Abstract
This paper deals with an extension of a previous work [Gravitation & Cosmology, Vol. 4, 1998, pp 107--113] to exact spherical symmetric solutions to the spinor field equations with nonlinear terms which are arbitrary functions of $S=\bar{\psi}\psi $, taking into account their own gravitational field. Equations with power and polynomial nonlinearities are studied in detail. It is shown that the initial set of the Einstein and spinor field equations with a power nonlinearity has regular solutions with spinor field localized energy and charge densities. The total energy and charge are finite. Besides, exact solutions, including soliton - like solutions, to the spinor field equations are also obtained in flat space-time.
Highlights
The unification of quantum mechanics and general relativity into a theory of quantum gravity remains a hard unsolved problem and physical phenomena requiring both general relativity and quantum theory for their description cannot be possibly completely understood
For a review on some recent progress in the investigation of solitons and black holes in non-Abelian gauge theories coupled to gravity, see [1] and references therein
The most fascinating offsprings of this union are undoubtedly, on the one hand, the cosmological theory of the history of our universe from its birth to its ultimate demise if ever, and on the other hand, the prediction for regions of space-time to be so much curled up by their energy-matter content that even light can no longer escape from such black holes
Summary
The unification of quantum mechanics and general relativity into a theory of quantum gravity remains a hard (as yet) unsolved problem and physical phenomena requiring both general relativity and quantum theory for their description cannot be possibly completely understood. For more explanation on these profound concepts, quantum theory and relativity, which have culminated into relativistic spacetime geometry and quantum gauge theory as the principles for gravity and the three other known fundamental interactions, see notes [2] on The quantum geometer’s universe: Particles, interactions and topology delivered in 2001 by Govaerts at the Second International Workshop on Contemporary Problems in Mathematical Physics All these activities, diverse and complementary, made in this field [1,2,3,4,5,6,7,8,9,10,11,12,13,14], are mainly motivated by the wide roles of Einstein and Dirac equations in modern physics, for example, for investigating the spin particle and for the necessity of analysis of synchrotronic radiation [11].
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