Neutrino-Antineutrino Asymmetry From The Space-time Noncommutativity
NNEUTRINO-ANTINEUTRINO ASYMMETRY FROM THESPACE-TIME NONCOMMUTATIVITY
Ouahiba MEBARKI , , N. MEBARKI and H. AISSAOUI ∗
2. Laboratoire de Physique Math´ematique et Subatomique (LPMS),Constantine 1 University, Constantine, Algeriamebarki [email protected]
A new mechanism having as an origin the space-time non commutativity has been shownto generate anisotropy and axial like interaction giving rise to a leptonic asymmetry forfermionic particles propagating in a curved non commutative
F RW universe. As a byproduct, for ultra relativistic particles like neutrinos, an analytical expression of thisasymmetry is derived explicitly. Constraints and bounds from the cosmological parame-ters are also discussed.
Keywords : Noncommutative Geometry; Modified Theories of Gravity; Leptogenesis.
1. Introduction
Leptogenesis is a very important observation and can affect the present energy den-sity and cosmic microwave background (
CM B ) etc... [1]. To explain the origin ofthis lepton number asymmetry, many mechanisms were proposed like the one ofAfflek-Dine [2], fermion propagation in a curved space-time [3,4], Lorentz and
CP T violating scenarios in the context of Riemanian-Cartan space-time [5, 6] etc... Onthe other hand, the non commutative nature of the space-time has been a subjectof a very active research [7–17]. The motivation was that the space-time non com-mutativity could be significant in the early universe where quantum gravity effectsbecome important and very sensitive to search for signatures in the cosmologicalobservations [18]. Moreover, the observed anisotropies of the
CM B may be causedby the non commutativity of space-time geometry [19, 20]. The main goal of thispaper is to show that a dynamical leptonic (including neutrinos) asymmetry can begenerated only by the non commutativity of a curved expanding space-time leadingto a new mechanism explaining matter-antimatter asymmetry in the universe. Insection 2, we present the mathematical formalism. In section 3,we derive the ana-lytical expression of the neutrino-antineutrino asymmetry. Finally, in section 4, wediscuss the numerical results and draw our conclusions. ∗ Permanent address 1 a r X i v : . [ h e p - t h ] F e b
2. Mathematical Formalism
Following the general approach of ref. [21], we present a deformed
F RW solutionin a non commutative (
N CG ) gauge gravity where the structure of the space-timeis affected by the commutation relations:[ (cid:98) x µ , (cid:98) x ν ] = i Θ µν ( (cid:126) = c = 1) (1)where Θ µν (cid:0) µ, ν = 1 , (cid:1) are antisymmetric canonical parameters. The N CG gaugefields (spin connection) are denoted by (cid:98) ω ABµ ( x, Θ µν ) and can be expanded in powerseries of the Θ µν as: (cid:98) ω ABµ = ω ABµ − i Θ υρ ω ABµνρ + Θ υρ Θ λτ ω ABµυρλτ + ... (2)where ω ABµνρ = 14 { ω ν , ∂ ρ ω µ + R ρµ } AB (3) ω ABµυρλτ = 132 { ω λ , ∂ τ { ω λ , ∂ ρ ω µ + R ρµ }} + 2 { ω λ , { R τν , R µρ }}− { ω λ , { ω ν , D ρ R τµ + ∂ ρ R τµ }} − {{ ω ν , ∂ ρ ω λ + R ρλ } , ∂ τ ω µ + R τµ } +2 [ ∂ ν ω λ , ∂ ρ ( ∂ τ ω µ + R τµ )] AB (4)with { α, β } AB = α AC β BC + β AC α BC (5)and the covariant derivative D µ R ABρσ is such that: D µ R ABρσ = ∂ µ R ABρσ + (cid:0) ω ACµ R DBρσ + ω BCµ R DAρσ (cid:1) η CD (6)here R ABµν = e Aρ e Bσ R µνρσ (7)where e Aρ , R µνρσ and ω ABµ are the ordinary commutative inverses of the vierbein(tetrad), Riemanian tensor and spin connection respectively (Greek and Latin in-dices are for curved and flat space-time respectively).Now, if we assume a vanishing commutative torsion, the components of the
N CG tetrad fields (cid:98) e Aµ read: (cid:98) e Aµ = e Aµ − i Θ νρ e Aµνρ + Θ νρ Θ λτ e Aµνρλτ + .... (8)where e Aµνρ = 1 / (cid:8) ω ACν ∂ ρ e Dµ + (cid:0) ∂ ρ ω ACµ + R ACρµ (cid:1) e Dν (cid:9) η CD (9)and e Aµνρλτ = 1 / − ω ABλ (cid:0) D ρ R CDτµ + ∂ ρ R CDτµ (cid:1) e mν η Dm − { ω ν , ∂ ρ ω λ + R ρλ } AB ∂ τ e Cµ − { ω ν , D ρ R τµ + ∂ ρ R τµ } AB e Cλ − ∂ τ { ω ν , ∂ ρ ω µ + R ρµ } AB e Cλ − ω ABλ ∂ τ ( ω CDν ∂ ρ e mµ + (cid:0) ∂ ρ ω CDµ + R CDρµ (cid:1) e mν ) η Dm + 2 ∂ ν ω ABλ ∂ ρ ∂ τ e Cµ − ∂ ρ (cid:0) ∂ τ ω ABµ + R ABτµ (cid:1) ∂ ν e Cλ + 2 { R τν , R µρ } AB e Cλ + (cid:0) ∂ τ ω ABµ + R ABτµ (cid:1) (cid:0) ω CDν ∂ ρ e mλ + ( ∂ ρ ω CDλ + R CDρµ (cid:1) e mν ) η Dm η BC (10)a real N CG metric (cid:98) g µν is given by [22, 23]: (cid:98) g µν = (1 / η AB (cid:0)(cid:98) e Aµ ∗ (cid:98) e B + ν + (cid:98) e B + ν ∗ (cid:98) e Aµ (cid:1) (11)here the superscript “+” denotes the complex conjugate and “ ∗ ” the Moyal-Weylstar product defined as: f ( x ) ∗ g ( y ) = f ( x ) exp (cid:34) i µν ←− ∂∂x µ −→ ∂∂y ν (cid:35) g ( y ) | x = y (12)In what follows, we take the signature of space-time (+ , + , + , − ), and in order tosimplify our calculations, we choose the component Θ as the only non vanishingspace-space components. It is worth to mention as it was pointed out in ref. [24] anon vanishing time-space components Θ i (cid:0) i = 1 , (cid:1) will affect unitarity and causal-ity [24–26].Now, let us start from a spherically, isotropic and homogeneous flat space( k = 0) F RW universe. To keep our results more transparent and simple, we usedimensionless spherical coordinates (cid:98) r = rr , θ, ϕ and time (cid:98) t = tt ( r and t aresome cosmological scales parameters which will be specified later), and a power lawformula for the scale factor a (cid:0)(cid:98) t (cid:1) of the form a (cid:0)(cid:98) t (cid:1) = (cid:98) t β . Because all the measuredproper distances r between co-moving points increase proportionally to a ( t ), thenone can write (cid:98) r ≈ a ( t ) a ( t ) = (cid:98) t β . Moreover, the time scale t is related to the parameter β and matter density ρ through the relation (see Appendix Appendix A): t = β (cid:18) χρ (cid:19) (13)where χ = πGc and G the Newton’s gravitational constant.Now, straightforward calculations using a Maple package, and writing Θ = (cid:101) Θ( (cid:101) Θ = ΘΛ and Λ is an N CG scale factor), give the following non-zero components ofthe deformed metric (cid:98) g µν up to the O (cid:0) Θ (cid:1) as it is shown in Appendix Appendix B.Notice that (cid:98) g µν is real and in general not symmetric. Moreover, N CG has gen-erated a non homogeneous and anisotropic universe, the corresponding componentsof the tetrad are in general complex (see Appendix Appendix B). Similarly, directbut tedious calculations give the non vanishing
N CG spin connection (cid:98) ω ABµ up tothe O (cid:0) Θ (cid:1) (see Appendix Appendix B).Notice that contrary to the ordinary commutative case, (cid:98) ω ABµ is in general acomplex and not completely antisymmetric with respect to A and B .
3. NCG Neutrino-Antineutrino Asymmetry
In an
N CG curved space-time, the generalized Dirac Lagrangian density L is as-sumed to have the form [7], [14] : L = (cid:112) − (cid:98) g ∗ (cid:98) Ψ ∗ (cid:16) − iγ µ ∗ (cid:98) D µ ∗ − m (cid:17) (cid:98) Ψ + c.c (14)where the
N CG covariant derivative (cid:98) D µ is given by: (cid:98) D µ = ∂ µ − i (cid:98) ω ABµ Σ AB (15)with γ µ = (cid:98) e µa γ a , Σ [ AB ] = i γ A , γ B ] , Σ ( AB ) = 12 { γ A , γ B } = η AB (16)( γ (cid:48) i s are the Dirac Gamma matrices in the flat Minkowski space-time and (cid:98) Ψ the
N CG c.c ” stands for complex conjugate and (cid:98) e µa the inverse of N CG vierbein. Using the fact that12 (cid:2) γ d Σ [ AB ] + Σ [ AB ] γ d (cid:3) = ε fdab γ e γ (17)and i (cid:2) γ d Σ [ AB ] − Σ [ AB ] γ d (cid:3) = g db γ a − g da γ b (18)as well as the N CG orthogonality relation (cid:98) e µa ∗ (cid:98) e µb = δ ab (19)where ε fdab is the 4-rank totally antisymmetric tensor, one can show that the cor-responding N CG
Dirac equation takes the form: (cid:104) γ f (cid:16) i∂ f + (cid:98) A f (cid:17) + γ f γ (cid:98) B f (cid:105) ∗ (cid:98) Ψ = 0 (20)where ∂ f = (cid:98) e µf ∂ µ (21) A f = (cid:61) ( (cid:98) e µf (cid:88) a =1 ω aaµ ) + (cid:60) (cid:0)(cid:98) e µd (cid:0)(cid:98) ω fdµ − (cid:98) ω dfµ (cid:1)(cid:1) + O (cid:0) Θ (cid:1) (22) B f = (cid:20) (cid:61) (cid:0)(cid:98) e µd (cid:98) ω abµ (cid:1) + 14 Θ ρσ Θ αβ (cid:0) ∂ ρ ∂ α e µd (cid:1) (cid:0) ∂ σ ∂ β ω abµ (cid:1)(cid:21) ε fdab + O (cid:0) Θ (cid:1) (23)one can show (as in ref. [27]) that the dispersion relations for the left and rightchiral fields (particles X and antiparticles X ) read: E X = (cid:114)(cid:16) −→ P − −→ Λ X (cid:17) + m + Λ X (24)and E X = (cid:114)(cid:16) −→ P − −→ Λ X (cid:17) + m + Λ X (25) where −→ Λ X = −→ B + −→ A , −→ Λ X = −−→ B + −→ A (26)and Λ X = B + A Λ X = − B + A (27)In fact, this result was expected because of CP T violation which affects the dis-persion relations of particles and antiparticles leading to a difference between theirenergies. In our case, the violation of
CP T can be induced by:1) Curvature in a certain non trivial anisotropic
N CG geometry. In fact,fermions in a curved space-time can interact via an axial vector currentdue to their spin with space-time curvature or torsion [27]. Therefore, thespace-time curvature background has the effect of inducing an axial vectorfield especially in certain anisotropic space-time geometries like in
N CG .2)
N CG : since it violates Lorentz invariance leading to a non conservation of
CP T .Thus, in our case the violation of
CP T is induced by both the curvature andnon commutativity of the space-time.Now, in order to derive the neutrino-antineutrino asymmetry in the context of
N CG , one has to have a thermal equilibrium background coming after the neutrinodecoupling. In fact, the synthetics of light elements depends strongly on the ratioof number of neutrinos / number of protons freezout abundance determined by theinterplay between the weak interaction and expansion rate of the universe. Bothare influenced by the neutrino decoupling temperature. The neutrino decouplingmeans that the neutrinos do not interact with baryonic matter and consequently donot influence the dynamics of the universe at early stage. The decoupling happenswhen the weak interaction rate of neutrinos is smaller than the expansion rate ofthe universe ( kT νdecoup ≈ M eV, t decoup ≈ seconde ) . After decoupling, a thermalequilibrium background of relativistic neutrinos is expected with an effective tem-perature at late time T ν ≈ . T γ . Now, at this equilibrium temperature and usingthe Fermi-Dirac distribution together with the result: (cid:90) ∞ dx x m z − e x = z Γ ( m + 1) Φ ( − z, m + 1 ,
1) (28)where Γ ( x ) is the Euler Gamma function and Φ ( z, s, a ) the Lerch transcendentfunction given by Φ ( z, s, α ) = ∞ (cid:88) m =0 z n ( n + α ) s (29)the analytical expression of the difference between the neutrino and antineutrino N CG number density ∆ n NCG is given by (in the system where the Boltzman con- stant k = 1): ∆ n NCG ≈ ( T ν ) π ∞ (cid:88) l =1 ( − l l sinh (cid:32) l (cid:98) B T ν (cid:33) (30)where (cid:98) B has as an expression (cid:99) B = ΘΛ t (cid:20) ˆ t − β − θ − β sin θ ˆ t β − (cid:21) (31)here t stands for the decoupling time t decoup .Notice that since (cid:99) B is propotional to Θ, as a first approximation, eq.(30) reducesto (cid:52) n NCG ≈ ( T ν ) π (cid:98) B ζ (2) (32)where ζ (2) is the Dirichlet zeta function. Now, using the CM B photon numberdensity n γ , the ratio R = (cid:52) n NCG n γ reads: R = π ζ (3) (cid:32) (cid:98) B T ν (cid:33) (cid:18) T ν T γ (cid:19) (33)where T γ is the CM B photon temperature. Now, it is clear from eq.(33) thatcontrary to the commutative
F RW universe (with isotropy, homogeneity and space-time spherical symmetry) where there is no geometrical contribution to the neutrino-antineutrino asymmetry ∆ n , the N CG space-time has generated anisotropy, nonhomogeneity and Broken spherical symmetry leading to a dynamical non vanishingnet asymmetry ∆ n NCG which has a non commutative geometrical origin.
4. Discussions and Conclusions
We have found that if neutrinos are propagating in a curved
N CG universe, wherea space-time anisotropy as well as Lorentz violation invariance are generated, anet asymmetry between neutrinos and antineutrinos arises at the thermodynamicalequilibrium. Thus, we have shown that a new source of a leptonic and antilep-tonic asymmetry which has as an origin the noncommutative geometrical structureof space-time can be generated leading to a new mechanism explaining matter-antimatter asymmetry. It is worth to mention that in the standard cosmology atpresent time, the ratio R = ∆ nn γ is given by [1], [28] R = π ζ (3) (cid:18) T ν T γ (cid:19) (cid:18) µ ν T ν (cid:19) (34) µ ν is the neutrino chemical potential. At the decoupling time where t ≈ s , T ν T γ ≈ .
71 and (cid:12)(cid:12)(cid:12) µ ν T ν (cid:12)(cid:12)(cid:12) (cid:46) . , the ratio R (cid:46) × − . Now, if we consider the θ -averaged (cid:68) (cid:98) B (cid:69) θ (cid:0) − π ≤ θ ≤ π (cid:1) at present time with the leading term ∼ (cid:98) t β − , R takes thesimple form: R | t = t ≈ π ζ (3) (cid:34) − (cid:101) Θ4 t β (cid:98) t β − T ν (cid:35) (cid:18) T ν T γ (cid:19) , (35)here (cid:98) t = t b t , using the fact that kT ν ≈ × − J and t b ≈ . × s, one gets: | R | = (cid:12)(cid:12)(cid:12)(cid:12) R R (cid:12)(cid:12)(cid:12)(cid:12) t = t b ≈ (cid:101) Θ β . × − (cid:0) × (cid:1) β − J (36)which means that if | R | ∼ O (1) , then Λ ∼ Θ β , × − (cid:0) × (cid:1) β − T eV.
Notice that the
N CG scale Λ and β = ω ) are intimately related. Further-more, for an accelerated expansion of the universe one has β > (cid:31) Θ20 , × − T eV.
This is a new lower bound of the non commutative scale Λ(Θ < . Fig. 1. Ratio R as a function of ΛΘ and (cid:101) t for β ≈ . . . Fig.1. shows the ratio R as a function of the N CG parameter ΛΘ ( denotedby Lambda and the reduced time in a T eV unit), the reduced time (cid:101) t (denoted byttilde, (cid:98) t = 10 (cid:101) t ) for a fixed β ≈ .
38 or equivalently the equation of state parameter(
EOS ) parameter ω ≈ − .
75. Notice that R is an increasing function of time fora fixed ΛΘ . Moreover, for smaller values of ΛΘ of O (1 T eV ), R ∼
10 when (cid:98) t ∼ t ob ( t ob is the observed time). This shows that R becomes important and N CG effectsdominate in comparison to the one of the standard cosmology. It is worth to noticethat if the
N CG effects become relevant (
R > , one gets an upper limit for ΛΘ of O (1 T eV ).Fig.2. displays the ratio R as a function of the β parameter (denoted by beta)and reduced time (cid:101) t for a fixed N CG parameter ΛΘ ∼ T eV.
Notice that the ratio R is very sensitive to the variations of β ; e.g. if β ∼ . − .
4, and (cid:101) t ∼ −
2, theratio R ∼ − . Fig. 2. Ratio R as a function of β and (cid:101) t for ΛΘ ≈ T eV.
Fig. 3. Ratio R as a function of ΛΘ and β at present time. Fig.3. shows the ratio R as a function of β and ΛΘ at the present time. For ΛΘ ∼ T eV and β ∼ , the ratio R reaches the value 1 . . Fig.4. displays the contour plots (cid:16) Λ( T eV )Θ , β (cid:17) for a fixed R at the present time.Notice that for a fixed β or ω as R increases; the value of the N CG parameter ΛΘ decreases and becomes relevant at a T eV scale. As an important and novel results,the
N CG ΛΘ parameter is strongly related to the neutrino-antineutrino asymmetrywhich has as an origin the space-time N CG structure, and the
EOS parameter ω (or equivalently β ) . Moreover, this asymmetry is dynamical in the sense that it is atime dependent quantity. It increases with time for an accelerated expansion of theuniverse. Finally, as a conclusion a new mechanism from the non commutativity ofspace-time generating the matter-antimatter asymmetry was found.
5. Acknowledgments
We are very grateful to the Algerian Ministry of higher education and scientificresearch and the
DGRSDT for the financial support.
Fig. 4. Contour plots (cid:16) ΛΘ , β (cid:17) at present time for a fixed R. Appendix A. Appendix
From Friedman’s equations: 2 .. aa + χ (cid:18) P + 13 ρ (cid:19) = 0 (A.1)and 2 .. aa + χ ( P + ρ ) − . a a = 0 (A.2)( ρ and P are the matter density and pressure respectively), one can get the followingcontinuity equation (in what follows, we use the system where χ = 1 and ” . ” standsfor time derivative) . ρ + 3 . aa ( P + ρ ) = 0 (A.3)leading to: ddt (cid:0) ρa (cid:1) + ωρ da dt = 0 (A.4)for a perfect fluid where P = ωρ. One can show that the solution of eq.(A.4) is ofthe form ρ = D a − ω ) . Moreover, from eqs.(A.1) and (A.2), one can obtain: . a = ρ a a ( t ) = (cid:18)
32 (1 + ω ) (cid:19) ω ) (cid:18) χD (cid:19) ω ) t ω ) (A.6) Now, it is clear that: β = 23 (1 + ω ) (A.7) D = ρ a ω )0 (A.8) t = β (cid:18) χρ (cid:19) (A.9) Appendix B. Appendix
Starting from eqs.(2), (8) and (11) and using maple 18 package, one obtains thefollowing complex
N CG vierbeins, the metric components and the spin connectioncomponents respectively; (We set : h = r t and (cid:101) Θ = ΘΛ t ), in what follow, the upper(respectively lower) indices stand for flat (respectively curved) space-time, (cid:98) e = (cid:98) t β (cid:34)
128 + (cid:101) Θ β (cid:98) r (cid:98) t β − h (cid:0) β (cid:98) r (cid:98) t − sin θ + 14 sin θ + 1 (cid:1)(cid:35) (B.1) (cid:98) e = − (cid:101) Θ (cid:98) rr (cid:98) t β sin 2 θ (cid:0) − (cid:98) t β − β (cid:98) r h (cid:1) (B.2) (cid:98) e = − i (cid:101) Θ2 (cid:98) rr (cid:98) t β (B.3) (cid:98) e = − (cid:101) Θ β ( β − (cid:98) r (cid:98) t β − h (B.4) (cid:98) e = − (cid:101) Θ β (cid:98) r (cid:98) t β − h sin 2 θ (B.5) (cid:98) e = − (cid:98) rr (cid:98) t β (cid:26) −
128 + (cid:101) Θ (cid:98) t β (cid:20) β (cid:98) r (cid:98) t β − h (cid:0) − θ + 18 cos θ (cid:1) − β (cid:98) r (cid:98) t β − h sin θ − (cid:98) t β (cid:0) θ + 1 (cid:1) (cid:21)(cid:27) (B.6) (cid:98) e = − i (cid:101) Θ (cid:98) rr (cid:98) t β (cid:2) cos 2 θ − β (cid:98) r (cid:98) t β − h sin θ (cid:3) (B.7) (cid:98) e = − (cid:101) Θ β ( β − (cid:98) r (cid:98) t β − h sin 2 θ (B.8) (cid:98) e = (cid:98) e = 0 (B.9) (cid:98) e = − i (cid:101) Θ β (cid:98) r r (cid:98) t β − h sin θ (B.10) (cid:98) e = 1128 (cid:98) rr (cid:98) t β sin θ (cid:26)
128 + (cid:101) Θ (cid:20) − β (cid:98) r (cid:98) t β − h (cid:0) θ (cid:1) +17 β (cid:98) r (cid:98) t β − h sin θ + 2 (cid:21)(cid:27) (B.11) (cid:98) e = 1128 (cid:101) Θ β (cid:98) r (cid:98) t β − h (cid:2) θ + 22 β (cid:98) r (cid:98) t β − h (cid:3) (B.12) (cid:98) e = − β (cid:101) Θ (cid:98) r r h (cid:98) t β − (cid:2) − β (cid:98) r (cid:98) t β − h (cid:3) (B.13) (cid:98) e = i (cid:101) Θ (cid:98) r r βh (cid:98) t β − sin 2 θ (B.14) (cid:98) e = − (cid:26) −
64 + (cid:101) Θ (cid:20) − β ( β − (cid:98) r (cid:98) t β − h sin θ − (cid:98) r h β ( β − (cid:98) t β − (cid:0) θ + 1 (cid:1) (cid:21)(cid:27) (B.15) • (cid:98) g µν components: (cid:98) g = (cid:98) t β (cid:110)
64 + (cid:101) Θ β (cid:98) t β − (cid:98) r h (cid:2) (cid:0) β (cid:98) t β − (cid:98) r h − (cid:1) sin θ − (cid:3)(cid:111) (B.16) (cid:98) g = − (cid:101) Θ (cid:98) rr (cid:98) t β sin 2 θ (cid:2)
12 + 25 β (cid:98) t β − (cid:98) r h (cid:3) (B.17) (cid:98) g = (cid:98) g = (cid:98) g = (cid:98) g = (cid:98) g = (cid:98) g = 0 (B.18) (cid:98) g = 2257 (cid:101) Θ β (cid:98) r (cid:98) t β − h (cid:2) β (19 β + 8) (cid:98) t β − (cid:98) r h sin θ −
10 cos 2 θ (cid:3) (B.19) (cid:98) g = − (cid:101) Θ (cid:98) rr (cid:98) t β sin 2 θ (cid:2)
12 + 25 β (cid:98) t β − (cid:98) r h (cid:3) (B.20) (cid:98) g = 164 (cid:98) r r (cid:98) t β (cid:26) −
64 + (cid:101) Θ (cid:20)
12 sin θ + 6 + 21 β (cid:98) r (cid:98) t β − × h (cid:0) θ −
18 cos θ (cid:1) β (cid:98) r h (cid:98) t β − (cid:21)(cid:27) (B.21) (cid:98) g = − β r h (cid:98) r (cid:98) t β − sin θ [25 β −
28] (B.22) (cid:98) g = (cid:98) r r (cid:98) t β (cid:16) (cid:101) Θ sin θ (cid:17) (B.23) (cid:98) g = − (cid:101) Θ β (cid:98) r (cid:98) t β − h (cid:2)(cid:0) β sin θ − (cid:1) (cid:98) t β − h β (cid:98) r − θ (cid:3) (B.24) (cid:98) g = (cid:101) Θ β r (cid:98) r (cid:98) t β − h [25 β −
28] (B.25) (cid:98) g = 132 (cid:26) −
32 + (cid:101) Θ h β (cid:98) r (cid:98) t β − (cid:20) − β ( β − (cid:98) t β − h (cid:98) r sin θ + ( β − (cid:0) θ − (cid:1) (cid:21)(cid:27) (B.26) • Spin connection components at order of Θ : (cid:98) ω = (cid:98) ω = (cid:98) ω = (cid:98) ω = (cid:98) ω = (cid:98) ω = (cid:98) ω = (cid:98) ω = (cid:98) ω = (cid:98) ω = 0 (B.27) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (B.28) (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) (cid:98) ω (cid:39) O (cid:16) (cid:101) Θ (cid:17) (B.29) (cid:98) ω = − (cid:98) ω = − β (cid:98) t β − t + O (cid:16) (cid:101) Θ (cid:17) (B.30) (cid:98) ω = − (cid:98) ω = − i (cid:101) Θ2 β (cid:98) rt h (cid:98) t β − sin θ + O (cid:16) (cid:101) Θ (cid:17) (B.31) (cid:98) ω = − (cid:98) ω = − i (cid:101) Θ2 β (cid:98) rt h (cid:98) t β − cos θ + O (cid:16) (cid:101) Θ (cid:17) (B.32) (cid:98) ω = − (cid:98) ω = − O (cid:16) (cid:101) Θ (cid:17) (B.33) (cid:98) ω = − (cid:98) ω = − i (cid:101) Θ4 (cid:0) − (cid:98) t β − β h (cid:98) r (cid:1) cos θ + O (cid:16) (cid:101) Θ (cid:17) (B.34) (cid:98) ω = − (cid:98) ω = − β (cid:98) rh (cid:98) t β − + O (cid:16) (cid:101) Θ (cid:17) (B.35) (cid:98) ω = i (cid:101) Θ4 (cid:0) (cid:98) t β − β h (cid:98) r (cid:1) sin θ + O (cid:16) (cid:101) Θ (cid:17) (B.36) (cid:98) ω = i (cid:101) Θ4 (cid:0) (cid:98) t β − β h (cid:98) r (cid:1) sin θ + O (cid:16) (cid:101) Θ (cid:17) (B.37) (cid:98) ω = − i (cid:101) Θ4 βh (cid:98) r (cid:98) t β − (cid:0) (cid:98) t β − β h (cid:98) r (cid:1) + O (cid:16) (cid:101) Θ (cid:17) (B.38) (cid:98) ω = − i (cid:101) Θ4 βh (cid:98) r (cid:98) t β − (cid:0) (cid:98) t β − β h (cid:98) r (cid:1) + O (cid:16) (cid:101) Θ (cid:17) (B.39) (cid:98) ω = i (cid:101) Θ sin 2 θ + O (cid:16) (cid:101) Θ (cid:17) (B.40) (cid:98) ω = i (cid:101) Θ4 (cid:0) cos θ − (cid:98) t β − β h (cid:98) r sin θ (cid:1) + O (cid:16) (cid:101) Θ (cid:17) (B.41) (cid:98) ω = − (cid:98) ω = − sin θ + O (cid:16) (cid:101) Θ (cid:17) (B.42) (cid:98) ω = (cid:98) ω = − i (cid:101) Θ βh (cid:98) r (cid:98) t β − sin 2 θ + O (cid:16) (cid:101) Θ (cid:17) (B.43) (cid:98) ω = − i (cid:101) Θ4 (cid:0) (cid:98) t β − β h (cid:98) r (cid:1) sin θ + O (cid:16) (cid:101) Θ (cid:17) (B.44) (cid:98) ω = − i (cid:101) Θ8 (cid:0) (cid:98) t β − β h (cid:98) r (cid:1) sin 2 θ + O (cid:16) (cid:101) Θ (cid:17) (B.45) (cid:98) ω = − (cid:98) ω = − cos θ + O (cid:16) (cid:101) Θ (cid:17) (B.46) (cid:98) ω = i (cid:101) Θ4 βh (cid:98) r (cid:98) t β − (1 + 2 (cid:98) t β − β h (cid:98) r ) sin θ + O (cid:16) (cid:101) Θ (cid:17) (B.47) (cid:98) ω = i (cid:101) Θ2 β h (cid:98) r (cid:98) t β − sin 2 θ + O (cid:16) (cid:101) Θ (cid:17) (B.48) (cid:98) ω = − (cid:98) ω = − βh (cid:98) r (cid:98) t β − sin θ + O (cid:16) (cid:101) Θ (cid:17) (B.49) (cid:98) ω = − i (cid:101) Θ4 βh (cid:98) r (cid:98) t β − (cid:2) cos θ − (cid:98) t β − β h (cid:98) r sin θ (cid:3) + O (cid:16) (cid:101) Θ (cid:17) (B.50) (cid:98) ω = i (cid:101) Θ8 β h (cid:98) r (cid:98) t β − sin 2 θ + O (cid:16) (cid:101) Θ (cid:17) (B.51) (cid:98) ω = − (cid:98) ω = − i (cid:101) Θ2 β ( β − (cid:98) r t h (cid:98) t β − sin θ + O (cid:16) (cid:101) Θ (cid:17) (B.52) (cid:98) ω = − i (cid:101) Θ2 β ( β − (cid:98) rt h (cid:98) t β − cos θ + O (cid:16) (cid:101) Θ (cid:17) (B.53) (cid:98) ω = 0 . (B.54) References
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