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=ψψ, 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.
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. Such a challenge stimulates intense research activities in various field-theoretical models with full non-perturbative account of gravity. Among all these activities, the investigations of solitons in these theories, with a special emphasis on flat space theories, attracted a particular importance due to their properties. Indeed, the soliton sector in the flat space gauge theories is quite well understood, the most notable example being the t’Hooft-Polyakov magnetic monopole. For a review on some recent progress in the investigation of solitons and black holes in non-Abelian gauge theories coupled to gravity, see [
On the other hand, the marriage of relativity and quantum theory leads naturally to the quantum field theory description of the elementary particles and their interactions, at the most intimate presently accessible scales of space and energy, a fact made manifest by the value of the product Mev.fm. In fact, one offspring of this second union is the unification of matter and radiation, namely of particles with their corpuscular propagating properties and fields with their wavelike propagating properties. Particles, characterized through their energy, momentum and spin values in correspondence with the Poincaré symmetries of Minkowski space-time in the absence of gravity, are nothing but the relativistic energy-momentum quanta of a field, thereby implying a tremendous economy in the description of the physical universe, accounting for instance at once in terms of a single field filling all of space-time for the indistinguishability of identical particles and their statistics. Furthermore, quantum relativistic interactions are then understood simply as couplings between the various quantum fields locally in space-time, which translate in terms of particles as diverse exchanges of the associated quanta. Such a picture lends itself most ideally to a perturbative understanding of the fundamental interactions, which has proved to be so powerful beginning with quantum electrodynamics, up to the modern Standard Model of the strong and electroweak interactions. 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 [
All these activities, diverse and complementary, made in this field [1-14], are also 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 [
Moreover, it is also worthy of attention a previous study, which will be referred to Part I of the present work, where Adomou and Shikin [
The present work, considered as Part II of all these investigated initiated in [
The paper is organized as follows. Section 2 addresses the model with fundamental equations. We consider a selfconsistent system to obtain spherical-symmetric solutions, taking into account the own gravitational field of particles. Section 3 deals with main results and their discussion; the solutions of the Einstein and nonlinear spinor field equations are derived. Besides, the regularity properties of the obtained solutions as well as the asymptotic behavior of the energy and charge densities are studied. Concluding remarks are outlined in Section 4.
We consider the Lagrangian of the self-consistent system of spinor and gravitational fields in the form [
where R is the scalar curvature; is Einstein’s gravitational constant and is an arbitrary function depending on.
Instead of the static plane-symmetric metric chosen in [
being some functions depending only onwhere r stands for the radial component of the spherical symmetric metric, and satisfying the coordinate condition
From the Lagrangian (2.1), through the variational principle and usual algebraic manipulations, one can readily deduce the Einstein equations for the metric (2.3) under the condition (2.4), the spinor field equations for the functions, , and the components of the metric spinor field energy-momentum tensor, respectively, in the form [
where is the covariant spinor derivative [
; are the spinor affine connection matrices. To define the matrices, let us use the equalities
where; are the Dirac’s matrices in flat space-time; are tetradic 4- vectors. Then we get:
The matrices are then determined as follows:
The matrices are chosen as in [
with the spinor
Taking into account (2.18), let us write explicitly the nonzero components of the tensor:
setting the condition,
Using the obtained expressions for in (2.15)- (2.17), we can expand (2.10) as
yielding the following set of equations:
From the set of Equations (2.22)-(2.25), we infer that the invariant function
satisfies a first order differential equation:
giving the evident solution
C being a constant. Combining the spinor field Equation (2.21) with its conjugate expression results the following expression for (2.20):
The difference of the Einstein equations with (2.19) leads to
which can be transformed into a Liouville equation (see [
where the quantity A is expressed in terms of the Newton’s gravitational constant G as:
h being an integration constant and another non zero integration constant. Taking into account (3.5) and (3.6), we get from (2.4) the following relations:
and
Substituting (3.8) into (2.6), we obtain the Einstein equation in the form
Since with the invariant, from (3.10), we get:
With the knowledge of and from the relations (3.5), (3.6) and (3.8), respectively, the invariant as well as the solutions of the Einstein equations can be completely determined. Furthermore, considering the concrete expression of the invariant, namely, we can establish the regularity properties of the obtained solutions. Studying the distribution of the energy per unit invariant volume, we can also deduce their localization properties.
We can get a concrete form of the functions by solving Equations (2.22)-(2.25) in a more compact form if we pass to the functions, ρ = 1, 2, 3, 4:
where
Re-express Equations (3.12)-(3.15) under forms depending on functions of the argument, i.e. , Then we get for the functions the following set of equations:
where
with determined by (3.11).
Differentiating now Equations (3.17)-(3.20) and substituting Equations (3.20) and (3.17) into the result, we obtain second-order differential equations obeyed by the functions and:
Summing (3.22) and (3.23) and setting afford the differential equation:
which, under the condition with, yields the solution
where. Substracting Equations (3.17) and (3.20) and taking into account (3.25), we obtain
It then follows, from the Equations (3.25) and (3.26), that
and
with.
Analogously operating on Equations (3.18) and (3.19), we arrive at
and
with,
As mentioned in [
In this case, the relation (3.32) giving becomes:
From (3.5), (3.6) and (3.8), we get:
showing that the invariant S and the functions, , , are regular. In the case under consideration we have, i.e. the energy density is localized.
Using (3.11), (3.21) and (3.31), we get:
with.
Let us find the explicit form of. To this end, we retrieve the expressions of and from (3.37), knowing that. Without loss of generality, let us set. Then,
Substituting from (3.33) into (3.38), we get
with.
We then replace the expressions of and from (3.39) into (3.27)-(3.30) and get an explicit form of, and subsequently the expressions of
:
which represent nothing but the regular localized solitonlike solutions.
In the sequel, we deal with a concrete type of nonlinear spinor field equations which have the virtue that, where is a nonlinearity parameter,. It is convenient to separately analyze the two cases and:
• : and we have the nonlinear spinor field equation
The equalities (3.33)-(3.36) remain valid. Let us find an explicit form of. For that, we deduce from (3.37) the function and:
that we substitute into (3.27)-(3.30) to get an explicit expression of and subsequently the initial functions,.
Let us compute the distribution of the spinor field energy density per unit invariant volume . From (2.19) and (3.33) we have the following expression for:
permiting to write
inferring that the quantities, , and are regular and, from (3.47), the total energy
is finite. Therefore, the equation
(3.44) possesses a soliton-like solution.
• : and the energy density is
From (3.33), the distribution of the spinor field energy density per unit invariant volume takes the form
i.e.
showing that the spinor field energy density per unit invariant volume f is localized and the total energy is finite. To compute, , we need the functions and:
where
and
that we substitute into (3.27)-(3.30) to get an explicit expression for, and then we readily compute the initial functions
for. Using the solutions (3.27)-(3.30), we deduce the components of the spinor current vector as follows:
Since the configuration is static, only the component is nonzero. The constants in the solution of the spinor field equation are obtained from the equations and, thus giving, and. The component defines the charge density of the spinor field whose the chronometric invariant form is characterized by:
where. The total charge of the spinor field is:
being the center of the field configuration.
The relations (3.33), (3.39), (3.45), (3.55) and (3.56) infer that the charge density of the spinor field is localized, and the total charge is a finite quantity, when, or, or.
In this paper, we have obtained exact spherical symmetric solutions to the spinor and gravitational field equations and studied their regularity properties as well as the localization properties of both the energy and charge densities in different configurations, when, and.
In all these cases, the solutions are regular; the energy and charge densities are localized. The total energy and charge of the spinor field are finite quantities. The study of the set of all regular spherical solutions with a possible criterion of their classification could deserve some interest. Such investigation will be in the core of the forthcoming paper.
This work is partially supported by the Abdus Salam International Centre for Theoretical Physics (ICTP, Trieste, Italy) through the OEA-ICMPA-Prj-15. The ICMPA is in partnership with the Daniel Iagolnitzer Foundation (DIF), France.