Formation and decay of Einstein-Yang-Mills black holes

We study various aspects of black holes and gravitational collapse in Einstein-Yang-Mills theory under the assumption of spherical symmetry. Numerical evolution on hyperboloidal surfaces extending to future null infinity is used. We begin by constructing colored and Reissner-Nordstrom black holes on surfaces of constant mean curvature and analyze their perturbations. These linearly perturbed black holes are then evolved into the nonlinear regime and the masses of the final Schwarzschild black holes are computed as a function of the initial horizon radius. We compare with an information-theoretic bound on the lifetime of unstable hairy black holes derived by Hod. Finally we study critical phenomena in gravitational collapse at the threshold between different Yang-Mills vacuum states of the final Schwarzschild black holes, where the n=1 colored black hole forms the critical solution. The work of Choptuik et al. (1999) is extended by using a family of initial data that includes another region in parameter space where the colored black hole with the opposite sign of the Yang-Mills potential forms the critical solution. We investigate the boundary between the two regions and discover that the Reissner-Nordstrom solution appears as a new approximate codimension-two attractor.Comment: 13 pages, 12 figures. Minor changes to agree with published versio

Critical phenomena in the general spherically symmetric Einstein-Yang-Mills system

We study critical behavior in gravitational collapse of a general spherically symmetric Yang-Mills field coupled to the Einstein equations. Unlike the magnetic ansatz used in previous numerical work, the general Yang-Mills connection has two degrees of freedom in spherical symmetry. This fact changes the phenomenology of critical collapse dramatically. The magnetic sector features both type I and type II critical collapse, with universal critical solutions. In contrast, in the general system type I disappears and the critical behavior at the threshold between dispersal and black hole formation is always type II. We obtain values of the mass scaling and echoing exponents close to those observed in the magnetic sector, however we find some indications that the critical solution differs from the purely magnetic discretely self-similar attractor and exact self-similarity and universality might be lost. The additional "type III" critical phenomenon in the magnetic sector, where black holes form on both sides of the threshold but the Yang-Mills potential is in different vacuum states and there is a mass gap, also disappears in the general system. We support our dynamical numerical simulations with calculations in linear perturbation theory; for instance, we compute quasi-normal modes of the unstable attractor (the Bartnik-McKinnon soliton) in type I collapse in the magnetic sector.Comment: 15 pages, 15 figures; v2: matches published versio

Numerical Brill-Lindquist initial data with a Schwarzschildean end at spatial infinity

We construct numerically time-symmetric initial data that are Schwarzschildean at spatial infinity and Brill-Lindquist in the interior. The transition between these two data sets takes place along a finite gluing region equipped with an axisymmetric Brill wave metric. The construction is based on an application of Corvino's gluing method using Brill waves due to Giulini and Holzegel. Here, we use a gluing function that includes a simple angular dependence. We also investigate the dependence of the ADM mass of our construction on the details of the gluing procedure.Comment: 6 pages, 1 figure, 1 table. Conference proceedings for the Spanish Relativity Meeting, Valencia 201

Spectral approach to axisymmetric evolution of Einstein's equations

We present a new formulation of Einstein's equations for an axisymmetric spacetime with vanishing twist in vacuum. We propose a fully constrained scheme and use spherical polar coordinates. A general problem for this choice is the occurrence of coordinate singularities on the axis of symmetry and at the origin. Spherical harmonics are manifestly regular on the axis and hence take care of that issue automatically. In addition a spectral approach has computational advantages when the equations are implemented. Therefore we spectrally decompose all the variables in the appropriate harmonics. A central point in the formulation is the gauge choice. One of our results is that the commonly used maximal-isothermal gauge turns out to be incompatible with tensor harmonic expansions, and we introduce a new gauge that is better suited. We also address the regularisation of the coordinate singularity at the origin.Comment: 6 pages, based on a talk given by one of the authors at the Spanish Relativity Meeting ERE14 in Valencia, published versio

Superradiance of a charged scalar field coupled to the Einstein-Maxwell equations

We consider the Einstein-Maxwell-Klein-Gordon equations for a spherically symmetric scalar field scattering off a Reissner-Nordstr\"om black hole in asymptotically flat spacetime. The equations are solved numerically using a hyperboloidal evolution scheme. For suitable frequencies of the initial data, superradiance is observed, leading to a substantial decrease of mass and charge of the black hole. We also derive a Bondi mass loss formula using the Kodama vector field and investigate the late-time decay of the scalar field.Comment: 12 pages, 9 figures; 3 references adde

Outer boundary conditions for Einstein's field equations in harmonic coordinates

We analyze Einstein's vacuum field equations in generalized harmonic coordinates on a compact spatial domain with boundaries. We specify a class of boundary conditions, which is constraint-preserving and sufficiently general to include recent proposals for reducing the amount of spurious reflections of gravitational radiation. In particular, our class comprises the boundary conditions recently proposed by Kreiss and Winicour, a geometric modification thereof, the freezing-Ψ0 boundary condition and the hierarchy of absorbing boundary conditions introduced by Buchman and Sarbach. Using the recent technique developed by Kreiss and Winicour based on an appropriate reduction to a pseudo-differential first-order system, we prove well posedness of the resulting initial-boundary value problem in the frozen coefficient approximation. In view of the theory of pseudo-differential operators, it is expected that the full nonlinear problem is also well posed. Furthermore, we implement some of our boundary conditions numerically and study their effectiveness in a test problem consisting of a perturbed Schwarzschild black hole

Gravitational Collapse of Prolate Brill Waves

It has been conjectured that the gravitational collapse of sufficiently prolate vacuum axisymmetric gravitational waves (Brill waves) may violate cosmic censorship. Improving on earlier work by Garfinkle and Duncan, I present a numerical evolution of such a prolate initial data set that does form an apparent horizon. Related advances in the construction of axisymmetric constrained evolution schemes are also discussed

A flow approach to Bartnik's static metric extension conjecture in axisymmetry

We investigate Bartnik's static metric extension conjecture under the additional assumption of axisymmetry of both the given Bartnik data and the desired static extensions. To do so, we suggest a geometric flow approach, coupled to the Weyl-Papapetrou formalism for axisymmetric static solutions to the Einstein vacuum equations. The elliptic Weyl-Papapetrou system becomes a free boundary value problem in our approach. We study this new flow and the coupled flow--free boundary value problem numerically and find axisymmetric static extensions for axisymmetric Bartnik data in many situations, including near round spheres in spatial Schwarzschild of positive mass.Comment: 60 pages, 13 figures. Expanded Section 3.3 to address longtime existence and uniqueness of solutions to the linearised flow equations. To appear in Pure and Applied Mathematics Quarterly, special issue in honour of Robert Bartni

Constructing Reference Metrics on Multicube Representations of Arbitrary Manifolds

Reference metrics are used to define the differential structure on multicube representations of manifolds, i.e., they provide a simple and practical way to define what it means globally for tensor fields and their derivatives to be continuous. This paper introduces a general procedure for constructing reference metrics automatically on multicube representations of manifolds with arbitrary topologies. The method is tested here by constructing reference metrics for compact, orientable two-dimensional manifolds with genera between zero and five. These metrics are shown to satisfy the Gauss-Bonnet identity numerically to the level of truncation error (which converges toward zero as the numerical resolution is increased). These reference metrics can be made smoother and more uniform by evolving them with Ricci flow. This smoothing procedure is tested on the two-dimensional reference metrics constructed here. These smoothing evolutions (using volume-normalized Ricci flow with DeTurck gauge fixing) are all shown to produce reference metrics with constant scalar curvatures (at the level of numerical truncation error).Comment: 37 pages, 16 figures; additional introductory material added in version accepted for publicatio

Gauge drivers for the generalized harmonic Einstein equations

The generalized harmonic representation of Einstein's equations is manifestly hyperbolic for a large class of gauge conditions. Unfortunately most of the useful gauges developed over the past several decades by the numerical relativity community are incompatible with the hyperbolicity of the equations in this form. This paper presents a new method of imposing gauge conditions that preserves hyperbolicity for a much wider class of conditions, including as special cases many of the standard ones used in numerical relativity: e.g., K freezing, Gamma freezing, Bona-Massó slicing, conformal Gamma drivers, etc. Analytical and numerical results are presented which test the stability and the effectiveness of this new gauge-driver evolution system