6,903 research outputs found
Solving Einstein's Equations With Dual Coordinate Frames
A method is introduced for solving Einstein's equations using two distinct
coordinate systems. The coordinate basis vectors associated with one system are
used to project out components of the metric and other fields, in analogy with
the way fields are projected onto an orthonormal tetrad basis. These field
components are then determined as functions of a second independent coordinate
system. The transformation to the second coordinate system can be thought of as
a mapping from the original ``inertial'' coordinate system to the computational
domain. This dual-coordinate method is used to perform stable numerical
evolutions of a black-hole spacetime using the generalized harmonic form of
Einstein's equations in coordinates that rotate with respect to the inertial
frame at infinity; such evolutions are found to be generically unstable using a
single rotating coordinate frame. The dual-coordinate method is also used here
to evolve binary black-hole spacetimes for several orbits. The great
flexibility of this method allows comoving coordinates to be adjusted with a
feedback control system that keeps the excision boundaries of the holes within
their respective apparent horizons.Comment: Updated to agree with published versio
Evolutions of Magnetized and Rotating Neutron Stars
We study the evolution of magnetized and rigidly rotating neutron stars
within a fully general relativistic implementation of ideal
magnetohydrodynamics with no assumed symmetries in three spatial dimensions.
The stars are modeled as rotating, magnetized polytropic stars and we examine
diverse scenarios to study their dynamics and stability properties. In
particular we concentrate on the stability of the stars and possible critical
behavior. In addition to their intrinsic physical significance, we use these
evolutions as further tests of our implementation which incorporates new
developments to handle magnetized systems.Comment: 12 pages, 8 figure
Turduckening black holes: an analytical and computational study
We provide a detailed analysis of several aspects of the turduckening
technique for evolving black holes. At the analytical level we study the
constraint propagation for a general family of BSSN-type formulation of
Einstein's field equations and identify under what conditions the turducken
procedure is rigorously justified and under what conditions constraint
violations will propagate to the outside of the black holes. We present
high-resolution spherically symmetric studies which verify our analytical
predictions. Then we present three dimensional simulations of single distorted
black holes using different variations of the turduckening method and also the
puncture method. We study the effect that these different methods have on the
coordinate conditions, constraint violations, and extracted gravitational
waves. We find that the waves agree up to small but non-vanishing differences,
caused by escaping superluminal gauge modes. These differences become smaller
with increasing detector location.Comment: Minor changes to match the final version to appear in PR
Boundary Conditions for the Einstein Evolution System
New boundary conditions are constructed and tested numerically for a general
first-order form of the Einstein evolution system. These conditions prevent
constraint violations from entering the computational domain through timelike
boundaries, allow the simulation of isolated systems by preventing physical
gravitational waves from entering the computational domain, and are designed to
be compatible with the fixed-gauge evolutions used here. These new boundary
conditions are shown to be effective in limiting the growth of constraints in
3D non-linear numerical evolutions of dynamical black-hole spacetimes.Comment: 21 pages, 12 figures, submitted to PR
Scalar field induced oscillations of neutron stars and gravitational collapse
We study the interaction of massless scalar fields with self-gravitating
neutron stars by means of fully dynamic numerical simulations of the
Einstein-Klein-Gordon perfect fluid system. Our investigation is restricted to
spherical symmetry and the neutron stars are approximated by relativistic
polytropes. Studying the nonlinear dynamics of isolated neutron stars is very
effectively performed within the characteristic formulation of general
relativity, in which the spacetime is foliated by a family of outgoing light
cones. We are able to compactify the entire spacetime on a computational grid
and simultaneously impose natural radiative boundary conditions and extract
accurate radiative signals. We study the transfer of energy from the scalar
field to the fluid star. We find, in particular, that depending on the
compactness of the neutron star model, the scalar wave forces the neutron star
either to oscillate in its radial modes of pulsation or to undergo
gravitational collapse to a black hole on a dynamical timescale. The radiative
signal, read off at future null infinity, shows quasi-normal oscillations
before the setting of a late time power-law tail.Comment: 12 pages, 13 figures, submitted to Phys. Rev.
Controlling the growth of constraints in hyperbolic evolution systems
Motivated by the need to control the exponential growth of constraint violations in numerical solutions of the Einstein evolution equations, two methods are studied here for controlling this growth in general hyperbolic evolution systems. The first method adjusts the evolution equations dynamically, by adding multiples of the constraints, in a way designed to minimize this growth. The second method imposes special constraint preserving boundary conditions on the incoming components of the dynamical fields. The efficacy of these methods is tested by using them to control the growth of constraints in fully dynamical 3D numerical solutions of a particular representation of the Maxwell equations that is subject to constraint violations. The constraint preserving boundary conditions are found to be much more effective than active constraint control in the case of this Maxwell system
Hamiltonian Relaxation
Due to the complexity of the required numerical codes, many of the new
formulations for the evolution of the gravitational fields in numerical
relativity are not tested on binary evolutions. We introduce in this paper a
new testing ground for numerical methods based on the simulation of binary
neutron stars. This numerical setup is used to develop a new technique, the
Hamiltonian relaxation (HR), that is benchmarked against the currently most
stable simulations based on the BSSN method. We show that, while the length of
the HR run is somewhat shorter than the equivalent BSSN simulation, the HR
technique improves the overall quality of the simulation, not only regarding
the satisfaction of the Hamiltonian constraint, but also the behavior of the
total angular momentum of the binary. The latest quantity agrees well with
post-Newtonian estimations for point-mass binaries in circular orbits.Comment: More detailed description of the numerical implementation added and
some typos corrected. Version accepted for publication in Class. and Quantum
Gravit
Impact of densitized lapse slicings on evolutions of a wobbling black hole
We present long-term stable and second-order convergent evolutions of an
excised wobbling black hole. Our results clearly demonstrate that the use of a
densitized lapse function extends the lifetime of simulations dramatically. We
also show the improvement in the stability of single static black holes when an
algebraic densitized lapse condition is applied. In addition, we introduce a
computationally inexpensive approach for tracking the location of the
singularity suitable for mildly distorted black holes. The method is based on
investigating the fall-off behavior and asymmetry of appropriate grid
variables. This simple tracking method allows one to adjust the location of the
excision region to follow the coordinate motion of the singularity.Comment: 10 pages, 8 figure
Optimal Constraint Projection for Hyperbolic Evolution Systems
Techniques are developed for projecting the solutions of symmetric hyperbolic
evolution systems onto the constraint submanifold (the constraint-satisfying
subset of the dynamical field space). These optimal projections map a field
configuration to the ``nearest'' configuration in the constraint submanifold,
where distances between configurations are measured with the natural metric on
the space of dynamical fields. The construction and use of these projections is
illustrated for a new representation of the scalar field equation that exhibits
both bulk and boundary generated constraint violations. Numerical simulations
on a black-hole background show that bulk constraint violations cannot be
controlled by constraint-preserving boundary conditions alone, but are
effectively controlled by constraint projection. Simulations also show that
constraint violations entering through boundaries cannot be controlled by
constraint projection alone, but are controlled by constraint-preserving
boundary conditions. Numerical solutions to the pathological scalar field
system are shown to converge to solutions of a standard representation of the
scalar field equation when constraint projection and constraint-preserving
boundary conditions are used together.Comment: final version with minor changes; 16 pages, 14 figure
Energy Norms and the Stability of the Einstein Evolution Equations
The Einstein evolution equations may be written in a variety of equivalent
analytical forms, but numerical solutions of these different formulations
display a wide range of growth rates for constraint violations. For symmetric
hyperbolic formulations of the equations, an exact expression for the growth
rate is derived using an energy norm. This expression agrees with the growth
rate determined by numerical solution of the equations. An approximate method
for estimating the growth rate is also derived. This estimate can be evaluated
algebraically from the initial data, and is shown to exhibit qualitatively the
same dependence as the numerically-determined rate on the parameters that
specify the formulation of the equations. This simple rate estimate therefore
provides a useful tool for finding the most well-behaved forms of the evolution
equations.Comment: Corrected typos; to appear in Physical Review
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