Cartan approach to Teleparallel Equivalent to General Relativity: a review
aa r X i v : . [ g r- q c ] J a n Cartan approach to Teleparallel Equivalent to General Relativity: a review
E. Huguet , M. Le Delliou , , and M. Fontanini - Universit´e de Paris, APC-Astroparticule et Cosmologie (UMR-CNRS 7164), F-75006 Paris, France. ∗ - Institute of Theoretical Physics, School of Physical Science and Technology,Lanzhou University, No.222, South Tianshui Road, Lanzhou, Gansu 730000, P R China and - Instituto de Astrof´ısica e Ciˆencias do Espa¸co, Universidade de Lisboa,Faculdade de Ciˆencias, Ed. C8, Campo Grande, 1769-016 Lisboa, Portugal † (Dated: January 19, 2021)In previous works, questioning the mathematical nature of the connection in the translationsgauge theory formulation of Teleparallel Equivalent to General Relativity (TEGR) Theory led us topropose a new formulation using a Cartan connection. In this review, we summarize the presentationof that proposal and discuss it from a gauge theoretic perspective. PACS numbers: 04.50.-h, 11.15.-q, 02.40.-k
CONTENTS
I. Introduction 1II. Why a Cartan connection to describe TEGR? 2A. Constitutive elements for gauge-TEGR 2B. Specific features of gauge TEGR connection 2III. Cartan TEGR 3A. The Cartan connection 31. Klein geometry 32. Cartan geometry 43. The TEGR connection 5B. Matter coupling with Cartan connection 61. From Cartan connection to Levi-Civitacoupling 62. A remark about the gravitational field inCartan TEGR 6C. Is Cartan TEGR a gauge theory? 7Frames as gauge fields 7IV. A note on Cartan GR 7V. Conclusion 8Acknowledgements 8A. Right (left) G-space, homogeneous space and cosetspace G/H 8B. A note on gravity as a gauge theory oftranslations: a not so obvious relation 8References 9 ∗ [email protected]@apc.univ-paris7.fr † ([email protected],)[email protected] I. INTRODUCTION
As a description of gravity, the Teleparallel Equivalentto General Relativity (TEGR) offers a symmetric andclassically equivalent way of presenting the physics de-scribed by General Relativity (GR): the main differencein perspective being that GR uses curvature of spacetimeto describe the effects of gravity requiring zero torsion,while TEGR requires zero spacetime curvature and fullyencodes gravity in the torsional part. This mirror ap-proach to the description of gravity is not proprietary toTEGR, Symmetric Teleparallel Gravity (STGR), intro-duced in Ref. [1], is another example of a possible equiv-alent description where curvature and torsion are sym-metrically taken to be null and non-metricity containsall the relevant physics. In the many years TEGR hasbeen around, its formal presentation has taken variousforms, from the translation-gauge approach of Ref. [2],the “pure tetrads formalism” point of view in Ref. [3], tothe tensorial formalism of Ref. [4]. Its alternative viewof gravity also created fertile ground for the proposal ofnaturally ghost free modified gravity developments suchas f ( T ) [5–7], f ( T, B ) [8], f ( R, T ) [9], or other generali-sations such as Lovelock Teleparallel Equivalent Gravity[10], or Conformal TEGR [3, 11, 12].One of the advantages presented by TEGR over GRis the claim that it can be formulated as a gauge the-ory for the translation group [see 2, 13, and referencesherein]; in such formulation the principal bundle frame-work of gauge theories is paralleled to extract the TEGRtorsion from the curvature of a connection defined on theprincipal bundle of translations. In this approach, the so-called canonical one-form, required to define the torsion,is indeed implicitly identified with the connection one-form. Since the definition of the canonical one form, as astructure element of the frame bundle, does not fit withthat of a connection, this identification becomes problem-atic [14–16]. The Cartan connection has therefore beenproposed as a solution to this difficulty, it has in factthe right geometrical properties while being able to offera bundle description and thus possibly support a gaugetheory interpretation. This paper reviews such proposal,its recent developments, as well as additional details onrelated topics.Differential geometry definitions and concepts can bereferred to in [17–20]. In Sec. II, we will overview therequirements emerging from TEGR that would be neces-sary to express it in gauge form. A review of the coherentframework using a Cartan connection for TEGR, withits possible opening to the gauge point of view, is spelledout in Sec. III, followed, in Sec. IV, by a small note onsuch framework’s application to GR, before concludingin Sec. V.
II. WHY A CARTAN CONNECTION TODESCRIBE TEGR?A. Constitutive elements for gauge-TEGR
In the context of TEGR and gauge theory, let us exam-ine how the constitutive elements of TEGR fit within theframework of a typical gauge theory, as found in particlephysics.On physical side, the main constitutive elements ofsuch theories are gauge fields and their associated fieldstrengths. They are dynamical fields whose free fieldequation (uncoupled to matter) exhibits gauge invari-ance. They mediate an interaction between matter fieldsand ensure that the matter field equations are locallyinvariant under some symmetry. On the mathematicalside, the gauge fields are sections of Ehresmann connec-tions defined on a principal G -bundle, G being a theglobal symmetry group of the free matter-field equations.The field strengths are (sections of) the curvature of theseconnection one-forms.In TEGR, spacetime is a metric manifold ( M, g ), grav-ity is carried by torsion, curvature is null, as well as non-metricity. Taking the Cartan view, the metric is inducedby orthonormal (co-)frames (tetrads) through η ( e, e ) = g , η being the metric of Minkowski tangent space. Here werecall some central structures related to these notions:1. the orthonormal frame bundle OM , a principalSO(1 , F M : its base manifold is the spacetime and itsfibers contains all the orthonormal frames e at abase manifold point, each frame being in one-to-one correspondence with an element of SO(1 ,
3) a(local) Lorentz transformation,2. the canonical one-form θ , an R -valued one-form,defined through:( θ ( e ) , V ) = ( e − , π ∗ V ) , (1)or in coordinates, θ a ( e )[ V ] = e a [ π ∗ V ] = V a where e is a frame at a point x of the base man-ifold M , e − its co-frame, V a vector field of the tangent bundle T OM , and π the projection on thebase. The canonical one-form and the co-framesare related through the important equality σ ∗ θ = e, (2)where σ is some section of the frame bundle .3. an Ehresmann connection ω (a Lorentz or spin con-nection in this context) defined on OM allowing usto define the torsion through the usual expressionon the frame bundle:Θ( ω ) = dθ + ω ∧ θ. (3)The two objects OM and θ are defined as soon as aspacetime is present. If in addition an Ehresmann con-nection is defined on the frame bundle, these structuresallow us to define the torsion through Eq. (3). Alongsome section σ , this torsion leads to the torsion on thespacetime (base manifold): T = σ ∗ Θ = de + e ω ∧ e , where e ω = σ ∗ ω .A first observation is that the canonical one-form, theframe bundle (on which it is defined) and a (Lorentz)connection are needed to obtain the torsion. By contrast,the curvature, which is defined through the expression:Ω( ω ) = dω + ω ∧ ω , can be defined as soon as an Ehres-mann connection ω is defined on a principal G -bundle, asis the case for particle physics gauge theories. A secondobservation is that in the TEGR framework the curva-ture of the connection is by definition zero, which singlesout the Weitzenb¨ock connection: ω W .From these considerations, we notice that the identi-fication of the basic objects of TEGR with those of agauge theory are far from obvious: although a principalbundle (the frame bundle) is present, the connection nat-urally associated with TEGR has a null curvature, andthe canonical one-form (entering in the definition of tor-sion) possesses no equivalent in the usual gauge theoreticframework.A way out this issue is to consider on physical groundsthat, since in TEGR gravity manifests itself throughtorsion, this torsion should be considered as the fieldstrength of a yet undetermined gauge theory. A firststep in finding a gauge formulation of TEGR can thenbe made by setting a connection, defined on the orthonor-mal frame bundle (or a principal bundle containing it),whose torsion is the curvature. B. Specific features of gauge TEGR connection
If torsion, as defined in Eq. (3), needs to be the curva-ture of some connection ω C , the sought for connection re- Note that, since θ is an element of structure this relation es-sentially states an equivalence between local sections and localframes. A change of gauge, which in the bundle formalism ofgauge theories is a change of section, here precisely correspondsto a change of frame. quires to be built out of both the Ehresmann connection ω appearing in the torsion and the canonical one-form θ .These two objects ω and θ are different in nature. Let usbriefly recall some of their properties, to finally motivatethe appearance of a Cartan connection.The one-form ω defines an Ehresmann connection inthe principal SO(1 , OM . In a general sense, anEhresmann connection is a mean to split in a unique wayeach tangent spaces of a fiber bundle into vertical andhorizontal subspaces. As vertical subspaces are alwaysdefined as tangent spaces to the fibers, the Ehresmannconnection uniquely specifies horizontal subspaces and,in this sense, defines uniquely the notion of horizontal-ity in the fiber bundle. To be meaningful in a princi-pal G -bundle, the splitting between vertical and horizon-tal subspaces must be invariant under the action of thesymmetry group G . This invariant splitting is realized inpractice through a connection one-form ω E such that:1. it takes its values in the Lie algebra g of the Liegroup G ,2. it satisfies: R ∗ g ω E = Ad g − ω E , R g being the rightaction of G on the bundle,3. it reduces to the Maurer-Cartan form ω G of thegroup G along the fibers: ω E ( V ) = ω G ( V ), for anyvertical vector V .Following these properties, at each point of the G -bundle,vertical vectors are mapped to the Lie algebra g and hori-zontal vectors belong to the kernel of ω E . In addition, anEhresmann connection allows one to construct paralleltransport and its related covariant derivative. The one-form θ Eq. (1), contrary to a connection one-form, is canonically and only defined in the frame bundle(hence on its restriction OM ). The content of Eq. (1)can be described as follows: for each frame bundle point,mapped to coordinates ( e, x ), the action of θ , througheach of its components θ a , consists in mapping a bundle’stangent space vector V to its corresponding component V a in the frame e at the base manifold tangent spacepoint x . In comparison with the Ehresmann connection, θ : 1. takes its values in R ,2. satisfies: R ∗ g θ = g − θ , R g being the right action ofthe matrix group GL(4 , R ) on F M (or SO(1 ,
3) on OM ), Note that since the fiber is isomorphic to the group in a PrincipalG-bundle, the tangent spaces to fibers are all isomorphic to theLie algebra g . The relation between these notions appears more evidently inthe frame bundle, where a parallel-transported frame (so-calledautoparallel frame field) along a base manifold curve ˜ γ induces ahorizontal curve γ , i.e. whose tangent vector is always horizontal,in the full bundle [see for instance 17, p. 544 19.5.1]
3. is horizontal in the sense that: θ ( V ) = 0, for anyvertical vector V .As is physically motivated, a connection promotes aglobal symmetry to a local symmetry. At the mathe-matical level, this local symmetry is encoded in the con-nection’s values taken in Lie algebra of the symmetrygroup. Thus, the orthonormal frame bundle OM ’s Ehres-mann connection ω takes values in the Lie algebra of theLorentz group. Similarly, the TEGR connection ω C weare seeking, with torsion as field strength, should imple-ment local symmetry as well as include the additional θ one-form. Since θ takes its values in R , which can beconsidered as the algebra for the translation group, thissuggests to build an ω C with values in the algebra of thePoincar´e group SO(1 , ⋊ R .The considerations above would appear to point to-wards a gauge theory with an affine connection definedon the principal Poincar´e bundle, an affine bundle de-noted hereafter P M . However, starting from the or-thonormal frame bundle, it suffices to enlarge its Lorentzconnection’s algebra, so we will consider a more mini-malistic generalization: a Cartan connection, valued inthe Poincar´e algebra while being defined on the originalorthonormal frame bundle OM . III. CARTAN TEGRA. The Cartan connection
A Cartan connection enters in the definition of a Car-tan geometry which can either be considered to generaliseRiemann geometry or Klein geometry, each themselvesoffering a generalization of Euclid geometry. An intro-duction to Cartan geometries in the context of gravitymay be found in [21, 22], while a comprehensive mathe-matical account is given in [23]. Since Riemann geometryis well known in GR, let us briefly discuss the notion ofKlein geometry, which is as important as Riemann’s, forthe Cartan generalization.
1. Klein geometry
Instead of introducing space curvature to get awayfrom Euclid geometry, as in Riemann geometry, Klein ge-ometry generalizes Euclid spaces to homogeneous spaces(also termed maximally symmetric, see App. A for areminder). The latter are always isomorphic (and conse-quently identified) to some coset space
G/H (also termed We emphasize that θ is defined independently from the presenceof a connection, which is not responsible here for the horizontal-ity. These generalizations applies, when a metric is present, to pos-itive definite metric (e.g. Riemannian geometry) or to non-positive definite metric (pseudo-Riemannian geometry). quotient space), where G is a Lie group and H one ofits (closed) subgroup. This basically defines a Klein ge-ometry ( G, H ), which is the homogeneous space
G/H .When
G/H is connected, a connected Klein geometry isobtained – disconnected homogeneous spaces are irrele-vant for Cartan geometry, so we’ll focus on connectedones. In this context the familiar three dimensional Eu-clidean space appears as a particular case of Klein geom-etry where G = SO(3) ⋊ R and H = SO(3) the rotationgroup.As they derive from the action of Lie groups on(differentiable) manifolds (see App. A), homogeneousspaces are (differentiable) manifolds . In four dimen-sions, Lorentzian homogeneous spaces are the familiarspaces of:Minkowski: G = SO(1 , ⋊ R , H = SO(1 , G = SO(1 , H = SO(1 , G = SO(2 , H = SO(1 , G can itself be seen as principal fiberbundle of fiber H , with the homogeneous space G/H asbase manifold, i.e. G(G/H,H) [see for instance 20, p.55]. The adjoint representation of H (the action of H onits Lie algebra) allows one to associate a vector bundle,with fiber g / h , to G(G/H,H). This associated bundle canbe shown to be isomorphic to the tangent bundle of thehomogeneous space G/H , T ( G/H ) [see 23, prop. 5.1 p.163, for details]. This important property means, in par-ticular, that each tangent space to a homogeneous spaceis isomorphic to the quotient g / h , namely that at eachpoint x of G/H , T x ( G/H ) ≃ g / h . (4)In addition to the above relation, the tangent spacesinherit, from the aforementioned bundle isomorphism,all the structures defined on g / h . In particular a non-degenerate H -invariant metric defined on g / h pulls-backto an H -invariant metric on the corresponding homoge-neous space. A paradigmatic example in GR is given bythe metric of the four dimensional homogeneous (maxi-mally symmetric) spacetimes of Minkowski, de Sitter andAnti-de Sitter [21], inherited from the symmetries of thecorresponding coset spaces. Here, the hypothesis that H is a (topologically) closed subgroupof the Lie group G is central for the space G/H to be a manifold[see for instance 23, p. 146].
2. Cartan geometry
We introduced Sec. III A recalling that Cartan geome-try represents at the same time a generalization of Kleinand Riemann geometries. Here we will give more de-tails on how it actually does merge both. In Sec. III A 1,we discussed how Klein geometry generalizes Euclid byconsidering the Euclidean space as a particular case ofhomogeneous spaces, and how these spaces are definedby the quotient of a Lie group G by one of its (closed)subgroups H , sharing the symmetries of the group G .Riemann geometry is historically an N -dimensional gen-eralisation of the two-dimensional Gaussian differentialgeometry of surfaces departing from the Euclid (plane)geometry by introducing curvature. In contemporary ter-minology Riemann geometries are metric orientable dif-ferentiable manifolds, equipped with the Levi-Civita con-nection, their curvature measures how they locally departfrom the flatness of their Euclidean tangent spaces, whilethe connection relates tangent spaces at different points.In Cartan geometry the Euclidean tangent space isgeneralized to a tangent homogeneous space. To simplifyvisualisation of such tangent spaces, the two-dimensionalcase can be examined: the Riemann view considers a tan-gent plane at each point of the surface, its contact pointwith the surface constituting its origin, when identifiedwith the (invariant) vector space R (under the rotationgroup SO(2)). By contrast, the Cartan view rolls, with-out slipping, a unique space on the surface, a space whichcan either be a plane, sphere or hyperbolo¨ıd, one of thethree homogeneous spaces in two dimensions. The con-tact point thus moves in the homogeneous space, puttingin evidence its affine nature. The case of a plane there-fore identifies with the affine plane R , invariant underthe affine group SO(2) ⋊ R .These two different descriptions of tangent spaces arereflected respectively in the Cartan and Levi-Civita (gen-eralised to affine Ehresmann) connections. They bothprovide methods of transfering objects between pointsjoined by some (continuous) path γ . The Cartan connec-tion effects the transfer of contact point, rolling withoutslipping, in the homogeneous space, while the Levi-Civitaconnection provides a rule to transfer objects along thepath γ from one plane to another.The defining properties of a Cartan geometry are gath-ered in the following definitions: A Cartan geometry , modeled over a Klein geometry(G, H), is a principal H -bundle P ( M, H ) equippedwith a Cartan connection.
A Cartan connection is defined through a one-form ω C such that:1. it takes values in the algebra g ⊃ h of G ⊃ H .2. it is, at each point p of the H -bundle, a linearisomorphism between the tangent space T p P at p and the Lie algebra g . This property re-quires that G and the tangent space T p P sharethe same dimension.3. it satisfies: R ∗ h ω C = Ad h − ω C , R h being theright action of H on the bundle,4. it reduces to the Maurer-Cartan form ω H ofthe group H along the fibers.The Cartan geometry definition sets its own mathe-matical framework. In the TEGR situation we are in-terested in, the H -bundle will be the orthonormal framebundle OM with H = SO(1 ,
3) and M , the spacetime. Inaccordance with the conclusion of Sec. II B, we will take,for the group G , the Poincar´e group and, consequently,for the homogeneous space (the Klein geometry), we willuse the Minkowski space.The case we are interested in then presents an addi-tional property: its Cartan geometry is reductive. For-mally [see 23, p. 197, for details], this means that theLie algebra decomposition as vector space g = h ⊕ g / h isAd( H )-invariant. In practice, among other results, it im-plies that any g -valued form defined on P splits along thisdecomposition and, in particular, the connection one-form ω C splits into h and g / h parts. For the Poincar´ealgebra, these correspond respectively to the Lorentz andtranslation parts, h = so (1 ,
3) and g / h = R . Conse-quently, the reductive Cartan connection one-form splitsinto ω C = ω + θ, (5)where ω is an h -valued Ehresmann type connection one-form, and θ a g / h -valued one-form, both defined on theprincipal H -bundle. This decomposition also follows forthe curvature of ω C ,Ω( ω C ) = Ω( ω ) + Θ( ω ) , (6)where the curvature of the ω part reads Ω( ω ) = dω + ω ∧ ω and both parts combine in the torsion Θ( ω ) = dθ + ω ∧ θ .The definition of the Cartan connection, through ω C , differs from that of the Ehresmann connection, inSec. II B, in the first two properties which we will nowdiscuss.From property 1, instead of being valued in the Lie al-gebra of the fiber h , ω C is valued in the whole g algebra.This agrees with the “minimal extension” mentioned atthe end of Sec. II B since, of the full algebra g , only ap-pears the translations part g / h .Property 2 relies both on the Klein geometry and thesoldering property. It imposes the tangent space, T p P of P ( M, H, π ) at p , and g to be isomorphic. In addition,its vertical part is always defined as the tangent space tothe fiber and is also isomorphic to h ; Eq. (4) implies thatthe orthogonal complement g / h of h in the decomposition g = h ⊕ g / h identifies with the tangent space T π ( p ) ( G/H )to the homogeneous space at π ( p ); therefore, it can beidentified with the tangent space T π ( p ) ( M ) to the basemanifold at π ( p ). In the same way as T π ( p ) ( M ) is a fiberof the tangent bundle T M , T π ( p ) ( G/H ) is the fiber of a “tangent homogeneous-space” bundle of M associatedto P : P × H G/H . Since the connection induces g / h ≃ T π ( p ) ( G/H ) ≃ T π ( p ) ( M ), that identifies a tangent spaceof M to a tangent space of a fiber of P × H G/H , it thuseffects a soldering [see for instance 14, App. 3] of the H -principal bundle to its base M , illustrating this “built-inproperty” of the Cartan connection.Finally, properties 3 and 4 correspond identically toproperties 2 and 3 for Ehresmann connection one-forms.Returning to the frame bundle we consider for TEGR(with its Cartan connection), and as anticipated inEq. (5) notation, its canonical one-form θ does indeedcorrespond to the g / h term in the Cartan connection.The interpretation of the Cartan geometry as a move-ment of the tangent space rolling without slipping on thebase, added with Eq. (2), allows us to interpret that termof the Cartan connection in link with the so-called mov-ing (co)frame.
3. The TEGR connection
From the discussion above and the requirement that forTEGR the curvature of the connection yields the torsion,Eq. (5) specializes to ω C = ω W + θ, (7)where the Ehressman term, ω W , stands for the curvature-less Weitzenb¨ock connection and where θ coincides withthe canonical one-form on OM . Then, through Eq. (6)or by direct calculation [see 16], the required relationΩ( ω C ) = Θ( ω W ) is obtained.As we previously pointed out while analyzing theirdifferences, the Cartan connection, that we suggest de-scribes TEGR, contrasts with the Ehresmann connection,that represents particle physics gauge theories. At thisstage it is premature to claim whether this frameworkallows TEGR to be seen as a gauge theory, however, inthe Cartan geometry, ω C does give the expected fieldstrength, in addition to relating gravitation to the trans-lation symmetry, in accordance with Noether’s theorem:while each term of ω C in Eq. (7) corresponds to Lorentzand translation symmetries respectively – through val-ues in the Poincar´e Lie algebra – and as the Weitzenb¨ockcurvature vanishes, the only contributing curvature (fieldstrength) term comes from the translation ( g / h = R )valued θ term. In contrast with the difficulties to extracta translation term from the usual formulation of GR (seeApp. B), the natural appearance of translation with θ inCartan TEGR counts as its achievement. Furthermore,to preserve the physical relation between field strengthand gauge field in the Cartan TEGR framework, Eq. (2)points toward identifying the co-frame as the gauge fieldof the theory. To settle the identification of a gauge fieldin the theory requires to examine the coupling to matter,since particle physics coupling is mediated by the gaugefield. B. Matter coupling with Cartan connection
In particle physics gauge theories, charged matterfields interact through the exchange of gauge bosons, me-diating the interaction. At the classical level, this inter-action is described thanks to a covariant derivative en-suring the minimal coupling. That covariant derivativeis directly related to an Ehresmann connection whosepullback on the base manifold (the Minkowski space) isthe gauge field. In fact, covariant derivative and paralleltransport are directly built from an Ehresmann connec-tion in a principal bundle. In classical GR the couplingto classical matter field is also realized through a covari-ant derivative: in Cartan (tetrads) formalism, the co-variant derivative is implemented from the Levi-Civitaconnection one-form ω LC in the frame bundle. So far,GR – and thus the Levi-Civita connection – reproducesdramatically well observational data and hence compelsany formulation of gravity to reproduce the Levi-Civitacoupling.
1. From Cartan connection to Levi-Civita coupling
In this section we will recall from [24] how to obtain theLevi-Civita part of coupling to matter from the Cartanconnection one-form ω C , proposed to describe TEGR.Starting from the Cartan geometry, the Levi-Civitacoupling is obtained in three main steps, commentedhereafter1. The Cartan connection one-form ω C on OM ismapped to an Ehresmann connection one-form ω E thanks to the Sharpe theorem [Prop. 3.1 p. 365 of23] (also reproduced in [24]).2. The Weitznb¨ock term, present in ω E : the first termof the r.h.s. of Eq. (7), is mapped to the Levi-Civita one-form thanks to the contorsion one-formdefined on OM [25, theorem 6.2.5 p. 79].3. The resulting one-form, an affine connection on thePoincar´e principal bundle P M , is then pulled backon OM with a map [20, proposition 3.1 p. 127] thatsplits it into the canonical and the Levi-Civita one-forms, parallel transport and covariant derivativeproceeding then from the Levi-Civita Ehresmannconnection, now in a Cartan (tetrads) formalismsetting.More explicitly: as the Cartan connection does notdefine canonically parallel transport, by contrast withEhresmann connection, the first step relates, in our pe-culiar case, the set of ω C to the set of affine Ehresmann Unfortunately, the interpretation of the Levi-Civita connectionas a gauge field mediating gravitation is not satisfying (seeApp. B). connections on the affine (Poincar´e) bundle, as obtainedin [24] with the help of Sharpe theorem . As the theorem,in our case, results in the identity of the connection one-form on the base manifold (the spacetime), the pullbackof the Weitzenb¨ock term ω W in Eq. (7) is still present.To obtain the correct matter coupling requires to mapthis term to the Levi-Civita one-form ω LC .This is done, in the second step, applying the theorem that relates any Ehresmann one-form from the orthonor-mal frame bundle OM to the Levi-Civita one-form viaits contorsion one-form.Two remarks are in order here. First, although any twoEhresmann connections are always related by some spe-cific one-form, the existence of the contorsion, and hencegetting the Levi-Civita one-form, relies on the existenceof a metric. That metric is present in OM , where thepossibility to map Weitzenb¨ock to Levi-Civita one-formscan be traced back to the choice of the Minkowski met-ric for tangent spaces to the base (the spacetime), andin this sense related to the Equivalence Principle. Thesecond remark concerns the building of the Levi-Civitaone-form in this context: as contorsion is composed fromtorsion, itself built out of the Weitzenb¨ock connection,and of the canonical one-form, the resulting Levi-Civitaone-form is a function of the Weitzenb¨ock one-form, thecanonical one-form, and its derivatives.This Weitzenb¨ock-to-Levi-Civita mapping allows us toredefine the Ehresmann one-form ω E , into e ω E , now thesum of a linear, Levi-Civita one-form, ω LC , and a trans-lation term, the canonical one form θ , also defined onthe affine bundle P M . The third step maps back e ω E into OM [20, proposition 3.1 p. 127], where it splits into ω LC ,and θ .Compositing these steps, the final result yields a mapon OM from Cartan to Levi-Civita connection: ω C ω E = ω W + θ e ω E = ω LC + θ ( ω LC , θ ) . (8)This map provides matter coupling to TEGR-gravity inagreement with the familiar coupling of GR, as well as theFock-Ivanenko derivative in Cartan (tetrads) formalism.
2. A remark about the gravitational field in Cartan TEGR
In classical GR, the gravitational interaction is carriedby the metric g or, in Cartan (tetrads) formalism, by theframe field e . The metric, or the frame field, also en-tirely determines the Levi-Civita connection. Althoughthe Cartan TEGR framework reaches these results froman entirely different perspective, e remains the dynamical A technical condition restricts the set of possible Ehresmannconnections. However, since it bears no influence on our results,we just point this out to the interested reader. The proof in the bundle formalism is given in [25, theorem 6.2.5p. 79]. field and coupling remains mediated through the Levi-Civita connection. The gravitational field thus remainscarried by the frame field in Cartan TEGR.
C. Is Cartan TEGR a gauge theory?
As discussed Sec. II A, TEGR requires some crucialstructural elements that are absent from particle physicsgauge theories. Therefore, adopting the usual particlephysics structure as gauge theory definition would ex-clude TEGR. In the wider context of gauge-gravity theo-ries [see 26, for a comprehensive, historical account] somerather deep change of paradigm were considered [27].Here, following [24] on physical grounds, we only min-imally propose to extend the usual gauge theory frame-work to include the required extra structure for TEGR.Although this extension hints at a possible interpretationof Cartan TEGR as a gauge theory, which we discuss be-low, we do not claim adhesion to it, leaving interpretationopen.
Frames as gauge fields
The Cartan connection ω C , as decomposed in Eq. (7),yields a curvature equal to the torsion. Moreover, sincethe curvature of the Weitzenb¨ock connection vanishes,the only contributing dynamical field to the field strengthremains the field of frames. This, as pointed out inSec. III A 3, hints at the field of frames behaving as agauge field. We also pointed out Sec. III B 2 that theframe can be considered as mediating the gravitationalfield. This is already the case in GR since the Levi-Civitaconnection, determined from the metric, thus the frames,in a Cartan (tetrads) formulation, mediates coupling totensorial and spinorial fields. Additionally, since classi-cal scalar fields also couple to gravity, while their covari-ant derivative reduces to a partial derivative, couplingshould be present in the partial derivative. As the tan-gent vector ∂ µ decomposes in the frame basis e a following ∂ µ = ( ∂ µ ) a e a , such coupling reinforces the interpretationof the frame field as mediating gravitational interaction.All the above arguments advocate for the interpreta-tion of e as the gauge field on the basis that it mediatesinteraction. At the same time, interpreting e as a gaugefield departs from the paradigmatic framework of gaugetheories, that would relate e to an Ehresmann connec-tion by a simple pullback on the base manifold: it re-lates instead to the Cartan connection ω C through thecomposite map combining Eqs. (8) and (2). To inter-pret TEGR as a gauge theory thus requires to abandonthe strict correspondence between gauge field and Ehres-mann connection. Note that the usual particle physicsgauge theories remain unaffected by this change, since itonly involves external ingredients.In addition to mediating interaction, particle physicsgauge fields also implement some local (i.e. spacetime point dependent) symmetry invariance in matter fieldequations. As a frame e , from Eq. (2), is clearly re-lated to the canonical one-form θ , valued in the trans-lation part of the Poincar´e algebra, it thus relates to lo-cal and infinitesimal translations. The discussion abovereveals that matter coupling involves two occurrences ofthe frame e : in the expansion of the partial derivativeand in the Levi-Civita connection, seen as the differencebetween the non-dynamical Weitzenb¨ock connection andthe frame ”induced” contorsion. However, the field e does not emerge from localisation of a global transla-tion symmetry. Instead, the symmetry is already lo-cal and infinitesimal since e only relates to the trans-lation algebra. The usual process involved in ”gauginga theory” by which a globally invariant matter field La-grangian becomes locally invariant with the introductionof the gauge field in the derivatives is not present in thiscontext. Moreover, the basic symmetry of GR, i.e. dif-feomorphism invariance (the invariance under spacetimecoordinate changes), does not proceed from translationsymmetry but is already a built-in property of differentialgeometry.From the above discussion, the frame e , on the basisthat it mediates the interaction with matter fields, canbe perceived to share physical properties of a gauge field.Nevertheless, such an interpretation requires to enlargethe notion of gauge field compared with its meaning inusual particle field theories. In particular, the frame field e is the pullback of the canonical one-form, rather thanthat of an Ehresmann connection, and is the translationpart of a Cartan connection whose curvature, still con-sidered as the field strength, is the torsion. IV. A NOTE ON CARTAN GR
As TEGR is equivalent to GR, the Cartan TEGRstructure to represent TEGR can be transposed to usea Cartan connection to describe GR. This transpositioncan be obtained by1. setting the Ehresmann term in the Cartan connec-tion one-form (5) to the Levi-Civita connection: ω C = ω LC + θ ,2. recognising the mapping using contortion reducesto identity, as torsion – and thus contorsion – van-ishes for the Levi-Civita connection, such that themap (8) becomes ω C ω E = ω LC + θ ( ω LC , θ ) . The Cartan curvature of the corresponding ω C then re-duces to the Riemann curvature term of the Levi-Civitaconnection, as expected in GR. Note that the paralleltransport for such reductive Cartan connection could bedirectly seen as induced, in the initial bundle, by theEhresmann part of its one-form ω C . The only differenceof this framework with the usual Cartan (tetrad) formal-ism of GR lies in the Cartan connection and the possibil-ity it awards GR, in a similar way as for TEGR above,to be conceivably interpreted as a gauge theory. Simi-lar modifications to particle physics type gauge theoryare required, in the lines described above. The possi-ble interpretation of the frame field as a gauge field fortranslations could follow similar arguments to the onespresented above, leading to a recognition of the resultingCartan GR as a gauge theory to be similarly left open.We note that the use of Cartan connection to describeGR has already been studied, along other lines, in [22,28]. V. CONCLUSION
In this paper we have reviewed our proposal formulat-ing TEGR from a Cartan type connection. The theorywe obtain, and refer to as Cartan TEGR, produces iden-tical predictions to TEGR and thus to GR. It provides aconsistent mathematical framework in which the torsionis obtained as the curvature of a reductive Cartan con-nection. This connection is the sum of the curvaturelessWeitzenb¨ock connection and the so-called canonical one-form whose pullback along some section on spacetimeis a frame. The usual (GR) coupling to matter fieldsis obtained thanks to Sharpe’s theorem relating Cartanand Ehresmann connections, and to the introduction ofthe contorsion one-form in relation with the equivalenceprinciple.Cartan TEGR is then examined from a gauge theoreticperspective. This is done without claiming it is a gaugetheory – this interpretation is left open – and we dis-cuss the mandatory extensions and the departures fromthe particle physics gauge theory paradigm. This per-spective allows us to specify to which extent the (local)field of frame could be recognized as a translation gaugefield mediating gravity, and how it could be associatedto torsion, as a field strength.
ACKNOWLEDGEMENTS
The authors wish to thanks D. Bennequin andT. Lawrence for helpful discussions, respectively, on ge-ometry and on the status of diffeomorphisms in groupsand gravity. The work of M. Le D. has been supportedby Lanzhou University starting fund, and the Fundamen-tal Research Funds for the Central Universities (GrantNo.lzujbky-2019-25).
Appendix A: Right (left) G-space, homogeneousspace and coset space G/H
We follow Fecko [17] (Sec. 13.1, 13.2 for details). Amanifold M on which a right (left) action of a Lie group G is available is a right (left) G-space. The orbit of somepoint x in M is the set of points which can be reachedfrom x by the group action. Note that orbits are disjoints.In the special case when M only contains a single orbit,that is when the action of G is transitive on M , the G-space is called a homogeneous space. The left (right) coset space G/H, where H is a topolog-ically closed subgroup of G, is the set of equivalent classes[ g ] for the relation noted ∼ . Since the left product in G gg ′ is transitive, the left action in G/H defined through L g [ g ′ ] := [ gg ′ ] is then transitive, and consequently G/His a homogeneous space.In this context, the group G does act, through L g , onG/H, and therefore this coset space can be consideredas a left G-space. In fact, one can show that all homo-geneous G-spaces are obtained in this way, which meansthat any homogeneous space manifold M with a groupaction from G is isomorphic to a G/H for some closedsubset H.Moreover, one can show that H can be taken as the sta-bilizer of some point x of M . Since M is homogeneous,any two points being in the same orbit share isomorphicstabilizers and the choice of the point x is unimportant.Finally, we note that G is a principal H-bundleover the (left) coset space G/H if G is a Lie groupand H, a (topologically) closed (not necessarily in-variant) subgroup [see for instance 20, Exemple 5.1p.66]. Here the action of H on G is just the rightmultiplication. The fibers are the left cosets of Hand since the fiber containing the identity is natu-rally isomorphic to H, all fibers are isomorphic to H. Appendix B: A note on gravity as a gauge theory oftranslations: a not so obvious relation
Since Noether theorem associates translations to thestress-energy tensor, it suggests that gravity, as describedby the Einstein equations of General Relativity (GR), isrelated to translations. Whereas translation is a globalsymmetry of physical equations in Minkowski space, itcan become local when gravity is present. This, in turn,open the possibility that gravity arises from a gauge the-ory of translations, where, following the usual gauge the-ory construction, a gauge field is minimally coupled to the(Minkowskian) matter fields in order to obtain a matterequation locally invariant under translations. G transitive on M means there exists for all pair of point (x, y)in M g ∈ G such that R g x = y ( L g x = y ). The map ∼ is defined by: g ∼ g iff there exists an h ∈ H suchthat g = g h ( g = hg ). The stabilizer of some point x of M with a group action from G isthe subset G x of G (which is indeed a subgroup) whose elementsleave x invariant. A point of terminology: by translation we mean an element of thetranslation group ( R , +), often abbreviated in R . The transla- Unfortunately, the situation is more complicated. Afirst difficulty concerns the structure of a gauge theory:contrary to the action of particle physics gauge groups,involving matter and gauge fields, translations also acton space-time (often termed “external” symmetry). Thisrelates to the soldering property [see for instance 14,App. 3]. Since gauge theories of particle physics’ stan-dard model are built on internal groups (U(1), SU(2), etc. ), they are not concerned with this property. On thecontrary, a translation gauge theory should take solderinginto account. This suggests to enlarge the mathemat-ical framework of gauge-theories to include this “new”ingredient. A second difficulty stems from the symme- tries under consideration: in gauge field theory the localinvariance is implemented thanks to the connection (val-ued in the Lie algebra of the symmetry group) appearingin the covariant derivative. In general relativity, in theCartan (tetrads) formalism, matter coupling involves theso-called spin or Lorentz connection. As it takes its valuein the Lorentz algebra so (1 , ,
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