61 research outputs found
The Algebraic-Hyperbolic Approach to the Linearized Gravitational Constraints on a Minkowski Background
An algebraic-hyperbolic method for solving the Hamiltonian and momentum
constraints has recently been shown to be well posed for general nonlinear
perturbations of the initial data for a Schwarzschild black hole. This is a new
approach to solving the constraints of Einstein's equations which does not
involve elliptic equations and has potential importance for the construction of
binary black hole data. In order to shed light on the underpinnings of this
approach, we consider its application to obtain solutions of the constraints
for linearized perturbations of Minkowski space. In that case, we find the
surprising result that there are no suitable Cauchy hypersurfaces in Minkowski
space for which the linearized algebraic-hyperbolic constraint problem is well
posed.Comment: Clarification adde
A New Way to Make Waves
I describe a new algorithm for solving nonlinear wave equations. In this
approach, evolution takes place on characteristic hypersurfaces. The algorithm
is directly applicable to electromagnetic, Yang-Mills and gravitational fields
and other systems described by second differential order hyperbolic equations.
The basic ideas should also be applicable to hydrodynamics. It is an especially
accurate and efficient way for simulating waves in regions where the
characteristics are well behaved. A prime application of the algorithm is to
Cauchy-characteristic matching, in which this new approach is matched to a
standard Cauchy evolution to obtain a global solution. In a model problem of a
nonlinear wave, this proves to be more accurate and efficient than any other
present method of assigning Cauchy outer boundary conditions. The approach was
developed to compute the gravitational wave signal produced by collisions of
two black holes. An application to colliding black holes is presented.Comment: In Proceeding of CIMENICS 2000, The Vth International Congress on
Numerical Methods in Engineering and Applied Science (Puerto La Cruz,
Venezuela, March 2000
Toward computing gravitational initial data without elliptic solvers
Two new methods have been proposed for solving the gravitational constraints
without using elliptic solvers by formulating them as either an
algebraic-hyperbolic or parabolic-hyperbolic system. Here, we compare these two
methods and present a unified computational infrastructure for their
implementation as numerical evolution codes. An important potential application
of these methods is the prescription of initial data for the simulation of
black holes. This paper is meant to support progress and activity in that
direction.Comment: Title and presentation change
Boosted Schwarzschild Metrics from a Kerr-Schild Perspective
The Kerr-Schild version of the Schwarzschild metric contains a Minkowski
background which provides a definition of a boosted black hole. There are two
Kerr-Schild versions corresponding to ingoing or outgoing principle null
directions. We show that the two corresponding Minkowski backgrounds and their
associated boosts have an unexpected difference. We analyze this difference and
discuss the implications in the nonlinear regime for the gravitational memory
effect resulting from the ejection of massive particles from an isolated
system. We show that the nonlinear effect agrees with the linearized result
based upon the retarded Green function only if the velocity of the ejected
particle corresponds to a boost symmetry of the ingoing Minkowski background. A
boost with respect to the outgoing Minkowski background is inconsistent with
the absence of ingoing radiation from past null infinity.Comment: 13 pages, matches published versio
Black hole initial data without elliptic equations
We explore whether a new method to solve the constraints of Einstein's
equations, which does not involve elliptic equations, can be applied to provide
initial data for black holes. We show that this method can be successfully
applied to a nonlinear perturbation of a Schwarzschild black hole by
establishing the well-posedness of the resulting constraint problem. We discuss
its possible generalization to the boosted, spinning multiple black hole
problem
Kerr Black Holes and Nonlinear Radiation Memory
The Minkowski background intrinsic to the Kerr-Schild version of the Kerr
metric provides a definition of a boosted spinning black hole. There are two
Kerr-Schild versions corresponding to ingoing or outgoing principal null
directions. The two corresponding Minkowski backgrounds and their associated
boosts differ drastically. This has an important implication for the
gravitational memory effect. A prior analysis of the transition of a
non-spinning Schwarzschild black hole to a boosted state showed that the memory
effect in the nonlinear regime agrees with the linearised result based upon the
retarded Green function only if the final velocity corresponds to a boost
symmetry of the ingoing Minkowski background. A boost with respect to the
outgoing Minkowski background is inconsistent with the absence of ingoing
radiation from past null infinity. We show that this results extends to the
transition of a Kerr black hole to a boosted state and apply it to set upper
and lower bounds for the boost memory effect resulting from the collision of
two spinning black holes.Comment: 17 pages, revised discussion sectio
Null cone evolution of axisymmetric vacuum spacetimes
We present the details of an algorithm for the global evolution of
asymptotically flat, axisymmetric spacetimes, based upon a characteristic
initial value formulation using null cones as evolution hypersurfaces. We
identify a new static solution of the vacuum field equations which provides an
important test bed for characteristic evolution codes. We also show how
linearized solutions of the Bondi equations can be generated by solutions of
the scalar wave equation, thus providing a complete set of test beds in the
weak field regime. These tools are used to establish that the algorithm is
second order accurate and stable, subject to a Courant-Friedrichs-Lewy
condition. In addition, the numerical versions of the Bondi mass and news
function, calculated at scri on a compactified grid, are shown to satisfy the
Bondi mass loss equation to second order accuracy. This verifies that numerical
evolution preserves the Bianchi identities. Results of numerical evolution
confirm the theorem of Christodoulou and Klainerman that in vacuum, weak
initial data evolve to a flat spacetime. For the class of asymptotically flat,
axisymmetric vacuum spacetimes, for which no nonsingular analytic solutions are
known, the algorithm provides highly accurate solutions throughout the regime
in which neither caustics nor horizons form.Comment: 25 pages, 6 figure
Some mathematical problems in numerical relativity
The main goal of numerical relativity is the long time simulation of highly
nonlinear spacetimes that cannot be treated by perturbation theory. This
involves analytic, computational and physical issues. At present, the major
impasses to achieving global simulations of physical usefulness are of an
analytic/computational nature. We present here some examples of how analytic
insight can lend useful guidance for the improvement of numerical approaches.Comment: 17 pages, 12 graphs (eps format
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