Twistor Characterization of Stationary Axisymmetric Solutions of Einstein's Equations

Author(s):  
J. Fletcher ◽  
N.M.J. Woodhouse
Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1469
Author(s):  
Cooper K. Watson ◽  
William Julius ◽  
Matthew Gorban ◽  
David D. McNutt ◽  
Eric W. Davis ◽  
...  

In the years 1917–1919 Tullio Levi-Civita published a number of papers presenting new solutions to Einstein’s equations. This work, while partially translated, remains largely inaccessible to English speaking researchers. In this paper we review these solutions, and present them in a modern readable manner. We will also compute both Cartan–Karlhede and Carminati–Mclenaghan invariants such that these solutions are invariantly characterized by two distinct methods. These methods will allow for these solutions to be totally and invariantly characterized. Because of the variety of solutions considered here, this paper will also be a useful reference for those seeking to learn to apply the Cartan–Karlhede algorithm in practice.


2002 ◽  
Vol 17 (20) ◽  
pp. 2762-2762
Author(s):  
E. GOURGOULHON ◽  
J. NOVAK

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.


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