5 research outputs found
A Model of Two Dimensional Turbulence Using Random Matrix Theory
We derive a formula for the entropy of two dimensional incompressible
inviscid flow, by determining the volume of the space of vorticity
distributions with fixed values for the moments Q_k= \int_w(x)^k d^2 x. This
space is approximated by a sequence of spaces of finite volume, by using a
regularization of the system that is geometrically natural and connected with
the theory of random matrices. In taking the limit we get a simple formula for
the entropy of a vortex field. We predict vorticity distributions of maximum
entropy with given mean vorticity and enstrophy; also we predict the
cylindrically symmetric vortex field with maximum entropy. This could be an
approximate description of a hurricane.Comment: latex, 12 pages, 2 figures, acknowledgement adde
The Vacuum Einstein Equations via Holonomy around Closed Loops on Characteristic Surfaces
We reformulate the standard local equations of general relativity for
asymptotically flat spacetimes in terms of two non-local quantities, the
holonomy around certain closed null loops on characteristic surfaces and
the light cone cut function , which describes the intersection of the future
null cones from arbitrary spacetime points, with future null infinity. We
obtain a set of differential equations for and equivalent to the vacuum
Einstein equations. By finding an algebraic relation between and this
set of equations is reduced to just two coupled equations: an
integro-differential equation for which yields the conformal structure of
the underlying spacetime and a linear differential equation for the ``vacuum''
conformal factor. These equations, which apply to all vacuum asymptotically
flat spacetimes, are however lengthy and complicated and we do not yet know of
any solution generating technique. They nevertheless are amenable to an
attractive perturbative scheme which has Minkowski space as a zeroth order
solution.Comment: 28 pages, RevTeX, 3 PostScript figure
Asymmetric Light Bending in the Equatorial Kerr Metric
The observation of the bending of light by mass, now known as gravitational
lensing, was key in establishing general relativity as one of the pillars of
modern physics. In the past couple of decades, there has been increasing
interest in using gravitational lensing to test general relativity beyond the
weak deflection limit. Black holes and neutron stars produce the strong
gravitational fields needed for such tests. For a rotating compact object, the
distinction between prograde and retrograde photon trajectories becomes
important. In this paper, we explore subtleties that arise in interpreting the
bending angle in this context and address the origin of seemingly contradictory
results in the literature. We argue that analogies that cannot be precisely
quantified present a source of confusion