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We review some well posed formulations of the evolution part of the Cauchy problem of General Relativity that we have recently obtained. We include also a new first order symmetric hyperbolic system based directly on the Riemann tensor and…
The principle part of Einstein equations in the harmonic gauge consists of a constrained system of 10 curved space wave equations for the components of the space-time metric. A new formulation of constraint-preserving boundary conditions of…
We have recently constructed a numerical code that evolves a spherically symmetric spacetime using a hyperbolic formulation of Einstein's equations. For the case of a Schwarzschild black hole, this code works well at early times, but…
The effectiveness of the hyperbolic relaxation method for solving the Einstein constraint equations numerically is studied here on a variety of compact orientable three-manifolds. Convergent numerical solutions are found using this method…
The ADM Hamiltonian formulation of general relativity with prescribed lapse and shift is a weakly hyperbolic system of partial differential equations. In general weakly hyperbolic systems are not mathematically well posed. For well…
We present new many-parameter families of strongly and symmetric hyperbolic formulations of Einstein's equations that include quite general algebraic and live gauge conditions for the lapse. The first system that we present has 30 variables…
We report on a new 3D numerical code designed to solve the Einstein equations for general vacuum spacetimes. This code is based on the standard 3+1 approach using cartesian coordinates. We discuss the numerical techniques used in developing…
We use an orthonormal frame approach to provide a general framework for the first order hyperbolic reduction of the Einstein equations coupled to a fairly generic class of matter models. Our analysis covers the special cases of dust and…
This paper is concerned with the Einstein equations in axisymmetric vacuum spacetimes. We consider numerical evolution schemes that solve the constraint equations as well as elliptic gauge conditions at each time step. We examine two such…
Equations of motion for general gravitational connection and orthonormal coframe from the Einstein-Hilbert type action are derived. Our formulation does not fix coframe to be tangential to spatial section hence Lorentz group is still…
In order to perform accurate and stable long-term numerical integration of the Einstein equations, several hyperbolic systems have been proposed. We here report our numerical comparisons between weakly hyperbolic, strongly hyperbolic, and…
We present in this paper a 4-dimensional formulation of the Newton equations for gravitation on a Lorentzian manifold, inspired from the 1+3 and 3+1 formalisms of general relativity. We first show that the freedom on the coordinate velocity…
We discuss the initial-boundary value problem of General Relativity. Previous considerations for a toy model problem in electrodynamics motivate the introduction of a variational principle for the lapse with several attractive properties.…
We review recent efforts to re-formulate the Einstein equations for fully relativistic numerical simulations. In order to complete a long-term and accurate simulations of binary compact objects, people seek a robust set of equations against…
A new gauge driver is introduced for the generalized harmonic (GH) representation of Einstein's equation. This new driver allows a rather general class of gauge conditions to be implemented in a way that maintains the hyperbolicity of the…
Four-dimensional spacetimes foliated by a two-parameter family of homologous two-surfaces are considered in Einstein's theory of gravity. By combining a 1+(1+2) decomposition, the canonical form of the spacetime metric and a suitable…
The demand to obtain answers to highly complex problems within strong-field gravity has been met with significant progress in the numerical solution of Einstein's equations - along with some spectacular results - in various setups. We…
Through averaging the Einstein equations over transverse gravitational perturbations it is obtained a closed system of two ordinary differential equations describing macroscopic cosmological evolution of the isotropic space-flat Universe…
A new approach to gravitational gauge-invariant perturbation theory begins from the fourth-order Einstein-Ricci system, a hyperbolic formulation of gravity for arbitrary lapse and shift whose centerpiece is a wave equation for curvature. In…
Upcoming gravitational wave-experiments promise a window for discovering new physics in astronomy. Detection sensitivity of the broadband laser interferometric detectors LIGO/VIRGO may be enhanced by matched filtering with accurate…