1,778 research outputs found
Coincidence of length spectra does not imply isospectrality
Penrose--Lifshits mushrooms are planar domains coming in nonisometric pairs
with the same geodesic length spectrum. Recently S. Zelditch raised the
question whether such billiards also have the same eigenvalue spectrum for the
Dirichlet Laplacian (conjecturing ``no''). Here we show that generically (in
the class of smooth domains) the two members of a mushroom pair have different
spectra.Comment: 8 pages, 5 figure
Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential
The Schwarz function has played an elegant role in understanding and in
generating new examples of exact solutions to the Laplacian growth (or "Hele-
Shaw") problem in the plane. The guiding principle in this connection is the
fact that "non-physical" singularities in the "oil domain" of the Schwarz
function are stationary, and the "physical" singularities obey simple dynamics.
We give an elementary proof that the same holds in any number of dimensions for
the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17]
(1989). A generalization is also given for the so-called "elliptic growth"
problem by defining a generalized Schwarz potential. New exact solutions are
constructed, and we solve inverse problems of describing the driving
singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n -
techniques can be used to locate the singularity set of the Schwarz potential.
One of our methods is to prolong available local extension theorems by
constructing "globalizing families". We make three conjectures in potential
theory relating to our investigation
Initial Data for General Relativity with Toroidal Conformal Symmetry
A new class of time-symmetric solutions to the initial value constraints of
vacuum General Relativity is introduced. These data are globally regular,
asymptotically flat (with possibly several asymptotic ends) and in general have
no isometries, but a group of conformal isometries. After
decomposing the Lichnerowicz conformal factor in a double Fourier series on the
group orbits, the solutions are given in terms of a countable family of
uncoupled ODEs on the orbit space.Comment: REVTEX, 9 pages, ESI Preprint 12
Time dependence of Bragg forward scattering and self-seeding of hard x-ray free-electron lasers
Free-electron lasers (FELs) can now generate temporally short, high power
x-ray pulses of unprecedented brightness, even though their longitudinal
coherence is relatively poor. The longitudinal coherence can be potentially
improved by employing narrow bandwidth x-ray crystal optics, in which case one
must also understand how the crystal affects the field profile in time and
space. We frame the dynamical theory of x-ray diffraction as a set of coupled
waves in order to derive analytic expressions for the spatiotemporal response
of Bragg scattering from temporally short incident pulses. We compute the
profiles of both the reflected and forward scattered x-ray pulses, showing that
the time delay of the wave is linked to its transverse spatial shift
through the simple relationship , where
is the grazing angle of incidence to the diffracting planes. Finally,
we apply our findings to obtain an analytic description of Bragg forward
scattering relevant to monochromatically seed hard x-ray FELs.Comment: 11 pages, 6 figure
Black holes, cosmological singularities and change of signature
There exists a widespread belief that signature type change could be used to
avoid spacetime singularities. We show that signature change cannot be utilised
to this end unless the Einstein equation is abandoned at the suface of
signature type change. We also discuss how to solve the initial value problem
and show to which extent smooth and discontinuous signature changing solutions
are equivalent.Comment: 14pages, Latex, no figur
Off-diagonal coefficients of the DeWitt-Schwinger and Hadamard representations of the Feynman propagator
Having in mind applications to gravitational wave theory (in connection with
the radiation reaction problem), stochastic semiclassical gravity (in
connection with the regularization of the noise kernel) and quantum field
theory in higher-dimensional curved spacetime (in connection with the Hadamard
regularization of the stress-energy tensor), we improve the DeWitt-Schwinger
and Hadamard representations of the Feynman propagator of a massive scalar
field theory defined on an arbitrary gravitational background by deriving
higher-order terms for the covariant Taylor series expansions of the
geometrical coefficients -- i.e., the DeWitt and Hadamard coefficients -- that
define them.Comment: 42 pages; v2: Incorrect claims suppressed, ref. added, typos
corrected, expansion of the Van Vleck-Morette determinant improved; v3: Minor
changes and refs. added to match the published versio
Two-Dimensional Hydrodynamic Simulations of Convection in Radiation-Dominated Accretion Disks
The standard equilibrium for radiation-dominated accretion disks has long
been known to be viscously, thermally, and convectively unstable, but the
nonlinear development of these instabilities---hence the actual state of such
disks---has not yet been identified. By performing local two-dimensional
hydrodynamic simulations of disks, we demonstrate that convective motions can
release heat sufficiently rapidly as to substantially alter the vertical
structure of the disk. If the dissipation rate within a vertical column is
proportional to its mass, the disk settles into a new configuration thinner by
a factor of two than the standard radiation-supported equilibrium. If, on the
other hand, the vertically-integrated dissipation rate is proportional to the
vertically-integrated total pressure, the disk is subject to the well-known
thermal instability. Convection, however, biases the development of this
instability toward collapse. The end result of such a collapse is a gas
pressure-dominated equilibrium at the original column density.Comment: 10 pages, 7 figures, accepted for publication in ApJ. Please send
comments to [email protected]
Steep sharp-crested gravity waves on deep water
A new type of steady steep two-dimensional irrotational symmetric periodic
gravity waves on inviscid incompressible fluid of infinite depth is revealed.
We demonstrate that these waves have sharper crests in comparison with the
Stokes waves of the same wavelength and steepness. The speed of a fluid
particle at the crest of new waves is greater than their phase speed.Comment: 4 pages, 2 figures, submitted to Phys. Rev. Let
Effective Viscosity of Dilute Bacterial Suspensions: A Two-Dimensional Model
Suspensions of self-propelled particles are studied in the framework of
two-dimensional (2D) Stokesean hydrodynamics. A formula is obtained for the
effective viscosity of such suspensions in the limit of small concentrations.
This formula includes the two terms that are found in the 2D version of
Einstein's classical result for passive suspensions. To this, the main result
of the paper is added, an additional term due to self-propulsion which depends
on the physical and geometric properties of the active suspension. This term
explains the experimental observation of a decrease in effective viscosity in
active suspensions.Comment: 15 pages, 3 figures, submitted to Physical Biolog
Uniqueness properties of the Kerr metric
We obtain a geometrical condition on vacuum, stationary, asymptotically flat
spacetimes which is necessary and sufficient for the spacetime to be locally
isometric to Kerr. Namely, we prove a theorem stating that an asymptotically
flat, stationary, vacuum spacetime such that the so-called Killing form is an
eigenvector of the self-dual Weyl tensor must be locally isometric to Kerr.
Asymptotic flatness is a fundamental hypothesis of the theorem, as we
demonstrate by writing down the family of metrics obtained when this
requirement is dropped. This result indicates why the Kerr metric plays such an
important role in general relativity. It may also be of interest in order to
extend the uniqueness theorems of black holes to the non-connected and to the
non-analytic case.Comment: 30 pages, LaTeX, submitted to Classical and Quantum Gravit
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