105 research outputs found
Fast and Slow solutions in General Relativity: The Initialization Procedure
We apply recent results in the theory of PDE, specifically in problems with
two different time scales, on Einstein's equations near their Newtonian limit.
The results imply a justification to Postnewtonian approximations when
initialization procedures to different orders are made on the initial data. We
determine up to what order initialization is needed in order to detect the
contribution to the quadrupole moment due to the slow motion of a massive body
as distinct from initial data contributions to fast solutions and prove that
such initialization is compatible with the constraint equations. Using the
results mentioned the first Postnewtonian equations and their solutions in
terms of Green functions are presented in order to indicate how to proceed in
calculations with this approach.Comment: 14 pages, Late
The Initial-Boundary Value Problem in General Relativity
In this article we summarize what is known about the initial-boundary value
problem for general relativity and discuss present problems related to it.Comment: 11 pages, 2 figures. Contribution to a special volume for Mario
Castagnino's seventy fifth birthda
Strongly hyperbolic second order Einstein's evolution equations
BSSN-type evolution equations are discussed. The name refers to the
Baumgarte, Shapiro, Shibata, and Nakamura version of the Einstein evolution
equations, without introducing the conformal-traceless decomposition but
keeping the three connection functions and including a densitized lapse. It is
proved that a pseudo-differential first order reduction of these equations is
strongly hyperbolic. In the same way, densitized Arnowitt-Deser-Misner
evolution equations are found to be weakly hyperbolic. In both cases, the
positive densitized lapse function and the spacelike shift vector are arbitrary
given fields. This first order pseudodifferential reduction adds no extra
equations to the system and so no extra constraints.Comment: LaTeX, 16 pages, uses revtex4. Referee corections and new appendix
added. English grammar improved; typos correcte
Geometrically motivated hyperbolic coordinate conditions for numerical relativity: Analysis, issues and implementations
We study the implications of adopting hyperbolic driver coordinate conditions
motivated by geometrical considerations. In particular, conditions that
minimize the rate of change of the metric variables. We analyze the properties
of the resulting system of equations and their effect when implementing
excision techniques. We find that commonly used coordinate conditions lead to a
characteristic structure at the excision surface where some modes are not of
outflow-type with respect to any excision boundary chosen inside the horizon.
Thus, boundary conditions are required for these modes. Unfortunately, the
specification of these conditions is a delicate issue as the outflow modes
involve both gauge and main variables. As an alternative to these driver
equations, we examine conditions derived from extremizing a scalar constructed
from Killing's equation and present specific numerical examples.Comment: 9 figure
Numerical stability of a new conformal-traceless 3+1 formulation of the Einstein equation
There is strong evidence indicating that the particular form used to recast
the Einstein equation as a 3+1 set of evolution equations has a fundamental
impact on the stability properties of numerical evolutions involving black
holes and/or neutron stars. Presently, the longest lived evolutions have been
obtained using a parametrized hyperbolic system developed by Kidder, Scheel and
Teukolsky or a conformal-traceless system introduced by Baumgarte, Shapiro,
Shibata and Nakamura. We present a new conformal-traceless system. While this
new system has some elements in common with the
Baumgarte-Shapiro-Shibata-Nakamura system, it differs in both the type of
conformal transformations and how the non-linear terms involving the extrinsic
curvature are handled. We show results from 3D numerical evolutions of a
single, non-rotating black hole in which we demonstrate that this new system
yields a significant improvement in the life-time of the simulations.Comment: 7 pages, 2 figure
Constructing hyperbolic systems in the Ashtekar formulation of general relativity
Hyperbolic formulations of the equations of motion are essential technique
for proving the well-posedness of the Cauchy problem of a system, and are also
helpful for implementing stable long time evolution in numerical applications.
We, here, present three kinds of hyperbolic systems in the Ashtekar formulation
of general relativity for Lorentzian vacuum spacetime. We exhibit several (I)
weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric
hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's
original equations form a weakly hyperbolic system. We discuss how gauge
conditions and reality conditions are constrained during each step toward
constructing a symmetric hyperbolic system.Comment: 15 pages, RevTeX, minor changes in Introduction. published as Int. J.
Mod. Phys. D 9 (2000) 1
Schwarzschild Tests of the Wahlquist-Estabrook-Buchman-Bardeen Tetrad Formulation for Numerical Relativity
A first order symmetric hyperbolic tetrad formulation of the Einstein
equations developed by Estabrook and Wahlquist and put into a form suitable for
numerical relativity by Buchman and Bardeen (the WEBB formulation) is adapted
to explicit spherical symmetry and tested for accuracy and stability in the
evolution of spherically symmetric black holes (the Schwarzschild geometry).
The lapse and shift which specify the evolution of the coordinates relative to
the tetrad congruence are reset at frequent time intervals to keep the
constant-time hypersurfaces nearly orthogonal to the tetrad congruence and the
spatial coordinate satisfying a kind of minimal rate of strain condition. By
arranging through initial conditions that the constant-time hypersurfaces are
asymptotically hyperbolic, we simplify the boundary value problem and improve
stability of the evolution. Results are obtained for both tetrad gauges
(``Nester'' and ``Lorentz'') of the WEBB formalism using finite difference
numerical methods. We are able to obtain stable unconstrained evolution with
the Nester gauge for certain initial conditions, but not with the Lorentz
gauge.Comment: (accepted by Phys. Rev. D) minor changes; typos correcte
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