scholarly journals Rotating dust solutions of Einstein’s equations with 3-dimensional symmetry groups. I. Two Killing fields spanned on uα and wα

1998 ◽  
Vol 39 (1) ◽  
pp. 380-400 ◽  
Author(s):  
Andrzej Krasiński
Keyword(s):  
Symmetry Groups ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
3 Dimensional ◽  
Killing Fields

Numerical relativity: challenges for computational science

Acta Numerica ◽  
1999 ◽  
Vol 8 ◽  
pp. 1-45 ◽  
Author(s):  
Gregory B. Cook ◽  
Saul A. Teukolsky
Keyword(s):  
Black Holes ◽  
General Relativity ◽  
Numerical Relativity ◽  
Hyperbolic Systems ◽  
Nonlinear Partial Differential Equations ◽  
Elliptic Systems ◽  
Computational Domain ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
3 Dimensional

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 even to 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.

scholarly journals Einstein's equations and the embedding of 3-dimensional CR manifolds

2008 ◽  
Vol 57 (7) ◽  
pp. 3131-3176 ◽  
Author(s):  
C. Denson Hill ◽  
Jerzy Lewandowski ◽  
Pawel Nurowski
Keyword(s):  
Einstein’S Equations ◽  
Cr Manifolds ◽  
Einstein's Equations ◽  
3 Dimensional

COVARIANT CONFORMAL DECOMPOSITION OF EINSTEIN EQUATIONS

2002 ◽  
Vol 17 (20) ◽  
pp. 2762-2762
Author(s):  
E. GOURGOULHON ◽  
J. NOVAK
Keyword(s):  
Numerical Simulations ◽  
Extrinsic Curvature ◽  
Einstein Equations ◽  
Numerical Implementation ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Tensor Density ◽  
Dirac Gauge ◽  
Ill Posed ◽  
Isolated Systems

It has been shown1,2 that the usual 3+1 form of Einstein's equations may be ill-posed. This result has been previously observed in numerical simulations3,4. We present a 3+1 type formalism inspired by these works to decompose Einstein's equations. This decomposition is motivated by the aim of stable numerical implementation and resolution of the equations. We introduce the conformal 3-"metric" (scaled by the determinant of the usual 3-metric) which is a tensor density of weight -2/3. The Einstein equations are then derived in terms of this "metric", of the conformal extrinsic curvature and in terms of the associated derivative. We also introduce a flat 3-metric (the asymptotic metric for isolated systems) and the associated derivative. Finally, the generalized Dirac gauge (introduced by Smarr and York5) is used in this formalism and some examples of formulation of Einstein's equations are shown.

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