerical methods are the only tool available to study them. Therefore, 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. I have two general objectives:

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

2) Find new types of equilibrium black holes.

Successfully achieving these two objectives will provide a deeper understanding of gravity as described by general relativity.

"

Initially, most of the work was dedicated to setup the group and to develop the code that would later be used to achieve the main scientific goals in the Description of Action (DoA).

In many of the systems of interest, newly relevant length scales are randomly generated during the evolution of certain instabilities. To accurately describe such systems numerically, we developed a code which has fully flexible adaptive mesh refinement. This allow 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, and is publicly available on our website: http://www.grchombo.org/.

Most of the research has been focused on Project 2 of my Objective 1 in the DoA. We have successfully evolved the various instabilities that afflict asymptotically flat 5-dimensional black rings and spherical black holes in 6 dimensions and found compelling evidence for the endpoints of such instabilities. Thin rings had been found to be unstable under a Gregory-Laflamme (GL) instability, which suggested that the weak cosmic censorship conjecture (WCC) could be violated in asymptotically flat spaces, similar to what famously happens to unstable black strings in 5 dimensions. By evolving these black ring instabilities, we discovered that not so thin rings are dominated by a new type of “elastic” instability, which results in collapse into a spherical black hole and no violation of the WCC. On other hand, for thin enough rings, the evolution is dominated by the GL instability and the endpoint is a naked singularity. This constitutes the first counter-example of the WCC in higher dimensional asymptotically flat spacetimes.

The results about the evolution of thin rings left a number of questions unanswered because of the computational cost of the simulations. For this reason, we studied the evolution of the ultraspinning instability of spherical rotating black holes in 6 spacetime dimensions, which reduces to an effective 2+1 problem. In this project we managed to follow the instability very close to the singularity and showed that: 1) the approach to the singularity is not self-similar and, 2) a singularity forms in finite asymptotic time. Therefore, the WWC is also violated in higher dimensional rotating spherical black holes.

We have also studied the evolution of non-axisymmetric instabilities of spherical black holes in 6 and higher dimensions. These instabilities were first discovered by Shibata and Yoshino, but only relatively small values of the spin parameter had been explored. In our work, we have been able to study these instabilities for much larger values of the spin parameter. We found that for sufficiently rapid rotations, these instabilities deform the black hole in a way such that local GL instabilities should kick in. Therefore, our work provides a complete picture of the black hole instabilities in higher dimensions and it implies that the WCC is generically violated. This culminates Project 2 in Objective 1.

We have also investigated the non-linear evolution of generic perturbations of global AdS. We studied, for the first time, gravitational collapse of a massless scalar field in AdS beyond spherical symmetry. Our results showed that breaking spherical symmetry facilitates gravitational collapse.

We have also developed a code to construct static Kaluza-Klein black holes and non-uniform black strings in 10 dimensions to an unprecedented level of accuracy. This has allowed us to understand black hole phases on a circle in supergravity and compare the results to lattice simulations of the dual gauge theory. The code development that we have performed to address this problem will allow us to address a number of other problems specified in the DoA.

In many of the systems of interest, newly relevant length scales are randomly generated during the evolution of certain instabilities. To accurately describe such systems numerically, we developed a code which has fully flexible adaptive mesh refinement. This allow 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, and is publicly available on our website: http://www.grchombo.org/.

Most of the research has been focused on Project 2 of my Objective 1 in the DoA. We have successfully evolved the various instabilities that afflict asymptotically flat 5-dimensional black rings and spherical black holes in 6 dimensions and found compelling evidence for the endpoints of such instabilities. Thin rings had been found to be unstable under a Gregory-Laflamme (GL) instability, which suggested that the weak cosmic censorship conjecture (WCC) could be violated in asymptotically flat spaces, similar to what famously happens to unstable black strings in 5 dimensions. By evolving these black ring instabilities, we discovered that not so thin rings are dominated by a new type of “elastic” instability, which results in collapse into a spherical black hole and no violation of the WCC. On other hand, for thin enough rings, the evolution is dominated by the GL instability and the endpoint is a naked singularity. This constitutes the first counter-example of the WCC in higher dimensional asymptotically flat spacetimes.

The results about the evolution of thin rings left a number of questions unanswered because of the computational cost of the simulations. For this reason, we studied the evolution of the ultraspinning instability of spherical rotating black holes in 6 spacetime dimensions, which reduces to an effective 2+1 problem. In this project we managed to follow the instability very close to the singularity and showed that: 1) the approach to the singularity is not self-similar and, 2) a singularity forms in finite asymptotic time. Therefore, the WWC is also violated in higher dimensional rotating spherical black holes.

We have also studied the evolution of non-axisymmetric instabilities of spherical black holes in 6 and higher dimensions. These instabilities were first discovered by Shibata and Yoshino, but only relatively small values of the spin parameter had been explored. In our work, we have been able to study these instabilities for much larger values of the spin parameter. We found that for sufficiently rapid rotations, these instabilities deform the black hole in a way such that local GL instabilities should kick in. Therefore, our work provides a complete picture of the black hole instabilities in higher dimensions and it implies that the WCC is generically violated. This culminates Project 2 in Objective 1.

We have also investigated the non-linear evolution of generic perturbations of global AdS. We studied, for the first time, gravitational collapse of a massless scalar field in AdS beyond spherical symmetry. Our results showed that breaking spherical symmetry facilitates gravitational collapse.

We have also developed a code to construct static Kaluza-Klein black holes and non-uniform black strings in 10 dimensions to an unprecedented level of accuracy. This has allowed us to understand black hole phases on a circle in supergravity and compare the results to lattice simulations of the dual gauge theory. The code development that we have performed to address this problem will allow us to address a number of other problems specified in the DoA.

The results that we have obtained so far 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, we studied the endpoint of the Gregory-Laflamme instability of asymptotically flat black rings in 5 dimensions, which is a (3+1)-dimensional problem and hence significantly more difficult and computationally expensive. Our work was a completely new development which shed new light into the weak cosmic censorship conjecture, especially because its original formulation involved asymptotically flat spaces. 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. These new methods are likely to find applications in other contexts, in particular, in the astrophysical simulations of black hole binaries and neutron stars.

Similarly, our work on the endpoint of the ultraspinning instability was novel and beyond the state of the art: in this case, we could determine the details of the approach to the singularity in asymptotically flat black hole spacetimes and uncover new physics. In particular, we found that in asymptotically flat spaces, the approach to the singularity is not self-similar and it happens faster. In additon, 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 numerical relativity.

Last but not least, in the context of the gravitational instability of AdS space, all previous numerical simulations assumed spherical symmetry. Therefore, our work was the first one to drop this symmetry assumption and reveal that gravitational collapse happens faster. In fact, the code that we developed is completely general and can handle the general case (with no symmetry assumptions).

To carry out the previously mentioned projects, we had to develop a new code, GRChombo, which has, among other features, fully flexible adaptive mesh refinement (AMR). Moreover, with this code we can also simulate dynamical spacetimes that are asymptotically AdS using Cauchy evolution. The latter is new since most of the existing codes using the characteristic formulation of the Einstein equations. Cauchy evolution has a number of technical advantages, most notoriously are AMR and the scalability to thousands of CPUs. This tool should put us in a great position to achieve our remaining scientific objectives as described in the Description of Action (DoA).

Similarly, our work on the endpoint of the ultraspinning instability was novel and beyond the state of the art: in this case, we could determine the details of the approach to the singularity in asymptotically flat black hole spacetimes and uncover new physics. In particular, we found that in asymptotically flat spaces, the approach to the singularity is not self-similar and it happens faster. In additon, 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 numerical relativity.

Last but not least, in the context of the gravitational instability of AdS space, all previous numerical simulations assumed spherical symmetry. Therefore, our work was the first one to drop this symmetry assumption and reveal that gravitational collapse happens faster. In fact, the code that we developed is completely general and can handle the general case (with no symmetry assumptions).

To carry out the previously mentioned projects, we had to develop a new code, GRChombo, which has, among other features, fully flexible adaptive mesh refinement (AMR). Moreover, with this code we can also simulate dynamical spacetimes that are asymptotically AdS using Cauchy evolution. The latter is new since most of the existing codes using the characteristic formulation of the Einstein equations. Cauchy evolution has a number of technical advantages, most notoriously are AMR and the scalability to thousands of CPUs. This tool should put us in a great position to achieve our remaining scientific objectives as described in the Description of Action (DoA).