Numerical simulations of instabilities in general relativity

General relativity, one of the pillars of our understanding of the universe, has been a remarkably successful theory. It has stood the test of time for more than 100 years and has passed all experimental tests so far. Most recently, the LIGO collaboration made the first-ever direct detection of gravitational waves, confirming a long-standing prediction of general relativity. Despite this, several fundamental mathematical questions remain unanswered, many of which relate to the global existence and the stability of solutions to Einstein’s equations. This thesis presents our efforts to use numerical relativity to investigate some of these questions. We present a complete picture of the end points of black ring instabilities in five dimensions. Fat rings collapse to Myers-Perry black holes. For intermediate rings, we discover a previously unknown instability that stretches the ring without changing its thickness and causes it to collapse to a Myers-Perry black hole. Most importantly, however, we find that for very thin rings, the Gregory-Laflamme instability dominates and causes the ring to break. This provides the first concrete evidence that in higher dimensions, the weak cosmic censorship conjecture may be violated even in asymptotically flat spacetimes. For Myers-Perry black holes, we investigate instabilities in five and six dimensions. In six dimensions, we demonstrate that both axisymmetric and non-axisymmetric instabilities can cause the black hole to pinch off, and we study the approach to the naked singularity in detail. Another question that has attracted intense interest recently is the instability of anti-de Sitter space. In this thesis, we explore how breaking spherical symmetry in gravitational collapse in anti-de Sitter space affects black hole formation. These findings were made possible by our new open source general relativity code, GRChombo, whose adaptive mesh capabilities allow accurate simulations of phenomena in which new length scales are produced dynamically. In this thesis, we describe GRChombo in detail, and analyse its performance on the latest supercomputers. Furthermore, we outline numerical advances that were necessary for simulating higher dimensional black holes stably and efficiently.My PhD was funded by an STFC studentship initially and by the European Research Council Grant No. ERC-2014-StG 639022-NewNGR in my final year. Furthermore, I received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant agreement No. 690904. The simulations presented in this thesis were carried out on the following supercomputers: *) The COSMOS Shared Memory system at DAMTP, University of Cambridge, operated on behalf of the STFC DiRAC HPC Facility. This sytem is funded by BIS National E-infrastructure capital Grant No.~ST/ J005673/1 and STFC Grants No.~ST/H008586/1, No.~ST/K00333X/1. *) MareNostrum III and MareNostrum IV at the Barcelona Supercomputing Centre through the grants FI-2016-3-0006 and PRACE Tier-0 PPFPWG respectively. *) Stampede and Stampede2 at the Texas Advanced Computing Center, University of Texas at Austin, through the NSF-XSEDE grant No.~PHY-090003 and an allocation provided by Intel for their Parallel Computing Centres. *) SuperMike-II at Louisiana State University under allocation NUMREL06. *) Cartesius, SURFsara, in the Netherlands through the PRACE DECI grant NRBA

End Point of Black Ring Instabilities and the Weak Cosmic Censorship Conjecture.

This is the author accepted manuscript. The final version is available at http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.071102#fulltext.We produce the first concrete evidence that violation of the weak cosmic censorship conjecture can occur in asymptotically flat spaces of five dimensions by numerically evolving perturbed black rings. For certain thin rings, we identify a new, elastic-type instability dominating the evolution, causing the system to settle to a spherical black hole. However, for sufficiently thin rings the Gregory-Laflamme mode is dominant, and the instability unfolds similarly to that of black strings, where the horizon develops a structure of bulges connected by necks which become ever thinner over time.We are very grateful to Garth Wells (Dept. Engineering, U. Cambridge) for suggesting to us the shock capturing technique which has proven so valuable in this work. We would like to thank J. Briggs, J. Camps, R. Emparan, J. Jäykkä, K. Kornet, L. Lehner, F. Pretorius, H. Reall, E. Schnetter, U. Sperhake, T. Wiseman and H. Witek for numerous stimulating discussions. P.F. would like to especially thank E. Schnetter and U. Sperhake for early collaboration in this project. We are very grateful to our collaborators and co-developers of the GRC HOMBO code, K. Clough, E. Lim and H. Finkel. We would also like to thank J. Santos and B. Way for allowing us to display their data in Fig. 1. A significant part of this work was undertaken on the COSMOS Shared Memory system at DAMTP, University of Cambridge, operated on behalf of the STFC DiRAC HPC Facility. This equipment is funded by BIS National E-infrastructure capital Grant No. ST/J005673/1 and STFC Grants No. ST/H008586/1, No. ST/K00333X/1. Further portions of this research were conducted with high performance computational resources provided by Louisiana State University [31] on its SuperMike-II cluster under allocation NUMREL06. The authors also acknowledge HPC resources from the NSF-XSEDE Grant No. PHY-090003, provided by the Texas Advanced Computing Center (TACC) at The University of Texas at Austin on its Stampede cluster, and by the San Diego Supercomputer Center (SDSC) at UC San Diego on its Comet cluster. P.F. and S.T. were supported by the European Research Council Grant No. ERC-2011-StG 279363- HiDGR. P.F. was also supported by the Stephen Hawking Advanced Research Fellowship from the Centre for Theoretical Cosmology, University of Cambridge. P.F. is currently supported by a Royal Society University Research Fellowship and by the European Research Council Grant No. ERC-2014-StG 639022-NewNGR. MK is supported by an STFC studentship. P.F. wants to thank Perimeter Institute and Princeton University for hospitality during various stages of this work

GRChombo : Numerical Relativity with Adaptive Mesh Refinement

In this work, we introduce GRChombo: a new numerical relativity code which incorporates full adaptive mesh refinement (AMR) using block structured Berger-Rigoutsos grid generation. The code supports non-trivial "many-boxes-in-many-boxes" mesh hierarchies and massive parallelism through the Message Passing Interface (MPI). GRChombo evolves the Einstein equation using the standard BSSN formalism, with an option to turn on CCZ4 constraint damping if required. The AMR capability permits the study of a range of new physics which has previously been computationally infeasible in a full 3+1 setting, whilst also significantly simplifying the process of setting up the mesh for these problems. We show that GRChombo can stably and accurately evolve standard spacetimes such as binary black hole mergers and scalar collapses into black holes, demonstrate the performance characteristics of our code, and discuss various physics problems which stand to benefit from the AMR technique.Comment: 48 pages, 24 figure

Dimensional reduction in numerical relativity: Modified cartoon formalism and regularization

We present in detail the Einstein equations in the Baumgarte–Shapiro–Shibata–Nakamura formulation for the case of D-dimensional spacetimes with SO(D−d)isometry based on a method originally introduced in Ref. 1. Regularized expressions are given for a numerical implementation of this method on a vertex centered grid including the origin of the quasi-radial coordinate that covers the extra dimensions with rotational symmetry. Axisymmetry, corresponding to the value d = D − 2, represents a special case with fewer constraints on the vanishing of tensor components and is conveniently implemented in a variation of the general method. The robustness of the scheme is demonstrated for the case of a black-hole head-on collision in D = 7 spacetime dimensions with SO(4) symmetry.U.S. is supported by the H2020 ERC Consolidator Grant “Matter and strong-field gravity: New frontiers in Einstein’s theory” grant agreement No. MaGRaTh–646597, the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904, the STFC Consolidator Grant No. ST/L000636/1, the SDSC Comet and TACC Stampede clusters through NSF-XSEDE Award Nos. PHY-090003, the Cambridge High Performance Computing Service Supercomputer Darwin using Strategic Research Infrastructure Funding from the HEFCE and the STFC, and DiRAC’s Cosmos Shared Memory system through BIS Grant No. ST/J005673/1 and STFC Grant Nos. ST/H008586/1, ST/K00333X/1. P.F. and S.T. are supported by the H2020 ERC Starting Grant “New frontiers in numerical general relativity” grant agreement No. NewNGR- 639022. P.F. is also supported by a Royal Society University Research Fellowship. W.G.C. and M.K. are supported by STFC studentships.This is the final version of the article. It first appeared from the World Scientific Publishing Company via http://dx.doi.org/10.1142/S021827181641013

Doing the right thing for the right reason: Evaluating artificial moral cognition by probing cost insensitivity

Is it possible to evaluate the moral cognition of complex artificial agents? In this work, we take a look at one aspect of morality: `doing the right thing for the right reasons.' We propose a behavior-based analysis of artificial moral cognition which could also be applied to humans to facilitate like-for-like comparison. Morally-motivated behavior should persist despite mounting cost; by measuring an agent's sensitivity to this cost, we gain deeper insight into underlying motivations. We apply this evaluation to a particular set of deep reinforcement learning agents, trained by memory-based meta-reinforcement learning. Our results indicate that agents trained with a reward function that includes other-regarding preferences perform helping behavior in a way that is less sensitive to increasing cost than agents trained with more self-interested preferences.Comment: 11 pages, 3 figure

GRChombo: An adaptable numerical relativity code for fundamental physics

GRChombo is an open-source code for performing Numerical Relativity time evolutions, built on top of the publicly available Chombo software for the solution of PDEs. Whilst GRChombo uses standard techniques in NR, it focusses on applications in theoretical physics where adaptability, both in terms of grid structure, and in terms of code modification, are key drivers

End point of nonaxisymmetric black hole instabilities in higher dimensions

We report on the end state of nonaxisymmetric instabilities of singly spinning asymptotically flat Myers-Perry black holes. Starting from a singly spinning black hole in D=5,6,7 dimensions, we introduce perturbations with angular dependence described by m=2, m=3, or m=4 azimuthal mode numbers about the axis of rotation. In D=5, we find that all singly spinning Myers-Perry black holes are stable, in agreement with the results from perturbation theory. In D=6 and 7, we find that these black holes are nonlinearly stable only for sufficiently low spins. For intermediate spins, although the m=2 bar mode becomes unstable and leads to large deformations, the black hole settles back down to another member of the Myers-Perry family via gravitational wave emission; surprisingly, we find that all such unstable black holes settle to the same member of the Myers-Perry family. The amount of energy radiated into gravitational waves can be very large, in some cases more than 30% of the initial total mass of the system. For high enough spins, the m=4 mode becomes the dominant unstable mode, leading to deformed black holes that develop local Gregory-Laflamme instabilities, thus forming a naked singularity in finite time, which is further evidence for the violation of the weak cosmic censorship conjecture in asymptotically flat higher-dimensional spacetimes

Your Policy Regularizer is Secretly an Adversary

Policy regularization methods such as maximum entropy regularization are widely used in reinforcement learning to improve the robustness of a learned policy. In this paper, we show how this robustness arises from hedging against worst-case perturbations of the reward function, which are chosen from a limited set by an imagined adversary. Using convex duality, we characterize this robust set of adversarial reward perturbations under KL and alpha-divergence regularization, which includes Shannon and Tsallis entropy regularization as special cases. Importantly, generalization guarantees can be given within this robust set. We provide detailed discussion of the worst-case reward perturbations, and present intuitive empirical examples to illustrate this robustness and its relationship with generalization. Finally, we discuss how our analysis complements and extends previous results on adversarial reward robustness and path consistency optimality conditions.Comment: Transactions on Machine Learning Researc