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

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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 lacks asymptotic flatness) and some examples with spinor field nonlinearities.

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Fig. 1


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    This singularity is related to an infinite value of the curvature invariants that involve the squared component of the Riemann tensor [11] \(R^{02}{}_{02} = -\mathrm{e}^{-2\alpha } \beta '\gamma '\), where the quantities \(\alpha \) given by (16) and \(\beta '=1/r\) are finite at \(r=r_0\) while \(\gamma ' = \infty \).


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The work of K.B. was partly performed within the framework of the Center FRPP supported by MEPhI Academic Excellence Project (Contract No. 02.a03.21.0005, 27.08.2013) and also partly funded by the RUDN University Program 5-100. The work of BS was supported in part by a joint Romanian-JINR, Dubna Research Project, Order No. 396/27.05.2019 p-71 and was also partly funded by the RUDN University Program 5-100.

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Correspondence to Bijan Saha.

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Bronnikov, K.A., Rybakov, Y.P. & Saha, B. Spinor fields in spherical symmetry: Einstein–Dirac and other space-times. Eur. Phys. J. Plus 135, 124 (2020).

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