699 research outputs found
The Cosmological Time Function
Let be a time oriented Lorentzian manifold and the Lorentzian
distance on . The function is the cosmological
time function of , where as usual means that is in the causal
past of . This function is called regular iff for all
and also along every past inextendible causal curve. If the
cosmological time function of a space time is regular it has
several pleasant consequences: (1) It forces to be globally hyperbolic,
(2) every point of can be connected to the initial singularity by a
rest curve (i.e., a timelike geodesic ray that maximizes the distance to the
singularity), (3) the function is a time function in the usual sense, in
particular (4) is continuous, in fact locally Lipschitz and the second
derivatives of exist almost everywhere.Comment: 19 pages, AEI preprint, latex2e with amsmath and amsth
Relativistic Two-stream Instability
We study the (local) propagation of plane waves in a relativistic,
non-dissipative, two-fluid system, allowing for a relative velocity in the
"background" configuration. The main aim is to analyze relativistic two-stream
instability. This instability requires a relative flow -- either across an
interface or when two or more fluids interpenetrate -- and can be triggered,
for example, when one-dimensional plane-waves appear to be left-moving with
respect to one fluid, but right-moving with respect to another. The dispersion
relation of the two-fluid system is studied for different two-fluid equations
of state: (i) the "free" (where there is no direct coupling between the fluid
densities), (ii) coupled, and (iii) entrained (where the fluid momenta are
linear combinations of the velocities) cases are considered in a
frame-independent fashion (eg. no restriction to the rest-frame of either
fluid). As a by-product of our analysis we determine the necessary conditions
for a two-fluid system to be causal and absolutely stable and establish a new
constraint on the entrainment.Comment: 15 pages, 2 eps-figure
Self-gravitating elastic bodies
Extended objects in GR are often modelled using distributional solutions of
the Einstein equations with point-like sources, or as the limit of
infinitesimally small "test" objects. In this note, I will consider models of
finite self-gravitating extended objects, which make it possible to give a
rigorous treatment of the initial value problem for (finite) extended objects.Comment: 16 pages. Based on a talk given at the 2013 WE-Heraeus seminar on
"Equations of motion in relativistic gravity
The area of horizons and the trapped region
This paper considers some fundamental questions concerning marginally trapped
surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation.
An area estimate for outermost marginally trapped surfaces is proved. The proof
makes use of an existence result for marginal surfaces, in the presence of
barriers, curvature estimates, together with a novel surgery construction for
marginal surfaces. These results are applied to characterize the boundary of
the trapped region.Comment: 44 pages, v3: small changes in presentatio
Linearized gravity and gauge conditions
In this paper we consider the field equations for linearized gravity and
other integer spin fields on the Kerr spacetime, and more generally on
spacetimes of Petrov type D. We give a derivation, using the GHP formalism, of
decoupled field equations for the linearized Weyl scalars for all spin weights
and identify the gauge source functions occuring in these. For the spin weight
0 Weyl scalar, imposing a generalized harmonic coordinate gauge yields a
generalization of the Regge-Wheeler equation. Specializing to the Schwarzschild
case, we derive the gauge invariant Regge-Wheeler and Zerilli equation directly
from the equation for the spin 0 scalar.Comment: 24 pages, corresponds to published versio
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Long-Term Prognostic Importance of Diabetes After a Myocardial Infarction Depends on Left Ventricular Systolic Function
Objective: This study was performed to understand how left ventricular function modulates the prognostic importance of diabetes after myocardial infarction (MI). Research Design and Methods: Consecutively hospitalized MI patients screened for three clinical trials were followed for a median of 7 years. Multivariable Cox regression models were used to assess the risk of mortality associated with diabetes, and the importance of diabetes was examined independently within defined left ventricular ejection fraction (LVEF) subgroups. Results: A total of 16,912 patients were included; 1,819 (11%) had diabetes. Diabetes and 15% unit depression in LVEF were of similar prognostic importance: hazard ratios (HRs) were 1.45 (95% CI 1.37â1.54) and 1.41 (1.37â1.45) for diabetes and LVEF depression, respectively. LVEF modified the outcomes associated with diabetes, with HRs being 1.29 (1.19â1.40) and 1.61 (1.49â1.74) in patients with LVEF <40% and LVEF â„40%, respectively (P = 0.03). Conclusions: Patients within the higher LVEF categories have a greater mortality risk attributable to diabetes than patients within the lower LVEF categories
Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior
We consider the Maxwell equation in the exterior of a very slowly rotating
Kerr black hole. For this system, we prove the boundedness of a positive
definite energy on each hypersurface of constant . We also prove the
convergence of each solution to a stationary Coulomb solution. We separate a
general solution into the charged, Coulomb part and the uncharged part.
Convergence to the Coulomb solutions follows from the fact that the uncharged
part satisfies a Morawetz estimate, i.e. that a spatially localised energy
density is integrable in time. For the unchanged part, we study both the full
Maxwell equation and the Fackerell-Ipser equation for one component. To treat
the Fackerell-Ipser equation, we use a Fourier transform in . For the
Fackerell-Ipser equation, we prove a refined Morawetz estimate that controls
3/2 derivatives with no loss near the orbiting null geodesics.Comment: 50 pages. v3 minor typographical change
Elastic Stars in General Relativity: II. Radial perturbations
We study radial perturbations of general relativistic stars with elastic
matter sources. We find that these perturbations are governed by a second order
differential equation which, along with the boundary conditions, defines a
Sturm-Liouville type problem that determines the eigenfrequencies. Although
some complications arise compared to the perfect fluid case, leading us to
consider a generalisation of the standard form of the Sturm-Liouville equation,
the main results of Sturm-Liouville theory remain unaltered. As an important
consequence we conclude that the mass-radius curve for a one-parameter sequence
of regular equilibrium models belonging to some particular equation of state
can be used in the same well-known way as in the perfect fluid case, at least
if the energy density and the tangential pressure of the background solutions
are continuous. In particular we find that the fundamental mode frequency has a
zero for the maximum mass stars of the models with solid crusts considered in
Paper I of this series.Comment: 22 pages, no figures, final version accepted for publication in
Class. Quantum Grav. The treatment of the junction conditions has been
improve
Notes on a paper of Mess
These notes are a companion to the article "Lorentz spacetimes of constant
curvature" by Geoffrey Mess, which was first written in 1990 but never
published. Mess' paper will appear together with these notes in a forthcoming
issue of Geometriae Dedicata.Comment: 26 page
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