Critical exact solutions for self-gravitating Dirac fields

TL;DR

This paper finds specific exact solutions to the Einstein-Dirac equations for a self-gravitating neutrino, revealing solutions with flat spacetime and unique spinor properties that challenge standard quantum field theory approaches.

Contribution

It identifies exact axially-symmetric solutions with critical features, highlighting a tension between classical gravity solutions and quantum field theory methods.

Findings

Solutions have flat spacetime curvature.

Spinor fields exhibit vanishing scalar bi-linears.

Solutions cannot be quantized using standard quantum field theory.

Abstract

We consider the Einstein-Dirac field equations describing a self-gravitating massive neutrino, looking for axially-symmetric exact solutions; in the search of general solutions, we find some that are specific and which have critical features, such as the fact that the space-time curvature turns out to be flat and the spinor field gives rise to a vanishing bi-linear scalar $\overline{\psi}\psi=0$ with non-vanishing bi-linear pseudo-scalar $i\overline{\psi}\gamma^5\psi\not=0$: because in quantum field theory general computational methods are built on plane-wave solutions, for which bi-linear pseudo-scalar vanishes while the bi-linear scalar does not vanish, then the solutions we found cannot be treated with the usual machinery of quantum field theory. This means that for the Einstein-Dirac system there exist admissible solutions which nevertheless cannot be quantized with the common prescriptions; we regard this situation as yet another issue of tension between Einstein gravity and quantum principles. Possible ways to quench this tension can be seen either in enlarging the validity of quantum field theory or by restricting the space of the solutions of the Einstein-Dirac system of field equations.