Exact solutions of the vacuum Einstein’s equations allowing for two noncommutative killing vectors (Type G2II of Petrov classification)

1992 ◽  
Vol 33 (7) ◽  
pp. 2567-2573 ◽  
Author(s):  
B. N. Aliev ◽  
A. N. Leznov
Keyword(s):  
Exact Solutions ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Killing Vectors ◽  
Petrov Classification

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.

scholarly journals Some axially symmetric zero mass meson solutions of Einstein's equations

1975 ◽  
Vol 19 (2) ◽  
pp. 140-145 ◽  
Author(s):  
L. K. Patel ◽  
V. M. Trivedi
Keyword(s):  
Exact Solutions ◽  
Electromagnetic Fields ◽  
Axially Symmetric ◽  
Einstein’S Equations ◽  
Field Equations ◽  
Einstein's Equations ◽  
Zero Mass ◽  
Oblate Spheroidal

AbstractAn axially symmetric metric in oblate spheroidal co-ordinates is considered. Two exact solutions of the field equations corresponding to zero mass meson fields are obtained. The details of the solutions are also discussed. These solutions are also generalized to include electromagnetic fields.

scholarly journals Solution-generating methods of Einstein’s equations by Hamiltonian reduction

2018 ◽  
Vol 33 (18) ◽  
pp. 1850101
Author(s):  
Seung Hun Oh ◽  
Kyoungtae Kimm ◽  
Yongmin Cho ◽  
Jong Hyuk Yoon
Keyword(s):  
Exact Solutions ◽  
Degrees Of Freedom ◽  
Hamiltonian Reduction ◽  
Einstein’S Equations ◽  
Schwarzschild Solution ◽  
Einstein's Equations ◽  
Constraint Equations ◽  
Dimensional Spacetime ◽  
Free Data ◽  
Privileged Coordinates

The purpose of this paper is to demonstrate a new method of generating exact solutions to Einstein’s equations obtained by the Hamiltonian reduction. The key element to the successful Hamiltonian reduction is finding the privileged spacetime coordinates in which physical degrees of freedom manifestly reside in the conformal two-metric, and all the other metric components are determined by the conformal two-metric. In the privileged coordinates, Einstein’s constraint equations become trivial; the Hamiltonian and momentum constraints are simply the defining equations of a nonvanishing gravitational Hamiltonian and momentum densities in terms of conformal two-metric and its conjugate momentum, respectively. Thus, given any conformal two-metric, which is a constraint-free data, one can construct the whole four-dimensional spacetime by integrating the first-order superpotential equations. As the first examples of using Hamiltonian reduction in solving Einstein’s equations, we found two exact solutions to Einstein’s equations in the privileged coordinates. Suitable coordinate transformations from the privileged to the standard coordinates show that they are just the Einstein–Rosen wave and the Schwarzschild solution. The local gravitational Hamiltonian and momentum densities of these spacetimes are also presented in the privileged coordinates.

scholarly journals Exact solutions of Einstein's equations

Scholarpedia ◽  
2013 ◽  
Vol 8 (12) ◽  
pp. 8584 ◽  
Author(s):  
Malcolm MacCallum
Keyword(s):  
Exact Solutions ◽  
Einstein’S Equations ◽  
Einstein's Equations

Exact solutions of Einstein's equations that are used in cosmology

2010 ◽  
pp. 11-54
Author(s):  
Krzysztof Bolejko ◽  
Andrzej Krasinski ◽  
Charles Hellaby ◽  
Marie-Noelle Celerier
Keyword(s):  
Exact Solutions ◽  
Einstein’S Equations ◽  
Einstein's Equations
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