Note about the spin connection in general relativity Renata Jora a∗ a National Institute of Physics and Nuclear Engineering PO Box MG-6, Bucharest-Magurele, Romania In general relativity the fermions are treated from the perspective of the gauged Lorentz group and by introducing the corresponding gauge fields the spin connection. This procedure is intimately related to the so-called ”vielbein” formalism and stems from the general structure that can be asso- ciated to a manifold, the affine connection. In this work we derive the correct spin connection based only on the general covariance of the theory, on the gauged Lorentz symmetry and on the known space-time properties of fermion bilinears generalized to the curved space. Our result coincides exactly with the spin connection obtain through the tetrad formalism. In general relativity the ordinary derivative of a tensor, It is considered that the gamma matrices in the curved in order to obtain the general behavior of a tensor, is re- space satisfy a generalized Clifford algebra with the anti placed by the covariant derivative written in terms of an commutation relation: affine connection. For example for a vector the covariant µ ν µν derivative is given by: {γ ,γ } =2g . (6) If one requires further that the operations of parallel ∂ V → D V = ∂ V − Γρ V . (1) µ ν µ ν µ ν µν ρ transport and projection on flat and curved indices com- ρ mute one arrives of the vielbein postulate: Here Γµν represents the affine connection which may be independent of a metric. In particular however it is al- m m ν m m n Dρe = ∂ρe (x) − Γ e − ω e =0, (7) ways more amenable to consider an affine connection that µ µ ρµ ν ρn mu m satisfies two main requirements [1]: where ωρn is the spin connection which can be extracted ρ ρ a) to be torsion free, i.e. Γµν =Γνµ. from Eq. (7) to be: b) to be metric compatible which amount to asking ωmn = emΓν eσn + em∂ eνn = that the covariant derivative of the metric tensor is zero: µ ν σµ ν µ m ν σn m νn eν Γσµe − ∂µeν e . (8) σ σ ∆ρgµν = ∂ρgµν − Γ gσν − Γ gσµ =0. (2) ρµ ρν Then the covariant derivative of a Dirac fermion in the . curved space time is written as: Having established how a derivative of a tensor field i ab must be modified in curved space time one needs to con- DρΨ= ∂ρΨ − ωρ σabΨ, (9) sider another type of fields of relevance in a quantum 4 i a b field theory, the fermion fields. In QFT the fermions lie where σab = 2 [γ ,γ ]. in a four dimensional representation of the Lorentz group Various attempts have been made in the literature [4] SO(3, 1) given by the gamma matrices which span a Clif- to introduce fermion covariant derivative without the use ford algebra with the anti-commutation rule: of the vielbein formalism in terms of only the curvilinear a b ab coordinates. These involved usually new and complicated {γ ,γ } =2η , (3) mathematical structures and an entire formalism of their own. where ηab is the Minkowski metric. The natural approach In the following we will derive the exact expression for in the presence of a curved space-time and of a general arXiv:1911.05283v2 [gr-qc] 22 Apr 2020 the fermion covariant derivative without the use of the coordinate transformation would be then to gauge the vielbein formalism by only making some natural assump- Lorentz group and to introduce the gauge fields associ- tions. ated to this, the spin connection. Then in a formalism Consider two Dirac fermions Ψ and Ψ¯ and the gamma introduced by Cartan and developed further in [2], [3] matrices γµ in the curved space time. The main assump- one defines the gamma matrices in the curved space as: tion is that the quantity Ψ¯ γµΨ transforms as a vector in µ a µ the curved space-time. Then the quantity, γ (x)= γ ea , (4) ¯ µ where γµ depend on the coordinate, µ is the index in the Dρ[Ψγ Ψ], (10) curved space and a is the index in the flat space. The D µ where ρ was introduced in Eq. (1) should transform as quantities ea are called a tetrad and satisfy the relation: a rank two tensor. gµν = eµeν ηab. (5) We are interested in writing a covariant derivative for a b the fermion fields such that the quantity in Eq. (10) transforms as a second rank tensor. Consider that this covariant derivative is expressed as, ∗ [email protected] DρΨ= ∂ρΨ+ XρΨ, (11) 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. [1] S. Caroll, arXiv:gr-qc/9712019 (1997). [2] H. Weyl, Z. Phys. 56, 330 (1929); T. W. Kibble, J. Math. 3 Phys. 2, 212 (1961). [3] S. Deser and C. J. Isham, Phys. Rev. D 14, 2505 (1976). [4] A. Weldon, Phys. Rev. D 63, 104010 (2010)..