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Register a new userAxially-Symmetric Exact Solutions for Flagpole Fermions with Gravity
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The Lounesto classification splits spinors in six classes: I, II, III are those for which at least one among scalar and pseudo-scalar bi-linear spinor quantities is non-zero, its spinors are called regular, and among them we find the usual Dirac spinor. IV, V, VI are those for which the scalar and pseudo-scalar bi-linear spinor quantities are identically zero, its spinors are called singular, and they are split in further sub-classes: IV has no further restrictions, its spinors are called flag-dipole; V is the one for which the spin axial-vector vanishes, its spinors are called flagpole, and among them we find the Majorana spinor; VI is the one for which the momentum antisymmetric-tensor vanishes, its spinors are called dipole, and among them we find the Weyl spinor. In the quest for exact solutions of fully-coupled systems of spinor fields in their own gravity, we have already given examples in the case of Dirac fields and Weyl fields but never in the case of Majorana or more generally flagpole spinor fields. Flagpole spinor fields in interaction with their own gravitational field, in the case of axial symmetry, will be considered. Exact solutions of the field equations will be given.
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In a recent paper (Beyer and Hennig, 2012 [9]), we have introduced a class of inhomogeneous cosmological models: the smooth Gowdy-symmetric generalized Taub-NUT solutions. Here we derive a three-parametric family of exact solutions within this class, which contains the two-parametric Taub solution as a special case. We also study properties of this solution. In particular, we show that for a special choice of the parameters, the spacetime contains a curvature singularity with directional behaviour that can be interpreted as a true spike in analogy to previously known Gowdy symmetric solutions with spatial T3-topology. For other parameter choices, the maximal globally hyperbolic region is singularity-free, but may contain false spikes.
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The complete sets of analytic solutions of the geodesic equation in Taub--NUT--(anti-)de Sitter, Kerr--(anti-)de Sitter and also in general Plebanski--Demianski space--times without acceleration are presented. The solutions are given in terms of the Kleinian sigma functions.
We find a new method for looking for the static and spherically symmetric solutions in $F(R)$ theory of gravity. With this method, a number of new solutions in terms of the analytic functions are obtained. We hope this investigation may be of some help in the searching for some other solutions in $F(R)$ theory of gravity.
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