scholarly journals A Nonaxisymmetric Solution of Einstein’s Equations Featuring Pure Radiation from a Rotating Source

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
William Davidson
Keyword(s):  
Ricci Tensor ◽  
Kerr Metric ◽  
Type Ii ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Pure Radiation ◽  
Tensor Density ◽  
Null Tetrad ◽  
Radiation Divergence ◽  
Special Case

A special nonaxisymmetric solution of Einstein’s equations is derived, representing pure radiation from a rotating isolated source. The spacetime is assumed to be algebraically special having a multiple null eigenvector of the Weyl tensor forming a geodesic, shear-free, diverging, and twisting congruence k. Employing a complex null tetrad involving the vector k, the Ricci tensor, density of the radiation, divergence, and twist are calculated for the derived metric. A particular (nonaxisymmetric) subcase is shown to be flat at infinity and to contain the axisymmetric radiating Kerr metric, derived by Kramer and separately by Vaidya and Patel, as a special case. The spacetime is of Petrov type II and without Killing vectors.

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.

Hodge Theory on Compact Two-Dimensional Spacetimes and the Uniqueness ofgijwith a SpecifiedRij

1987 ◽  
Vol 39 (6) ◽  
pp. 1459-1474
Author(s):  
Edwin Ihrig
Keyword(s):  
Ricci Tensor ◽  
Energy Tensor ◽  
Main Question ◽  
Two Dimensional ◽  
Stress Energy Tensor ◽  
Riemann Curvature ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Stress Energy ◽  
Curvature Tensors

The main question we wish to address in this paper is to what extent does the Ricci curvature of a spacetime determine the metric of that spacetime. Although it is relatively easy to see that the full Riemann curvature uniquely determines the metric for a generic choice of curvature tensors (see[4], [10], [11], [14] and [15], and the references contained therein), very little has been discovered about whether, if ever, Ric (or the stress energy tensor in Einstein's equations for that matter) determinesg. Most exact solution techniques for Einstein's equations look only for solutions that have the same symmetries as Ric. It is not true in general thatgmust inherit the symmetries of Ric. It is not even clear that there is a Ric such that everygwith this Ricci tensor is known.

Ricci inheritance collineations in Bianchi type II spacetime

2016 ◽  
Vol 31 (17) ◽  
pp. 1650102 ◽  
Author(s):  
Tahir Hussain ◽  
Sumaira Saleem Akhtar ◽  
Ashfaque H. Bokhari ◽  
Suhail Khan
Keyword(s):  
Lie Algebra ◽  
Ricci Tensor ◽  
Bianchi Type ◽  
Complete Classification ◽  
Type Ii

In this paper, we present a complete classification of Bianchi type II spacetime according to Ricci inheritance collineations (RICs). The RICs are classified considering cases when the Ricci tensor is both degenerate as well as non-degenerate. In case of non-degenerate Ricci tensor, it is found that Bianchi type II spacetime admits 4-, 5-, 6- or 7-dimensional Lie algebra of RICs. In the case when the Ricci tensor is degenerate, majority cases give rise to infinitely many RICs, while remaining cases admit finite RICs given by 4, 5 or 6.

scholarly journals On the Bean critical-state model in superconductivity

1996 ◽  
Vol 7 (3) ◽  
pp. 237-247 ◽  
Author(s):  
L. Prigozhin
Keyword(s):  
Variational Inequalities ◽  
Critical State ◽  
Existence And Uniqueness ◽  
Critical State Model ◽  
Type Ii ◽  
Two Dimensional ◽  
Axially Symmetric ◽  
Uniqueness Of The Solution ◽  
Type Ii Superconductivity ◽  
Special Case

We consider two-dimensional and axially symmetric critical-state problems in type-II superconductivity, and show that these problems are equivalent to evolutionary quasi-variational inequalities. In a special case, where the inequalities become variational, the existence and uniqueness of the solution are proved.

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