Some Thoughts on Geometries and on the Nature of the Gravitational Field

In this paper we show how a gravitational field generated by a given energy-momentum distribution (for all realistic cases) can be represented by distinct geometrical structures (Lorentzian, teleparallel and non null nonmetricity spacetimes) or that we even can dispense all those geometrical structures and simply represent the gravitational field as a field in the Faraday's sense living in Minkowski spacetime. The explicit Lagrangian density for this theory is given and the field equations (which are Maxwell's like equations) are shown to be equivalent to Einstein's equations. Some examples are worked in detail in order to convince the reader that the geometrical structure of a manifold (modulus some topological constraints) is conventional as already emphasized by Poincare long ago, and thus the realization that there are disctints geometrical representations (and a physical model related to a deformation of the continuum supporting Minkowski spacetime) for any realistic gravitational field strongly suggests that we must investigate the origin of its physical nature. We hope that this paper will convince readers that this is indeed the case.


Introduction
Physics students learn General Relativity (GR) as the modern theory of gravitation.In that theory each gravitational field generated by a given energymomentum tensor is represented by a Lorentzian spacetime, i.e., a structure (M, D, g, τ g , ↑) where M is a non compact (locally compact) 4-dimensional Hausdorff manifold, g is a Lorentzian metric on M and D is its Levi-Civita connection.Moreover M is supposed oriented by the volume form τ g and the symbol ↑ means that the spacetime is time orientable 1 .From the geometrical objects in the structure (M, D, g, τ g , ↑) we can calculate the Riemann curvature tensor R of D and a nontrivial GR model is one in which R = 0.In that way textbooks often say that in GR spacetime is curved.Unfortunately many people mislead the curvature of a connection D on M with the fact that M can eventually be a bent surface in an (pseudo) Euclidean space with a sufficient number of dimensions 2 .This confusion leads to all sort of wishful thinking because many forget that GR does not fix the topology 3 of M that often must be put "by hand" when solving a problem, and thus think that they can bend spacetime if they have an appropriate kind of some exotic matter.Worse, the insistence in supposing that the gravitational field is geometry lead the majority of physicists to relegate the search for the real physical nature of the gravitational field as not important at all 4 .Instead, students are advertised that GR is considered by may physicists as the most beautiful physical theory [23].However textbooks with a few exceptions (see, e.g., the excellent book by Sachs and Wu [39]) forget to say to their readers that in GR there are no genuine conservation laws of energy-momentum and angular momentum unless spacetime has some additional structure which is not present in a general Lorentzian spacetime [27].Only a few people tried to develop consistently theories where the gravitational field (at least from the classical point of view) is a field in the Faraday's sense living in Minkowski spacetime (see below).
In this paper we want to recall two important results that hopefully will lead people to realize that eventually it is time to disclose the real nature of the gravitational field 5 .The first result is that the representation of gravitational fields by Lorentzian spacetimes is eventually no more than an consequence of to the differential geometry knowledge of Einstein and Grossmann when they where struggling to find a consistent way to describe the gravitational field 6 .Indeed, there are some geometrical structures different from (M, D, g, τ g , ↑) that can equivalently represent such a field.The second result is that the gravitational field (in all known situations) can also be nicely represented as a field in the Faraday's sense [26] living in a fixed background spacetime 7 .Concerning the alternative geometrical models, the particular cases where the connection is teleparallel (i.e., it is metrical compatible, has null Riemann curvature tensor and non null torsion tensor) and the one where the connection is not metrical compatible (i.e., its nonmetricity tensor A η = 0) will be addressed below.However to understand how those alternative geometrical models (and the physical model) can be constructed and why the Lorentzian spacetime model was Einstein's first choice, it is eventually worth to recall some historical facts concerning attempts by Einstein (and others) to build a geometrical unified theory of the gravitational and electromagnetic fields.
We start with the word torsion.Although such a word seems to have been introduced by Cartan [2] in 1922 the fact is that the concept behind the name already appeared in a Ricci's paper [31] from 1895 and was also used in [32]!In those papers Ricci introduced what is now called the Cartan's moving frames and the teleparallel geometry 8 .
Moreover in 1901 Ricci and Levi-Civita 9 published a joy [33], which has become the bible of tensor calculus and which has been extensively studied by Einstein and Grossmann in their search for the theory of the gravitational field 10 .However Einstein and Grossmann seems to have studied only the first part of reference 4 and so missed the "Cartan's moving method" and the concept of torsion.It seems also that only after 1922 Einstein become interested in the second chapter of the joy, titled La Géométrie Intrinseque Comme Instrument de Calcul and discovered torsion and the teleparallel geometry 11 .As it is well known he tried to identify a certain contraction of the torsion tensor of a teleparallel geometry with the electromagnetic potential, but after sometime he discovered that the idea did not work.Einstein's first papers on the subject 12 are [11, 12, 13].Also in a paper which Einstein wrote in 1925 [10] the torsion tensor concept already appeared, since he considered as one of his field variables the antisymmetric part of a non symmetric connection.All those papers by Einstein have been translated into English by Unzicker and Case [44] and can be downloaded from the arXiv.We also can learn in [7, 18] that Cartan tried to explain the teleparallel geometry to Einstein when he visited Paris in 1922 using the example of what we call the Nunes connection (or navigator connection) on the punctured sphere.Since this example illustrates in a crystal clear way the fact that one must not confound the Riemann curvature of a given connection defined on a manifold M with the fact that M may be viewed as a 7 The preferred one is, of course, Minkowski spacetime, the simple choice.But, the true background spacetime may be eventually a more complicated one, since that manifold must represent the global topological structure of the universe, something that is not known at the time of this writing [53].
8 Also known as Weintzböck geometry [55]. 9A Ricci's student at that time. 10An english translation of the joy with very useful comments has been done by the mathematical physicist Robert Hermann [21] in 1975 and that text (and many others books by Hermann) can be downloaded from http://books.google.com.br/books?q=robert+hermann 11Some interesting historical details may be found in [18, 40]  12 For a complete list of Einstein's papers on the subject see [18].
bent hypersurface embedded in an Euclidean space (with appropriate number of dimensions) it will be presented in Appendix A. A comparison of the parallel transport according to the Nunes connection and according to the usual Levi-Civita connection is done 13 , and it is shown that the Nunes connection the Riemann curvature of the punctured sphere is null.In this sense the geometry of the punctured sphere is conventional as emphasized by Poincaré [30] long ago.
As we already said the main objective of the present paper is to clarify the fact that there are different ways of geometrically representing a gravitational field, such that the field equations in each representation result equivalent to Einstein's field equations.Explicitly we mean by this statement the following: any model of a gravitational field in GR represented by a Lorentzian spacetime (with non null Riemann curvature tensor and null torsion tensor which is also parallelizable 14 ) is equivalent to a teleparallel spacetime (i.e., a spacetime structure equipped with a metrical compatible teleparallel connection, which has null Riemann curvature tensor and non null torsion tensor) 15 or equivalent to a special spacetime structure, where the manifold M is equipped with a Minkowski metric, and where there is also defined a connection such that its nonmetricity tensor is not null.The teleparallel possibility is described in details in Section 2 using the modern theory of differential forms and we claim that our presentation leaves also clear that we can even dispense with the concept of a connection in the description of a gravitational field 16 , it is only necessary for such a representation to exists that the manifold M representing the set of all possible events be parallelizable, admitting four global (not all exact) 1-form fields coupled in a specific way (see below).The second possibility is illustrated with an example in Section 3. In Section 4 we present the conclusions.
2 Torsion as a Description of Gravity

Some Notation
Suppose that a 4-dimensional M manifold is parallelizable, thus admitting a set of four global linearly independent vector e a ∈ sec T M , a = 0, 1, 2, 3 fields 17 such {e a } is a basis for T M and let {θ a }, θ a ∈ sec T * M be the corresponding dual basis (θ a (e b ) = δ a b ).Suppose also that not all the θ a are closed, i.e., 13 The material of Appendix A follows the presentation in Section 4.7.7 of [36]. 14A manifold M is said to be parallelizable if it admits four global linearly independent vector fields. 15There are hundreds of papers (as e.g., [6]) on the subject. 16Explicitly, we mean that the gravitational field may be interpreted as a field in the sense of Faraday, as it is the case of the electromagnetic field. 17We recall that sec T M means section of the tangent bundle and sec T * M means section of the cotangent bundle.Also sec T r s M means the bundle of tensors of type (r, s) and sec ^rT * M a section of the bundle of r-forms fields.
Then, according to g the {e a } are orthonormal, i.e., Remark 1 Since according to Eq.(3) e 0 is a global time like vector field it follows it defines a time orientation in M which we denote by ↑.It follows that that the 4-tuple (M, g, τ g , ↑) is part of a structure defining a Lorentzian spacetime and can serve as a substructure to model a gravitational field in GR.
For future use we also introduce g ∈ sec T 2 0 M by and we write: Due to the hypothesis given by Eq.( 1) the vector fields e a , a = 0, 1, 2, 3 will in general satisfy [e a , e b ] = c k ab e k , where the c k ab are the structure coefficients of the basis {e a }.It can been easily shown that18 Now, we introduce two different metric compatible connections on M , namely D (the Levi-Civita connection of g) and a teleparallel connection ∇, such that The objects ω c ab are called the connection coefficients of the connection D in the {e a } basis and the objects ω a b ∈ sec T * M defined by are called the connection 1-forms in the {e a } basis.

Remark 2
The connection coefficients b ac of ∇ and the connection 1-forms of ∇ in the basis {e a } are null according to the second line of Eq.( 8) and thus the basis {e a } is called teleparallel.So, the connection ∇ defines an absolute parallelism on M .We recall that as said in the introduction that idea has been introduced by Ricci.
Remark 3 Of course, as it is well known the Riemann curvature tensor of D, is in general non null in all points of M , but the torsion tensor of D is zero in all points of M .On the other hand the Riemann curvature tensor of ∇ is null in all points of M , whereas the torsion tensor of ∇ is non null in all points of M .
We recall also in order to fix notation that for a general connection, say D on M (not necessarily metric compatible) the torsion and curvature operations and the torsion and curvature tensors of a given general connection, say D, are respectively the mappings: and for every u, v, w ∈ sec T M and α ∈ sec 1 T * M .In particular we write and define the Ricci tensor by We shall need also in order to fix our conventions to briefly recall the definitions of the scalar product and left and right contractions on the so called Hodge bundle ( T * M, g) where The left and right contractions of X , Y ∈ sec T * M are defined for arbitrary (nonhomogeneous) multiforms as the mappings where the tilde means the operation of reversion, e.g., if . The Hodge star operator (or Hodge dual) is the linear mapping for every Hodge star operator is given by g where sgn g = det g/| det g| denotes the sign of the determinant of the matrix with entries g ij = g(e i , e j ).The Hodge coderivative operator δ g (associated to

Cartan's Structure Equations
Given that we introduced two different connections D and ∇ defined in the manifold M we can write two different pairs of Cartan's structure equations, each one of the pairs describing respectively the geometry of the structures (M, D, g, τ g , ↑) and (M, ∇, g, τ g , ↑) which will be called respectively a Lorentzian spacetime and a teleparallel spacetime.

Cartan's Structure Equations for D
In this case we write where the Θ a ∈ sec

Cartan's Structure Equations for ∇
In this case since a b = 0 we have Θa : where the Θa ∈ sec T * M , a, b = 0, 1, 2, 3 and are respectively the torsion and the curvature 2-forms of ∇ given by formulas analogous to the ones in Eq. (22).
We next suppose that the {θ a } are the basic variables representing a gravitation field.We postulate for the {θ a } interacts with the matter fields though the following Lagrangian density19 where L m is the matter Lagrangian density and Remark 4 This Lagrangian is not invariant under arbitrary point dependent Lorentz rotations of the basic cotetrad fields.In fact, if θ a → θ a = Λ a b θ b , where for each x ∈ M , Λ a b (x) ∈ L ↑ + (the homogeneous and orthochronous Lorentz group) we get that differs from L g by an exact differential.So, the field equations derived by the variational principle results invariant under a change of gauge. 20ow, to derive the field equations directly from Eq.( 25) is a nontrivial and laborious exercise, whose details the interested reader may find in [36].The result is: where g and the 21 g are the energy-momentum 3-forms of the matter fields 22 .
Recalling that from Eq.( 23) it is Θa := dθ a , the field equations (Eq.( 27)) can be written as where Finally recalling the definition of the Hodge coderivative operator ( Eq.( 20)) we can write Eq.( 31) as δ g with the t d ∈ sec 1 T * M given by legitimate energy-momenta 23 1-form fields for the gravitational field.Note that the total energy momentum tensor of matter plus the gravitational field is trivially conserved in our theory, i.e., Remark 5 In [26] a theory of the gravitational field in Minkowski spacetime has been presented where a nontrivial gravitational field configuration was interpreted as generating an effective Lorentzian spacetime (M R 4 , g, D, τ g , ↑) where g satisfies Einstein equations and where probe particles and/or fields move.It was assumed there that the gravitational field g = η ab θ a ⊗θ b is a field in Faraday sense 24 , i.e., the fields θ a have their dynamics described by a (postulated ) Lagrangian density like the one in Eq.( 25).Moreover, it was postulated that the θ a couple universally with the matter fields and that the 21 We suppose that Lm does not depend explicitly on the dθ a . 22In reality, due the conventions used in this paper the true energy-momentum 3-forms are presence of energy-momentum due to matter fields in some region of Minkowski spacetime distorts the Lorentz vacuum in much the same way that stresses in an elastic body distorts it.Now, distortions (or deformations) in the theory of dislocations according to [56] can be of the elastic or plastic type.An elastic distortion is described by a diffeomorphism h: M → M .In this case the induced metric is g = h * η (analogous to the Cauchy-Green tensor [16] of elasticity theory) and according to Remark 250 in [36] its Levi-Civita connection is h * Ḋ.This implies that the structure (hM R 4 , g, h * D, τ g , ↑) is again Minkowski spacetime.In the original versions of [26, 36] this was the type of deformation considered, but this has been corrected in improved versions of those manuscripts, respectively at the arXiv and at http://ime.unicamp.br/walrod/recenteswhere an errata to Chapter 10 of [36] at .In the quoted errata, the deformation is taken to be of the plastic type and represented by a distortion extensor field M , i.e., a linear mapping h : 1 T * M → 1 T * M such that g = h † h, where g is the extensor field associated with g (see below ) 25 .A paper describing a detailed theory of gravitation as a plastic deformation of the Lorentz vacuum is in preparation [14].
where F ∈ sec 2 T * M is the electromagnetic field and J ∈ sec 1 T * M is the electric current.The two Maxwell equations in Eq.( 36) can be written as a single equation using the Clifford bundle formalism [36].In this formalism T * M → C (M, g) .Then it can be shown that in this case is the Dirac operator (acting on sections of C (M, g)) and we can write Maxwell equation as Now, if you fell uncomfortable in needing four distinct potentials θ a for describing the gravitational field you can put then together defining a vector valued differential form and in this case the gravitational field equations are where Θ= Θa ⊗e a , T = T a ⊗e a , t = t a ⊗e a .By considering the bundle C (M, g)⊗ T M we can even write the two equations in Eq.( 39) as a single equation, i.e.,

Relation with Einstein's Theory
At this point the reader may be asking: which is the relation of the theory just presented with Einstein's GR theory?The answer is that recalling that the connection 1-forms ω cd of D are given by we can show through a laborious (but standard) exercise (see [36] for details) that the first member of Eq.( 27) is exactly − g G d (the Einstein 3-forms).So, we have with R d = R d a θ a the Ricci 1-forms and R the scalar curvature.Then Eq.( 27) results equivalent to and taking the dot product of both members with θ a we get which is the usual tensorial form of Einstein's equations.
Remark 7 When the θ a and the dθ a are packed in the form of the connection 1-forms the Lagrangian density L g becomes where (with R cd given by Eq.( 21)) is the Einstein-Hilbert Lagrangian density.

A Comment on Einstein Most Happy Though
The exercises presented above indicates that a geometrical interpretation for the gravitational field is no more than an option among many ones.Indeed, it is not necessary to introduce any connection D or ∇ on M to have a perfectly well defined theory of the gravitational field whose field equations are equivalent to the Einstein field equations.Note that we have not give until now any details on the global topology of the world manifold M .However, since we admitted that M carries four global (not all closed) 1-form fields θ a which defines the object g it follows that (M, D, g, τ g , ↑) is a spin manifold [17, 36], i.e., admit spinor fields.This, of course, is necessary if the theory is to be useful in the real world since fundamental matter fields are spinor fields.The most simple spin manifold is clearly Minkowski spacetime which is represented by a structure (M, D, η, τ η , ↑) where M R 4 , and D is the Levi-Civita connection of the Minkowski metric η.In that case it is possible to interpret g as a field in the Faraday sense living in (M, D, η, τ η , ↑), or to work directly with the θ a which has a well defined dynamics and coupling to the matter fields.
At last we want to comment that as well known in Einstein's GR one can easily distinguish in any real physical laboratory [28] (despite some claims on the contrary) a true gravitational field from an acceleration field of a given reference frame in Minkowski spacetime.This is because in GR the mark of a real gravitational field is the non null Riemann curvature tensor of D, and the Riemann curvature tensor of the Levi-Civita connection of D (present in the definition of Minkowski spacetime) is null.However if we interpret a gravitational field as the torsion 2-forms on the structure (M, ∇, g, τ g , ↑) viewed as a deformation of Minkowski spacetime then one can also interpret an acceleration field of an accelerated reference frame in Minkowski spacetime as generating an effective teleparallel spacetime (M, e ∇, η, τ η , ↑).This can be done as follows.Let Z ∈ sec T U , U ⊂ M with η(Z, Z) = 1 an accelerated reference frame on Minkowski spacetime.This means (see, e.g., [36] for details) that Call e 0 = Z and define an accelerated reference frame as non trivial if ϑ 0 = η(Z, ) is not an exact differential.Next recall that in U ⊂ M there always exist three other η-orthonormal vector fields e i , i = 1, 2, 3 such that {e a } is an η-orthonormal basis for T U , i.e., where {ϑ a } be the dual basis 26 of {e a }.We then have Dea e b = ωc ab e c , Dea What remains in order to be possible to interpret an acceleration field as a kind of 'gravitational field' is to introduce on M a η-metrical compatible connection e ∇ such that the {e a } is teleparallel according to it.We have With this connection the structure (M R 4 , e ∇, η, τ η , ↑) has null Riemann curvature tensor but a non null torsion tensor, which an easy calculation shows to be related with the acceleration and the other coefficients ωc ab of the connection D in that basis, which describe the motion on Minkowski spacetime of a grid represented by the orthonormal frame {e a }.Schücking [40] thinks that such a description of the gravitational field makes Einstein most happy though, i.e., the equivalence principle (understood as equivalence between acceleration and gravitational field) a legitimate mathematical idea.However, a true gravitational field must satisfy (at least with good approximation) Eq.( 31), whereas there is no single reason for an acceleration field to satisfy that equation.

4
A Model for the Gravitational Field Represented by the Nonmetricity of a Connection In this section we suppose that the world manifold M is a 4-dimensional manifold diffeomorphic to R 4 .Let moreover (t, x, y, z) = (x 0 , x 1 , x 2 , x 3 ) be global Cartesian coordinates for M .Next, introduce on M two metric fields: and In Eq. ( 51) Now, introduce (t, r, ϑ, ϕ) = (x 0 , x 1 , x 2 , x 3 ) as the usual spherical coordinates for M .Recall that and the range of these coordinates in η are r > 0, 0 < ϑ < π, 0 < ϕ < 2π.For g the range of the r variable must be (0, 2m) ∪ (2m, ∞).
As can be easily verified, the metric g in spherical coordinates is: which we immediately recognize as the Schwarzschild metric of GR.Of course, η is a Minkowski metric on M .As next step we introduce two distinct connections, D and D on M .We assume that D is the Levi-Civita connection of η in M and D is the Levi-Civita connection of g in M .Then, by definition (see, e.g., [36] for more details) the ammetricities tensors of D relative to η and of D relative to g are null, i.e., Dη = 0, However, the nonmetricity tensor and also the nonmetricity tensor We now calculate the components of A η in the coordinated bases {∂ µ } for T M and {dx ν } for T * M associated with the coordinates (x 0 , x 1 , x 2 , x 3 ) of M .Since D is the Levi-Civita connection of the Minkowski metric η we have that i.e., the connection coefficients L ρ µν of D in this basis are null.Then, To fix ideas, recall that for Q 100 it is, which is non null for x 1 = 0. Note that also that From what has been said it is obvious that since (M, η) and (M, g) are both orientable and time orientable, then (M, η, D), (M, g, D) are part of the structures representing respectively Minkowski spacetime and Schwarzschild spacetime.More precisely, (M, g, D, τ g , ↑) represents in GR the gravitational field of a point mass with world line given by (t, 0, 0, 0).As usual in GR this world line is left out of the effective manifold 27 .We claim that (M, D, g) or (M, η, D) are legitimate equivalent representations for the gravitational field described in GR by the substructure (M, g, D).To find, e.g., the relation between the models (M, g, D) and (M, g, D) it is necessary to recall that if in the bases {∂ µ } for T M and {dx ν } for T * M , we have and the Christoffel symbols are not all null.Moreover, in the spherical coordinates introduced above and the L ρ µν and Γ ρ µν are not all null.Now, L ρ µν and Γ ρ µν are related by 28 : where S ρ αβ are the components of the so called strain tensor of the connection D relative to the connection D. For the present case it is Now, since in the Cartesian coordinates L ρ αβ = 0, but not all Γ ρ αβ are null we get and thus, e.g., 4.2 A η as the Gravitational Field Note that using coordinates (Riemann normal coordinates {ξ µ } covering V ⊂ U ⊂ M ) naturally adapted to a reference frame Z∈ sec T V 29 in free fall according to GR (D Z Z = 0, dα ∧ α = 0, α = g(Z, )) it is possible to put the 27 The manifold where Schwarzschild solution is obtained is one with boundary, i.e., it is The reason for that is that almost all mathematical physicists use manifolds with boundary in order to avoid the use of distributions (generalized functions).Indeed, for a rigorous point of view, taking into account that Einstein's equations are non linear we cannot solve it using Schwartz distributions.To solve problems involving singular distributions in GR in a rigorous way it is necessary to use Colombeau theory of generalized functions as described, e.g., in [20]. 28See, e.g., Section 4.5.8 of [36]. 29For the mathematical definitions of reference frames, naturally adapted coordinates to a reference frame and observers, see, e.g., Chapter 5 of [36].
connection coefficients of the Levi-Civita connection D of g equal to zero in all points of the world line of a free fall observer (an observer is here modelled as an integral line σ of a reference frame Z, where Z is a time like vector field pointing to the future such that Z| σ = σ * ).
In the Riemann normal coordinates system covering U ⊂ M , it is obvious that not all the connection coefficients of the connection D (that relative to g is a non metrical one) are null.Moreover, the nonmetricity tensor A η is not null and it represents in our model the true gravitational field.Indeed, an observer following σ does not fell any force along its world line because the gravitational force represented by the nonmetricity field A η is compensated by an inertial force represented by the non null connection coefficients 30 L ρ µν of D in the basis The situation is somewhat analogous to what happens in any non inertial reference frame which, of course, may be conveniently used in any Special Relativity problem (as e.g., in a rotating disc [37]), where the connection coefficients of the Levi-Civita connection of η are not all null.
Remark 8 The theoretical definition of standard clocks of GR are reasonably well realized by atomic clocks, i.e., under certain limits atomic clocks behave as theoretically predicted (see however [35]) Note however that atomic clocks are not the standard clocks of the model proposed here.We would say that the gravitational field distorts the period of the atomic clocks relative to the standard clocks of the proposed model where gravity is represented by a nonmetricity tensor 31 .But, who are the devices that now materialize those concepts?Well, they may are paper concepts, like the notion of time in some Newtonian theories.They are defined and calculated in order to make correct predictions.However, given the status of present technology we can easily imagine how to build devices for directly realizing the standard clocks (and rulers) of the proposed model.

Conclusions
In this paper we recalled two important results.The first is that a gravitational field generated by a given energy-momentum distribution can be represented by distinct geometrical structures (Lorentzian, teleparallel and non null nonmetricity spacetimes).The second important result is that we can even dispense all those geometrical structures and simply represent the gravitational field as a field in the Faraday's sense living in Minkowski spacetime.The explicit Lagrangian density for this theory has been discussed and the field equations have been shown to be equivalent to Einstein's equations.We hope that our study clarifies the real difference between mathematical models and physical reality 30 The explicit form of the coefficients L ρ µν may be found in Chapter 5 of [36]. 31Schwinger [41] showed with very simple arguments how the gravitational field distorts the period of atomic clocks making then to register the proper time predicted by GR.His arguments can be easily adapted for the alternative models studied in this paper, because once g is known experimentally we can determine η with the mathematical techniques described in [36].
and leads people to think about the real physical nature of the gravitational field (and also of the electromagnetic field 32 ) As a final remark, we want to leave clear that after studying Einstein's papers (and also papers by many others authors) on the use Riemann-Cartan 33 to describe a classical unified theory of gravitation and electromagnetism we became convinced that it seems impossible to represent the electromagnetic field using a contraction of the torsion tensor (or the torsion tensor) without introducing ad hoc hypothesis.Having said that we recall that from time to time some authors return to the embryo of Einstein's original idea claiming to have obtained an unified theory of gravitation and electromagnetism using that tool.Among those theories that appeared in the last few years some are completely worthless, since based in a very bad use of Mathematical concepts, but some looks at least at a first sight interesting enough (at least from the mathematical point of view) to deserve some comments, which will be discussed elsewhere 34 .

A The Levi-Civita and the Nunes Connection on S2
Consider S 2 , an sphere of radius R = 1 embedded in R 3 .Let (x 1 , x 2 ) = (ϑ, ϕ) 0 < ϑ < π, 0 < ϕ < 2π, be the standard spherical coordinates of S 2 , which covers all the open set U which is S 2 with the exclusion of a semi-circle uniting the north and south poles.
Introduce the coordinate bases for T U and T * U .Next introduce the orthonormal bases {e a }, {θ a } for T U and T * U with Then, Moreover the metric g ∈ sec T 0 2 S 2 inherited form the ambient Euclidean metric is: The Levi-Civita connection D of g has the following non null connections coefficients Γ ρ µν in the coordinate basis (just introduced): Also, in the basis {e a }, D e i e j = ω k ij e k and the non null coefficients are: The torsion and the (Riemann) curvature tensors of D (recall Eq.( 12) and Eq.( 13 which results in T = 0 and that the non null components of R are R Since the Riemann curvature tensor is non null the parallel transport of a given vector depends on the path to be followed.We say that a vector (say v 0 ) is parallel transported along a generic path R ⊃I → γ(s) ∈ R 3 (say, from A = γ(0) to B = γ(1)) with tangent vector γ * (s) (at γ(s)) if it determines a vector field V along γ satisfying and such that V(γ(0)) = v 0 .When the path is a geodesic 35   i.e., γ * • g V = constant.This is clearly illustrated in Figure 1 (from [1]).
Consider next the manifold S2 = {S 2 \north pole + south pole} ⊂ R 3 , which is our sphere of radius R = 1 but this time excluding the north and south poles.Let again g ∈ sec T 0 2 S2 be the metric field on S2 inherited from the ambient space R 3 and introduce on S2 the Nunes (or navigator) connection 36 ∇ defined by the following parallel transport rule: a vector at an arbitrary point of S2 is parallel transported along a curve γ, if it determines a vector field on γ such that at any point of γ the angle between the transported vector and the vector tangent to the latitude line passing through that point is constant during the 35 We recall that a geodesic of D also determines the minimal distance (as given by the metric g) between any two points on S 2 .
36 Pedro Salacience Nunes (1502-1578) was one of the leading mathematicians and cosmographers of Portugal during the Age of Discoveries.He is well known for his studies in Cosmography, Spherical Geometry, Astronomic Navigation, and Algebra, and particularly known for his discovery of loxodromic curves and the nonius.Loxodromic curves, also called rhumb lines, are spirals that converge to the poles.They are lines that maintain a fixed angle with the meridians.In other words, loxodromic curves directly related to the construction of the Nunes connection.A ship following a fixed compass direction travels along a loxodromic, this being the reason why Nunes connection is also known as navigator connection.Nunes discovered the loxodromic lines and advocated the drawing of maps in which loxodromic spirals would appear as straight lines.This led to the celebrated Mercator projection, constructed along these recommendations.Nunes invented also the Nonius scales which allow a more precise reading of the height of stars on a quadrant.The device was used and perfected at the time by several people, including Tycho Brahe, Jacob Kurtz, Christopher Clavius and further by Pierre Vernier who in 1630 constructed a practical device for navigation.For some centuries, this device was called nonius.During the 19 th century, many countries, most notably France, started to call it vernier.More details in http://www.mlahanas.de/Stamps/Data/Mathematician/N.htm.transport.This is clearly illustrated in Figure 2. and to distinguish the Nunes transport from the Levi-Civita transport we ask also for the reader to study with attention the caption of Figure  We recall that from the calculation of the Riemann tensor R it follows that the structures ( S2 , g, D, τ g ) and also (S 2 , g, D, τ g ) are Riemann spaces of constant curvature.We now show that the structure ( S2 , g, ∇, τ g ) is a teleparallel space 37 , with zero Riemamn curvature tensor, but non zero torsion tensor.Indeed, from Figure 2 it is clear that (a) if a vector is transported along the infinitesimal quadrilateral pqrs composed of latitudes and longitudes, first starting from p along pqr and then starting from p along psr the parallel transported vectors that result in both cases will coincide (study also the caption of Figure (1).Now, the vector fields e 1 and e 2 in Eq.(68a) define a basis for each point p of T p S2 and ∇ is clearly characterized by: ∇ e j e i = 0. (77) The components of curvature operator are: 37 As recalled in Section 1, a teleparallel manifold M is characterized by the existence of global vector fields which is a basis for TxM for any x ∈ M .The reason for considering S2 for introducing the Nunes connection is that as well known (see, e.g., [8]) S 2 does not admit a continuous vector field that is nonnull at on points of it.

Remark 6 1 T
In electromagnetic theory on a Lorentzian spacetime we have only one potential A ∈ sec * M and the field equations are

4. 1
(M, η, D), (M, η, D), (M, g, D) and (M, g, D) ) are T (θ k , e i , e j ) = θ k (τ (e i , e j )) = θ k D e j e i − D e i e j − [e i , e j ] , (73) R(e k , θ a , e i , e j ) = θ a D e i D e j − D e j D e i − D [e i , e j ] e k , of the connection D, i.e.,a curve R ⊃I → c(s) ∈ R 3 with tangent vector c * (s) (at c(s)) satisfying D c * c * = 0, (76) the parallel transported vector along a c forms a constant angle with c * .Indeed, from Eq.(75) it is γ * • g D γ * V = 0. Then taking into account Eq.(76) it follows that D γ * (γ * • g V) = 0.

Figure 1 :
Figure 1: Levi-Civita and Nunes transport of a vector v 0 satarting at p through the paths psr and pqr.Levi-Civita tranport through psr leads to v 1 whereas Nunes transport leads to v 2 .Along pqr both Levi-Civita and Nunes transport agree and leads to v 2 . (1).

Figure 2 :
Figure 2: Characterization of the Nunes connection.