Traditionally, frontiers represent a treacherous terrain to venture into, where hidden obstacles are present and uncharted territories lie ahead. At the same time, frontiers are also a place where new perspectives can be appreciated and have often been the cradle of new and thriving developments. With this in mind, the numerical-relativity group at the Albert Einstein Institute (AEI) organised a workshop with the goal of exploring and understanding these “New Frontiers”. The workshop took place from July 17-21, 2006 at the AEI campus in Golm, Germany. The meeting was focussed on the numerous issues that occur in numerical relativity, such as: formulations of the Einstein equations, initial data, multiblock approaches, boundary and gauge conditions, and of course relativistic fluids and plasmas.
Almost 20 years since the homonymous meeting held at Urbana-Champaign (``Frontiers in Numerical Relativity'', 1988), this meeting saw the enthusiastic participation of a great part of the community, with 127 participants present (in 1988 there were 55) and with a large majority being represented by students and postdocs, a reassuring sign of good health for the community. The program was organised so as to have few talks with ample time dedicated to discussions, which were then continued over breaks, meals and late evenings. In addition, a whole session spanning the last afternoon was dedicated to an ``unconstrained'' discussion which covered some of the most controversial issues that emerged during the conference. During this discussion, led by E. Seidel, particular emphasis was placed on the need for systematic comparisons between waveforms generated by different codes, as well as on the connection to the data-analysis community.
A good overview of the conference can be found on the webpage of the conference http://numrel.aei.mpg.de/nfnr, which contains the list of the participants, a copy of the program and downloadable version of the talks. Because of this, in what follows I will simply report the highlights of the different thematic sessions which composed the program.
This session saw talks covering issues that go from pointing out clues about ``why do codes crash'' (C. Bona), over to the generalized harmonic gauge conditions in use by the Caltech/Cornell group (L. Lindblom), to the well-posedness and equivalence of different formulations and their relations when ``live'' gauge conditions are used (J.M. Martin-Garcia), to conclude with a prescription on how to deal with constraint violations in first-order evolution systems. Particularly interesting was also the progress report on the ability to perform numerical simulations of the tensor wave equation with pseudospectral methods and which represents the first step towards the solution of a maximally-constrained formulation of the Einstein equations (J. Novak).
This session covered a classical topic in numerical relativity: the construction of initial data for binary black hole systems. The talks focussed on solutions found with a parallel multigrid solver for binary systems with non-trivial spin combination (S. Hawley), on how to improve the Bowen-York prescription the initial data with spinning black holes (M. Hannam), or on how to use matched asymptotic expansions to obtain approximate but hopefully more realistic binary black hole initial data (W. Tichy). Particularly interesting were also the progress reports about the use of ingenious coordinate transformations to build quasi-equilibrium configurations of arbitrary binaries (M. Ansorg) or on how to take properly into account spin in the construction of initial data for binary black holes with spins and in circular orbits (H. Pfeiffer). Both approaches showed the impressive accuracy of pseudospectral methods for this type of problems.
A lot of excitement preceded this session and it was all well motivated. A number of impressive results were in fact presented, some of them simply beyond (a realistic) imagination only a couple of years ago. Some of the results on the ``moving-punctures'' prescription, which have been recently published, were presented in great detail (C. Lousto, J. Baker) and led to a lively discussion. Equally interesting were the talks of other groups reporting their ability to now perform multiple orbits simulations of binary black hole systems when treated using moving punctures and a conformal traceless formulation of the Einstein equations (P. Marronetti, B. Brügmann, F. Hermann, D. Pollney). Of topical relevance to the community engaged in puncture evolutions, was the recent work which studied the stationary slicing of puncture spacetimes and the behavior of fields at the puncture (Brügmann, Pollney). Also rather impressive were the results on binary inspiral and merger carried out within a harmonic formulation of the equations either as second-order systems with finite-difference techniques (F. Pretorius) or as a first-order system with pseudospectral methods (M. Scheel). While the latter approach still needs to find an effective management of the domains at the time of the merger, the quality of the results presented for the inspiral has provided additional evidence of the accuracy of spectral methods. In addition, a useful comparison between the harmonic and conformal-traceless formulations was also presented as a first application of a newly developed code (B. Szilágyi). Very interesting work is also being done in areas beyond the binary black hole problem, with simulations of general singularities and the apparent validity of the BKL conjecture (D. Garfinkle), or the formation of naked singularities in the collapse of an ultrarelativistic fluid (M. Snajdr), or on a new prescription to smooth-out a singularity and perform stable and accurate simulations (E. Schnetter).
The large number of abstracts submitted to this session is an important indication that numerical relativity is not interested only in evolutions of pure-black-hole spacetimes and that a wider bridge towards numerical relativistic-astrophysics can be built. The session saw talks over a wide range of topics, from the analysis of the dynamical barmode instability and which provided a conceptual framework to determine why and when the instability is suppressed (G. Manca), over to the use of a spectral-methods code to study the behaviour of rotating and magnetized stars in quasi-equilibrium (S. Bonazzola), and to the modelling of radio images of Sgr A* using accretion disk simulations from a General Relativistic Magnetohydrodynamics (GRMHD) code on a Kerr background (S. Noble). Focus of a lot of attention were also simulations of gravitational collapse with talks on either the collapse of stellar cores to proto-neutron stars or of dynamically unstable neutron stars to black holes. More specifically, results were presented of 3D simulations of realistic stellar cores employing a finite-temperature equation of state and an approximate treatment of deleptonization (C. Ott, H. Dimmelmeier) as well as of 2D simulations of magnetized stellar cores in the test-field approximation (T. Font). Also, results were presented of 3D simulations of uniformly rotating neutron stars in which a novel technique avoided the use of excision and has allowed calculation of the first complete waveform of the process (L. Baiotti), as well as of 2D simulations in full GRMHD of differentially rotating neutron stars, whose dynamics could be of help in modelling the engines powering short gamma-ray bursts (B. Stephens). Two newly developed codes were also presented which solve the equations of GRMHD either on a fixed black hole background and developed to model jet formation (Y. Mizuno), or on an arbitrary background and developed to extend the applications of the Whisky code to scenarios in which magnetic fields play an important role (B. Giacomazzo). Last but not least, a critical assessment was made of present techniques to handle surfaces and interfaces in relativistic hydrodynamics, and which will need to be improved for a future description of multiple fluids (I. Hawke).
This session saw talks on an area of numerical relativity which has grown rapidly in recent years and is expected to be of increasing importance. In particular, details were given about the multidomain pseudospectral collocation methods used by the Caltech/Cornell group to evolve spacetimes with black holes (L. Kidder) as well on 6-patches schemes with either overlapping or simply touching patches. In the first case details were presented on the way in which the patch ghost zones are ``synchronized'' by interpolation, on the tensor basis used in each patch, and on the handling of non-tensor field variables (J. Thornburg). Similarly, the main ingredients for the touching patches approach, such as high-order summation by parts finite-differencing operators and compatible dissipation operators, the use of penalty methods for the inter-block boundaries and adaptive time stepping, were also discussed in detail (P. Diener). Finally, results were presented for a new class of analytic solutions to the linearized Einstein equations for the Bondi-Sachs metric to be used as a testbed in a code employing stereographic coordinates and six angular patches (N. Bishop) and for an AMR code with fourth-order discretization in space and time and exploting compactification in space. Examples were given on the use of this code for the study of non-interacting massive Klein-Gordon field together with Yang-Mills-Higgs systems (P. Csizmadia).
Recent work on another of the classical areas of research in numerical relativity, that which is interested in the definition of mathematically consistent and numerically accurate boundary conditions, was presented in this session. More specifically, a presentation was made on how to use trapping horizons to provide simple and satisfactory inner boundary conditions in black-hole spacetimes making use of the excision technique (E. Gourgoulhon). In addition, outer boundary conditions were the focus of several talks which addressed issues such as the definition of well-posed radiation-controlling boundary conditions for the harmonic formulation of the Einstein equations (O. Rinne), or the use of absorbing boundary conditions through a geometric approach (O. Sarbach) and the application of absorbing boundary conditions in examples of the linearized form of the Einstein equations (L. Buchman). Finally, results were presented on how to use black-hole perturbation theory and effective-one-body ideas to determine the gravitational-wave signal for the inspiral and merger of binary black-hole systems in the extreme mass-ratio limit (A. Nagar).
In recognition of his important work in the field, the conference hosted a public lecture by Jimmy York on ``Dynamical Principles of General Relativity'', held at the picturesque Schlosstheater im Neuen Palais, within the premises of the Sans Souci Park in Potsdam.
Talks given at the conference will appear as regular refereed articles in a special issue of CQG to be published in 2007, with M. Campanelli and L. Rezzolla acting as editors.