Note About the Spin Connection in General Relativity

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Consider as, in tensor. expressed quantity is rank derivative the second covariant a that as such transforms fields fermion the tensor. two rank a rmE.()t be: to (7) Eq. from hntecvratdrvtv faDrcfrini the in as: fermion written Dirac is time a space of curved derivative covariant the Then osdrtoDrcfrin and Ψ fermions Dirac two Consider for expression exact the derive will we following the In [4] literature the in made been have attempts Various eaeitrse nwiigacvratdrvtv for derivative covariant a writing in interested are We a ∗ h eea tutr htcnb asso- be can that structure general the m h uvdsae u eutcoincides result Our space. curved the D ω σ D ab ρn m ormalism. ncin hspoeuei intimately is procedure This nnection. ρ γ ρ G6 uhrs-auee Romania Bucharest-Magurele, MG-6, x etv ftegue oet group Lorentz gauged the of pective v h orc pncneto based connection spin correct the ive a nrdcdi q 1 hudtasomas transform should (1) Eq. in introduced was µ e = stesi oncinwihcnb extracted be can which connection spin the is et ymtyado h known the on and symmetry rentz µ m ω e ntecre pc ie h anassump- main The time. space curved the in 2 i ν m µ mn = [ γ Γ D a ∂ σµ ν = ρ γ , ρ D = Ψ e e e µ m b { σn ρ ]. ν m γ = Ψ ( x µ Γ D − ∂ γ , ) σµ ν ρ ρ Ψ − ∂ ¯ Ψ [ ν ∂ e µ Ψ γ ¯ } ρ Γ σn e − µ γ + Ψ ν m ρµ ν 2 = rnfrsa etrin vector a as transforms Ψ µ + e 4 i Ψ] e νn ω ν m g e X , ρ ab ν m µν . − ρ σ ∂ Ψ . ω µ ab , e ρn m n h gamma the and Ψ Ψ ¯ νn e , mu n = 0 = , (10) (11) (9) (6) (7) (8) 2 where Xρ may contain in it gamma matrices in the curved behaves as a second rank tensor. Since the term in the space. first line of Eq. (21) behaves like a tensor then also the We make the assumption that the covariant derivative terms on the second plus the third line must behave as a operator is linear, although acting with different expres- second rank tensor. Then, sion on various fields. Consider the quantity: µ αβ αβ µ Ψ¯ iγ Aραβ σ Ψ − Ψ¯ iAραβσ γ Ψ+ ¯ Y µ, ¯ µ µ ¯ σ µ ΨΨ (12) Ψ(∂ργ )Ψ+Γρσ Ψγ Ψ= Tρ , (22) µ µ where Tρ is an arbitrary tensor expressed in terms of the where Y is a vector in the curved space. Then, fermion fields. Since there is not such tensor besides those µ introduced at this point with the correct mass dimension Dρ[ΨΨ¯ Y ], (13) one can consider this tensor zero. is a rank two tensor. The linearity of the operator implies One may rewrite Eq. (22) as: that the quantity, µ αβ µ σ µ iAραβ[γ , σ ]= −Γρσγ − ∂ργ . (23) [D (ΨΨ)]¯ Y µ + ΨΨ¯ D Y µ, (14) ρ ρ In the flat space we know that: is also a second rank tensor. Further on then, 1 [γa, σbc]= i(ηabγc − ηacγb). (24) 2 Dρ[ΨΨ]¯ = [DρΨ]Ψ¯ + Ψ¯ DρΨ, (15) Since we consider in the curved space a similar Clifford ¯ is also a second rank tensor. Since ΨΨ is a scalar the algebra this time with the gamma matrices space time only second rank tensor that one may form from it is: dependent the same relation should work if the flat in- dices would be replaced by the curved indices. Then Eq. ∂ [ΨΨ]¯ . (16) ρ (23) becomes: Eqs. (11), (15) and (16) then imply that: µ β µ σ µ −4Aρβγ = −Γρσγ − ∂ργ . (25) D Ψ=¯ ∂ Ψ¯ − Ψ¯ X . (17) ρ ρ ρ We multiply Eq. (25) by γλ and take the trace to obtain:

In order to determine the composition of Xρ in the µ 1 µ 1 λ µ 16 dimensional Lorentz space associated to the gamma A = Γ + Tr[γ ∂ργ ]. (26) ρ λ 4 ρλ 16 matrices we take into account the fact that the covari- ant derivative is a result of gauging the Lorentz symme- Finally the covariant derivative for the fermion fields is try and therefore should be an expansion in σµν . Then written in terms of only quantities in the curved space without loss of generality one may write: as: αβ D ∂ iAα σ β , Xρ = −iAραβσ , (18) ρΨ= ρΨ − ρβ α Ψ (27) α In summary one has: where Aρβ is given in Eq. (26). Next we will show that the spin connection introduced αβ DρΨ= ∂ρΨ − iAραβ σ in Eq. (27) is identical to that in Eq. (9). For that we βα write: DρΨ=¯ ∂ρΨ − iAραβ σ . (19) −iAµ σ λ = One can expand Eq. (10) which leads to: ρ λ µ 1 µ 1 λ µ λ µ −i [Γρλ + Tr[γ ∂ργ ]]σµ = Dρ[Ψ¯γ Ψ] = 4 4 µ µ (∂ρΨ)¯ γ Ψ+ Ψ[¯ ∂ργ ]Ψ + 1 µ µ λ ab λ cd −i [Γ + ∂ρea eb η ]eµced σ = ¯ µ µ ¯ σ 4 ρλ Ψγ ∂ρΨ+ΓρσΨγ Ψ. (20) 1 µ λ µ cd −i [Γ eµce + ∂ρe eµc]σ . (28) If we introduce the expression in Eq. (19) into Eq. 4 ρλ d d (20) one obtains that the quantity, Eq. (28) shows that the exact expression of the spin µ µ connection obtained through the vielbein formalism can DρΨ¯ γ Ψ+ Ψ¯ γ DρΨ+ be obtained by using only quantities defined in the curved ¯ µ αβ ¯ αβ µ +Ψiγ Aραβ σ Ψ − ΨiAραβσ γ Ψ+ space with the gamma matrices in the curved space sat- ¯ µ µ ¯ σ Ψ(∂ργ )Ψ+ΓρσΨγ Ψ, (21) isfying a similar Clifford algebra.

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