2,132 research outputs found
Asymptotic properties of the development of conformally flat data near spatial infinity
Certain aspects of the behaviour of the gravitational field near null and
spatial infinity for the developments of asymptotically Euclidean, conformally
flat initial data sets are analysed. Ideas and results from two different
approaches are combined: on the one hand the null infinity formalism related to
the asymptotic characteristic initial value problem and on the other the
regular Cauchy initial value problem at spatial infinity which uses Friedrich's
representation of spatial infinity as a cylinder. The decay of the Weyl tensor
for the developments of the class of initial data under consideration is
analysed under some existence and regularity assumptions for the asymptotic
expansions obtained using the cylinder at spatial infinity. Conditions on the
initial data to obtain developments satisfying the Peeling Behaviour are
identified. Further, the decay of the asymptotic shear on null infinity is also
examined as one approaches spatial infinity. This decay is related to the
possibility of selecting the Poincar\'e group out of the BMS group in a
canonical fashion. It is found that for the class of initial data under
consideration, if the development peels, then the asymptotic shear goes to zero
at spatial infinity. Expansions of the Bondi mass are also examined. Finally,
the Newman-Penrose constants of the spacetime are written in terms of initial
data quantities and it is shown that the constants defined at future null
infinity are equal to those at past null infinity.Comment: 24 pages, 1 figur
Can one detect a non-smooth null infinity?
It is shown that the precession of a gyroscope can be used to elucidate the
nature of the smoothness of the null infinity of an asymptotically flat
spacetime (describing an isolated body). A model for which the effects of
precession in the non-smooth null infinity case are of order is
proposed. By contrast, in the smooth version the effects are of order .
This difference should provide an effective criterion to decide on the nature
of the smoothness of null infinity.Comment: 6 pages, to appear in Class. Quantum Gra
Painleve-Gullstrand Coordinates for the Kerr Solution
We construct a coordinate system for the Kerr solution, based on the zero
angular momentum observers dropped from infinity, which generalizes the
Painleve-Gullstrand coordinate system for the Schwarzschild solution. The Kerr
metric can then be interpreted as describing space flowing on a (curved)
Riemannian 3-manifold. The stationary limit arises as the set of points on this
manifold where the speed of the flow equals the speed of light, and the
horizons as the set of points where the radial speed equals the speed of light.
A deeper analysis of what is meant by the flow of space reveals that the
acceleration of free-falling objects is generally not in the direction of this
flow. Finally, we compare the new coordinate system with the closely related
Doran coordinate system.Comment: 6 pages; v2: new section, matches final published version; v3: sign
error in the expression of the function delta correcte
Extreme Bowen-York initial data
The Bowen-York family of spinning black hole initial data depends essentially
on one, positive, free parameter. The extreme limit corresponds to making this
parameter equal to zero. This choice represents a singular limit for the
constraint equations. We prove that in this limit a new solution of the
constraint equations is obtained. These initial data have similar properties to
the extreme Kerr and Reissner-Nordstrom black hole initial data. In particular,
in this limit one of the asymptotic ends changes from asymptotically flat to
cylindrical. The existence proof is constructive, we actually show that a
sequence of Bowen-York data converges to the extreme solution.Comment: 21 page
Soluble two-species diffusion-limited Models in arbitrary dimensions
A class of two-species ({\it three-states}) bimolecular diffusion-limited
models of classical particles with hard-core reacting and diffusing in a
hypercubic lattice of arbitrary dimension is investigated. The manifolds on
which the equations of motion of the correlation functions close, are
determined explicitly. This property allows to solve for the density and the
two-point (two-time) correlation functions in arbitrary dimension for both, a
translation invariant class and another one where translation invariance is
broken. Systems with correlated as well as uncorrelated, yet random initial
states can also be treated exactly by this approach. We discuss the asymptotic
behavior of density and correlation functions in the various cases. The
dynamics studied is very rich.Comment: 28 pages, 0 figure. To appear in Physical Review E (February 2001
Gravitational waves and dragging effects
Linear and rotational dragging effects of gravitational waves on local
inertial frames are studied in purely vacuum spacetimes. First the linear
dragging caused by a simple cylindrical pulse is investigated. Surprisingly
strong transversal effects of the pulse are exhibited. The angular momentum in
cylindrically symmetric spacetimes is then defined and confronted with some
results in literature. In the main part, the general procedure is developed for
studying weak gravitational waves with translational but not axial symmetry
which can carry angular momentum. After a suitable averaging the rotation of
local inertial frames due to such rotating waves can be calculated explicitly
and illustrated graphically. This is done in detail in the accompanying paper.
Finally, the rotational dragging is given for strong cylindrical waves
interacting with a rotating cosmic string with a small angular momentum.Comment: Scheduled to appear in Class. Quantum Grav. July 200
Symmetry and species segregation in diffusion-limited pair annihilation
We consider a system of q diffusing particle species A_1,A_2,...,A_q that are
all equivalent under a symmetry operation. Pairs of particles may annihilate
according to A_i + A_j -> 0 with reaction rates k_{ij} that respect the
symmetry, and without self-annihilation (k_{ii} = 0). In spatial dimensions d >
2 mean-field theory predicts that the total particle density decays as n(t) ~
1/t, provided the system remains spatially uniform. We determine the conditions
on the matrix k under which there exists a critical segregation dimension
d_{seg} below which this uniformity condition is violated; the symmetry between
the species is then locally broken. We argue that in those cases the density
decay slows down to n(t) ~ t^{-d/d_{seg}} for 2 < d < d_{seg}. We show that
when d_{seg} exists, its value can be expressed in terms of the ratio of the
smallest to the largest eigenvalue of k. The existence of a conservation law
(as in the special two-species annihilation A + B -> 0), although sufficient
for segregation, is shown not to be a necessary condition for this phenomenon
to occur. We work out specific examples and present Monte Carlo simulations
compatible with our analytical results.Comment: latex, 19 pages, 3 eps figures include
On the construction of a geometric invariant measuring the deviation from Kerr data
This article contains a detailed and rigorous proof of the construction of a
geometric invariant for initial data sets for the Einstein vacuum field
equations. This geometric invariant vanishes if and only if the initial data
set corresponds to data for the Kerr spacetime, and thus, it characterises this
type of data. The construction presented is valid for boosted and non-boosted
initial data sets which are, in a sense, asymptotically Schwarzschildean. As a
preliminary step to the construction of the geometric invariant, an analysis of
a characterisation of the Kerr spacetime in terms of Killing spinors is carried
out. A space spinor split of the (spacetime) Killing spinor equation is
performed, to obtain a set of three conditions ensuring the existence of a
Killing spinor of the development of the initial data set. In order to
construct the geometric invariant, we introduce the notion of approximate
Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the
initial hypersurface and satisfy a certain second order elliptic equation
---the approximate Killing spinor equation. This equation arises as the
Euler-Lagrange equation of a non-negative integral functional. This functional
constitutes part of our geometric invariant ---however, the whole functional
does not come from a variational principle. The asymptotic behaviour of
solutions to the approximate Killing spinor equation is studied and an
existence theorem is presented.Comment: 36 pages. Updated references. Technical details correcte
Persistence in the One-Dimensional A+B -> 0 Reaction-Diffusion Model
The persistence properties of a set of random walkers obeying the A+B -> 0
reaction, with equal initial density of particles and homogeneous initial
conditions, is studied using two definitions of persistence. The probability,
P(t), that an annihilation process has not occurred at a given site has the
asymptotic form , where is the
persistence exponent (``type I persistence''). We argue that, for a density of
particles , this non-trivial exponent is identical to that governing
the persistence properties of the one-dimensional diffusion equation, where
. In the case of an initially low density, , we find asymptotically. The probability that a site
remains unvisited by any random walker (``type II persistence'') is also
investigated and found to decay with a stretched exponential form, , provided . A heuristic argument
for this behavior, based on an exactly solvable toy model, is presented.Comment: 11 RevTeX pages, 19 EPS figure
- âŠ