skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Generalized harmonic spatial coordinates and hyperbolic shift conditions

Abstract

We propose a generalization of the condition for harmonic spatial coordinates analogous to the generalization of the harmonic time slices introduced by Bona et al., and closely related to dynamic shift conditions recently proposed by Lindblom and Scheel, and Bona and Palenzuela. These generalized harmonic spatial coordinates imply a condition for the shift vector that has the form of an evolution equation for the shift components. We find that in order to decouple the slicing condition from the evolution equation for the shift it is necessary to use a rescaled shift vector. The initial form of the generalized harmonic shift condition is not spatially covariant, but we propose a simple way to make it fully covariant so that it can be used in coordinate systems other than Cartesian. We also analyze the effect of the shift condition proposed here on the hyperbolicity of the evolution equations of general relativity in 1+1 dimensions and 3+1 spherical symmetry, and study the possible development of blowups. Finally, we perform a series of numerical experiments to illustrate the behavior of this shift condition.

Authors:
; ; ;  [1];  [2];  [1];  [3]
  1. Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, A.P. 70-543, Mexico D.F. 04510 (Mexico)
  2. Theoretical Physics Institute, University of Jena, Max-Wien-Platz 1, 07743, Jena (Germany)
  3. (Germany)
Publication Date:
OSTI Identifier:
20774539
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. D, Particles Fields; Journal Volume: 72; Journal Issue: 12; Other Information: DOI: 10.1103/PhysRevD.72.124018; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; COORDINATES; COSMOLOGY; EQUATIONS; GENERAL RELATIVITY THEORY; SPACE-TIME; SYMMETRY; VECTORS

Citation Formats

Alcubierre, Miguel, Corichi, Alejandro, Nunez, Dario, Salgado, Marcelo, Gonzalez, Jose A., Reimann, Bernd, and Max Planck Institut fuer Gravitationsphysik, Albert Einstein Institut, Am Muehlenberg 1, 14476 Golm. Generalized harmonic spatial coordinates and hyperbolic shift conditions. United States: N. p., 2005. Web. doi:10.1103/PhysRevD.72.124018.
Alcubierre, Miguel, Corichi, Alejandro, Nunez, Dario, Salgado, Marcelo, Gonzalez, Jose A., Reimann, Bernd, & Max Planck Institut fuer Gravitationsphysik, Albert Einstein Institut, Am Muehlenberg 1, 14476 Golm. Generalized harmonic spatial coordinates and hyperbolic shift conditions. United States. doi:10.1103/PhysRevD.72.124018.
Alcubierre, Miguel, Corichi, Alejandro, Nunez, Dario, Salgado, Marcelo, Gonzalez, Jose A., Reimann, Bernd, and Max Planck Institut fuer Gravitationsphysik, Albert Einstein Institut, Am Muehlenberg 1, 14476 Golm. Thu . "Generalized harmonic spatial coordinates and hyperbolic shift conditions". United States. doi:10.1103/PhysRevD.72.124018.
@article{osti_20774539,
title = {Generalized harmonic spatial coordinates and hyperbolic shift conditions},
author = {Alcubierre, Miguel and Corichi, Alejandro and Nunez, Dario and Salgado, Marcelo and Gonzalez, Jose A. and Reimann, Bernd and Max Planck Institut fuer Gravitationsphysik, Albert Einstein Institut, Am Muehlenberg 1, 14476 Golm},
abstractNote = {We propose a generalization of the condition for harmonic spatial coordinates analogous to the generalization of the harmonic time slices introduced by Bona et al., and closely related to dynamic shift conditions recently proposed by Lindblom and Scheel, and Bona and Palenzuela. These generalized harmonic spatial coordinates imply a condition for the shift vector that has the form of an evolution equation for the shift components. We find that in order to decouple the slicing condition from the evolution equation for the shift it is necessary to use a rescaled shift vector. The initial form of the generalized harmonic shift condition is not spatially covariant, but we propose a simple way to make it fully covariant so that it can be used in coordinate systems other than Cartesian. We also analyze the effect of the shift condition proposed here on the hyperbolicity of the evolution equations of general relativity in 1+1 dimensions and 3+1 spherical symmetry, and study the possible development of blowups. Finally, we perform a series of numerical experiments to illustrate the behavior of this shift condition.},
doi = {10.1103/PhysRevD.72.124018},
journal = {Physical Review. D, Particles Fields},
number = 12,
volume = 72,
place = {United States},
year = {Thu Dec 15 00:00:00 EST 2005},
month = {Thu Dec 15 00:00:00 EST 2005}
}
  • A nonlinear model representing the quantum harmonic oscillator on the sphere and the hyperbolic plane is solved in polar coordinates (r,{phi}) by making use of a curvature-dependent formalism. The curvature {kappa} is considered as a parameter and then the radial Schroedinger equation becomes a {kappa}-dependent Gauss hypergeometric equation. The energy spectrum and the wave functions are exactly obtained in both the sphere S{sup 2} ({kappa}>0) and the hyperbolic plane H{sup 2} ({kappa}<0). A comparative study between the spherical and the hyperbolic quantum results is presented.
  • We present a time-dependent generalized pseudospectral (TDGPS) approach in hyperspherical coordinates for fully ab initio nonperturbative treatment of multiphoton dynamics of atomic systems in intense laser fields. The procedure is applied to the investigation of high-order-harmonic generation (HHG) of helium atoms in ultrashort laser pulses at a KrF wavelength of 248.6 nm. The six-dimensional coupled hyperspherical-adiabatic-channel equations are discretized and solved efficiently and accurately by means of the TDGPS method. The effects of electron correlation and doubly excited states on HHG are explored in detail. A HHG peak with Fano line profile is identified which can be attributed to amore » broad resonance of doubly excited states. Comparison of the HHG spectra of the ab initio two-electron and the single-active-electron model calculations is also presented.« less
  • A nonlinear model of the quantum harmonic oscillator on two-dimensional space of constant curvature is exactly solved. This model depends on a parameter {lambda} that is related with the curvature of the space. First, the relation with other approaches is discussed and then the classical system is quantized by analyzing the symmetries of the metric (Killing vectors), obtaining a {lambda}-dependent invariant measure d{mu}{sub {lambda}} and expressing the Hamiltonian as a function of the Noether momenta. In the second part, the quantum superintegrability of the Hamiltonian and the multiple separability of the Schroedinger equation is studied. Two {lambda}-dependent Sturm-Liouville problems, relatedmore » with two different {lambda}-deformations of the Hermite equation, are obtained. This leads to the study of two {lambda}-dependent families of orthogonal polynomials both related with the Hermite polynomials. Finally the wave functions {psi}{sub m,n} and the energies E{sub m,n} of the bound states are exactly obtained in both the sphere S{sup 2} and the hyperbolic plane H{sup 2}.« less
  • In this paper, scaling properties of hyperbolic and nonhyperbolic model systems are discussed by using the generalized thermodynamic formalism. The central quantity for the investigation is the generalized entropy function. With the help of this approach, insight into the possible occurrence of phase trnasitions in the various entropy-like scaling functions can be gained. It is shown how this effect is determined by the existence of a critical line in the surface described by the generalized entropy function.