Related papers: Mixed Hyperbolic - Second-Order Parabolic Formulat…
In a recent paper the first coauthor presented a new parabolic extension (PADM) of the standard 3+1 Arnowitt, Deser, Misner formulation of the equations of general relativity. By parabolizing first-order ADM in a certain way, the PADM…
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…
The Arnowitt-Deser-Misner (ADM) evolution equations for the induced metric and the extrinsic-curvature tensor of the spacelike surfaces which foliate the space-time manifold in canonical general relativity are a first-order system of…
We present two families of first-order in time and second-order in space formulations of the Einstein equations (variants of the Arnowitt-Deser-Misner formulation) that admit a complete set of characteristic variables and a conserved energy…
We study asymptotically constrained systems for numerical integration of the Einstein equations, which are intended to be robust against perturbative errors for the free evolution of the initial data. First, we examine the previously…
We consider two strongly hyperbolic Hamiltonian formulations of general relativity and their numerical integration with a free and a partially constrained symplectic integrator. In those formulations we use hyperbolic drivers for the shift…
We introduce a proposal to modify Einstein's equations by embedding them in a larger symmetric hyperbolic system. The additional dynamical variables of the modified system are essentially first integrals of the original constraints. The…
There is a tendency to write the equations of general relativity as a first order symmetric system of time dependent partial differential equations. However, for numerical reasons, it might be advantageous to use a second order formulation…
Einstein's system of equations in the ADM decomposition involves two subsystems of equations: evolution equations and constraint equations. For numerical relativity, one typically solves the constraint equations only on the initial time…
The evolution equations of Einstein's theory and of Maxwell's theory---the latter used as a simple model to illustrate the former--- are written in gauge covariant first order symmetric hyperbolic form with only physically natural…
We obtain the Arnowitt-Deser-Misner formulation of general relativity in $n$ dimensions ($n \geq 3$) from its either $SO(n-1,1)$ [$SO(n)$] or $SO(n-1)$ Palatini Hamiltonian formulations and vice versa [we recall that $SO(n-1,1)$ [$SO(n)$]…
Two geometrical well-posed hyperbolic formulations of general relativity are described. One admits any time-slicing which preserves a generalized harmonic condition. The other admits arbitrary time-slicings. Both systems have only the…
The well-posedness of the gravitational equations of $f(R)$ gravity are studied in this paper. Three formulations of the $f(R)$ gravity with dynamical shifts (which are all based on the Arnowitt-Deser-Misner (ADM) formalism of the…
Motivated by the initial-boundary value problem for the Einstein equations, we propose a definition of symmetric hyperbolicity for systems of evolution equations that are first order in time but second order in space. This can be used to…
We study the transformation leading from Arnowitt, Deser, Misner (ADM) Hamiltonian formulation of General Relativity (GR) to the $\Gamma\Gamma$ metric Hamiltonian formulation derived from the Lagrangian density which was firstly proposed by…
The authors present concepts and mathematical developments which give rise to the Hamiltonian formulation of Einstein's general relativity, first introduced by Arnowitt, Deser, and Misner. All the geometrical quantities needed for the…
Our attempts to find an explanation for quantum behavior of the Early Universe appeal, as a rule, to the Wheeler - DeWitt Quantum Geometrodynamics which relies upon Hamiltonian formulation of General Relativity proposed by Arnowitt, Deser…
Einstein's equations for general relativity, when viewed as a dynamical system for evolving initial data, have a serious flaw: they cannot be proven to be well-posed (except in special coordinates). That is, they do not produce unique…
Strong coupling expansion is computed for the Einstein equations in vacuum in the Arnowitt-Deser-Misner (ADM) formalism. The series is given by the duality principle in perturbation theory as presented in [M.Frasca, Phys. Rev. A 58, 3439…
$3+1$ formulations of the Einstein field equations have become an invaluable tool in Numerical relativity, having been used successfully in modeling spacetimes of black hole collisions, stellar collapse and other complex systems. It is…