5,658 research outputs found
Initial boundary value problems for Einstein's field equations and geometric uniqueness
While there exist now formulations of initial boundary value problems for
Einstein's field equations which are well posed and preserve constraints and
gauge conditions, the question of geometric uniqueness remains unresolved. For
two different approaches we discuss how this difficulty arises under general
assumptions. So far it is not known whether it can be overcome without imposing
conditions on the geometry of the boundary. We point out a natural and
important class of initial boundary value problems which may offer
possibilities to arrive at a fully covariant formulation.Comment: 19 page
The Lundgren-Monin-Novikov Hierarchy: Kinetic Equations for Turbulence
We present an overview of recent works on the statistical description of
turbulent flows in terms of probability density functions (PDFs) in the
framework of the Lundgren-Monin-Novikov (LMN) hierarchy. Within this framework,
evolution equations for the PDFs are derived from the basic equations of fluid
motion. The closure problem arises either in terms of a coupling to multi-point
PDFs or in terms of conditional averages entering the evolution equations as
unknown functions. We mainly focus on the latter case and use data from direct
numerical simulations (DNS) to specify the unclosed terms. Apart from giving an
introduction into the basic analytical techniques, applications to
two-dimensional vorticity statistics, to the single-point velocity and
vorticity statistics of three-dimensional turbulence, to the temperature
statistics of Rayleigh-B\'enard convection and to Burgers turbulence are
discussed.Comment: Accepted for publication in C. R. Acad. Sc
3D simulations of Einstein's equations: symmetric hyperbolicity, live gauges and dynamic control of the constraints
We present three-dimensional simulations of Einstein equations implementing a
symmetric hyperbolic system of equations with dynamical lapse. The numerical
implementation makes use of techniques that guarantee linear numerical
stability for the associated initial-boundary value problem. The code is first
tested with a gauge wave solution, where rather larger amplitudes and for
significantly longer times are obtained with respect to other state of the art
implementations. Additionally, by minimizing a suitably defined energy for the
constraints in terms of free constraint-functions in the formulation one can
dynamically single out preferred values of these functions for the problem at
hand. We apply the technique to fully three-dimensional simulations of a
stationary black hole spacetime with excision of the singularity, considerably
extending the lifetime of the simulations.Comment: 21 pages. To appear in PR
Exponential Decay for Small Non-Linear Perturbations of Expanding Flat Homogeneous Cosmologies
It is shown that during expanding phases of flat homogeneous cosmologies all
small enough non-linear perturbations decay exponentially. This result holds
for a large class of perfect fluid equations of state, but notably not for very
``stiff'' fluids as the pure radiation case
First-order symmetrizable hyperbolic formulations of Einstein's equations including lapse and shift as dynamical fields
First-order hyperbolic systems are promising as a basis for numerical
integration of Einstein's equations. In previous work, the lapse and shift have
typically not been considered part of the hyperbolic system and have been
prescribed independently. This can be expensive computationally, especially if
the prescription involves solving elliptic equations. Therefore, including the
lapse and shift in the hyperbolic system could be advantageous for numerical
work. In this paper, two first-order symmetrizable hyperbolic systems are
presented that include the lapse and shift as dynamical fields and have only
physical characteristic speeds.Comment: 11 page
Conformal loop ensembles and the stress-energy tensor
We give a construction of the stress-energy tensor of conformal field theory
(CFT) as a local "object" in conformal loop ensembles CLE_\kappa, for all
values of \kappa in the dilute regime 8/3 < \kappa <= 4 (corresponding to the
central charges 0 < c <= 1, and including all CFT minimal models). We provide a
quick introduction to CLE, a mathematical theory for random loops in simply
connected domains with properties of conformal invariance, developed by
Sheffield and Werner (2006). We consider its extension to more general regions
of definition, and make various hypotheses that are needed for our construction
and expected to hold for CLE in the dilute regime. Using this, we identify the
stress-energy tensor in the context of CLE. This is done by deriving its
associated conformal Ward identities for single insertions in CLE probability
functions, along with the appropriate boundary conditions on simply connected
domains; its properties under conformal maps, involving the Schwarzian
derivative; and its one-point average in terms of the "relative partition
function." Part of the construction is in the same spirit as, but widely
generalizes, that found in the context of SLE_{8/3} by the author, Riva and
Cardy (2006), which only dealt with the case of zero central charge in simply
connected hyperbolic regions. We do not use the explicit construction of the
CLE probability measure, but only its defining and expected general properties.Comment: 49 pages, 3 figures. This is a concatenated, reduced and simplified
version of arXiv:0903.0372 and (especially) arXiv:0908.151
Numerical stability of the AA evolution system compared to the ADM and BSSN systems
We explore the numerical stability properties of an evolution system
suggested by Alekseenko and Arnold. We examine its behavior on a set of
standardized testbeds, and we evolve a single black hole with different gauges.
Based on a comparison with two other evolution systems with well-known
properties, we discuss some of the strengths and limitations of such simple
tests in predicting numerical stability in general.Comment: 16 pages, 12 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
Generic metrics and the mass endomorphism on spin three-manifolds
Let be a closed Riemannian spin manifold. The constant term in the
expansion of the Green function for the Dirac operator at a fixed point is called the mass endomorphism in associated to the metric due to
an analogy to the mass in the Yamabe problem. We show that the mass
endomorphism of a generic metric on a three-dimensional spin manifold is
nonzero. This implies a strict inequality which can be used to avoid
bubbling-off phenomena in conformal spin geometry.Comment: 8 page
On the universality of small scale turbulence
The proposed universality of small scale turbulence is investigated for a set
of measurements in a cryogenic free jet with a variation of the Reynolds number
(Re) from 8500 to 10^6. The traditional analysis of the statistics of velocity
increments by means of structure functions or probability density functions is
replaced by a new method which is based on the theory of stochastic Markovian
processes. It gives access to a more complete characterization by means of
joint probabilities of finding velocity increments at several scales. Based on
this more precise method our results call in question the concept of
universality.Comment: 4 pages, 4 figure
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