31 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 Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions
We derive explicit formulae for a set of constraints for the Einstein
equations on a null hypersurface, in arbitrary dimensions. We solve these
constraints and show that they provide necessary and sufficient conditions so
that a spacetime solution of the Cauchy problem on a characteristic cone for
the hyperbolic system of the reduced Einstein equations in wave-map gauge also
satisfies the full Einstein equations. We prove a geometric uniqueness theorem
for this Cauchy problem in the vacuum case.Comment: 83 pages, 1 figur
A large class of non constant mean curvature solutions of the Einstein constraint equations on an asymptotically hyperbolic manifold
We construct solutions of the constraint equation with non constant mean
curvature on an asymptotically hyperbolic manifold by the conformal method. Our
approach consists in decreasing a certain exponent appearing in the equations,
constructing solutions of these sub-critical equations and then in letting the
exponent tend to its true value. We prove that the solutions of the
sub-critical equations remain bounded which yields solutions of the constraint
equation unless a certain limit equation admits a non-trivial solution.
Finally, we give conditions which ensure that the limit equation admits no
non-trivial solution.Comment: remark on the equivalence between the existence of a solution to the
Lichnerowicz equation and to the prescribed scalar curvature equation added,
reference [BPB09] added, to appear in Commun. Math. Phy
The Cauchy problems for Einstein metrics and parallel spinors
We show that in the analytic category, given a Riemannian metric on a
hypersurface and a symmetric tensor on , the metric
can be locally extended to a Riemannian Einstein metric on with second
fundamental form , provided that and satisfy the constraints on
imposed by the contracted Codazzi equations. We use this fact to study the
Cauchy problem for metrics with parallel spinors in the real analytic category
and give an affirmative answer to a question raised in B\"ar, Gauduchon,
Moroianu (2005). We also answer negatively the corresponding questions in the
smooth category.Comment: 28 pages; final versio
Radiative multipole moments of integer-spin fields in curved spacetime
Radiative multipole moments of scalar, electromagnetic, and linearized
gravitational fields in Schwarzschild spacetime are computed to third order in
v in a weak-field, slow-motion approximation, where v is a characteristic
velocity associated with the motion of the source. To zeroth order in v, a
radiative moment of order l is given by the corresponding source moment
differentiated l times with respect to retarded time. At second order in v,
additional terms appear inside the spatial integrals. These are near-zone
corrections which depend on the detailed behavior of the source. At third order
in v, the correction terms occur outside the spatial integrals, so that they do
not depend on the detailed behavior of the source. These are wave-propagation
corrections which are heuristically understood as arising from the scattering
of the radiation by the spacetime curvature surrounding the source. Our
calculations show that the wave-propagation corrections take a universal form
which is independent of multipole order and field type. We also show that in
general relativity, temporal and spatial curvatures contribute equally to the
wave-propagation corrections.Comment: 34 pages, ReVTe
Local well-posedness for membranes in the light cone gauge
In this paper we consider the classical initial value problem for the bosonic
membrane in light cone gauge. A Hamiltonian reduction gives a system with one
constraint, the area preserving constraint. The Hamiltonian evolution equations
corresponding to this system, however, fail to be hyperbolic. Making use of the
area preserving constraint, an equivalent system of evolution equations is
found, which is hyperbolic and has a well-posed initial value problem. We are
thus able to solve the initial value problem for the Hamiltonian evolution
equations by means of this equivalent system. We furthermore obtain a blowup
criterion for the membrane evolution equations, and show, making use of the
constraint, that one may achieve improved regularity estimates.Comment: 29 page
Notes on a paper of Mess
These notes are a companion to the article "Lorentz spacetimes of constant
curvature" by Geoffrey Mess, which was first written in 1990 but never
published. Mess' paper will appear together with these notes in a forthcoming
issue of Geometriae Dedicata.Comment: 26 page
Newtonian Cosmology in Lagrangian Formulation: Foundations and Perturbation Theory
The ``Newtonian'' theory of spatially unbounded, self--gravitating,
pressureless continua in Lagrangian form is reconsidered. Following a review of
the pertinent kinematics, we present alternative formulations of the Lagrangian
evolution equations and establish conditions for the equivalence of the
Lagrangian and Eulerian representations. We then distinguish open models based
on Euclidean space from closed models based (without loss of generality)
on a flat torus \T^3. Using a simple averaging method we show that the
spatially averaged variables of an inhomogeneous toroidal model form a
spatially homogeneous ``background'' model and that the averages of open
models, if they exist at all, in general do not obey the dynamical laws of
homogeneous models. We then specialize to those inhomogeneous toroidal models
whose (unique) backgrounds have a Hubble flow, and derive Lagrangian evolution
equations which govern the (conformally rescaled) displacement of the
inhomogeneous flow with respect to its homogeneous background. Finally, we set
up an iteration scheme and prove that the resulting equations have unique
solutions at any order for given initial data, while for open models there
exist infinitely many different solutions for given data.Comment: submitted to G.R.G., TeX 30 pages; AEI preprint 01
Exploring new physics frontiers through numerical relativity
The demand to obtain answers to highly complex problems within strong-field gravity has been met with significant progress in the numerical solution of Einstein's equations - along with some spectacular results - in various setups. We review techniques for solving Einstein's equations in generic spacetimes, focusing on fully nonlinear evolutions but also on how to benchmark those results with perturbative approaches. The results address problems in high-energy physics, holography, mathematical physics, fundamental physics, astrophysics and cosmology