2,391 research outputs found
A model problem for conformal parameterizations of the Einstein constraint equations
We investigate the possibility that the conformal and conformal thin sandwich
(CTS) methods can be used to parameterize the set of solutions of the vacuum
Einstein constraint equations. To this end we develop a model problem obtained
by taking the quotient of certain symmetric data on conformally flat tori.
Specializing the model problem to a three-parameter family of conformal data we
observe a number of new phenomena for the conformal and CTS methods. Within
this family, we obtain a general existence theorem so long as the mean
curvature does not change sign. When the mean curvature changes sign, we find
that for certain data solutions exist if and only if the transverse-traceless
tensor is sufficiently small. When such solutions exist, there are generically
more than one. Moreover, the theory for mean curvatures changing sign is shown
to be extremely sensitive with respect to the value of a coupling constant in
the Einstein constraint equations.Comment: 40 pages, 4 figure
A study of electronic packages environmental control systems and vehicle thermal systems integration Quarterly report, Nov. 1966 - Jan. 1967
Heat balances of combined astrionic equipment and thermal conditioning subsystem of environmental control system, and vehicle configuration
A non-existence result for a generalization of the equations of the conformal method in general relativity
The constraint equations of general relativity can in many cases be solved by
the conformal method. We show that a slight modification of the equations of
the conformal method admits no solution for a broad range of parameters. This
suggests that the question of existence or non-existence of solutions to the
original equations is more subtle than could perhaps be expected.Comment: minor changes from previous versio
Asymptotically Hyperbolic Non Constant Mean Curvature Solutions of the Einstein Constraint Equations
We describe how the iterative technique used by Isenberg and Moncrief to
verify the existence of large sets of non constant mean curvature solutions of
the Einstein constraints on closed manifolds can be adapted to verify the
existence of large sets of asymptotically hyperbolic non constant mean
curvature solutions of the Einstein constraints.Comment: 19 pages, TeX, no figure
The constraint equations for the Einstein-scalar field system on compact manifolds
We study the constraint equations for the Einstein-scalar field system on
compact manifolds. Using the conformal method we reformulate these equations as
a determined system of nonlinear partial differential equations. By introducing
a new conformal invariant, which is sensitive to the presence of the initial
data for the scalar field, we are able to divide the set of free conformal data
into subclasses depending on the possible signs for the coefficients of terms
in the resulting Einstein-scalar field Lichnerowicz equation. For many of these
subclasses we determine whether or not a solution exists. In contrast to other
well studied field theories, there are certain cases, depending on the mean
curvature and the potential of the scalar field, for which we are unable to
resolve the question of existence of a solution. We consider this system in
such generality so as to include the vacuum constraint equations with an
arbitrary cosmological constant, the Yamabe equation and even (all cases of)
the prescribed scalar curvature problem as special cases.Comment: Minor changes, final version. To appear: Classical and Quantum
Gravit
Ricci flows, wormholes and critical phenomena
We study the evolution of wormhole geometries under Ricci flow using
numerical methods. Depending on values of initial data parameters, wormhole
throats either pinch off or evolve to a monotonically growing state. The
transition between these two behaviors exhibits a from of critical phenomena
reminiscent of that observed in gravitational collapse. Similar results are
obtained for initial data that describe space bubbles attached to
asymptotically flat regions. Our numerical methods are applicable to
"matter-coupled" Ricci flows derived from conformal invariance in string
theory.Comment: 8 pages, 5 figures. References added and minor changes to match
version accepted by CQG as a fast track communicatio
Resonant Interactions Between Protons and Oblique Alfv\'en/Ion-Cyclotron Waves
Resonant interactions between ions and Alfv\'en/ion-cyclotron (A/IC) waves
may play an important role in the heating and acceleration of the fast solar
wind. Although such interactions have been studied extensively for "parallel"
waves, whose wave vectors are aligned with the background magnetic
field , much less is known about interactions between ions and
oblique A/IC waves, for which the angle between and is nonzero. In this paper, we present new numerical results on resonant
cyclotron interactions between protons and oblique A/IC waves in collisionless
low-beta plasmas such as the solar corona. We find that if some mechanism
generates oblique high-frequency A/IC waves, then these waves initially modify
the proton distribution function in such a way that it becomes unstable to
parallel waves. Parallel waves are then amplified to the point that they
dominate the wave energy at the large parallel wave numbers at which the waves
resonate with the particles. Pitch-angle scattering by these waves then causes
the plasma to evolve towards a state in which the proton distribution is
constant along a particular set of nested "scattering surfaces" in velocity
space, whose shapes have been calculated previously. As the distribution
function approaches this state, the imaginary part of the frequency of parallel
A/IC waves drops continuously towards zero, but oblique waves continue to
undergo cyclotron damping while simultaneously causing protons to diffuse
across these kinetic shells to higher energies. We conclude that oblique A/IC
waves can be more effective at heating protons than parallel A/IC waves,
because for oblique waves the plasma does not relax towards a state in which
proton damping of oblique A/IC waves ceases
A rigidity theorem for nonvacuum initial data
In this note we prove a theorem on non-vacuum initial data for general
relativity. The result presents a ``rigidity phenomenon'' for the extrinsic
curvature, caused by the non-positive scalar curvature.
More precisely, we state that in the case of asymptotically flat non-vacuum
initial data if the metric has everywhere non-positive scalar curvature then
the extrinsic curvature cannot be compactly supported.Comment: This is an extended and published version: LaTex, 10 pages, no
figure
The ground state and the long-time evolution in the CMC Einstein flow
Let (g,K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold M
with non-positive Yamabe invariant (Y(M)). As noted by Fischer and Moncrief,
the reduced volume V(k)=(-k/3)^{3}Vol_{g(k)}(M) is monotonically decreasing in
the expanding direction and bounded below by V_{\inf}=(-1/6)Y(M))^{3/2}.
Inspired by this fact we define the ground state of the manifold M as "the
limit" of any sequence of CMC states {(g_{i},K_{i})} satisfying: i. k_{i}=-3,
ii. V_{i} --> V_{inf}, iii. Q_{0}((g_{i},K_{i}))< L where Q_{0} is the
Bel-Robinson energy and L is any arbitrary positive constant. We prove that (as
a geometric state) the ground state is equivalent to the Thurston
geometrization of M. Ground states classify naturally into three types. We
provide examples for each class, including a new ground state (the Double Cusp)
that we analyze in detail. Finally consider a long time and cosmologically
normalized flow (\g,\K)(s)=((-k/3)^{2}g,(-k/3))K) where s=-ln(-k) is in
[a,\infty). We prove that if E_{1}=E_{1}((\g,\K))< L (where E_{1}=Q_{0}+Q_{1},
is the sum of the zero and first order Bel-Robinson energies) the flow
(\g,\K)(s) persistently geometrizes the three-manifold M and the geometrization
is the ground state if V --> V_{inf}.Comment: 40 pages. This article is an improved version of the second part of
the First Version of arXiv:0705.307
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