On Einstein’s Equations with Two Commuting Killing Vectors

1990 ◽  
pp. 219-223
Author(s):  
B. Gaffet
Keyword(s):  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Killing Vectors

scholarly journals Observables in terms of connection and curvature variables for Einstein’s equations with two commuting Killing vectors

2015 ◽  
Vol 471 (2183) ◽  
pp. 20150350
Author(s):  
P. Kordas
Keyword(s):  
Transition Matrix ◽  
Momentum Density ◽  
Transition Matrices ◽  
Einstein’S Equations ◽  
Integrals Of Motion ◽  
Einstein's Equations ◽  
Killing Vectors ◽  
Klein Gordon ◽  
Quantum Operators ◽  
Energy And Momentum

Einstein’s equations with two commuting Killing vectors and the associated Lax pair are considered. The equations for the connection A ( ς , η , γ )= Ψ , γ Ψ −1 , where γ the variable spectral parameter are considered. A transition matrix T = A ( ς , η , γ ) A −1 ( ξ , η , γ ) for A is defined relating A at ingoing and outgoing light cones. It is shown that it satisfies equations familiar from integrable PDE theory. A transition matrix on ς = constant is defined in an analogous manner. These transition matrices allow us to obtain a hierarchy of integrals of motion with respect to time, purely in terms of the trace of a function of the connections g , ς g −1 and g , η g −1 . Furthermore, a hierarchy of integrals of motion in terms of the curvature variable B = A , γ A −1 , involving the commutator [ A (1), A (−1)], is obtained. We interpret the inhomogeneous wave equation that governs σ = lnN , N the lapse, as a Klein–Gordon equation, a dispersion relation relating energy and momentum density, based on the first connection observable and hence this first observable corresponds to mass. The corresponding quantum operators are ∂/∂ t , ∂/∂ z and this means that the full Poincare group is at our disposal.

COVARIANT CONFORMAL DECOMPOSITION OF EINSTEIN EQUATIONS

2002 ◽  
Vol 17 (20) ◽  
pp. 2762-2762
Author(s):  
E. GOURGOULHON ◽  
J. NOVAK
Keyword(s):  
Numerical Simulations ◽  
Extrinsic Curvature ◽  
Einstein Equations ◽  
Numerical Implementation ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Tensor Density ◽  
Dirac Gauge ◽  
Ill Posed ◽  
Isolated Systems

It has been shown1,2 that the usual 3+1 form of Einstein's equations may be ill-posed. This result has been previously observed in numerical simulations3,4. We present a 3+1 type formalism inspired by these works to decompose Einstein's equations. This decomposition is motivated by the aim of stable numerical implementation and resolution of the equations. We introduce the conformal 3-"metric" (scaled by the determinant of the usual 3-metric) which is a tensor density of weight -2/3. The Einstein equations are then derived in terms of this "metric", of the conformal extrinsic curvature and in terms of the associated derivative. We also introduce a flat 3-metric (the asymptotic metric for isolated systems) and the associated derivative. Finally, the generalized Dirac gauge (introduced by Smarr and York5) is used in this formalism and some examples of formulation of Einstein's equations are shown.

scholarly journals LOCAL COSMIC STRING AND C-FIELD

2006 ◽  
Vol 21 (18) ◽  
pp. 3727-3732 ◽  
Author(s):  
F. RAHAMAN ◽  
R. MONDAL ◽  
M. KALAM
Keyword(s):  
Cosmic String ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Einstein’S Equations ◽  
Einstein's Equations

We investigate a local cosmic string with a phenomenological energy–momentum tensor as prescribed by Vilenkin, in the presence of C-field. The solutions of full nonlinear Einstein's equations for exterior and interior regions of such a string are presented.

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