1,065 research outputs found
Sufficient Conditions for Apparent Horizons in Spherically Symmetric Initial Data
We establish sufficient conditions for the appearance of apparent horizons in
spherically symmetric initial data when spacetime is foliated extrinsically.
Let and be respectively the total material energy and the total
material current contained in some ball of radius . Suppose that the
dominant energy condition is satisfied. We show that if then
the region must possess a future apparent horizon for some non -trivial closed
subset of such gauges. The same inequality holds on a larger subset of gauges
but with a larger constant of proportionality which depends weakly on the
gauge. This work extends substantially both our joint work on moment of time
symmetry initial data as well as the work of Bizon, Malec and \'O Murchadha on
a maximal slice.Comment: 16 pages, revtex, to appear in Phys. Rev.
Geometric Bounds in Spherically Symmetric General Relativity
We exploit an arbitrary extrinsic time foliation of spacetime to solve the
constraints in spherically symmetric general relativity. Among such foliations
there is a one parameter family, linear and homogeneous in the extrinsic
curvature, which permit the momentum constraint to be solved exactly. This
family includes, as special cases, the extrinsic time gauges that have been
exploited in the past. These foliations have the property that the extrinsic
curvature is spacelike with respect to the the spherically symmetric superspace
metric. What is remarkable is that the linearity can be relaxed at no essential
extra cost which permits us to isolate a large non - pathological dense subset
of all extrinsic time foliations. We identify properties of solutions which are
independent of the particular foliation within this subset. When the geometry
is regular, we can place spatially invariant numerical bounds on the values of
both the spatial and the temporal gradients of the scalar areal radius, .
These bounds are entirely independent of the particular gauge and of the
magnitude of the sources. When singularities occur, we demonstrate that the
geometry behaves in a universal way in the neighborhood of the singularity.Comment: 16 pages, revtex, submitted to Phys. Rev.
Necessary Conditions for Apparent Horizons and Singularities in Spherically Symmetric Initial Data
We establish necessary conditions for the appearance of both apparent
horizons and singularities in the initial data of spherically symmetric general
relativity when spacetime is foliated extrinsically. When the dominant energy
condition is satisfied these conditions assume a particularly simple form. Let
be the maximum value of the energy density and the radial
measure of its support. If is bounded from above by some
numerical constant, the initial data cannot possess an apparent horizon. This
constant does not depend sensitively on the gauge. An analogous inequality is
obtained for singularities with some larger constant. The derivation exploits
Poincar\'e type inequalities to bound integrals over certain spatial scalars. A
novel approach to the construction of analogous necessary conditions for
general initial data is suggested.Comment: 15 pages, revtex, to appear in Phys. Rev.
Directivity enhancement and deflection of the beam emitted from a photonic crystal waveguide via defect coupling
Cataloged from PDF version of article.We experimentally and numerically investigate the spatial distribution of the emission from a photonic crystal waveguide, coupled with defects, that are located at the output edge. Two defects that are located symmetrically enhance the directivity of the beam compared to that of a plain waveguide, as was reported in recently conducted theoretical work. We further demonstrate that a single defect deflects of the beam. By choosing the defect resonance that is close to the edge of the pass band of the waveguide, where the group velocity of the beam within the waveguide is slow, a significant amount of deflection can be achieved. (c) 2007 Optical Society of Americ
The Constraints in Spherically Symmetric General Relativity II --- Identifying the Configuration Space: A Moment of Time Symmetry
We continue our investigation of the configuration space of general
relativity begun in I (gr-qc/9411009). Here we examine the Hamiltonian
constraint when the spatial geometry is momentarily static (MS). We show that
MS configurations satisfy both the positive quasi-local mass (QLM) theorem and
its converse. We derive an analytical expression for the spatial metric in the
neighborhood of a generic singularity. The corresponding curvature singularity
shows up in the traceless component of the Ricci tensor. We show that if the
energy density of matter is monotonically decreasing, the geometry cannot be
singular. A supermetric on the configuration space which distinguishes between
singular geometries and non-singular ones is constructed explicitly. Global
necessary and sufficient criteria for the formation of trapped surfaces and
singularities are framed in terms of inequalities which relate appropriate
measures of the material energy content on a given support to a measure of its
volume. The strength of these inequalities is gauged by exploiting the exactly
solvable piece-wise constant density star as a template.Comment: 50 pages, Plain Tex, 1 figure available from the authors
Hamiltonian Frenet-Serret dynamics
The Hamiltonian formulation of the dynamics of a relativistic particle
described by a higher-derivative action that depends both on the first and the
second Frenet-Serret curvatures is considered from a geometrical perspective.
We demonstrate how reparametrization covariant dynamical variables and their
projections onto the Frenet-Serret frame can be exploited to provide not only a
significant simplification of but also novel insights into the canonical
analysis. The constraint algebra and the Hamiltonian equations of motion are
written down and a geometrical interpretation is provided for the canonical
variables.Comment: Latex file, 14 pages, no figures. Revised version to appear in Class.
Quant. Gra
Hamilton's equations for a fluid membrane: axial symmetry
Consider a homogenous fluid membrane, or vesicle, described by the
Helfrich-Canham energy, quadratic in the mean curvature. When the membrane is
axially symmetric, this energy can be viewed as an `action' describing the
motion of a particle; the contours of equilibrium geometries are identified
with particle trajectories. A novel Hamiltonian formulation of the problem is
presented which exhibits the following two features: {\it (i)} the second
derivatives appearing in the action through the mean curvature are accommodated
in a natural phase space; {\it (ii)} the intrinsic freedom associated with the
choice of evolution parameter along the contour is preserved. As a result, the
phase space involves momenta conjugate not only to the particle position but
also to its velocity, and there are constraints on the phase space variables.
This formulation provides the groundwork for a field theoretical generalization
to arbitrary configurations, with the particle replaced by a loop in space.Comment: 11 page
ADM Worldvolume Geometry
We describe the dynamics of a relativistic extended object in terms of the
geometry of a configuration of constant time. This involves an adaptation of
the ADM formulation of canonical general relativity. We apply the formalism to
the hamiltonian formulation of a Dirac-Nambu-Goto relativistic extended object
in an arbitrary background spacetime.Comment: 4 pages, Latex. Uses espcrc2.sty To appear in the proceedings of the
Third Conference on Constrained Dynamics and Quantum Gravity, September,
1999. To appear in Nuclear Physics B (Proceedings Supplement
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