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PP-waves with torsion: a metric-affine model for the massless neutrino. (English) Zbl 1305.83054

Summary: In this paper we deal with quadratic metric-affine gravity, which we briefly introduce, explain and give historical and physical reasons for using this particular theory of gravity. We then introduce a generalisation of well known spacetimes, namely pp-waves. A classical pp-wave is a 4-dimensional Lorentzian spacetime which admits a nonvanishing parallel spinor field; here the connection is assumed to be Levi-Civita. This definition was generalised in our previous work to metric compatible spacetimes with torsion and used to construct new explicit vacuum solutions of quadratic metric-affine gravity, namely generalised pp-waves of parallel Ricci curvature. The physical interpretation of these solutions we propose in this article is that they represent a conformally invariant metric-affine model for a massless elementary particle. We give a comparison with the classical model describing the interaction of gravitational and massless neutrino fields, namely Einstein-Weyl theory and construct pp-wave type solutions of this theory. We point out that generalised pp-waves of parallel Ricci curvature are very similar to pp-wave type solutions of the Einstein-Weyl model and therefore propose that our generalised pp-waves of parallel Ricci curvature represent a metric-affine model for the massless neutrino.

MSC:

83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
83C35 Gravitational waves
83C15 Exact solutions to problems in general relativity and gravitational theory
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
53Z05 Applications of differential geometry to physics
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