International Journal of Modern Physics A

We give an introduction to the Einstein Toolkit, a mature, open-source computational infrastructure for numerical relativity based on the Cactus Framework, for the target group of new users. This toolkit is composed of several different modules, is developed by researchers from different institutions throughout the world and is in active continuous development. Documentation for the toolkit and its several modules is often scattered across different locations, a difficulty new users may at times have to struggle with. Scientific papers exist describing the toolkit and its methods in detail, but they might be overwhelming at first. With these lecture notes we hope to provide an initial overview for new users. We cover how to obtain, compile and run the toolkit, and give an overview of some of the tools and modules provided with it.

These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of evolution partial differential equations. I show how strong hyperbolicity guarantees well-posedness of the initial value problem. Symmetric hyperbolic systems are shown to render the initial boundary value problem well-posed with maximally dissipative boundary conditions. I discuss the Laplace–Fourier method for analyzing the initial boundary value problem. Finally, I state how these notions extend to systems that are first-order in time and second-order in space. In the second lecture I discuss the effect that the gauge freedom of electromagnetism has on the PDE status of the initial value problem. I focus on gauge choices, strong-hyperbolicity and the construction of constraint preserving boundary conditions. I show that strongly hyperbolic pure gauges can be used to build strongly hyperbolic formulations. I examine which of these formulations is additionally symmetric hyperbolic and finally demonstrate that the system can be made boundary stable.

Numerical relativity became a powerful tool to investigate the dynamics of binary problems with black holes or neutron stars as well as the very structure of General Relativity. Although public numerical relativity codes are available to evolve such systems, a proper understanding of the methods involved is quite important. Here, we focus on the numerical solution of elliptic partial differential equations. Such equations arise when preparing initial data for numerical relativity, but also for monitoring the evolution of black holes. Because such elliptic equations play an important role in many branches of physics, we give an overview of the topic, and show how to numerically solve them with simple examples and sample codes written in C++ and Fortran90 for beginners in numerical relativity or other fields requiring numerical expertise.

Black holes are among the most exciting phenomena predicted by General Relativity and play a key role in fundamental physics. Many interesting phenomena involve dynamical black hole configurations in the high curvature regime of gravity. In these lecture notes I will summarize the main numerical relativity techniques to explore highly dynamical phenomena, such as black hole collisions, in generic D-dimensional space–times. The present notes are based on my lectures given at the NR/HEP2 spring school at IST/Lisbon (Portugal) from March 11–14, 2013.

Black-hole perturbation theory is a useful tool to investigate issues in astrophysics, high-energy physics, and fundamental problems in gravity. It is often complementary to fully-fledged nonlinear evolutions and instrumental to interpret some results of numerical simulations. Several modern applications require advanced tools to investigate the linear dynamics of generic small perturbations around stationary black holes. Here, we present an overview of these applications and introduce extensions of the standard semianalytical methods to construct and solve the linearized field equations in curved space–time. Current state-of-the-art techniques are pedagogically explained and exciting open problems are presented.

We present a pedagogical introduction to the problem of evolving a head on collision of two Aichelburg–Sexl gravitational shock waves in D-dimensions, using perturbative techniques. We follow a constructive approach with examples, going in some detail through: the set up of the exact initial conditions and their properties; perturbative methods in flat space–time with Green function solutions; and numerical strategies to evaluate the integral solutions. We also discuss, briefly, radiation extraction methods adapted to this problem, together with some of the results for this system.

In these lecture notes, we discuss recently conjectured instability of anti-de Sitter space, resulting in gravitational collapse of a large class of arbitrarily small initial perturbations. We uncover the technical details used in the numerical study of spherically symmetric Einstein-massless scalar field system with negative cosmological constant that led to the conjectured instability.

High Performance Computing (HPC) has become an essential tool in every researcher's arsenal. Most research problems nowadays can be simulated, clarified or experimentally tested by using computational simulations. Researchers struggle with computational problems when they should be focusing on their research problems. Since most researchers have little-to-no knowledge in low-level computer science, they tend to look at computer programs as extensions of their minds and bodies instead of completely autonomous systems. Since computers do not work the same way as humans, the result is usually Low Performance Computing where HPC would be expected.