Computer Algebra Applications for Numerical Relativity (original) (raw)

Computer Algebra Systems in General Relativity

2007

This paper presents a review of the use of Computer Algebra Systems in General Relativity research and teaching. On one hand, the impact of using Computer Algebra Systems in General Relativity research is illustrated by pointing out some important achievements in the field. In particular, by using Computer Algebra Systems, the present author has been able to obtain results that would have been almost impossible otherwise. On the other hand, Computer Algebra Systems can be a very helpful tool in teaching and learning GR. Some reports on using Computer Algebra Systems in teaching General Relativity are outlined.

Symbolic and Numerical Analysis in General Relativity with Open Source Computer Algebra Systems

We study three computer algebra systems, namely SageMath (with SageManfolds package), Maxima (with ctensor package) and Python language (with GraviPy module), which allow tensor manipulation for general relativity calculations. We present simple examples and give a benchmark of these systems. After the general analysis, we focus on the SageMath+SageManifolds system to analyze and visualize the solutions of the massless Klein–Gordon equation and geodesic motion with Hamilton–Jacobi formalism.

Algebraic computing and tensors in general relativity

2019

Albert Einstein's field equations for general relativity have proven to be revolutionary in modern physics. There remains the possibility that many predictions may still lie within these equations. The thesis examines the use of software in assisting the exploration of the potential wealth of information in the equations.

Numerical relativity: challenges for computational science

Acta Numerica, 1999

We describe the burgeoning field of numerical relativity, which aims to solve Einstein's equations of general relativity numerically. The field presents many questions that may interest numerical analysts, especially problems related to nonlinear partial differential equations: elliptic systems, hyperbolic systems, and mixed systems. There are many novel features, such as dealing with boundaries when black holes are excised from the computational domain, or how to even pose the problem computationally when the coordinates must be determined during the evolution from initial data. The most important unsolved problem is that there is no known general 3-dimensional algorithm that can evolve Einstein's equations with black holes that is stable. This review is meant to be an introduction that will enable numerical analysts and other computational scientists to enter the field. No previous knowledge of special or general relativity is assumed.

Introduction to numerical relativity through examples

Revista Mexicana De Fisica, 2007

In these notes some examples of how to apply finite differencing to the solution of partial differential equations are presented and analyzed. The aim of this manuscript is to offer the reader a first step toward the numerical solution of sufficiently complicated and interesting problems within general relativity. The topics include the solution of the wave equation in one spatial dimension and the solution of real and complex self-gravitating scalar fields with spherical symmetry.

Investigations in Numerical Relativity

1993

Numerical relativity has come a long way in the last three decades and is now reaching a state of maturity. We are gaining a deeper understanding of the fundamental theoretical issues related to the field, from the well posedness of the Cauchy problem, to better gauge conditions, improved boundary treatment, and more realistic initial data. There has also been important work both in numerical methods and software engineering. All these developments have come together to allow the construction of several advanced fully three-dimensional codes capable of dealing with both matter and black holes. In this manuscript I make a brief review the current status of the field.

New Formalism for Numerical Relativity

Physical Review Letters, 1995

We present a new formulation of the Einstein equations that casts them in an explicitly rst order, ux-conservative, hyperbolic form. We show that this now can be done for a wide class of time slicing conditions, including maximal slicing, making it potentially very useful for numerical relativity. This development permits the application to the Einstein equations of advanced numerical methods developed to solve the uid dynamic equations, without overly restricting the time slicing, for the rst time. The full set of characteristic elds and speeds is explicitly given.