1,542 research outputs found
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.
Yang-Mills theory a la string
A surface of codimension higher than one embedded in an ambient space
possesses a connection associated with the rotational freedom of its normal
vector fields. We examine the Yang-Mills functional associated with this
connection. The theory it defines differs from Yang-Mills theory in that it is
a theory of surfaces. We focus, in particular, on the Euler-Lagrange equations
describing this surface, introducing a framework which throws light on their
relationship to the Yang-Mills equations.Comment: 7 page
Quantitative Perturbation Theory for Compact Operators on a Hilbert Space
This thesis makes novel contributions to a problem of practical and theoretical importance, namely how to determine explicitly computable upper bounds for the Hausdorff distance of the spectra of two compact operators on a Hilbert space in terms of the distance of the two operators in operator norm. It turns out that the answer depends crucially on the speed of decay of the sequence of singular values of the two operators. To this end, ‘compactness classes’, that is, collections of operators the singular values of which decay at a certain speed, are introduced and their functional analytic properties studied in some detail. The main result of the thesis is an explicit formula for the Hausdorff distance of the spectra of two operators belonging to the same compactness class. Along the way, upper bounds for the resolvents of operators belonging to a particular compactness class are established, as well as novel bounds for determinants of trace class operators
Whirling skirts and rotating cones
Steady, dihedrally symmetric patterns with sharp peaks may be observed on a
spinning skirt, lagging behind the material flow of the fabric. These
qualitative features are captured with a minimal model of traveling waves on an
inextensible, flexible, generalized-conical sheet rotating about a fixed axis.
Conservation laws are used to reduce the dynamics to a quadrature describing a
particle in a three-parameter family of potentials. One parameter is associated
with the stress in the sheet, aNoether is the current associated with
rotational invariance, and the third is a Rossby number which indicates the
relative strength of Coriolis forces. Solutions are quantized by enforcing a
topology appropriate to a skirt and a particular choice of dihedral symmetry. A
perturbative analysis of nearly axisymmetric cones shows that Coriolis effects
are essential in establishing skirt-like solutions. Fully non-linear solutions
with three-fold symmetry are presented which bear a suggestive resemblance to
the observed patterns.Comment: two additional figures, changes to text throughout. journal version
will have a wordier abstrac
Tensor Renormalization Group: Local Magnetizations, Correlation Functions, and Phase Diagrams of Systems with Quenched Randomness
The tensor renormalization-group method, developed by Levin and Nave, brings
systematic improvability to the position-space renormalization-group method and
yields essentially exact results for phase diagrams and entire thermodynamic
functions. The method, previously used on systems with no quenched randomness,
is extended in this study to systems with quenched randomness. Local
magnetizations and correlation functions as a function of spin separation are
calculated as tensor products subject to renormalization-group transformation.
Phase diagrams are extracted from the long-distance behavior of the correlation
functions. The approach is illustrated with the quenched bond-diluted Ising
model on the triangular lattice. An accurate phase diagram is obtained in
temperature and bond-dilution probability, for the entire temperature range
down to the percolation threshold at zero temperature.Comment: Added comment. Published version. 8 pages, 7 figures, 1 tabl
Dipoles in thin sheets
A flat elastic sheet may contain pointlike conical singularities that carry a
metrical "charge" of Gaussian curvature. Adding such elementary defects to a
sheet allows one to make many shapes, in a manner broadly analogous to the
familiar multipole construction in electrostatics. However, here the underlying
field theory is non-linear, and superposition of intrinsic defects is
non-trivial as it must respect the immersion of the resulting surface in three
dimensions. We consider a "charge-neutral" dipole composed of two conical
singularities of opposite sign. Unlike the relatively simple electrostatic
case, here there are two distinct stable minima and an infinity of unstable
equilibria. We determine the shapes of the minima and evaluate their energies
in the thin-sheet regime where bending dominates over stretching. Our
predictions are in surprisingly good agreement with experiments on paper
sheets.Comment: 20 pages, 5 figures, 2 table
Deformations of extended objects with edges
We present a manifestly gauge covariant description of fluctuations of a
relativistic extended object described by the Dirac-Nambu-Goto action with
Dirac-Nambu-Goto loaded edges about a given classical solution. Whereas
physical fluctuations of the bulk lie normal to its worldsheet, those on the
edge possess an additional component directed into the bulk. These fluctuations
couple in a non-trivial way involving the underlying geometrical structures
associated with the worldsheet of the object and of its edge. We illustrate the
formalism using as an example a string with massive point particles attached to
its ends.Comment: 17 pages, revtex, to appear in Phys. Rev. D5
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
Quantitative spectral perturbation theory for compact operators on a Hilbert space
We introduce compactness classes of Hilbert space operators by grouping
together all operators for which the associated singular values decay at a
certain speed and establish upper bounds for the norm of the resolvent of
operators belonging to a particular compactness class. As a consequence we
obtain explicitly computable upper bounds for the Hausdorff distance of the
spectra of two operators belonging to the same compactness class in terms of
the distance of the two operators in operator norm.Comment: 26 page
Force dipoles and stable local defects on fluid vesicles
An exact description is provided of an almost spherical fluid vesicle with a
fixed area and a fixed enclosed volume locally deformed by external normal
forces bringing two nearby points on the surface together symmetrically. The
conformal invariance of the two-dimensional bending energy is used to identify
the distribution of energy as well as the stress established in the vesicle.
While these states are local minima of the energy, this energy is degenerate;
there is a zero mode in the energy fluctuation spectrum, associated with area
and volume preserving conformal transformations, which breaks the symmetry
between the two points. The volume constraint fixes the distance , measured
along the surface, between the two points; if it is relaxed, a second zero mode
appears, reflecting the independence of the energy on ; in the absence of
this constraint a pathway opens for the membrane to slip out of the defect.
Logarithmic curvature singularities in the surface geometry at the points of
contact signal the presence of external forces. The magnitude of these forces
varies inversely with and so diverges as the points merge; the
corresponding torques vanish in these defects. The geometry behaves near each
of the singularities as a biharmonic monopole, in the region between them as a
surface of constant mean curvature, and in distant regions as a biharmonic
quadrupole. Comparison of the distribution of stress with the quadratic
approximation in the height functions points to shortcomings of the latter
representation. Radial tension is accompanied by lateral compression, both near
the singularities and far away, with a crossover from tension to compression
occurring in the region between them.Comment: 26 pages, 10 figure
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