867 research outputs found
On the well posedness of Robinson Trautman Maxwell solutions
We show that the so called Robinson-Trautman-Maxwell equations do not
constitute a well posed initial value problem. That is, the dependence of the
solution on the initial data is not continuous in any norm built out from the
initial data and a finite number of its derivatives. Thus, they can not be used
to solve for solutions outside the analytic domain.Comment: 9 page
On the well posedness of the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein's field equations
We give a well posed initial value formulation of the
Baumgarte-Shapiro-Shibata-Nakamura form of Einstein's equations with gauge
conditions given by a Bona-Masso like slicing condition for the lapse and a
frozen shift. This is achieved by introducing extra variables and recasting the
evolution equations into a first order symmetric hyperbolic system. We also
consider the presence of artificial boundaries and derive a set of boundary
conditions that guarantee that the resulting initial-boundary value problem is
well posed, though not necessarily compatible with the constraints. In the case
of dynamical gauge conditions for the lapse and shift we obtain a class of
evolution equations which are strongly hyperbolic and so yield well posed
initial value formulations
Stable radiation-controlling boundary conditions for the generalized harmonic Einstein equations
This paper is concerned with the initial-boundary value problem for the
Einstein equations in a first-order generalized harmonic formulation. We impose
boundary conditions that preserve the constraints and control the incoming
gravitational radiation by prescribing data for the incoming fields of the Weyl
tensor. High-frequency perturbations about any given spacetime (including a
shift vector with subluminal normal component) are analyzed using the
Fourier-Laplace technique. We show that the system is boundary-stable. In
addition, we develop a criterion that can be used to detect weak instabilities
with polynomial time dependence, and we show that our system does not suffer
from such instabilities. A numerical robust stability test supports our claim
that the initial-boundary value problem is most likely to be well-posed even if
nonzero initial and source data are included.Comment: 27 pages, 4 figures; more numerical results and references added,
several minor amendments; version accepted for publication in Class. Quantum
Gra
Constraint-preserving boundary treatment for a harmonic formulation of the Einstein equations
We present a set of well-posed constraint-preserving boundary conditions for
a first-order in time, second-order in space, harmonic formulation of the
Einstein equations. The boundary conditions are tested using robust stability,
linear and nonlinear waves, and are found to be both less reflective and
constraint preserving than standard Sommerfeld-type boundary conditions.Comment: 18 pages, 7 figures, accepted in CQ
Binary neutron-star mergers with Whisky and SACRA: First quantitative comparison of results from independent general-relativistic hydrodynamics codes
We present the first quantitative comparison of two independent
general-relativistic hydrodynamics codes, the Whisky code and the SACRA code.
We compare the output of simulations starting from the same initial data and
carried out with the configuration (numerical methods, grid setup, resolution,
gauges) which for each code has been found to give consistent and sufficiently
accurate results, in particular in terms of cleanness of gravitational
waveforms. We focus on the quantities that should be conserved during the
evolution (rest mass, total mass energy, and total angular momentum) and on the
gravitational-wave amplitude and frequency. We find that the results produced
by the two codes agree at a reasonable level, with variations in the different
quantities but always at better than about 10%.Comment: Published on Phys. Rev.
Improved outer boundary conditions for Einstein's field equations
In a recent article, we constructed a hierarchy B_L of outer boundary
conditions for Einstein's field equations with the property that, for a
spherical outer boundary, it is perfectly absorbing for linearized
gravitational radiation up to a given angular momentum number L. In this
article, we generalize B_2 so that it can be applied to fairly general
foliations of spacetime by space-like hypersurfaces and general outer boundary
shapes and further, we improve B_2 in two steps: (i) we give a local boundary
condition C_2 which is perfectly absorbing including first order contributions
in 2M/R of curvature corrections for quadrupolar waves (where M is the mass of
the spacetime and R is a typical radius of the outer boundary) and which
significantly reduces spurious reflections due to backscatter, and (ii) we give
a non-local boundary condition D_2 which is exact when first order corrections
in 2M/R for both curvature and backscatter are considered, for quadrupolar
radiation.Comment: accepted Class. Quant. Grav. numerical relativity special issue; 17
pages and 1 figur
Numerical evolutions of a black hole-neutron star system in full General Relativity: I. Head-on collision
We present the first simulations in full General Relativity of the head-on
collision between a neutron star and a black hole of comparable mass. These
simulations are performed through the solution of the Einstein equations
combined with an accurate solution of the relativistic hydrodynamics equations
via high-resolution shock-capturing techniques. The initial data is obtained by
following the York-Lichnerowicz conformal decomposition with the assumption of
time symmetry. Unlike other relativistic studies of such systems, no limitation
is set for the mass ratio between the black hole and the neutron star, nor on
the position of the black hole, whose apparent horizon is entirely contained
within the computational domain. The latter extends over ~400M and is covered
with six levels of fixed mesh refinement. Concentrating on a prototypical
binary system with mass ratio ~6, we find that although a tidal deformation is
evident the neutron star is accreted promptly and entirely into the black hole.
While the collision is completed before ~300M, the evolution is carried over up
to ~1700M, thus providing time for the extraction of the gravitational-wave
signal produced and allowing for a first estimate of the radiative efficiency
of processes of this type.Comment: 16 pages, 12 figure
Introduction to dynamical horizons in numerical relativity
This paper presents a quasi-local method of studying the physics of dynamical
black holes in numerical simulations. This is done within the dynamical horizon
framework, which extends the earlier work on isolated horizons to
time-dependent situations. In particular: (i) We locate various kinds of
marginal surfaces and study their time evolution. An important ingredient is
the calculation of the signature of the horizon, which can be either spacelike,
timelike, or null. (ii) We generalize the calculation of the black hole mass
and angular momentum, which were previously defined for axisymmetric isolated
horizons to dynamical situations. (iii) We calculate the source multipole
moments of the black hole which can be used to verify that the black hole
settles down to a Kerr solution. (iv) We also study the fluxes of energy
crossing the horizon, which describes how a black hole grows as it accretes
matter and/or radiation.
We describe our numerical implementation of these concepts and apply them to
three specific test cases, namely, the axisymmetric head-on collision of two
black holes, the axisymmetric collapse of a neutron star, and a
non-axisymmetric black hole collision with non-zero initial orbital angular
momentum.Comment: 20 pages, 16 figures, revtex4. Several smaller changes, some didactic
content shortene
Existence of families of spacetimes with a Newtonian limit
J\"urgen Ehlers developed \emph{frame theory} to better understand the
relationship between general relativity and Newtonian gravity. Frame theory
contains a parameter , which can be thought of as , where
is the speed of light. By construction, frame theory is equivalent to general
relativity for , and reduces to Newtonian gravity for .
Moreover, by setting \ep=\sqrt{\lambda}, frame theory provides a framework to
study the Newtonian limit \ep \searrow 0 (i.e. ). A number of
ideas relating to frame theory that were introduced by J\"urgen have
subsequently found important applications to the rigorous study of both the
Newtonian limit and post-Newtonian expansions. In this article, we review frame
theory and discuss, in a non-technical fashion, some of the rigorous results on
the Newtonian limit and post-Newtonian expansions that have followed from
J\"urgen's work
Hyperboloidal slices for the wave equation of Kerr-Schild metrics and numerical applications
We present new results from two open source codes, using finite differencing
and pseudo-spectral methods for the wave equations in (3+1) dimensions. We use
a hyperboloidal transformation which allows direct access to null infinity and
simplifies the control over characteristic speeds on Kerr-Schild backgrounds.
We show that this method is ideal for attaching hyperboloidal slices or for
adapting the numerical resolution in certain spacetime regions. As an example
application, we study late-time Kerr tails of sub-dominant modes and obtain new
insight into the splitting of decay rates. The involved conformal wave equation
is freed of formally singular terms whose numerical evaluation might be
problematically close to future null infinity.Comment: 15 pages, 12 figure
- …