Spin-spin interaction in general relativity and induced geometries with nontrivial topology
aa r X i v : . [ g r- q c ] D ec Spin-spin interaction in general relativityand induced geometries with nontrivial topology
V. G. Krechet and D. V. Sadovnikov Yaroslavl State Pedagogical University
We consider the dynamics of a self-gravitating spinor field and a self-gravitating rotating perfect fluid. Itis shown that both these matter distributions can induce a vortex field described by the curl 4-vector ofa tetrad: ω i = ε iklm e ( a ) k e ( a ) l ; m , where e ( a ) k are components of the tetrad. The energy-momentum tensor T ik ( ω ) of this field has been found and shown to violate the strong and weak energy conditions whichleads to possible formation of geometries with nontrivial topology like wormholes. The corresponding exactsolutions to the equations of general relativity have been found. It is also shown that other vortex fields,e.g., the magnetic field, can also possess such properties. As we have shown earlier [1, 2], a self-gravitatingDirac spinor field with the Lagrangian L ( ψ ) = ~ c (cid:2) ∇ i ¯ ψγ i ψ − ¯ ψγ i ∇ i ψ − F ( ¯ ψψ ) (cid:3) (1)can interact with the vortex component of the grav-itational field, which results in the appearance of amore general Lagrangian: L ( ψ ) = ~ c h ∂ i ¯ ψγ i ψ − ¯ ψγ i ∇ i ψ + ω i · ( ¯ ψγ i γ ψ ) − F ( ¯ ψψ ) i . (2)Here, ω i is the curl 4-vector of a tetrad: ω i = ε iklm e ( a ) k e ( a ) l ; m , i.e., the 4-vector of the gravita-tional field vortex; ∇ α ψ is the covariant derivativeof the spinor function ψ ( x k ): ∇ k ψ = ∂ k ψ − Γ k ψ ,where Γ k are the matrix spinor connection coef-ficients; γ k are the curved-space Dirac matricesdefined by the fundamental relation between thespace-time metric and spin, γ i γ k + γ k γ i = 2 g ik · I, and the axial vector ¯ ψγ i γ ψ is proportional to theproper angular momentum (spin) of the spinor field S k ( ψ ) = ~ c ¯ ψγ k γ ψ ; the function F ( ¯ ψψ ) is thespinor field potential depending on the invariant¯ ψψ . In particular, for a massive spinor field wehave F ( ¯ ψψ ) = 2 m ¯ ψψ .Variation of the total Lagrangian of the grav-itational and spinor fields L = − R/ (2 κ ) + L ( ψ )with respect to ω i leads to a relation between thegravitational field vortex and the spin density ofthe spinor field: ω i = κ ~ c ψγ i γ ψ. (3) e-mail: [email protected] Taking into account this relation, the spinor fieldLagrangian (2) takes the form L ( ψ ) = ~ c h ∂ k ¯ ψγ k ψ − ¯ ψγ k ∂ k ψ + κ hc ψγ k γ ψ )( ¯ ψγ k γ ψ ) − F ( ¯ ψψ ) i , (4)i.e., we have obtained the Lagrangian of a nonlinearspinor field with a quadratic pseudovector nonlin-earity.Interaction of such a nonlinear spinor field withgravity, even if the latter has no vortex component,e.g., in the case of spherical symmetry, leads toan interesting result. Spherically symmetric spinorfield configurations have a radially polarized spindensity vector S i ( ψ ) = ~ c ψγ i γ ψ = δ i ¯ ψγ γ ψ ~ c , distributed like the lines of force of a point elec-tric charge. Let us choose the metric of a static,spherically symmetric space-time in the form ds = e ν ( r ) dt − e λ ( r ) dr − r ( dθ + sin θ dϕ ) . (5)The components of the energy-momentum tensorof the nonlinear spinor field (4) in the absence ofthe potential F ( ¯ ψψ ) take the form T ik ( ψ )= hc (cid:2) ∇ i ¯ ψγ k ψ + ∇ k ¯ ψγ i ψ − ¯ ψγ i ∇ k ψ − ¯ ψγ k ∇ i ψ (cid:3) − hc κ hc ψγ s γ s ψ )( ¯ ψγ s γ ψ ) g ik . (6)Solving the set of Einstein-spinor equations due tothe Lagrangian (4) with (6), R ik − Rg ik = κ T ik ( ψ ) ,γ k ∇ k ψ − κ hc ψγ k γ ψ ) γ k γ ψ = 0 (7)in a space-time with the metric (5), we find thefunction ψ ( r ) and the metric coefficients e λ ( r ) and e ν ( r ) [3, 4]. In particular, for the coefficients e λ ( r ) and e ν ( r ) we obtain the expressions e ν = 1 and e λ = r / ( r − a ) ( a = const ), and to keep thesignature unchanged we must put r − a > r − a = x ( −∞
0) and theweak one ( ε + ( p + p + p ) / > T ik ( ω ) = [ p ( ω ) + ε ( ω )] U i U k − ( p − p ) χ i χ k − pg ik , (22)where χ i is the anisotropy vector directed along therotation axis and satisfying the conditions χ i U i =0, χ i χ i = − T ik ( ω ) obeys the local conservationlaw: T ik ( ω ) ; i = 0. The solution for the vortex in-tensity ω = ω / ( AD / ) in Eqs. (13) is just an in-tegral of this equation. In the case of a stationaryvortex gravitational field in a space-time describedby the metric (9), the energy-momentum tensor ofthis field T ik ( ω ) has the following components: T ik ( ω ) = ω κ c · diag (1 , , , . (23)It is seen that the negative pressure p z = − ω / ( κ c )along the rotation axis is three times as large asthe radial and transversal pressures, p r = p α = − ω / ( κ c ), and the sum ε ( ω ) + ( p r + p α + p z ) = − ω / (2 κ c ) <
0, i.e., the weak energy conditionis violated, thus leading to possible wormhole ex-istence. Besides, since the axial negative pressureis three times as large as the other components, atgravitational collapse of very massive rotating as-trophysical objects (the most massive stars, galac-tic nuclei), in which process the matter density ex-tremely grows along with rotation velocity, forminga vortex gravitational field like (23), such an ob-ject will stretch along its rotation axis. As a result,a stable, rapidly rotating astrophysical object canform, having a maximally stiff equation of stateand stretched along its rotation axis, which can bea wormhole described by the above solution (21)for a stationary rotating perfect fluid configuration.We can conclude that there can be a fourth finalstate of evolution of astrophysical objects (stars ofvarious masses and galactic nuclei), in addition tothree known states — a white dwarf, a neutron star(pulsar), and a black hole. Namely, a very massive rotating object can possibly form a wormhole withintense rotation and an equation of state close tothe limiting one.Thus we have shown that vortex fields (spinor,gravitational and magnetic) can form wormholes.A source of a vortex gravitational field can be aspinor field with polarized spin or a rapidly rotatingcontinuous medium. This leads to one more possi-ble final state of astrophysical object evolution, awormhole.The authors thank K.A. Bronnikov for atten-tion to the work and helpful comments.
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