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Title: Further applications of harmonic mappings of Riemannian manifolds to gravitational fields

Abstract

We consider static, spherically-symmetric and stationary, axially-symmetric gravitational fields in the framework of harmonic mappings of Riemannian manifolds. In this approach the emphasis is on a correspondence between the solution of the Einstein field equations and the geodesics in an appropriate Riemannian configuration space. Using Hamilton--Jacobi techniques, we obtain the geodesics and construct the resulting space--time geometries. For static space--times with spherical symmetry we obtain the well-known solutions of Schwarzschild, Reissner--Nordstroem, and Janis--Newman--Winicour, where in the latter two cases the source for the geometry is the static, source-free Maxwell field and the massless scalar field respectively. When we consider the source to be a triad of massless scalar fields with the internal symmetry SU(2) x SU(2), we find that the space--time geometry is once again that of Janis--Newman--Winicour. Finally we formulate the stationary, axisymmetric gravitational field problem in terms of composite mappings and obtain the Weyl, Papapetrou, and the generalized Lewis and van Stockum classes of axisymmetric solutions.

Authors:
Publication Date:
Research Org.:
Department of Physics, Middle East Technical University, Ankara, Turkey
OSTI Identifier:
7119998
Resource Type:
Journal Article
Journal Name:
J. Math. Phys. (N.Y.); (United States)
Additional Journal Information:
Journal Volume: 18:4
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; GRAVITATIONAL FIELDS; RIEMANN SPACE; TOPOLOGICAL MAPPING; EINSTEIN FIELD EQUATIONS; GENERAL RELATIVITY THEORY; GEODESICS; SU-2 GROUPS; EQUATIONS; FIELD EQUATIONS; FIELD THEORIES; LIE GROUPS; MATHEMATICAL SPACE; SPACE; SU GROUPS; SYMMETRY GROUPS; TRANSFORMATIONS; 657003* - Theoretical & Mathematical Physics- Relativity & Gravitation

Citation Formats

Eris, A. Further applications of harmonic mappings of Riemannian manifolds to gravitational fields. United States: N. p., 1977. Web. doi:10.1063/1.523311.
Eris, A. Further applications of harmonic mappings of Riemannian manifolds to gravitational fields. United States. https://doi.org/10.1063/1.523311
Eris, A. Fri . "Further applications of harmonic mappings of Riemannian manifolds to gravitational fields". United States. https://doi.org/10.1063/1.523311.
@article{osti_7119998,
title = {Further applications of harmonic mappings of Riemannian manifolds to gravitational fields},
author = {Eris, A},
abstractNote = {We consider static, spherically-symmetric and stationary, axially-symmetric gravitational fields in the framework of harmonic mappings of Riemannian manifolds. In this approach the emphasis is on a correspondence between the solution of the Einstein field equations and the geodesics in an appropriate Riemannian configuration space. Using Hamilton--Jacobi techniques, we obtain the geodesics and construct the resulting space--time geometries. For static space--times with spherical symmetry we obtain the well-known solutions of Schwarzschild, Reissner--Nordstroem, and Janis--Newman--Winicour, where in the latter two cases the source for the geometry is the static, source-free Maxwell field and the massless scalar field respectively. When we consider the source to be a triad of massless scalar fields with the internal symmetry SU(2) x SU(2), we find that the space--time geometry is once again that of Janis--Newman--Winicour. Finally we formulate the stationary, axisymmetric gravitational field problem in terms of composite mappings and obtain the Weyl, Papapetrou, and the generalized Lewis and van Stockum classes of axisymmetric solutions.},
doi = {10.1063/1.523311},
url = {https://www.osti.gov/biblio/7119998}, journal = {J. Math. Phys. (N.Y.); (United States)},
number = ,
volume = 18:4,
place = {United States},
year = {1977},
month = {4}
}