This talk is meant to give a fairly self-contained, quite formal, and very succinct introduction to the Einstein equations, including the required differential geometry and a choice of notational conventions. The equations will be first presented in an entirely coordinate-free manner emphasizing their geometric content, and then the corresponding PDEs satisfied by the metric components with respect to some coordinate system will be derived. The gauge freedom in the equations will be discussed both in the coordinate-free and the coordinatized context.
The talk introduces some aspects of the Einstein equations which are of direct interest in numerical relativity. Topics covered include the geometry of the 3+1 formalism, the constraint equations and the conformal method for solving the constraints, the classical linearized equations, and the ADM energy-momentum.
This talk will consist of a very broad, mainly non-technical overview of some of the basic issues which arise in the study of numerical solutions of Einstein's equations. Following a very brief review of the target physics, I will touch on a variety of topics including: the nature of solutions of the Einstein field equations, black hole singularities and black hole excision, discretization strategies, convergence and stability, resolution and adaptive mesh refinement, and the role of model problems in numerical relativity. The viewpoints espoused in the talk are not universally accepted in the field; indeed, one of the aims of the presentation is to stimulate lively discussion amongst the workshop participants concerning the basic approaches which have been used in numerical relativity the far, and the identification of particularly promising avenues for ongoing research.
Gregory B. Cook (Department of Physics, Wake Forest University)
Computation of initial data, I slides.pdf
In the first of two talks on the computation of initial data, we will look at some of the formalisms used for posing the constraint equations of general relativity as a boundary value problem. This process requires making well-motivated choices for which of the initial-data quantities are constrained and which can be freely specified. After looking at the general formalisms used to construct initial data, we will review the approaches that have been used to date in constructing black-hole and neutron-star initial data. Finally we will look at some of the current issues, from a physicist perspective, that are at the forefront of initial-data research.
Finite element approximation methods are described for model hyperbolic problems. These include the wave equation written as a second order scalar problem and also as a first order system. The talk emphasizes the derivation of approximation schemes which preserve discrete versions of important properties of the partial differential equation. These properties are often very useful in establishing stability and error estimates for the approximation scheme.
Computation of initial data, II slides.pdf
In this second of two talks on the computation of initial data, we will focus on the boundary value problem arising in the York conformal decomposition of the initial data. We examine in some detail the resulting coupled Hamiltonian and momentum constraints, focusing first on some fundamental issues such as well-posedness and approximation theory. We then review some of the numerical methods which have been used previously to solve the equations under various simplifying assumptions. Finally, we discuss the treatment of the general coupled nonlinear elliptic system using error-driven adaptive finite element discretization, Gummel decoupling methods, and Newton-multilevel iterative methods. We finish by outlining some of the open research questions, from the perspective of a mathematician and numerical analyst.
State of the art of numerical relativity slides.pdf
I will present an overview of the main approaches presently in use in numerical relativity. I will focus attention to 2D and 3D numerical evolutions of non-linear systems involving black holes and/or neutron stars, as well as numerical evolutions of gravitational collapse.
Luis Lehner (Department of Physics & Astronomy, University of British Columbia) email@example.com
Outer boundary conditions
This talk will review issues found when dealing with outer boundary conditions for Einstein equations. We will discuss the need for them in light of recent results and present some alternatives.
In the case boundary conditions are needed, we will present the status of current techniques and discuss how recent understanding of the problem can be employed to remove possible inconsistencies of `standard approaches.'
Finally, we will refer to the open issues and their relevance to current efforts.
Alan D. Rendall (Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut) firstname.lastname@example.org
The Cauchy problem for the Einstein equations has a number of special features when compared with that for other partial differential equations. These issues are briefly discussed, starting with model equations which illustrate some of these features in a less complicated context. Next the Cauchy problem for the Einstein equations is formulated. The standard approach to proving local existence is presented. Finally, some remarks are made on the 3+1 decomposition.
Formulations of GR for computation, I slides.pdf
After introduction of the relevant concepts, We analyze, in a systematic way, the hyperbolic type of several formulation of Einstein's evolution equations. In particular we concentrate in second and second-first order systems, like the ADM and BSSN formulations.
Mark A. Scheel (Department of Physics, Mathematics, and Astronomy, California Institute of Technology) email@example.com
Spectral methods and excision Slides.pdf
Spectral methods are considered for use in numerical relativity. A particular implementation of a pseudospectral collocation method for Cauchy evolution is described. The treatment of boundary conditions in this method automatically handles black hole excision, given an appropriate system of evolution equations and an appropriate gauge choice. Some numerical results are shown.
Deirdre Shoemaker (Department of Astronomy and Astrophysics, Pennsylvania State University) firstname.lastname@example.org
Computation of horizons and excision slides.pdf
One of the crucial aspects of numerically solving the Einstein equation for a space-time containing one or more black holes in a fully general relativistic manner, is handling the singularities intrinsic to the black holes. One of the most successful methods for handling the black-hole singularity in three-dimensional simulations is removing a region of the computational domain containing the singularity and interior to the black-hole horizon. This procedure is called excision and relies on knowledge of the black hole's horizon. I will discuss how the horizon of the black hole is located during numerical simulations and what the procedure is for excising the black-hole singularity in a code implementing finite differencing methods.
Computational Methods for Hyperbolic Systems. Preservation of Global and Local Invariants (pdf)
Numerical relativity as an initial-boundary value problem slides.pdf
I will discuss current efforts related to well posed initial-boundary value problems for Einstein's equations and their numerical implementation. I will concentrate on well posedness, constraint preservation and numerical stability.
Jeffrey Winicour (Physics Department, University of Pittsburgh) email@example.com
I describe the properties of the the horizon of an observer and the event horizon of a black hole spacetime, with emphasis on those aspects important for numerical simulation. The boundary of the spacetime region which can causally effect a given spacetime point P constitutes the event horizon of the observer at P. Thus the horizon is determined by the maximum signal propagation speed, i.e. by the characteristics of the partial differential equations underlying the theory. In relativity, these are the light rays. A black hole event horizon is the boundary of the causal past of the collection of observers at all "distant" spacetime points. In the flat spacetime of special relativity this boundary is empty and there are no black holes. In the curved spacetime of general relativity, black holes are produced by the lensing effect of a body undergoing gravitational collapse to a singularity. The final singularity is an impediment to numerical simulation. Fortunately, theoretical arguments suggest that the singularity lies inside the black hole and does not affect observations by distant astronomers. This allows singularity-avoiding strategies for computing the gravitational radiation emitted in the formation of black holes. This has been achieved with some success for a single black hole. Current efforts concentrate on handling the inspiral and merger of a binary black hole, which has a more complicated horizon structure.