4,655 research outputs found
The geometric Cauchy problem for developable submanifolds
Given a smooth distribution of -dimensional planes along a
smooth regular curve in , we consider the following
problem: To find an -dimensional developable submanifold of
, that is, a ruled submanifold with constant tangent space
along the rulings, such that its tangent bundle along coincides with
. In particular, we give sufficient conditions for the local
well-posedness of the problem, together with a parametric description of the
solution.Comment: 15 page
A scattering approach to some aspects of the Schwarzschild Black Hole
In this paper, we consider a massless field, with spin j, in interaction with
a Schwarzschild black hole in four dimensions, focusing mainly our study on the
s-wave scattering. First, using a Fourier analysis, we show that one can have a
simple and natural description of the Physics near the event horizon without
using any conformal field approaches. Then, within the same "scattering
picture", we derive analytically the imaginary part of the highly damped
quasinormal complex frequencies and, as a natural consequence of our analysis,
we show that thermal effects and in particular Hawking radiation, can be
understood through the scattering of an ingoing s-wave by the non null barrier
of the Regge-Wheeler potential associated with the Schwarzschild black hole.
Finally, with the help of the well-known expression of the highly damped
quasinormal complex frequencies, we propose a heuristic extension of the
"tripled Pauli statistics" suggested by Motl, some years ago.Comment: 18 pages, 1 figure, JHEP accepted articl
Cartan Ribbonization of Surfaces and a Topological Inspection
We develop the concept of Cartan ribbons and a method by which they can be
used to ribbonize any given surface in space by intrinsically flat ribbons. The
geodesic curvature along the center curve on the surface agrees with the
geodesic curvature of the corresponding Cartan development curve, and this
makes a rolling strategy successful. Essentially, it follows from the
orientational alignment of the two co-moving Darboux frames during the rolling.
Using closed center curves we obtain closed approximating Cartan ribbons that
contribute zero to the total curvature integral of the ribbonization. This
paves the way for a particular simple topological inspection -- it is reduced
to the question of how the ribbons organize their edges relative to each other.
The Gauss-Bonnet theorem leads to this topological inspection of the vertices.
Finally, we display two examples of ribbonizations of surfaces, namely of a
torus using two ribbons, and of an ellipsoid using its closed curvature lines
as center curves for the ribbons. The topological inspection of the torus
ribbonization is particularly simple as it has no vertex points, giving
directly the Euler characteristic . The ellipsoid has vertices --
corresponding to the umbilical points -- each of degree one and each
therefore contributing one-half to the Euler characteristic
Fine structure of high-energy absorption cross sections for black holes
The high-energy absorption cross section of the Schwarzschild black hole is
well approximated, in the eikonal regime, by the sum of two terms: the
geometrical cross section of the black hole photon sphere and the contribution
of a sinc function involving the geometrical characteristics (orbital period
and Lyapunov exponent) of the null unstable geodesics lying on this photon
sphere. From a numerical analysis, we show that, beyond the eikonal
description, this absorption cross section presents a simple fine structure. We
then describe it analytically by using Regge pole techniques and interpret it
in geometrical terms. We naturally extend our analysis to arbitrary static
spherically symmetric black holes endowed with a photon sphere and we then
apply our formalism to Schwarzschild-Tangherlini and Reissner-Nordstr\"om black
holes. Finally, on the example of the Schwarzschild black hole, we show
numerically that a complicated hyperfine structure lying beyond the fine
structure can also be observed.Comment: v2: Minor corrections; v3: Minor changes to match the published
versio
Marshall on Custom and Competition
Entry for the Elgar Companion to Alfred Marshall, edited by Tiziano Raffaelli, Marco Dardi, and Giacomo Becatini, Cheltenham: Edward Elgar 200
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