### Abstract

Gravitational effects of a *c* number neutrino field in spatially homogeneous geometries are analyzed. At the basis of the description of the geometries is an arbitrary timelike congruence with aligned tetrad reference frames, i.e., one basis vector of the tetrad frame at each point of space–time is aligned along the congruence. The remaining three basis vectors of the tetrad lie in orthogonal spacelike hypersurfaces, and for these there are three alternative choices: (1) *i* *n* *v* *a* *r* *i* *a* *n* *t* *r* *e* *f* *e* *r* *e* *n* *c* *e* *t* *r* *i* *a* *d* *s*, i.e., which conform to the isometry group of the space with structure constants G^{ c } _{ g h }; (2) *o* *r* *t* *h* *o* *n* *o* *r* *m* *a* *l* *r* *e* *f* *e* *r* *e* *n* *c* *e* *t* *r* *i* *a* *d* *s* simulating a Euclidean structure at each point; (3) *h* *o* *l* *o* *n* *o* *m* *i* *c* *c* *o* *o* *r* *d* *i* *n* *a* *t* *e* *t* *r* *i* *a* *d* *s* in terms of which metrics are usually given. The relations between (1), (2), and (3) for all nine Bianchi–Behr types are reviewed. The relatively compact vector–dyadic formalism used by Estabrook, Wahlquist, and Behr is introduced; it has the advantage that it gives directly the Bianchi–Behr classification of cosmologic solutions, and represents these solutions in terms of the orthonormal frame components of interesting physical quantities such as shear rates, volume expansion rates, and rotation with respect to inertially stabilized directions. The coupled Einstein–Dirac equations for the gravitational neutrino field and the neutrino stress‐energy tensor are derived, and the coupling equations are replaced by a set in which the neutrino amplitude is eliminated, and instead there appear the neutrinocurrent (J, *J* _{0}=‖J‖) (vector bilinear covariant) and the ’’eikonal momentum’’ P=∇ *f*, the gradient of the phase function *f*(*x*) which appears in the neutrino spinor amplitude ψ=χ(*t*)*e* ^{ i f(x)}. Solutions for all nine Bianchi types are classified. Further, with the two assumptions: (1) axial symmetry of the geometry, (2) alignment of the neutrino momentum density t with the Bianchi vector n (in the symmetry direction), a new exact solution is found explicitly. It is of type VII_{ h }. This solution goes over into a solution for type V when a parameter in the solution is set equal to zero. This is to be expected from the fact that type V may be regarded as a limiting case of type VII_{ h } when *a*=*b*→0.

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