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Spinor fields in spherical symmetry. Einstein-Dirac and other space-times

We discuss the static spherically symmetric Einstein-spinor field system in the possible presence of various spinor field nonlinearities. We take into account that the spinor field energy-momentum tensor (EMT) has in general some off-diagonal components, whose vanishing due to the Einstein equations substantially affects the form of the spinor field itself and the space-time geometry. In particular, the EMT structure with any spinor field nonlinearities turns out to be the same as that of the EMT of a minimally coupled scalar field with a self-interaction potential. Therefore many results previously obtained for systems with such scalar fields are directly extended to the Einstein-spinor field system. Some special solutions are obtained and discussed, in particular, a solution for the Einstein-Dirac system (which lack asymptotic flatness) and some examples with spinor field nonlinearities.