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Title: Well-posed constrained evolution of 3+1 formulations of general relativity

Abstract

We present an analysis of well-posedness of constrained evolution of 3+1 formulations of general relativity. In this analysis we explicitly take into account the energy and momentum constraints as well as possible algebraic constraints on the evolution of high-frequency perturbations of solutions of Einstein's equations. In this respect, our approach is principally different from standard analyses of well-posedness of free evolution in general relativity. Our study reveals the existence of subsets of the linearized Einstein's equations that control the well-posedness of constrained evolution. It is demonstrated that the well-posedness of Arnowitt-Deser-Misner (ADM), Baumgarte-Shapiro-Shibata-Nakamura and other 3+1 formulations derived from the ADM formulation by adding combinations of constraints to the right-hand side of the ADM formulation and/or by linear transformation of the dynamical ADM variables depends entirely on the properties of the gauge. For certain classes of gauges we formulate conditions for well-posedness of constrained evolution. This provides a new basis for constructing stable numerical integration schemes for a classical ADM and many other 3+1 formulations of general relativity.

Authors:
 [1];  [1];  [2];  [3];  [4];  [5]
  1. Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637 (United States)
  2. (United States)
  3. Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637 (United States)
  4. (Denmark)
  5. (Russian Federation)
Publication Date:
OSTI Identifier:
21010896
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevD.75.024026; (c) 2007 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; COSMOLOGY; DISTURBANCES; EINSTEIN FIELD EQUATIONS; FOUR-DIMENSIONAL CALCULATIONS; GAUGE INVARIANCE; GENERAL RELATIVITY THEORY; MATHEMATICAL SOLUTIONS; TRANSFORMATIONS

Citation Formats

Paschalidis, Vasileios, Khokhlov, Alexei, Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, Novikov, Igor, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, and Astro Space Center of P.N. Lebedev Physical Institute, Profsoyuznaya 84/32, Moscow, 117810. Well-posed constrained evolution of 3+1 formulations of general relativity. United States: N. p., 2007. Web. doi:10.1103/PHYSREVD.75.024026.
Paschalidis, Vasileios, Khokhlov, Alexei, Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, Novikov, Igor, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, & Astro Space Center of P.N. Lebedev Physical Institute, Profsoyuznaya 84/32, Moscow, 117810. Well-posed constrained evolution of 3+1 formulations of general relativity. United States. doi:10.1103/PHYSREVD.75.024026.
Paschalidis, Vasileios, Khokhlov, Alexei, Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, Novikov, Igor, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, and Astro Space Center of P.N. Lebedev Physical Institute, Profsoyuznaya 84/32, Moscow, 117810. Mon . "Well-posed constrained evolution of 3+1 formulations of general relativity". United States. doi:10.1103/PHYSREVD.75.024026.
@article{osti_21010896,
title = {Well-posed constrained evolution of 3+1 formulations of general relativity},
author = {Paschalidis, Vasileios and Khokhlov, Alexei and Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637 and Novikov, Igor and Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen and Astro Space Center of P.N. Lebedev Physical Institute, Profsoyuznaya 84/32, Moscow, 117810},
abstractNote = {We present an analysis of well-posedness of constrained evolution of 3+1 formulations of general relativity. In this analysis we explicitly take into account the energy and momentum constraints as well as possible algebraic constraints on the evolution of high-frequency perturbations of solutions of Einstein's equations. In this respect, our approach is principally different from standard analyses of well-posedness of free evolution in general relativity. Our study reveals the existence of subsets of the linearized Einstein's equations that control the well-posedness of constrained evolution. It is demonstrated that the well-posedness of Arnowitt-Deser-Misner (ADM), Baumgarte-Shapiro-Shibata-Nakamura and other 3+1 formulations derived from the ADM formulation by adding combinations of constraints to the right-hand side of the ADM formulation and/or by linear transformation of the dynamical ADM variables depends entirely on the properties of the gauge. For certain classes of gauges we formulate conditions for well-posedness of constrained evolution. This provides a new basis for constructing stable numerical integration schemes for a classical ADM and many other 3+1 formulations of general relativity.},
doi = {10.1103/PHYSREVD.75.024026},
journal = {Physical Review. D, Particles Fields},
number = 2,
volume = 75,
place = {United States},
year = {Mon Jan 15 00:00:00 EST 2007},
month = {Mon Jan 15 00:00:00 EST 2007}
}