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Rigorous Formulation of Duality in Gravitational Theories

In this paper we evince a rigorous formulation of duality in gravitational theories where an Einstein like equation is valid, by providing the conditions under which the Hodge duals (with respect to the metric tensor g) of T^a and R_b^a may be considered as the torsion and curvature 2-forms associated with a connection D', part of a Riemann-Cartan structure (M,g',D'), in the cases g = g' and g does not equal g', once T^a and R_b^a are the torsion and curvature 2-forms associated with a connection D part of a Riemann-Cartan structure (M,g,D). A new form for the Einstein equation involving the dual of the Riemann tensor of D is also provided, and the result is compared with others appearing in the literature.


Introduction
There has been a number of papers trying to put in evidence a possible analogy between electromagnetism and gravitation, in order to elicit a gravitational analogue for the magnetic monopole that appears in the generalized Maxwell equations with magnetic and electric currents 1 . Among the old ones we quote 2 [9,17,18]. Ten years ago Nieto [19] developed an analogue of S-duality 3 for linearized gravity in (3+1) dimensions (see, also [12,20,13]) and generalizations of that idea of duality for gravitational theories in more dimensions appear, e.g., in [1,2,3,5,6,7,11]. In particular for the case of gravity in (3 + 1) dimensions a set of equations has been proposed for Einstein equations, Bianchi identities, and their duals, although mainly used in the linear approximation. The main aim of this note is to derive exact equations that must be satisfied by the dual Einstein equations and for the duals of the torsion and curvature 2-forms of a general Riemann-Cartan structure (M, g, D). We study in which conditions the dualized objects realize a Riemann-Cartan structure (M, g, D ′ ), or even a (M, g ′ , D ′ ) one. In so doing we find that the correct field equations for a dual theory (in a precise mathematical sense defined below), are at variance with ones proposed in some of the above mentioned papers. In so doing we hope that the present note be useful for those pursuing the interesting ideas of duality in gravitational theories.
The paper, which uses an intrinsic formulation of the theories presented, is organized as follows. In Section 2 we present some necessary preliminaries that serve, besides the proposal of introducing our notation, also the one of presenting what it is understood here by a Riemann-Cartan gravitational theory. In this Section we review also the Bianchi identities for the torsion and curvature 2-forms T α and R α β of (M, g, D) in intrinsic and component forms, because those formulas for a Riemann-Cartan theory are not well known as they deserve to be, and sometimes concealed from the formalism. In Section 3 we introduce the Ricci 1-form fields R µ and the Einstein 1-forms fields G µ [23], and further prove a Proposition containing a formula that relates the dual ⋆ g R µ of R µ to a sum, involving the dual of the Riemann tensor and an important formula for the dual ⋆ g G µ of G µ , that permits us to write Einstein equations in a suggestive way concerning duality structures. In Section 4 we provide the correct dual of Einstein equation in Riemann-Cartan theory. In Section 5 we delve into the formalism under which conditions ⋆ g T α and ⋆ g R α β may be considered as the torsion and curvature 2-forms associated with a connection D ′ part of a Riemann-Cartan structure (M, g, D ′ ). Our result is then compared in Section 6 with the ones, e.g., in [1], which are then commented and analyzed in the present context. In Section 7 we study the same problem as in Section 5 but this time asking the conditions under which ⋆ g T α and ⋆ g R α β may be considered as the torsion and curvature 2-forms associated to a connection D ′ part of a Riemann-Cartan structure (M, g ′ , D ′ ) with g ′ = g. In Section 8 we present our conclusions. The paper contains some Appendices reviewing the definition of the exterior covariant derivative of indexed form fields, the decomposition of the Riemann and Ricci tensors of a general Riemann-Cartan structure (M, g, D), together with their respective similars for a Lorentzian structure (M,g,D), needed to perceive some statements in the main text. There is also an Appendix containing a collection of identities involving the contraction of differential forms and Hodge duals used in the derivations hereon.

Some Necessary Preliminaries
We start this Section by recalling some germane facts concerning the Riemann-Cartan structures and a particular and outstanding case of those structures, the Lorentzian one, which serves for the purpose of fixing our notations, besides other relevant properties and prominent applications. In what follows a general Riemann-Cartan structure will be denoted by (M, g, D). Here M is a 4-dimensional Hausdorff, paracompact, connected, and noncompact manifold, g ∈ sec T 0 2 M a metric tensor field of signature (1,3), D is a connection on M . Also the connection D is metric compatible, i.e., Dg = 0 and, moreover, for a general Riemann-Cartan structure the torsion and curvature tensors 4 of D -denoted by T and R -are non null. When T = 0 and R = 0, a Riemann-Cartan structure is called a Lorentzian structure and will be denoted by (M, g,D) 5 . When R = 0 a Lorentzian structure is called Minkowski structure. To present the definition of T and R, and the conventions used in this paper, first the torsion and curvature operations are introduced.
Definition 1 Let u, v ∈ sec T M . The torsion and curvature operations of a connection D, are respectively the mappings: τ : sec T M ⊗ sec T M → sec T M and ρ : sec T M ⊗ sec T M → sec T M given by Definition 2 Let u, v, w ∈ sec T M and α ∈ sec Λ 1 T * M . The torsion and curvature tensors of a connection D are the mappings T : sec( Given an arbitrary moving frame {e α } on T M , let {θ ρ } be the dual frame of {e α } (i.e., θ ρ (e α ) = δ ρ α ). Let also {e α } be the reciprocal basis of {e β }, i.e., g(e α , e β ) = δ α β and let {θ α } be the reciprocal basis of {θ ρ }, i.e., θ α (e β ) = δ β α . We write [e α ,e β ] = c ρ αβ e ρ , D e α e β = L ρ αβ e ρ , where c ρ αβ are the structure coefficients of the frame {e α } and L ρ αβ are the connection coefficients in this frame. Then, the components of the torsion and curvature tensors are given, respectively, by: 4 For the conventions used for those tensors in this paper see the Appendix. 5 The connection satisfyingDg = 0 and T =0 is unique and is called the Levi-Civita connection of g.
We can easily verify that defining R µναβ := g µρ R µ ρ αβ (7) it follows that Remark 3 When the torsion tensor of D is null, besides the symmetries given in Eq. (8), also the symmetry holds.
Now, taking into account Eq.(8) we introduce also a "physically equivalent" Riemann tensor R by In addition, where ω ρ β ∈ sec Λ 1 T * M are the connection 1-forms, T ρ ∈ sec Λ 2 T * M are the torsion 2-forms and R ρ β ∈ sec Λ 2 T * M are the curvature 2-forms, given respectively by Multiplying Eqs.(6) by 1 2 θ α ∧ θ β and using Eqs. (11) and (12), the Cartan's structure equations are derived: Definition 4 A Riemann-Cartan spacetime is a pentuple (M, g, D, τ g , ↑) where (M, g, D) is a Riemann-Cartan structure, and we suppose the existence of a global τ g ∈ sec Λ 4 T * M (which as well known defines an orientation for M ). Moreover, ↑ denotes that the Riemann-Cartan structure is time oriented. See, e.g., [23,22] for details.
Pentuples (M, g, D, τ g , ↑) represent gravitational fields in the so called Riemann-Cartan theories. In the theory presented, e.g., in [14], the equations of motion are the Einstein equation, where G ∈ sec T 0 2 M is the Einstein tensor, T ∈ sec T 0 2 M is the canonical energy-momentum tensor of the matter fields, and the algebraic identity where the Υ αβ ∈ sec Λ 1 T * M are such that their components are the so called modified torsion tensor components, and the ⋆ g J αβ ∈ sec Λ 3 T * M are the spin angular momentum densities of the matter fields 6 . Also, the symbol ⋆ g denotes the Hodge star operator associated to the metric g.

Remark 5
It is crucial to observe that for a general Riemann-Cartan structure G = G µν θ µ ⊗ θ ν and T = T µν θ µ ⊗ θ ν are not symmetric, i.e., G µν = G νµ and T µν = T νµ . We recall that where R µν are the components of the Ricci tensor (which, as G µν are not symmetric) and R = g µν R µ ρ ρν is the curvature scalar.
It is also well known that in GRT a gravitational field generated by a given matter distribution (represented by a given energy-momentum tensorT ∈ sec T 0 2 M ) is represented by a pentuple (M,g,D, τg, ↑) 7 and the equation of motion (Einstein equation) is given byG =T (17) and in this case the tensorsG andT are symmetric.

Remark 6
In the Appendix we review how to write the Riemann curvature tensor (respectively the Einstein tensor) of a Riemann-Cartan structure (M, g, D) in terms of the Riemann curvature tensor (respectively the Einstein tensor) of a Lorentzian structure (M,g,D). Those results are important for a proper understanding of this paper.

The Bianchi Identities
Given a general Riemann-Cartan structure (M, g, D) we have the following identities known respectively as the first and second Bianchi identities (see e.g., [8,23]). In the above equations, D is the exterior covariant derivative of indexed form fields [4,23], whose precise definition is recalled in Appendix A. Now, the coordinate expressions of Eqs. (18) and (19) can easily be found and are respectively [8,24] written as where (µνρ) denotes (as usual) the cyclic sum. For future use we observe that Remark 7 For a Lorentz structure (M,g,D) the Bianchi identities reduce to or in components:

Ricci and Einstein 1-form fields
Given R µν and G µν , respectively the components of the Ricci and Einstein tensors (in the general basis introduced above) we define the Ricci (R µ ∈ sec Λ 1 T * M ) and the Einstein (G µ ∈ sec Λ 1 T * M ) 1-form fields by 7 In fact a gravitational field is defined by an equivalence class of pentuples, where (M, g, D, τg, ↑) and where h * here denotes the pullback mapping). For more details, see, e.g., [22,23]. With the above definition we exclude from our considerations models with closed timelike curves, which according to our view are pure science fiction.
For future use we introduce also the energy-momentum 1-form fields T µ ∈ sec Λ 1 T * M by Also Proposition 8 The dual of the Ricci and Einstein 1-form fields, i.e., ⋆ can be written as: Proof. Using some of the identities in Appendix E we can write immediately and Eq.(26) is proved. Now Eq.(27) is evinced. By taking some of the identities in Appendix E, we can immediately write: and Eq.(27) is proved.
and so it follows that and taking into account Eq.(26) it reads:

The Dual of Einstein Equation in Riemann-Cartan Theory
We now return to Eq.(23) which in components can read 8 Multiplying this equation on both sides by θ ν and recalling the definitions of the Ricci, Einstein, and the energy-momentum 1-form fields given above we have Taking the dual of this equation we get Taking Eq.(32) and Eq.(25) into account, then Eq.(35) can be expressed as or equivalently

The Field and Structure Equations
We now summarize the field and Bianchi identities for a Riemann-Cartan theory where an Einstein-like equation holds. Those equations can be written conveniently in intrinsic and component forms respectively as: In a GRT model it follows that 5 Are ⋆ g T α and ⋆ g R α β the Torsion and Curvature 2-Forms of any Connection?
Despite the fact aforementioned in the last Remark, we may pose the question: can ⋆ g T α and ⋆ g R α β be the torsion and curvature 2-forms of a g-metric compatible connection, say D ′ , which defines on M the Riemann-Cartan structure (M, g, D ′ ) where also an Einstein like equation is valid? If the answer is positive, the following set of equations must hold: D ′ R ′α β = 0, and since by hypothesis or in component form (and obvious notation) (µαβ) 9 In particular, a correct expression for D ⋆ g T α has been found in [24].
Consequently, among the possible constraints in order to have a positive answer concerning the question in the head of the Section, the following two non trivial constraints are derived: (a) Using Eq.(40) and Eq.(49), it follows that (b) Using Eq.(48) and Eq.(41) we must have Let us analyze what those constraints imply if we start with (M, g,D), a Lorentzian structure (part of a Lorentzian spacetime structure) representing a gravitational field in GRT. In this case the second member of Eq.(52) must equal zero, and taking into account thatR =T ′ :=T κ κ andR * = −T ′ :=T ′κ κ we get that the structure (M, g,D ′ ) must also be torsion free and the following constraints must hold:

A Particular Case
It is obvious that we must then have: and comparing Eq.(52) with Eq.(57) we get So a Riemann-Cartan structure satisfying Eq.(54), Eq.(55) and Eq.(56) is possible only for matter distributions with T = T κ κ = 0 and which obey very stringent constraints. Also, [1] choose as "dual equations" the following set: 10 It is obvious from our previous considerations that the equation = ǫ ρλσκ T κ µ presented in [1] as an identity is in general wrong and invalidates most of the conclusions of that paper. Take also notice that in [1] it is defined a Hodge dual with respect to the first pair of indices. However, since they start from a Lorentzian structure (where torsion is null) we have the validity of Eq.(9) and so it does not matter in deriving Eq.(40) taking the dual with respect to the first or second pair of indices.
which, of course must imply Comparing Eq.(64) to Eq.(51) implies again that R = 0. So we end with the following constraints, necessary for the validity of the equations proposed in [1]: Such constraints are clearly violated by the examples in [1].
7 Is there a metric g ′ and a metric connection D ′ such that ⋆ g ′ T α and ⋆ g ′ R α β are their Torsion and Curvature Forms?
Now, we can also put the question: in which conditions may we conceive that ⋆ g T α and ⋆ g ′ R α β are the torsion and curvature 2-forms of a g ′ -metric compatible connection, say D ′ which defines on M the Riemann-Cartan structure (M, g ′ , D ′ ) where an Einstein like equation holds, i.e., (with obvious notation) the validity of the following set of equations (R ′ = g ′µβ R ′ ρ µρβ ): Since by hypothesis we must have ⋆ g T α = T ′α and ⋆ g R α β = R ′α β , calling R * = g ′µβ R * ρ µρβ the set of Eqs.(66) must be equal to: which are similar but not identical to the set given by Eq.(47). Due to their complexity we shall not inspect the nature of those equations solutions, a problem postponed for another publication.

Remark 11
The constraints concerned in this case are more involved than in the previous case, but we want to emphasize here that if we start with (M, g,D), a Lorentzian structure (part of a Lorentzian spacetime structure) representing a gravitational field in GRT, the structure (M, g ′ ,D ′ ) will be also torsion free. Here we recalled that [9] investigated long ago a similar problem (but only in the linear approximation) and found a positive answer for the question at the head of this Section.

Conclusions
In this paper we present the correct constraints that must be satisfied by any theory (in a 4-dimensional manifold) that intends to provide a dual presentation of the gravitational field equations for a general Riemann-Cartan theory. We compare our results with some ones proposed by authors quoted in the introduction and present some constructive criticisms. We hope that since the subject of duality becomes each day more important in non, e.g., abelian gauge theories, gravity and M -theory that our results shall become appreciated.
(c) For any metric-compatible connection D if g = g µν θ µ ⊗ θ ν then, Dg µν = 0. 11 Sometimes also called exterior covariant differential. 12 Which is not the case of the connection 1-forms ω α β , despite the name. More precisely, the ω α β are not true indexed forms, i.e., there does not exist a tensor field ω such that ω(e i , e β , ϑ α ) = ω α β (e i ). 13 As usual the inverted hat over a symbol (in Eq.(70)) means that the corresponding symbol is missing in the expression. 14 Multi indices are here represented by J and K.
and Q αβσ = −D αgβσ . Define the components of the strain tensor of the connection D (associated withD) by It is trivially established that whereΓ ρ αβ are the components of the Levi-Civita connection of g and T ρ αβ are the components of the torsion tensor of D 16 .
Eq.(77) can be used to relate the covariant derivatives with respect to the connectionsD and D of any tensor field on the manifold. In particular, recalling thatD αgβσ = e α (g βσ ) −g µσΓ µ αβ −g βµΓ µ ασ = 0, we get the expression of the nonmetricity tensor of D in terms of the torsion and the strain, namely, Eq.(78) can be inverted to yield the expression of the strain in terms of the torsion and the nonmetricity. We get: From Eq.(78) and Eq.(79) it is clear that nonmetricity and strain can be used interchangeably in the description of the geometry of a Riemann-Cartan-Weyl space. In particular, we have the relation: In order to simplify our next equations, let us introduce the notation: From Eq.(79) it follows that: Note also that for Dg = 0, K ρ αβ is the so-called contorsion tensor. Returning to Eq.(77), we obtain now the relation between the curvature tensor R µ ρ αβ associated with the connection D and the Riemann curvature tensorR µ ρ αβ of the Levi-Civita connection D associated with the metric g. We get, by a straightforward calculation: where: Multiplying both sides of Eq.(83) by 1 2 θ α ∧ θ β we get: From Eq.(83) we also get the relation between the Ricci tensors of the connections D andD. The Ricci tensor is defined by Then, we have with Observe that since the connection D is arbitrary, its Ricci tensor will be not be symmetric in general. Then, since the Ricci tensorR µα ofD is necessarily symmetric, we can split Eq.(87) into:

C Some Important Identities
Let (M, g) be a manifold and a Lorentzian metric as defined in Section 1. Let moreover Λ p T * M (p = 0, 1, 2, 3, 4) be the bundle of homogeneous p-form fields and ΛT * M = ⊕ 4 p=0 Λ p T * M the bundle of non homogeneous forms fields. We define in T * M a metric field g ∈ sec T 2 0 M such that concerning the general bases {e µ } and {θ µ } introduced in Section 1, if g = g µν θ µ ⊗ θ ν and g = g µν e µ ⊗ e ν then g µα g αν = δ µ ν . In ΛT * M we introduce a scalar product such that if A, B ∈ sec Λ r T * M are simple homogeneous r-forms with A = u 1 ∧ · · · ∧ u r and B = v 1 ∧ · · · ∧ v r , u i , v j ∈ sec Λ 1 T * M then A·B = det(g(u i , v j )), where (g(u i , v j )) means the matrix with entries (g(u i , v j )). This scalar product is then extended by linearity and orthogonality to all ΛT * M , and A · B = 0 if A ∈ sec Λ r T * M , and B ∈ sec Λ s T * M with r = s. Also, if a, b ∈ sec Λ 0 T * M then a · b = ab, the product of functions. If the metric manifold (M, g) is also endowed with an orientation, i.e., a volume 4-vector τg ∈ Λ 4 T * M such that τg · τg = −1, then a natural isomorphism between sections of Λ r T * M and Λ 4−r T * M (r = 0, . . . , 4) can be introduced. The Hodge star operator (or Hodge dual ) is the linear mapping ⋆ for every A, B ∈ Λ r T * M . Of course, this operator is naturally extended to an isomorphism ⋆ whereÃ means the reverse of A. If A = u 1 ∧ · · · ∧ u r thenÃ = u r ∧ · · · ∧ u 1 and 0 : sec ΛT * M → sec Λ 0 T * M is the projection of a general non homogeneous form into the Λ 0 T * M part.
We define the right and left contractions of non homogeneous differential forms as follows. For arbitrary multiforms X, Y, Z ∈ sec ΛT * M , the left ( (ii) If a, b 1 , . . . , b k ∈ sec ΛT * M then a g (b 1 ∧ · · · ∧ b k ) = k j=1 (−1) j+1 (a · b j )b 1 ∧ · · · ∧b j ∧ · · · ∧ b k , where the symbolb j means that the b j factor does not appear in the j-term of the sum.
(iii) For any Y j ∈ sec Λ j T * M and Y k ∈ sec Λ k T * M with j ≤ k (iv) For any Y j ∈ sec Λ j T * M and Y k ∈ sec Λ k T * M (v) For any X k , Y k ∈ sec Λ k T * M , then X k (vi) For any v ∈ sec Λ 1 T * M and X, Y ∈ sec ΛT * M , then v Finally we present some important identities involving contractions and the Hodge dual. Let A r ∈ sec Λ r T * M and B s ∈ sec Λ s T * M , r, s ≥ 0: