20 research outputs found
Bounds on the dragging rate and on the rotational mass-energy in slowly and differentially rotating relativistic stars
For relativistic stars rotating slowly and differentially with a positive
angular velocity, some properties in relation to the positiveness of the rate
of rotational dragging and of the angular momentum density are derived. Also, a
new proof for the bounds on the rotational mass-energy is given.Comment: 23 pages, latex. Submitted to J. Math. Phy
Relativistic stars in differential rotation: bounds on the dragging rate and on the rotational energy
For general relativistic equilibrium stellar models (stationary axisymmetric
asymptotically flat and convection-free) with differential rotation, it is
shown that for a wide class of rotation laws the distribution of angular
velocity of the fluid has a sign, say "positive", and then both the dragging
rate and the angular momentum density are positive. In addition, the "mean
value" (with respect to an intrinsic density) of the dragging rate is shown to
be less than the mean value of the fluid angular velocity (in full general,
without having to restrict the rotation law, nor the uniformity in sign of the
fluid angular velocity); this inequality yields the positivity and an upper
bound of the total rotational energy.Comment: 23 pages, no figures, LaTeX. Submitted to J. Math. Phy
Dirichlet Boundary Value Problems of the Ernst Equation
We demonstrate how the solution to an exterior Dirichlet boundary value
problem of the axisymmetric, stationary Einstein equations can be found in
terms of generalized solutions of the Backlund type. The proof that this
generalization procedure is valid is given, which also proves conjectures about
earlier representations of the gravitational field corresponding to rotating
disks of dust in terms of Backlund type solutions.Comment: 22 pages, to appear in Phys. Rev. D, Correction of a misprint in
equation (4
Exact relativistic treatment of stationary counter-rotating dust disks I: Boundary value problems and solutions
This is the first in a series of papers on the construction of explicit
solutions to the stationary axisymmetric Einstein equations which describe
counter-rotating disks of dust. These disks can serve as models for certain
galaxies and accretion disks in astrophysics. We review the Newtonian theory
for disks using Riemann-Hilbert methods which can be extended to some extent to
the relativistic case where they lead to modular functions on Riemann surfaces.
In the case of compact surfaces these are Korotkin's finite gap solutions which
we will discuss in this paper. On the axis we establish for general genus
relations between the metric functions and hence the multipoles which are
enforced by the underlying hyperelliptic Riemann surface. Generalizing these
results to the whole spacetime we are able in principle to study the classes of
boundary value problems which can be solved on a given Riemann surface. We
investigate the cases of genus 1 and 2 of the Riemann surface in detail and
construct the explicit solution for a family of disks with constant angular
velocity and constant relative energy density which was announced in a previous
Physical Review Letter.Comment: 32 pages, 1 figure, to appear in Phys. Rev.
(In)finiteness of Spherically Symmetric Static Perfect Fluids
This work is concerned with the finiteness problem for static, spherically
symmetric perfect fluids in both Newtonian Gravity and General Relativity. We
derive criteria on the barotropic equation of state guaranteeing that the
corresponding perfect fluid solutions possess finite/infinite extent. In the
Newtonian case, for the large class of monotonic equations of state, and in
General Relativity we improve earlier results
Time-Independent Gravitational Fields
This article reviews, from a global point of view, rigorous results on time
independent spacetimes. Throughout attention is confined to isolated bodies at
rest or in uniform rotation in an otherwise empty universe. The discussion
starts from first principles and is, as much as possible, self-contained.Comment: 47 pages, LaTeX, uses Springer cl2emult styl
Physically Realistic Solutions to the Ernst Equation on Hyperelliptic Riemann Surfaces
We show that the class of hyperelliptic solutions to the Ernst equation (the
stationary axisymmetric Einstein equations in vacuum) previously discovered by
Korotkin and Neugebauer and Meinel can be derived via Riemann-Hilbert
techniques. The present paper extends the discussion of the physical properties
of these solutions that was begun in a Physical Review Letter, and supplies
complete proofs. We identify a physically interesting subclass where the Ernst
potential is everywhere regular except at a closed surface which might be
identified with the surface of a body of revolution. The corresponding
spacetimes are asymptotically flat and equatorially symmetric. This suggests
that they could describe the exterior of an isolated body, for instance a
relativistic star or a galaxy. Within this class, one has the freedom to
specify a real function and a set of complex parameters which can possibly be
used to solve certain boundary value problems for the Ernst equation. The
solutions can have ergoregions, a Minkowskian limit and an ultrarelativistic
limit where the metric approaches the extreme Kerr solution. We give explicit
formulae for the potential on the axis and in the equatorial plane where the
expressions simplify. Special attention is paid to the simplest non-static
solutions (which are of genus two) to which the rigidly rotating dust disk
belongs.Comment: 32 pages, 2 figures, uses pstricks.sty, updated version (October 7,
1998), to appear in Phys. Rev.
Static perfect fluids with Pant-Sah equations of state
We analyze the 3-parameter family of exact, regular, static, spherically
symmetric perfect fluid solutions of Einstein's equations (corresponding to a
2-parameter family of equations of state) due to Pant and Sah and
"rediscovered" by Rosquist and the present author. Except for the Buchdahl
solutions which are contained as a limiting case, the fluids have finite radius
and are physically realistic for suitable parameter ranges. The equations of
state can be characterized geometrically by the property that the 3-metric on
the static slices, rescaled conformally with the fourth power of any linear
function of the norm of the static Killing vector, has constant scalar
curvature. This local property does not require spherical symmetry; in fact it
simplifies the the proof of spherical symmetry of asymptotically flat solutions
which we recall here for the Pant-Sah equations of state. We also consider a
model in Newtonian theory with analogous geometric and physical properties,
together with a proof of spherical symmetry of the asymptotically flat
solutions.Comment: 32 p., Latex, minor changes and correction
Nonlinear stability analysis of the Emden-Fowler equation
In this paper we qualitatively study radial solutions of the semilinear
elliptic equation with and on the
positive real line, called the Emden-Fowler or Lane-Emden equation. This
equation is of great importance in Newtonian astrophysics and the constant
is called the polytropic index. By introducing a set of new variables, the
Emden-Fowler equation can be written as an autonomous system of two ordinary
differential equations which can be analyzed using linear and nonlinear
stability analysis. We perform the study of stability by using linear stability
analysis, the Jacobi stability analysis (Kosambi-Cartan-Chern theory) and the
Lyapunov function method. Depending on the values of these different
methods yield different results. We identify a parameter range for where
all three methods imply stability.Comment: 12 pages; new reference added; 3 new references added; fully revised
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Evaluation of urban local-scale aerodynamic parameters: implications for the vertical profile of wind speed and for source areas
Nine methods to determine local-scale aerodynamic roughness length (z0) and zero-plane displacement (zd) are compared at three sites (within 60 m of each other) in London, UK. Methods include three anemometric (single-level high frequency observations), six morphometric (surface geometry) and one reference-based approach (look-up tables). A footprint model is used with the morphometric methods in an iterative procedure. The results are insensitive to the initial zd and z0 estimates. Across the three sites, zd varies between 5 – 45 m depending upon the method used. Morphometric methods that incorporate roughness-element height variability agree better with anemometric methods, indicating zd is consistently greater than the local mean building height. Depending upon method and wind direction, z0 varies between 0.1 and 5 m with morphometric z0 consistently being 2 – 3 m larger than the anemometric z0. No morphometric method consistently resembles the anemometric methods. Wind-speed profiles observed with Doppler lidar provide additional data with which to assess the methods. Locally determined roughness parameters are used to extrapolate wind-speed profiles to a height roughly 200 m above the canopy. Wind-speed profiles extrapolated based on morphometric methods that account for roughness-element height variability are most similar to observations. The extent of the modelled source area for measurements varies by up to a factor of three, depending upon the morphometric method used to determine zd and z0