Ghost neutrinos in plane−symmetric spacetimes

Abstract

An exact solution to the Einstein−Dirac equations is presented for a plane−symmetric spacetime generated by neutrinos. The neutrino field is nonzero and corresponds to a neutrino current along the symmetry axis of the space. The neutrinos yield a nonzero energy−momentum tensor, which we specialize to Tij = 0 for ’’ghost’’ neutrinos. We show that, since the energy−momentum tensor vanishes, the time−dependent ’’ghost’’ neutrino metric reduces to the static case. The time−dependent ’’ghost’’ current is then reduced to the static current through a Lorentz transformation and the ’’ghost’’ wavefunction reduced to the static wavefunction through a spinor transformation. The ’’ghost’’ neutrino current is geodesic and the spacetime is classified by the expansion, rotation, and shear of these geodesics. From previous results it follows that our plane−symmetric ’’ghost’’ solution is the most general solution to the Einstein−Dirac equations for a vanishing energy−momentum tensor and a neutrino current that is expanding. The solution is Petrov type D.