Critical Collapse of the Massless Scalar Field in Axisymmetry

We present results from a numerical study of critical gravitational collapse of axisymmetric distributions of massless scalar field energy. We find threshold behavior that can be described by the spherically symmetric critical solution with axisymmetric perturbations. However, we see indications of a growing, non-spherical mode about the spherically symmetric critical solution. The effect of this instability is that the small asymmetry present in what would otherwise be a spherically symmetric self-similar solution grows. This growth continues until a bifurcation occurs and two distinct regions form on the axis, each resembling the spherically symmetric self-similar solution. The existence of a non-spherical unstable mode is in conflict with previous perturbative results, and we therefore discuss whether such a mode exists in the continuum limit, or whether we are instead seeing a marginally stable mode that is rendered unstable by numerical approximation.Comment: 11 pages, 8 figure

Generalized harmonic formulation in spherical symmetry

In this pedagogically structured article, we describe a generalized harmonic formulation of the Einstein equations in spherical symmetry which is regular at the origin. The generalized harmonic approach has attracted significant attention in numerical relativity over the past few years, especially as applied to the problem of binary inspiral and merger. A key issue when using the technique is the choice of the gauge source functions, and recent work has provided several prescriptions for gauge drivers designed to evolve these functions in a controlled way. We numerically investigate the parameter spaces of some of these drivers in the context of fully non-linear collapse of a real, massless scalar field, and determine nearly optimal parameter settings for specific situations. Surprisingly, we find that many of the drivers that perform well in 3+1 calculations that use Cartesian coordinates, are considerably less effective in spherical symmetry, where some of them are, in fact, unstable.Comment: 47 pages, 15 figures. v2: Minor corrections, including 2 added references; journal version

Tips for implementing multigrid methods on domains containing holes

As part of our development of a computer code to perform 3D `constrained evolution' of Einstein's equations in 3+1 form, we discuss issues regarding the efficient solution of elliptic equations on domains containing holes (i.e., excised regions), via the multigrid method. We consider as a test case the Poisson equation with a nonlinear term added, as a means of illustrating the principles involved, and move to a "real world" 3-dimensional problem which is the solution of the conformally flat Hamiltonian constraint with Dirichlet and Robin boundary conditions. Using our vertex-centered multigrid code, we demonstrate globally second-order-accurate solutions of elliptic equations over domains containing holes, in two and three spatial dimensions. Keys to the success of this method are the choice of the restriction operator near the holes and definition of the location of the inner boundary. In some cases (e.g. two holes in two dimensions), more and more smoothing may be required as the mesh spacing decreases to zero; however for the resolutions currently of interest to many numerical relativists, it is feasible to maintain second order convergence by concentrating smoothing (spatially) where it is needed most. This paper, and our publicly available source code, are intended to serve as semi-pedagogical guides for those who may wish to implement similar schemes.Comment: 18 pages, 11 figures, LaTeX. Added clarifications and references re. scope of paper, mathematical foundations, relevance of work. Accepted for publication in Classical & Quantum Gravit

Numerical Relativity Using a Generalized Harmonic Decomposition

A new numerical scheme to solve the Einstein field equations based upon the generalized harmonic decomposition of the Ricci tensor is introduced. The source functions driving the wave equations that define generalized harmonic coordinates are treated as independent functions, and encode the coordinate freedom of solutions. Techniques are discussed to impose particular gauge conditions through a specification of the source functions. A 3D, free evolution, finite difference code implementing this system of equations with a scalar field matter source is described. The second-order-in-space-and-time partial differential equations are discretized directly without the use first order auxiliary terms, limiting the number of independent functions to fifteen--ten metric quantities, four source functions and the scalar field. This also limits the number of constraint equations, which can only be enforced to within truncation error in a numerical free evolution, to four. The coordinate system is compactified to spatial infinity in order to impose physically motivated, constraint-preserving outer boundary conditions. A variant of the Cartoon method for efficiently simulating axisymmetric spacetimes with a Cartesian code is described that does not use interpolation, and is easier to incorporate into existing adaptive mesh refinement packages. Preliminary test simulations of vacuum black hole evolution and black hole formation via scalar field collapse are described, suggesting that this method may be useful for studying many spacetimes of interest.Comment: 18 pages, 6 figures; updated to coincide with journal version, which includes some expanded discussions and a new appendix with a stability analysis of a simplified problem using the same discretization scheme described in the pape

Gravitational collapse in 2+1 dimensional AdS spacetime

We present results of numerical simulations of the formation of black holes from the gravitational collapse of a massless, minimally-coupled scalar field in 2+1 dimensional, axially-symmetric, anti de-Sitter (AdS) spacetime. The geometry exterior to the event horizon approaches the BTZ solution, showing no evidence of scalar `hair'. To study the interior structure we implement a variant of black-hole excision, which we call singularity excision. We find that interior to the event horizon a strong, spacelike curvature singularity develops. We study the critical behavior at the threshold of black hole formation, and find a continuously self-similar solution and corresponding mass-scaling exponent of approximately 1.2. The critical solution is universal to within a phase that is related to the angle deficit of the spacetime.Comment: 31 pages, 20 figures, LaTeX. Replaced with version to be published in Phys. Rev.

An Axisymmetric Gravitational Collapse Code

We present a new numerical code designed to solve the Einstein field equations for axisymmetric spacetimes. The long term goal of this project is to construct a code that will be capable of studying many problems of interest in axisymmetry, including gravitational collapse, critical phenomena, investigations of cosmic censorship, and head-on black hole collisions. Our objective here is to detail the (2+1)+1 formalism we use to arrive at the corresponding system of equations and the numerical methods we use to solve them. We are able to obtain stable evolution, despite the singular nature of the coordinate system on the axis, by enforcing appropriate regularity conditions on all variables and by adding numerical dissipation to hyperbolic equations.Comment: 19 pages, 9 figure

Modified general relativity as a model for quantum gravitational collapse

We study a class of Hamiltonian deformations of the massless Einstein-Klein-Gordon system in spherical symmetry for which the Dirac constraint algebra closes. The system may be regarded as providing effective equations for quantum gravitational collapse. Guided by the observation that scalar field fluxes do not follow metric null directions due to the deformation, we find that the equations take a simple form in characteristic coordinates. We analyse these equations by a unique combination of numerical methods and find that Choptuik's mass scaling law is modified by a mass gap as well as jagged oscillations. Furthermore, the results are universal with respect to different initial data profiles and robust under changes of the deformation.Comment: 22 pages, 4 figure

AMR, stability and higher accuracy

Efforts to achieve better accuracy in numerical relativity have so far focused either on implementing second order accurate adaptive mesh refinement or on defining higher order accurate differences and update schemes. Here, we argue for the combination, that is a higher order accurate adaptive scheme. This combines the power that adaptive gridding techniques provide to resolve fine scales (in addition to a more efficient use of resources) together with the higher accuracy furnished by higher order schemes when the solution is adequately resolved. To define a convenient higher order adaptive mesh refinement scheme, we discuss a few different modifications of the standard, second order accurate approach of Berger and Oliger. Applying each of these methods to a simple model problem, we find these options have unstable modes. However, a novel approach to dealing with the grid boundaries introduced by the adaptivity appears stable and quite promising for the use of high order operators within an adaptive framework

Are moving punctures equivalent to moving black holes?

When simulating the inspiral and coalescence of a binary black-hole system, special care needs to be taken in handling the singularities. Two main techniques are used in numerical-relativity simulations: A first and more traditional one ``excises'' a spatial neighbourhood of the singularity from the numerical grid on each spacelike hypersurface. A second and more recent one, instead, begins with a ``puncture'' solution and then evolves the full 3-metric, including the singular point. In the continuum limit, excision is justified by the light-cone structure of the Einstein equations and, in practice, can give accurate numerical solutions when suitable discretizations are used. However, because the field variables are non-differentiable at the puncture, there is no proof that the moving-punctures technique is correct, particularly in the discrete case. To investigate this question we use both techniques to evolve a binary system of equal-mass non-spinning black holes. We compare the evolution of two curvature 4-scalars with proper time along the invariantly-defined worldline midway between the two black holes, using Richardson extrapolation to reduce the influence of finite-difference truncation errors. We find that the excision and moving-punctures evolutions produce the same invariants along that worldline, and thus the same spacetimes throughout that worldline's causal past. This provides convincing evidence that moving-punctures are indeed equivalent to moving black holes.Comment: 4 pages, 3 eps color figures; v2 = major revisions to introduction & conclusions based on referee comments, but no change in analysis or result

Scalar field spacetimes and the AdS/CFT conjecture

We describe a class of asymptotically AdS scalar field spacetimes, and calculate the associated conserved charges for three, four and five spacetime dimensions using the conformal and counter-term prescriptions. The energy associated with the solutions in each case is proportional to $\sqrt{M^2 - k^2}$, where $M$ is a constant and $k$ is a scalar charge. In five spacetime dimensions, the counter-term prescription gives an additional vacuum (Casimir) energy, which agrees with that found in the context of AdS/CFT correspondence. We find a surprising degeneracy: the energy of the ``extremal'' scalar field solution $M=k$ equals the energy of pure AdS. This result is discussed in light of the AdS/CFT conjecture.Comment: 5 pages, Latex, additional commentary on results, version to appear in Phys. Rev.