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Integrals of motion in the two Killing vector reduction of general relativity

Technical Report:

Abstract

We apply the inverse scattering method to the midi-superspace models that are characterized by a two-parameter Abelian group of motions with two spacelike Killing vectors. We present a formulation that simplifies the construction of the soliton solutions of Belinskii and Zakharov. Furthermore, it enables us to obtain the zero curvature formulation for these models. Using this, and imposing periodic boundary conditions corresponding to the Gowdy models when the spatial topology is a three torus T{sup 3}, we show that the equation of motion for the monodromy matrix is an evolution equation of the Heisenberg type. Consequently, the eigenvalues of the monodromy matrix are the generating functionals for the integrals of motion. Furthermore, we utilise a suitable formulation of the transition matrix to obtain explicit expressions for the integrals of motion. This involves recursion relations which arise in solving an equation of Riccati type. In the case when the two Killing vectors are hypersurface orthogonal the integrals of motion have a particularly simple form. (authors). 19 refs.
Authors:
Manojlovic, N; [1]  Spence, B [2] 
  1. Syracuse Univ., NY (United States). Dept. of Physics
  2. Melbourne Univ., Parkville, VIC (Australia). School of Physics
Publication Date:
Aug 01, 1993
Product Type:
Technical Report
Report Number:
UM-P-93/77; SU-GP-93/7-8.
Reference Number:
SCA: 662110; PA: AIX-26:017894; EDB-95:032200; SN: 95001330004
Resource Relation:
Other Information: PBD: Aug 1993
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; GENERAL RELATIVITY THEORY; EQUATIONS OF MOTION; INVERSE SCATTERING PROBLEM; VECTOR FIELDS; BOUNDARY CONDITIONS; HAMILTONIANS; HEISENBERG MODEL; INTEGRALS; MATHEMATICAL MODELS; SOLITONS; SPACE-TIME; 662110; THEORY OF FIELDS AND STRINGS
OSTI ID:
10114029
Research Organizations:
Melbourne Univ., Parkville, VIC (Australia). School of Physics
Country of Origin:
Australia
Language:
English
Other Identifying Numbers:
Other: ON: DE95616281; TRN: AU9414197017894
Availability:
OSTI; NTIS (US Sales Only); INIS
Submitting Site:
INIS
Size:
19 p.
Announcement Date:
Jun 30, 2005

Technical Report:

Citation Formats

Manojlovic, N, and Spence, B. Integrals of motion in the two Killing vector reduction of general relativity. Australia: N. p., 1993. Web.
Manojlovic, N, & Spence, B. Integrals of motion in the two Killing vector reduction of general relativity. Australia.
Manojlovic, N, and Spence, B. 1993. "Integrals of motion in the two Killing vector reduction of general relativity." Australia.
@misc{etde_10114029,
title = {Integrals of motion in the two Killing vector reduction of general relativity}
author = {Manojlovic, N, and Spence, B}
abstractNote = {We apply the inverse scattering method to the midi-superspace models that are characterized by a two-parameter Abelian group of motions with two spacelike Killing vectors. We present a formulation that simplifies the construction of the soliton solutions of Belinskii and Zakharov. Furthermore, it enables us to obtain the zero curvature formulation for these models. Using this, and imposing periodic boundary conditions corresponding to the Gowdy models when the spatial topology is a three torus T{sup 3}, we show that the equation of motion for the monodromy matrix is an evolution equation of the Heisenberg type. Consequently, the eigenvalues of the monodromy matrix are the generating functionals for the integrals of motion. Furthermore, we utilise a suitable formulation of the transition matrix to obtain explicit expressions for the integrals of motion. This involves recursion relations which arise in solving an equation of Riccati type. In the case when the two Killing vectors are hypersurface orthogonal the integrals of motion have a particularly simple form. (authors). 19 refs.}
place = {Australia}
year = {1993}
month = {Aug}
}