Projective symmetries and induced electromagnetism in metric-affine gravity
Bert Janssen Alejandro Jimenez-Cano






                                         

                      
                        



                    
                    
                     
                     
                     





                                         













    







     

                                 






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        December 30, 2023 
        
         Projective symmetries and induced electromagnetism
         in metric-affine gravity
         Bert Janssen and Alejandro Jiménez-Cano[E-mails:  bjanssen@ugr.es,
                                                              alejandrojc@ugr.es]
         Departamento de Física Teórica y del Cosmos and Centro Andaluz de Física de Partículas Elementales Facultad de Ciencias, Avda Fuentenueva s/n, Universidad de Granada, 18071 Granada, Spain
        
         ABSTRACT
        
        13cm
        We present a framework in which the projective symmetry of the Einstein-Hilbert action in
        metric-affine gravity is used to induce an effective coupling between the Dirac lagrangian 
        and the Maxwell field. The effective U(1) gauge potential arises as the trace of the
        non-metricity tensor Q_μ a^a and couples in the appropriate way to the Dirac fields
        to in order to allow for local phase shifts. On shell, the obtained theory is
        equivalent to Einstein-Cartan-Maxwell theory in presence of Dirac spinors. 
        
        
        
        
        
        
        
        
        
        
        
         1. Introduction.2cm
        
        Metric-affine gravity is an extension of the usual (metric) theories of gravity, where the affine
        connection Γ_^ρ is considered as an independent (in general, dynamical) degree
        of freedom, whose expression is ultimately determined by its own equation of motion.
        In particular, it has been shown <cit.> that for the N-dimensional
        Einstein-Hilbert action,
        S=1/2κ∫^Nx (Γ) (N>2), the most general expression for the
        affine connection is of the form
        _^ρ =  Γ_^ρ +  V_μ δ_ν^ρ,
          
        where _^ρ is the Levi-Civita connection and V_μ an arbitrary vector field.
        It turns out, however, that this vector field  V_μ does not have any physically measurable
        influence: both the Einstein equation
        and the geodesic equation turn out to be identical to the equations in the metric formalism.
        In this sense, the metric and the Palatini formalism are equivalent for the Einstein-Hilbert
        action.
        
        In <cit.> it was argued
        that the vector field V_μ is related to the reparametrisation freedom of geodesics:
        affine geodesics of the connection _^ρ turn out to be pre-geodesics of the
        Levi-Civita connection _^ρ, through the reparametrisation
        τ/λ(λ)  =  exp[∫_0^λ x^ρ/λ' V_ρ λ'],
        
        where λ is the affine parameter for the _^ρ geodesics
        and τ the proper time along the Levi-Civita ones.
        
        In a certain way, the solution (<ref>) reflects the projective symmetry
        Γ_^ρ→Γ_^ρ + V_μδ_ν^ρ, under which the
        Einstein-Hilbert is known to be invariant<cit.>. Indeed, the Riemann tensor transforms
        under the projective transformation as
        _ ^λ→_ ^λ  +    F_(V)   δ_ρ^λ
        and the Ricci scalar = g^μρ δ_λ^ν _^λ is
        easily seen to be invariant.
        
        In more general theories, which include matter terms that couple to the connection, the
        Palatini connection (<ref>) might not be a solution and the metric and the
        Palatini formalism will in general no longer
        be equivalent. However, as we will show, there is a way to restore the symmetry and at
        the same time give a physical meaning to the Palatini vector V_μ. The previously
        arbitrary vector field will start to play the role of the electromagnetic potential of
        Maxwell theory and the projective symmetry of the entire action will be related to
        local U(1) transformations. The resulting theory will turn out to be on-shell equivalent
        to Einstein-Cartan-Maxwell theory, which describes General Relativity with a torsionful
        connection, coupled to the electromagnetic field. Einstein-Cartan theory is considered
        <cit.> to be the most likely classical gauge theory of gravity. See for example
        <cit.> for interesting phenomenology in Einstein-Cartan theory.
        
        
        We will present our model in the context of Einstein-Dirac theory in the metric-affine
        set-up, but it can equally well be done in presence of complex scalar fields. Since
        we will be working with Dirac spinors, we will switch to the tangent space description, 
        where the metric degrees of freedom are represented by the Vielbeinse^a_μ,
        which are the components of a local orthonormal coframe. The metric in this basis is
        then Minkowski, η_ab = e^μ_a e^ν_b g_, in any point of the manifold.
        Additionally, the affine connection is substituted by the components of the connection
        one-form, ω_μ a^b, through the appropriate basis transformation (sometimes
        called Vielbein Postulate). We will refer to  ω_μ a^b simply as connection
        from now on. Note that no antisymmetry in the last two indices is assumed, as the affine
        connection is not necessarily metric-compatible.
        
        .4cm
         2. Spinor covariant derivatives.2cm  
        
        The first issue to address is the construction of the covariant derivative for the
        Dirac spinor in the metric-affine context. A popular choice is the natural extension of
        the Lorentz covariant derivative to arbitrary (non-symmetric)
        connections:[We use the notation
          ^ab...c≡^[a^b...^c].]∇_μψ  =  ∂_μψ  -  14ω_μ ab^abψ.
        
        However, as the Lorentz generator 12^ab is antisymmetric, only the
        antisymmetric part of the connection, ω_μ [ab], will couple to the spinors. 
        Another possibility is to enhance the Lorentz generators to the general product of two
        gamma matrices and write a covariant derivative of the form 
        ∇_μψ  =  ∂_μψ  -  14ω_μ ab^a ^b ψ.
         
        Notice however that, even though ω_μ ab^a ^b  contains more
        terms than ω_μ ab^ab, most of the symmetric terms are projected out,
        due to the anti-commutation relations of the gamma matrices,
        {^a, ^b} = 2η^ab. The difference between the two covariant
        derivatives therefore reduces to the trace 2ω_μ abη^abψ, and the expression
        (<ref>) can be written without loss of generality as
        ∇_μψ  =  ∂_μψ  -  14ω_μ ab^abψ  -  18 Q_μ a^a ψ,
        
        where we have used the definition of the non-metricity tensor
        Q_μ ab≡-∇_μη_ab to relate its trace to the trace of the connection,
        ω_μ a^a=12 Q_μ a^a. There is actually no way the spinor can couple
        to the traceless symmetric part of the connection. Inspired by <cit.>
        (see also <cit.>), we will
        write down a generalised expression for the spinor covariant derivative,
        ∇_μψ  =  ∂_μψ  -  14ω_μ ab^abψ  -   k Q_μ a^a ψ,
        
        (with k in principle an arbitrary complex parameter), try to identify the gauge group 
        this connection corresponds to and determine its dynamics through the first-order
        formalism. 
        
        It is not difficult to see that if k is real-valued, the extra term in (<ref>)
        is the gauge term for local rescalings ψ→^kΩψ, provided that
        the trace of the non-metricity transforms
        as Q_μ a^a → Q_μ a^a + ∂_μΩ<cit.>.
        On the other hand, for
        purely imaginary values of k =  e, (<ref>) is a covariant derivative
        for local U(1) transformations, ψ→^ eΛψ, provided
        that Q transforms as Q_μ a^a → Q_μ a^a + ∂_μΛ. 
        
        Of course the trace of the non-metricity is not an independent field, but a part of the
        full connection ω_μ a^b. We should therefore embed the transformation of
        Q_μ a^a in a general transformation rule for ω_μ a^b. The simplest way
        of doing this are the transformation rules, respectively for a dilatation and a U(1)
        transformation,
        ω_μ a^b  → ω_μ a^b   +  1/2N ∂_μΩ δ_a^b, 2cmω_μ a^b  → ω_μ a^b   +  1/2N ∂_μΛ δ_a^b,
         
        which turn out to be projective transformations under which the Einstein-Hilbert action is
        invariant <cit.>. For the rest of this letter, we will focus on the U(1) case
        and consider only the covariant derivative
        ∇_μψ  =  ∂_μψ -  14ω_μ ab^abψ  -   e Q_μ a^a ψ.
        
        together with the transformation rule (<ref>b). 
        
        
        .4cm
         3. Induced electromagnetism from non-metricity.2cm
        
        We are now in the position to write down an action for Einstein-Dirac theory that is
        invariant under the combined local U(1) phase shifts and  projective transformation
        (<ref>b):
        
        S  =∫^Nx   |e|  [12κ(ω)
                           -  14 F_(Q) F^(Q) 
                           +  ħ2(ψ̅^μ∇_μψ +  ∇_μψ̅^μψ )
         -  mψ̅ψ].
        
        Here (ω) is the Ricci scalar of the full connection,
        (ω) = e^ν_b e^μ_c η^ac_ a^b(ω) with
        _ a^b (ω)= 2∂_[μω_ν] a^b
        - 2ω_[μ |a^c ω_|ν] c^b and F_(Q) = 2∂_[μ Q_ν]a^a.
        Note that we have added explicitly a gauge invariant kinetic term for the trace
        of the non-metricity, since these degrees of freedom do not appear in (ω),
        as a consequence of the projective symmetry. This term was also introduced in <cit.>,
        though without explicitly relating it to the U(1) symmetry of the matter
        action.[Including this term is equivalent to adding the quadratic curvature
        term _ a^a^_b^b (see for example <cit.>). ]
        
        It is important to realise that the only degrees of freedom of the action (<ref>) are the 
        Vielbeins e^a_μ, the connection ω_μ a^b and the Dirac spinor ψ. 
        In particular, at this stage Q_μ a^a is not an independent field and there is no a 
        priori reason to think of Q_μ a^a as the Maxwell potential. 
        The dynamics of Q_μ a^a should in fact be fully derived from the dynamics of
        ω_μ a^b.
         
        The equation of motion of the connection,
        
        0  =  κ/|e| e^b_σ  e^ρ_a δ S /δω_μ a^b   =   12T_λσ^μg^ρλ -  δ_σ^[μQ_λ^λ]ρ +  δ_σ^[μg^λ]ρ(12Q_λτ^τ-T_λτ^τ)        +  κ[2(∇_ν F^νμ(Q)
                            -   eħ ψ̅^μψ)δ_σ^ρ -  ħ4 ψ̅^μρ_σψ],
            
        can be solved in full generality. Indeed, taking the δ_ρ^σ trace, the equation
        reduces to
        ∇_ν F^νμ(Q)   =   eħ ψ̅^μψ,
        
        with ∇ the Levi-Civita covariant derivative. This is clearly the Maxwell
        equation with the vector Dirac current as source term, where the trace of the non-metricity
        is playing the role of the Maxwell potential, Q_μ a^a≡ A_μ
        (see also <cit.>).
        Once (<ref>) is taken into account, the equation of motion (<ref>) simplifies
        considerably: if we define 
        S_μ a^b≡ħ4ψ̅_μ a^bψ,
        then the most general solution of the traceless part of (<ref>) is given by
        <cit.>ω_μ a^b  = ω_μ a^b  +   V_μ δ_a^b  +  κ S_μ a^b,
        
        where ω_μ a^b is the Levi-Civita connection in the anholonomic
        frame and V_μ is a one-form, which is basically the vector field encountered in
        (<ref>). While for the Einstein-Hilbert action the vector field V_μ
        was completely arbitrary and void of physical content, in this context it takes up
        the role of the Maxwell potential A_μ. Indeed, it is straightforward to see that
        V_μ represents the trace Q_μ a^a=2NV_μ of the non-metricity
        Q_μ ab =  2V_μη_ab of the solution (<ref>) and hence not only is its
        dynamics dictated by the Maxwell equation (<ref>), but also the projective
        symmetry (<ref>) makes it behave as a U(1) gauge field. For future reference,
        we note that the torsion of the solution (<ref>) is given by
        T_^ρ = 2 V_[μδ_ν]^ρ + 2κ S_^ρ.
        
        On the other hand, the equations of motion for Dirac spinor ψ can be easily computed.
        Upon substituting (<ref>), its takes the form of the inhomogeneous Dirac equation coupled
        to the electromagnetic field, with S_μ a^b acting as its source term:
        ħ ^μ[∇_μ -  eA_μ] ψ -  mψ =   14ħκ ^μνρ S_μνρ ψ.
        
        Note that the torsion and non-metricity terms Q_[λμ]^λ + T_μλ^λ
        which appear in <cit.> cancel in our case, as they are both proportional to A_μ.
        
        Finally, after taking in account the solution (<ref>), the Vielbein equation of motion
        splits into its symmetric and antisymmetric parts,
        _μν-12g_μν   =   12κ^2 g_μνS^ρλσS_ρλσ +  κ [F_μ^λ(A)F_νλ(A)
                          - 14g_μνF_λσ(A)F^λσ(A)]       1cm
        -ħκ2[
          ψ̅_(ν(∇_μ) - e A_μ))ψ  -  (∇_(μ + e A_(μ)ψ̅_ν)ψ],
                            ∇_λS_μν^λ   =   ħ2[
          ψ̅_[ν(∇_μ] - e A_μ])ψ  -  (∇_[μ + e A_[μ)ψ̅_ν]ψ],
        
        which play the role of the Einstein equation and an equation that renders the torsion
        dynamical. Note that the total energy-momentum tensor on the right-hand side of the
        Einstein equation does not only contain the standard contributions of the electromagnetic
        and the Dirac field, but also an additional contribution of S_μ a^b,
        quadratic in κ. The set of equations (<ref>), (<ref>) and
        (<ref>) are the usual equations of motion of Einstein-Cartan gravity, coupled
        to the electromagnetic field and to Dirac spinors.
        
        
        .4cm
         4. Discussion.2cm
        
        We have shown that in the framework of metric-affine gravity, the Maxwell field can be
        geometrised and be interpreted as part of a more general connection ω_μ a^b.
        In particular, the Maxwell equation is encoded in (the trace of) the equation of motion
        of the connection. In order to obtain this it is necessary to relate the U(1) transformation
        of the matter fields with the projective transformation of the connection that leaves the
        Einstein-Hilbert term, and hence the entire action,  invariant. The equations of motion,
        after substituting the solution for the connection, turn out to be equivalent to the
        equations of motion of the Einstein-Cartan theory (i.e. the Einstein theory with a
        metric-compatible torsionful connection) in the presence of electromagnetism and a Dirac
        field.
        
        Actually, it is not that surprising that the action (<ref>) is on-shell equivalent
        to Einstein-Cartan gravity, coupled to the Maxwell and Dirac fields. Consider the
        decomposition of the general connection ω_μ a^b into its antisymmetric
        (and hence metric-compatible), its traceless symmetric and its trace parts,
        ω_μ a^b = ω̅_μ a^b +  V_μ δ_a^b +  Q̃_μ a^b,
        
        where  ω̅_μ^(ab)= Q̃_μ^[ab] = Q̃_μ c^c=0.
        Under this decomposition, the Ricci scalar (ω) takes the form
        (ω)  =  (ω̅)  +  Q̃_μνλQ̃^νμλ -  Q̃_μ^μλQ̃^ν_νλ,
        
        where (ω̅) is the Ricci scalar of  ω̅_μ ab, which 
        is a connection in its own right. Note that the trace V_μ does not contribute to the
        Ricci scalar, due to the projective symmetry.
        
        With this decomposition, the action (<ref>) can be written as
        
        S  =∫^Nx   |e|  [12κ(ω̅)
                   +  12κ(Q̃_μνλQ̃^νμλ  -  Q̃_μ^μλQ̃^ν_νλ)
                   -  14 F_(V) F^(V)
                   +  ℒ_Dirac].
        
        In this set-up it is clear  from its kinetic term and its coupling to the Dirac spinors
        that the trace of the connection V_μ already plays the role of the Maxwell potential.
        However the action (<ref>) is not yet equivalent to the standard
        Einstein-Cartan-Maxwell-Dirac theory, due to the presence of the traceless symmetric part
        of the connection Q̃_.
        
        The equation of motion (<ref>) of the original connection ω_μ a^b must be
        equivalent to the combined equations of motion of its three parts: on the one hand, the
        equation for the metric-compatible connection ω̅_μ a^b yields the standard
        Einstein-Cartan-Dirac solution ω̅_μ a^b = ω_μ a^b 
        + κ S_μ a^b and the equation for V_μ gives rise directly to the Maxwell
        equation coupled to the spinors. On the other hand, the traceless symmetric part
        Q̃_ appears quadratically, implying its own triviality, 
        Q̃_μλν =  0.
        
        So, with Q̃_ on-shell, we easily see that the action (<ref>) becomes
        the action of the Einstein-Cartan theory coupled to electromagnetism and a Dirac field.
        Consequently, the tensor S_μ a^b turns out to be the spin-density current
        associated to the Lorentz connection ω̅_μ a^b.
        
        Up to a certain point, the philosophy of this construction is similar to the construction
        made in <cit.> and references therein, where standard General Relativity with non-linear
        matter terms is reproduced from a class of modified metric-affine theories, called
        Ricci-based gravities. However we wish to emphasise that our action (<ref>) does not
        belong to this class of Ricci-based gravities, as it contains terms proportional to
        non-Ricci-like quadratic curvature invariant 
        _ a^a^_b^b, as pointed out in footnote <ref>. This suggests that the
        procedure of <cit.> might apply to a more general class of theories.
        
        
        
        
        
        
        
         Acknowledgements
        
        The authors would like to thank José Ignacio Illana, José Alberto Orejuela and Javier
        Martínez Lizana for useful discussions. This work was partially supported by the Junta
        de Andalucía (FQM101), the Universidad de Granada (PP2015-03) and the Spanish Ministry of
        Economy and Competitiveness (FIS2016-78198-P). AJC is supported by a PhD contract of the
        program FPU 2015 with reference FPU15/02864 (Spanish Ministry of Economy and Competitiveness).
        
        
        
        
        99Pons  N. Dadhich and J.M. Pons,  Gen. Rel. Grav.  44, (2012) 2337, 
        https://arxiv.org/abs/1010.0869arXiv:1010.0869 [gr-qc].
        
        BJJOSS A.N. Bernal, B. Janssen, A. Jiménez-Cano, J.A. Orejuela, M. Sánchez
          and P. Sánchez-Moreno, Phys. Lett. B768 (2017) 280-287, 
          https://arxiv.org/abs/1606.08756arXiv:1606.08756 [gr-qc].  
        
        Eisenh L.P. Eisenhart, Non-riemannian geometry,
          American Mathematical Society New York (1927).
        
        JS B. Julia and S. Silva, Class. Quant. Grav.  15 (1998) 2173, 
        https://arxiv.org/abs/gr-qc/9804029arXiv:9804029 [gr-qc].
        
        Hehl-book M. Blagojević and F. Hehl,  Gauge Theories of Gravitation,
          Singapore: World Scientific (2013) (See also 
          https://arxiv.org/abs/gr-qc/1210.3775arXiv:1210.3775 [gr-qc]).
        
        Poplawski N.J. Popławski, Phys. Lett. B694  (2010) 181-185.
          Erratum-ibid.B701 (2011) 672.
            https://arxiv.org/abs/1007.0587arXiv:1007.0587 [astro-ph.CO].
        
        Adak M. Adak, T. Dereli and L.H. Ryder, Int. J. Mod. Phys. D12 (2003) 145-156, 
        https://arxiv.org/abs/gr-qc/0208042v1arXiv:0208042 [gr-qc].
          
        Hurtley D.J. Hurley and M.A. Vandyck, J. Phys. A: Math. Gen.  27 (1994) 4569. 
        http://iopscience.iop.org/article/10.1088/0305-4470/27/13/030/metaDOI:10.1088/0305-4470/27/13/030.
        
        Koivisto T. Koivisto,  On an integrable geometrical foundation of gravity,
        https://arxiv.org/abs/gr-qc/1802.00650 arXiv:1802.00650 [gr-qc].
        
        Berredo G. de Berredo-Peixoto,  Non-metricities, torsion and fermions,
           https://arxiv.org/abs/1807.03270arXiv:1807.03270 [gr-qc]. 
        
        TW R.W. Tucker and C. Wang,  	Class. Quant. Grav. 12 (1995) 2587-2605, 
          https://arxiv.org/abs/gr-qc/9509011 arXiv:9509011 [gr-qc].
        
          
        HM  F.W. Hehl, and A. Macias,  Int. J. Mod. Phys. D8 (1999) 399-416,
        https://arxiv.org/abs/gr-qc/9902076 arXiv:9902076 [gr-qc]. 
          
        TFM A. Jiménez-Cano, Master Thesis (2016), University of Granada.
           
        FR J.B. Formiga, and C. Romero,  Int. J. Geom. Meth. Mod. Phys. 10 (2013) 1320012,
          https://arxiv.org/abs/gr-qc/1210.1615 arXiv:1210.1615 [gr-qc].
        
        DOR  A. Delhom-Latorre, G. J. Olmo, and M. Ronco, Phys. Lett. B780 (2018) 294,
           https://arxiv.org/abs/gr-qc/1709.04249 arXiv:1709.04249 [gr-qc].
        
        AOR  V.I. Afonso, G.J. Olmo, and D. Rubiera-Garcia, Phys. Rev. D97 (2018) 021503,
           https://arxiv.org/abs/gr-qc/1801.10406 arXiv:1801.10406 [gr-qc].
