Reduction of Einstein's equations for vacuum space-times with spacelike U(1) isometry groups

Journal Article · Sat Mar 01 04:00:00 EST 1986 · Ann. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1016/S0003-4916(86)80009-4· OSTI ID:5706659

Moncrief, V

We consider vacuum space-times with spacelike U(1) isometry groups which are defined on manifolds of the form R x B/sub n/, where B/sub n/ is an arbitrary S/sup 1/-bundle over the two-sphere. We reduce the Einstein equations for this problem to a system of ''harmonic map'' equations defined over the base manifold R x S/sup 2/ equipped with a Lorentzian metric determined uniquely by the solution of an associated nonlinear elliptic system. The harmonic map fields (which have a range space diffeomorphic to the hyperbolic two-plane) represent the unconstrained, dynamical degrees of freedom of the gravitational field. We give a complete discussion of the existence and uniqueness of solutions of the associated elliptic system and also display a Geroch-type solution generating technique for globally transforming the space of solutions associated with any one non-trivial bundle, R x B/sub n/..-->..R x S/sup 2/, to that of any other such bundle. The basic techniques could readily be generalized to treat S/sup 1/-bundles over R x M where M is a compact two-manifold of arbitrary genus. In the higher genus cases the Teichmueller space of conformally diffeomorphic Riemannian metrics over M arises as an additional component of the configuration manifold. For Mroughly-equalS/sup 2/ this space collapses to a point, which slightly simplifies the analysis.

• Research Organization: Department of Mathematics and Department of Physics, Yale University, New Haven, Connecticut 06511
• OSTI ID: 5706659
• Journal Information: Ann. Phys. (N.Y.); (United States), Journal Name: Ann. Phys. (N.Y.); (United States) Vol. 167:1; ISSN APNYA
• Country of Publication: United States
• Language: English

Related Subjects

657003* -- Theoretical & Mathematical Physics-- Relativity & Gravitation
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANALYTICAL SOLUTION
EINSTEIN FIELD EQUATIONS
EQUATIONS
FIELD EQUATIONS
FIELD THEORIES
FUNCTIONS
GENERAL RELATIVITY THEORY
HAMILTONIAN FUNCTION
LIE GROUPS
SPACE-TIME
SYMMETRY GROUPS
U GROUPS
U-1 GROUPS
VACUUM STATES