Teleparallel gravity (TEGR) as a gauge theory: Translation or Cartan connection?

M. Fontanini1, E. Huguet1, and M. Le Delliou2

1 - Universite Paris Diderot-Paris 7, APC-Astroparticule et Cosmologie (UMR-CNRS 7164),Batiment Condorcet, 10 rue Alice Domon et Leonie Duquet, F-75205 Paris Cedex 13, France.∗ and

2 - Institute of Theoretical Physics, Physics Department, Lanzhou University,No.222, South Tianshui Road, Lanzhou, Gansu 730000, P R China †

(Dated: November 12, 2018)

In this paper we question the status of TEGR, the Teleparallel Equivalent of General Relativity,as a gauge theory of translations. We observe that TEGR (in its usual translation-gauge view) doesnot seem to realize the generally admitted requirements for a gauge theory for some symmetry groupG: namely it does not present a mathematical structure underlying the theory which relates to aprincipal G-bundle and the choice of a connection on it (the gauge field). We point out that, while itis usually presented as absent, the gauging of the Lorentz symmetry is actually present in the theory,and that the choice of an Erhesmann connection to describe the gauge field makes the translationsdifficult to implement (mainly because there is in general no principal translation-bundle). Wefinally propose to use the Cartan Geometry and the Cartan connection as an alternative approach,naturally arising from the solution of the issues just mentioned, to obtain a more mathematicallysound framework for TEGR.

PACS numbers: 04.50.-h, 11.15.-q, 02.40.-k

CONTENTS

I. Introduction 1

II. Some preliminary notions 2

III. Some questions about the usual translationgauge formulation 4

IV. A conventional gauging of the Translationgroup 4

V. Some comments about the connections 6

VI. Approaching TEGR with the Cartanconnection 7

VII. Conclusion 8

Acknowledgements 9

A. Definitions of, and comments on, somemathematical structures 91. Tetrads 92. Comment on Ehresmann connection 93. Solder form 94. Associated (vector) bundle. 105. On the affine connection 11

References 12

∗ michele.fontanini@gmail.comhuguet@apc.univ-paris7.fr† (delliou@lzu.edu.cn,)morgan.ledelliou.ift@gmail.com

I. INTRODUCTION

In the present paper we are interested in the formula-tion of the Teleparallel Equivalent to General Relativity(TEGR) as a gauge theory. Let us recall that TEGR isa theory in which all the effects of gravity are encodedin the torsion tensor, the curvature being equal to zero:a feat achieved by choosing the Weitzenbock connectioninstead of the Levi-Civita connection of General Relativ-ity (GR) [1]. The dynamical equations for TEGR can beobtained from its action as usual (without reference toa gauge theory), thus displaying a classical equivalencewith GR thanks to the fact that the Einstein-Hilbert andTEGR actions only differ by a boundary term [see for in-stance 2]. A very important point is that TEGR is oftenpresented as the gauge theory of the translation group[3], the main motivation for the gauge approach being ,as for many other works [see 4, for a detailed account],to describe gravity consistently with the three other fun-damental forces of Nature which are mediated by gaugefields related to fundamental symmetries, namely (at ourenergy scale), U(1), SU(2) and SU(3) for the electromag-netic, weak an strong interactions respectively. By con-trast with gauge theories of particle physics, in which asymmetry group acts in a purely internal way, the trans-lation group, subgroup of Poincare group and part of thesymmetries underlying gravity, acts directly on spacetimeand thus corresponds to an external symmetry. This as-pect is reflected in the presence of the so-called solderingproperty1, which requires adapting the structure of thetranslations gauge theory to account for it. Indeed, such

1 A notion first formulated mathematically by C. Ehresmann inthe theory of connections [5], a first comprehensive exposition ofwhich can be found in Kobayashi [6].

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adjustment is far from trivial, it requires some adapta-tion of the underlying mathematics and is also present inthe larger perspective of gauge theories of gravitation. Inthe latter theories, different proposals for gauging Grav-ity using different symmetry groups and connections havebeen built without reaching a complete consensus on thestatus of these proposals [see for instance 4, 7–12].

The purpose of the present work is twofold: first, wewill point out some difficulties in interpreting TEGR as agauge theory of translations alone from a mathematicalpoint of view, and connect these difficulties to the choiceof the gauge field as an Ehresmann type connection; sec-ond, we will propose the introduction of another typeof connection, known as Cartan connection, to obtain aconsistent framework.

As physicists, we realize that the mathematical no-tions involved in the treatment of the topics above couldbe outside the common background in differential geom-etry. We thus made our goal to keep a pedagogical viewthroughout this work, especially when sharp distinctionsbetween related notions are required (in particular, thedistinction between the soldering and the canonical one-forms).

The paper is organized as follows. We begin in Sec.II with a review of the useful mathematical structures.This section can be skipped at first reading by geomet-rically informed readers. In Sec. III we motivate ourquestioning about the usual formulation of TEGR as agauge theory of the translation group. Then, in Sec.IV, we make a comparison between the usual translationgauge theory, as exposed in [3], and a “naive” attempt togauge the translation group following the standard math-ematical point of view of connections in a principal fiberbundle, and conclude that the interpretation of the the-ory described in [3] as a gauge theory of the translationsis difficult to defend. The role of connections is examinedin Sec. V in order to motivate the use of the Cartan con-nection. The latter is introduced in Sec. VI, in particularthrough its differences with the Ehresmann connection.As a conclusion in Sec. VII we propose to use the Cartanconnection for TEGR and discuss its status as a gaugetheory compared to other works. Various technicalitiesare described in Appendix A, and for definitions not ex-plicitly stated we refer to [13–15].

II. SOME PRELIMINARY NOTIONS

Throughout the paper we denote by P (F,M, π) a fiberbundle with total space P , typical fiber F , four dimen-sional differentiable base manifold M and projection π.Most of the time we will consider a principal G-bundle,that is a bundle whose fibers are identical to the structuregroup G of the bundle, which in turn is a Lie group [14].In fact, in order to be principal, a fiber bundle has to bedefined along with an action of the group over the totalspace [15, p. 50]. It can then be shown that equivalentlyone can build a principal bundle from a Lie group G (the

fiber) and its transitions functions [15, prop. 5.2].The geometrical framework of usual gauge theories of

particle physics or of Einstein-Cartan Theory (in termsof tetrads, App. A 1), of which General Relativity is aspecial case, is a principal bundle P (G,M, π) (see Fig.1) with a connection one-form ωE taking values in the

G

xTxM

g

M

P (G,M, π)

FIG. 1. Generic Fiber Bundle structure

Lie algebra g of the group G. The g-valued one-formωE, the realization of a so-called Ehresmann connection,allows us to define the notion of parallel transport and ofthe curvature two-form (the latter therefore is a propertyof the connection), reading:

Ω := dωE + ωE ∧ ωE. (1)

In gauge theories of particle physics, the group G is agauge group (U(1), SU(2), . . . ), and the connection andits curvature are respectively the gauge potential and thefield strength of the theory. These are defined on the totalspace P of the fiber bundle2, their corresponding quanti-ties A,F on the base manifold M are obtained through(the pullback of) a local section σ, which corresponds toa choice of gauge. Explicitly : A = σ∗ωE, F = σ∗Ω.

In the Einstein-Cartan theory the bundle consideredcorresponds to the orthonormal frame bundle, i.e. thebundle of orthonormal frames3: each fiber above somepoint x ∈M , of the base manifold is constituted by all or-thonormal basis of the tangent space TxM . These fibersare therefore isomorphic to the Lorentz group4, the iso-morphism being realized by choosing a specific standardframe e in a neighborhood of each point x and by identi-fying the transformed frame e′ with the unique element ofthe group g realizing the transformation from e to e′. Thepresence of such isomorphism is a necessary condition in

2 they are often denoted by A ≡ ωE and F ≡ Ω.3 Throughout the paper we assume the theory metric, the frames

are always orthonormalized with respect to this unspecified met-ric, and so are the well known tetrads.

4 Since we are using orthonormal frames, the bundle used is arestricted frame bundle in which the fiber is the Lorentz groupinstead of the general linear group.

3

order that the frame bundle be a principal fiber bun-dle.The connection ωE is in this context called the spinor Lorentz connection and we will denote it hereafter byωL. This Lorentz connection is related to the connectioncoefficients Γρµν through its pullback along some (local)section σ by [see for instance, 13, Sec. 15.6 and 19.2]:

(σ∗ωL)abµ = eaρ∂µeρb + eaρΓρµνe

νb = eaρ∇µe

ρb .

We will denote hereafter ωW the Lorentz connection cor-responding to TEGR, i.e. the so-called Weitzenbock con-nection, satisfying: (σ∗ωW )abµ = Λ(x)cb∂µΛac , where Λ in-dicates a Local Lorentz transformation. Note that Gen-eral Relativity is obtained choosing the usual Levi-Civitaconnection.

Let us recall that, although the Cartan (tetrads) for-malism uses a Lorentz connection in a principal fiber bun-dle to describe GR, it cannot be considered a propergauge theory of gravity. This is because, while localLorentz invariance is described by the Lorentz connec-tion, the so called diffeomorphism invariance of GR (theinvariance under R4 diffeomorphisms, in other words thecoordinate change invariance) is not encoded in thatLorentz connection. Indeed, the description of diffeo-morphism invariance is the main difficulty for gauge the-ories of gravity, which, for the most part, differ by thetreatment of local Lorentz invariance and diffeomorphisminvariance.

It is worth noting that there is an important speci-ficity in the choice of the frame bundle used in Tetradsformulation of General Relativity or Einstein-Cartan the-ory, compared to the principal bundles of gauge theoriesof particle physics, which comes from a structural differ-ence of the theory and leads to the definition of torsion. Apoint p in the frame bundle is basically a point x = π(p)on the base M together with a particular frame e of thetangent space Tπ(p)M : p = (x, e). As a consequence, ateach point p wecan obtain the components of a vector ofTπ(p)M in that specific frame e at p. This map is real-ized by the canonical form θ (also named fundamentalor tautological), a one-form defined on the frame bundlewith values in R4 relating a vector of TpP to the com-ponents of its horizontal part5 in the particular framee = ea : 〈 θa(x, e), v 〉 := (π∗v)a (see Fig. 2).

On the contrary, in the principal G-bundle of gaugetheories of particle physics the frame e is replaced by a“generalized” frame which has nothing to do with theframes of the tangent space Tπ(p)M (defined in the usualway), so then there is no natural (canonical) correspon-dence between the two sets of frames, although we canillustrate their similarities as in Fig.3.

Therefore, when an Ehresmann connection ωE is de-fined on the frame bundle6, the canonical form leads to

5 Here, “horizontal” means that θ(v) = 0 for v ∈ VerpP (thetangent space of the fiber at p), no connection is defined yet.

6 When the considered bundle is the frame bundle the connectionis said to be linear [15] p. 119.

SO0(1, 3)

xTxM

e

M

p = (x, e)

θ

Horp(LM) ⊂ TpLM

LM

FIG. 2. Frame Bundle structure and the canonical form

G

x

g

M

P (G,M, π)

SO0(1, 3)

x

e

M

p = (x, e)

LM

TxM TxM

FIG. 3. Frame Bundle structure (left), and a usual G-bundleof particle physics (right)

the torsion Θ which is defined as its exterior covariantderivative relative to ωE:

Θ := dθ + ωE ∧ θ, (2)

d being the exterior derivative on the frame bundle. Thetorsion T on the base manifold is again obtained by (thepullback of) a local section σ, corresponding to a framechoice. Explicitly : T = σ∗Θ. Note finally that, sincechoosing a section in the frame bundle corresponds tochoosing a frame field, one can show [see for instance 13,Sec.21.7] that

σ∗θa = ea. (3)

The role of Eq. (3) is to show that the canonicalform θ realizes the so-called “soldering” between thebase manifold and the fibers. It is important though todistinguish the canonical form θ from the soldering formθ which is a different mathematical object (see appendixA 3). Basically, the soldering for a principal bundleidentifies each tangent space TxM of the base manifoldat x with a corresponding space Tσ(x)V , tangent toa fiber of an associated vector bundle – that is, inshort, a bundle in which the principal bundle fiber Gis replaced by a representation of G on a vector spaceV (see appendix A 4) – along a global section σ, asdiscussed in appendix A 3. In the case of the framebundle formulation of GR or Einstein-Cartan theory,the tangent bundle TM is itself an associated vectorbundle, and as such, the tangent space to a fiber of the

4

associated vector bundle and the tangent space of thebase manifold of the principal bundle’s base are thesame, and the soldering form is from this point of viewthe identity7as we illustrate in Fig.4. It is worth notingthere that the solder and canonical forms are distinctand that the principal bundle soldering, effected throughthe isomorphism ξ, uses the composition of the two, bywhich we can henceforth understand that, since here thesoldering form is the identity, the canonical form realizesthe soldering.

III. SOME QUESTIONS ABOUT THE USUALTRANSLATION GAUGE FORMULATION

It is often claimed that TEGR can be formulated asa gauge theory for the translation group. Neverthe-less, as noted by Aldrovandi and Pereira [3, p.41], since“The gauge bundle will then present the soldering prop-erty . . .Teleparallelism will be necessarily a non-standardgauge theory”. In the present section we do not enterin the mathematical details of the gauge formulation ofTEGR but, closely following [3], in which this gauge ap-proach is comprehensively described, we point out someof these “non-standard” aspects which, in our opinion,raise questions about the interpretation of TEGR as agauge theory of the translation group. Although we fol-low the presentation of [3], we prefer to use the languageof differential geometry, which from our point of view al-lows us to express our arguments in a more concise way.

Similarly to other gauge theories, in TEGR the ef-fects of the local symmetry (gauge) group, the transla-tion group, are implemented through the gauge potentialdenoted B from here on. The field strength of B, here-

after denoted by•T , is given in Eq. (4.52) of [3]. Using

the definition of exterior derivative and wedge product itreads

•T = dB + ωL ∧B, (4)

ωL being the Lorentz connection for the theory. Thatexpression is recognized as being equal to the torsion ofthe connection ωL.

In Ref.[3] we find another expression for•T that con-

nects it to tetrads (Eq. 4.62)

•T = dh+ ωL ∧ h, (5)

where h is a, possibly nonholonomic, tetrad. Moreover,the two expressions (4) and (5) are linked by a relationbetween B and h [3, Eq.4.47]

h = e+B, (6)

7 Hence the name “tautological form” for the canonical form θ.

where e is a tetrad satisfying the so–called Maurer–Cartan formula: de+ ωL ∧ e = 0.

Interpreting the field strength•T as curvature of the

gauge field B is definitely non–standard for gauge the-ories. In fact, comparing (4) and (2) we note that the

field strength•T matches the general definition of tor-

sion of the Lorentz connection ωL, provided B plays therole of a tetrad, and the geometrical structure describ-ing the theory includes the frame bundle on which ωL isdefined. Let us stress the fact that the point of this “non-standardness” is not to make the curvature of B equalto the torsion of ωL, but rather, in order to ensure thisequality, to make B, which as a gauge field of the trans-lation group is a connection one form (up to a pull back)in the translation-bundle (whatever it may be), also bea tetrad in the frame bundle.

What we just described exemplifies well the kind ofdifficulties we wish to point out. We stress out that [3]uses mostly the tensorial formalism, and in particularthe “dual” definition of the field strength, given by thecommutator of the covariant derivatives related to the

gauge field:•Tµν = [hµ, hν ], the tetrad h being there,

as a differential operator, the covariant derivative. As aconsequence, the non–standard use of the objects seenabove appears more natural.

Finally, another question raised by the translationgauge approach is that: whilst here the Lorentz group isdescribed as left ungauged, the expression of the covari-ant derivative h is the same as that appearing in theoriesobtained from a very different geometrical structure inwhich the whole Poincare group is taken as the gaugegroup , thus including the Lorentz local symmetry. Forinstance in Eq. (72) of Tresguerres’s work [9].

We conclude from the above remarks that the linkbetween the physical quantities and their mathematicalcounterpart is at least puzzling. For instance, one mayask if, in the “non-standard” translation gauge approach,the field strength must always be described by the cur-vature of some connection? On the other hand, what isthe meaning of gauging two different groups, with twodifferent (connections) gauge fields, leading to the samecovariant derivative (the tetrad h)?

IV. A CONVENTIONAL GAUGING OF THETRANSLATION GROUP

We now compare the standard objects (connection,curvature, . . . ) and their physical counterparts (gaugefield, field strength, . . . ) that appear in the mathemati-cal framework of principal fiber bundles used to describegauge theories [see for instance Ch. 21 of 13], with the“non-standard” translation gauge approach described in[3], and discussed above. To make the comparison ex-plicit, we build a “naive” translation gauge theory fol-lowing the general framework applied to the translationgroup. As a word of caution, we stress out that this the-

5

SO0(1, 3)

x

ep = (x, e)

θ

Horp(LM) ⊂ TpLM

LM

TxM

x

σ(x)

M

LM ×Id V = TM

u = (x, σ(x))V eru (TM) = TxM

θ = Id

M

TxM

ξ = θ θ

FIG. 4. The solder form θ in the frame bundle LM is the identity between the base’s tangent planes and the vertical spaces ofthe associated vector bundle, the tangent bundle, which is again the tangent plane. Note that the solder form is a map betweenbundles (TM → V er(P ) = TM here).

General G = T4 TEGR from Ref. [3]

P (G,M, π) PT (T4,M, π) TM

(x, f) (xµ, vαg ) (xµ, xa)ω = ωIEI ωT = ωaTPa

Ω = dω ΩT = dωT

+ω ∧ ω +ωT ∧ ωT

σ vg(x) xa(x)A = σ∗ω AT = ωT (vg) B

F = σ∗Ω FT = dAT

•T = dB + ωL ∧B

TABLE I. Comparison of fiber bundle/gauge theory objects.The first column contains the main elements in a general fiberbundle approach. In the second column the gauge group isspecified to translations (T4). The last column shows theequivalent objects for the “non-standard” approach of [3].

ory does not pretend to describe a viable gauge theoryof translations, in particular because, as we will see, thetranslations bundle cannot in general be principal. In-deed, this is merely an exercise allowing us to pinpointwhere the standard treatment and the works in Ref. [3]diverge, to explain why, and how, in our point of view,the latter theory status as a gauge theory of translationis difficult to defend.

Table I displays the general mathematical ingredientsof the models in the cases of a general fiber bundle the-ory, its specifications to the hypothetic principal trans-lation group bundle (whatever it may be), and (if any)the equivalent objects in the “non-standard” translationgauge approach of [3]. In the first line we have the spec-ifications of the corresponding fiber bundle. The fiber

bundle of [3] is identified, following the authors, as thetangent bundle. The local trivializations, in the generalcase (x, f), are presented in the second line. The trans-lation group case can be paralleled to the frame bundleidentification between a frame and a Lorentz group ele-ment, by identifying a translation g with the correspond-ing vector vg in R4. For the authors in [3], this corre-sponds to tangent plane coordinates xa. The connectionsω and their associated curvature Ω are presented for thegeneral case, with Lie algebra basis EI and, for theT4 case, with the abelian translation algebra basis Pa.The section σ is a vector field vg(x) for T4, while as in [3]it is given (in components) by xa(xµ). The connection’scurvature on the spacetime (base) is noted F , given bythe curvature Ω’s pullback along σ, and reduces to dAT

for our “naive” translation theory since the translationgroup is abelian, while the “non-conventional” approachuses Eq. (4). We thus see that, apart from a change innotation and the use of the definition of curvature in thetotal space not presented in [3], the differences betweenthe “naive” and the usual theories are on the nature ofthe bundle of translations itself [identified with TM in

3], and the expression of the field strength (•T vs FT ).

These two points deserve a deeper comment based onsome important mathematical details [see 13–15].

The usual mathematical description of a gauge the-ory of translations would adopt a principal fiber bundle.That entails the bundle should employ the translationsgroup in two roles: as structure group and as (being iso-morphic to) fibers. Now, the tangent bundle withoutadditional specification is a vector bundle whose struc-ture group is, for a n-dimensional real base manifold,

6

GL(n,R), the general linear group8. However, the lineargroup GL(n,R) does not even contain the group Tn ofn-dimensional translations. In addition, as a vector bun-dle, each fiber of TM is the vector space Rn, which isalso, as a manifold, the translation group Tn. However,in order to identify the translation group with the fiberin the principal translations bundle one would first haveto exhibit a right action of that group on the total space,an action which is not defined in the tangent bundle andin any case would not use the structure group GL(n,R).

One could ask if there is a possibility to associate in anatural way a principal bundle with the tangent bundle.The answer is yes but it turns out that this bundle is pre-cisely the frame bundle, and no translations are presentthere. Indeed, the problem of defining an action on thewhole manifold forbids the tangent bundle, viewed as atranslation-bundle - that is, whose fibers are the trans-lation group - to be principal. This is because, if oneconsiders an arbitrary vector of Rn, viewed as the (mani-fold of the) translations group, one must define its actionon the total space of TM , viewed as the bundle of trans-lations. To this end, one has to specify how to identifyRn to each tangent space Tx of TM , that is to specify aframe for each Tx. Put in another way we need a fieldof frame on the whole base manifold, which is precisely aglobal section of the frame bundle. Now, there is a the-orem which states that a principal bundle (as the framebundle) admitting a global section is trivial [see for in-stance, 13, Sec. 20.1]. Thus, the bundle of translationsis principal if and only if the associated frame bundle istrivial. Note that, this argument says at once that thetangent bundle is not in general a principal Tn-bundleand that the hypothetic principal T4-bundle is not de-fined in general.

Let us now consider the other main “non–standardness”, namely the expression for the curvaturethat in [3] contains an additional term ωL ∧ B withrespect to the “naive” version. We first note thatthe latter, built by gauging translations alone, cannotaccount for the local Lorentz invariance and thus failsto properly describe gravity. Therefore a heuristicway to implement local Lorentz invariance would beto use a minimal coupling procedure, leading to thereplacement of the exterior derivative d by its Lorentzcovariant counterpart (D := d + ωL∧) in the expres-sion for the curvature, eventually giving precisely the“non-standard” approach expression (4). Indeed, thiswould be coherent with the view adopted in [3] in whichone considers B as the gauge field for the translationswhile the local Lorentz invariance is related to thenon-holonomic frames. Moreover, this approach pointstowards the interpretation of the “dual” role of the fieldB as a tetrad, which in this view is somehow forced bythe expression of the torsion (2), related to the canonical

8 We consider the general n-dimensional case in the present para-graph.

form through (3). The only concern with this heuristicview is that the introduction of a covariant derivativecorresponds, on mathematical grounds, to gauging theLorentz symmetry.

The discussion above allows us to clarify the doubtsraised at the end of the previous section about the factthat the theory of translations presented by Aldrovandiand Pereira in [3] and the gauge theory of Poincare sym-metries proposed by Tresguerres in [9] lead to the sameexpression for the covariant derivative. Indeed, what ishappening is that in [3] the Lorentz symmetry is im-plicitly gauged. The fact that the Lorentz connectionis introduced in order to take into account the most gen-eral orthonormal frames (including non-holonomic ones),does not allow us to ignore its mathematical nature,namely, a connection in the orthonormal frames bundle.From this point of view, the heuristic introduction ofthe Lorentz covariant derivative (the replacement of d byd + ωL∧) in the previous paragraph to account for localLorentz invariance in a gauge theory of translations issomehow reminiscent of the use of the composite connec-tion in Tresguerres works [9].

To summarize, although the “non-standard” gaugetheory described in [3] reproduces TEGR, in our opinionit is difficult to consider it a gauge theory of the transla-tion group. Our view is mainly motivated by the fact thatthe bundle of translations, whatever it may be, cannotbe identified, as a principal T4-bundle, with the tangentbundle. In addition, the claim that the Lorentz symme-try is left ungauged seems, on mathematical grounds, atodds with the definition of a covariant derivative thatincludes a term for a Lorentz connection. In our view,the usual “non-standard” gauge theory of translationspresented as TEGR is the Einstein-Cartan (tetrad) for-mulation of gravity, which takes place in the principalorthonormal frame bundle, with the Lorentz connectionchosen to be the Weitzenbock connection; the transla-tional part, which does not arise from a gauge in theEinstein-Cartan formulation, is given there by the canon-ical form θ, viewed as the translational (part of the) con-nection whose pullback on the base manifold through Eq.(3) is the gauge field B.

V. SOME COMMENTS ABOUT THECONNECTIONS

The commonly accepted mathematical framework forgauge theories is that of principal bundles in which localgauge symmetries correspond to connections. In the caseof gravitation, the local symmetries are

1. the local Lorentz invariance. and

2. the invariance under the local (R4) diffeomorphism,which corresponds to local translational invariance.

As already mentioned in Sec. II, to account for these dif-feomorphisms from a gauge perspective is a central dif-ficulty of gauge theory of gravity, and translates into an

7

equally difficult choice of the connection. The presentpaper being devoted to the gauge version of TEGR, wedo not aim to discuss general gauge theories of gravity.We nevertheless comment about the connections with theaim to motivate our proposal to use the Cartan connec-tion in next section.

On one hand, as shown in Sec. IV, [3]’s expositionof a “non-standard” gauging of translations does not fitwell in the principal fiber bundle mathematical frame-work that should describe a translations gauge theory, asthere

1. translations are not properly taken into account asa connection in a principal bundle,

2. the Lorentz symmetry is implicitly gauged.

On the other hand, a more straightforward approach inwhich only a gauge field for the translations would beconsidered fails because

1. the local Lorentz invariance is not satisfied by atranslational field alone

2. a fiber bundle for the translations moreover fails tobe principal except if it is trivial.

It thus seems that the two invariances, local Lorentzand local translations (R4 diffeomorphisms) should beconsidered together in a connection allowing TEGR tobe accounted for as a gauge theory. Since the force fieldstrength for TEGR, i.e. the bundle’s curvature (up to apullback), is the torsion, such a connection would haveto yield torsion as its curvature. Moreover, being thatTEGR lives in manifolds with no curvature, a corre-sponding Weitzenbock connection should be the choicefor its Lorentz connection. One can propose a simpleansatz for a connection, say ω, satisfying the propertiesjust discussed. It can be written as (omitting all indicesto keep matter simple)

ω = ωL + θT ,

where ωL is a Lorentz connection and θT embodies thetranslational part. The corresponding curvature thenreads

Ω := dω + ω ∧ ω = ΩωL+ ΘωL

+ θT ∧ ωL, (7)

where ΩωL:= dωL+ωL∧ωL and ΘωL

:= dθT +ωL∧θT , arerespectively the curvature and the torsion of the Lorentzconnection. Note that the term θT ∧ θT is zero since thegroup of translations is abelian.

The curvature of the connection ω, as shown in Eq.(7), reduces to the torsion of the Lorentz connection if

1. ωL = ωW , the Weitzenbock connection,

2. the last term θT ∧ ωL vanishes.

Note that the same is true for the curvature on the basemanifold by pulling back (7) along some local section.

Now, since an Ehresmann connection on a principalG-bundle, takes it values in the Lie algebra of the wholegroup G, it is difficult to see how to make the cross termθT ∧ ωL vanish. For instance, the gauge theory of thePoincare group using a composite connection proposedin [9] leads to a connection on the base manifold whichis the sum of a Lorentz and a translational part [see Eq.(70) of 9], but the cross term forbids identification of thatconnection’s curvature with the torsion of the Lorentzconnection.

A (perhaps?) more natural way to implement trans-lations would point to the use of an affine connection(see appendix A 5). However the affine connection us-ing Lorentz and translations is not the simple sum ofthe Lorentz and translations connections and neither isits curvature yielding the sum of Lorentz curvature andtorsion directly [15, Sec.3.3 p125].

Lastly, the Cartan connection, which is not of Ehres-mann type, reduces exactly to take the form of the con-nection ω. In the following section therefore, we willdescribe the main features of this connection and thenuse it to obtain TEGR.

VI. APPROACHING TEGR WITH THECARTAN CONNECTION

The Cartan connection appears in the context of Car-tan geometry which can be seen as a generalization ofRiemannian geometry in which the tangent space is re-placed by a tangent homogeneous (i.e., maximally sym-metric) space. This geometry and the properties of theCartan connection, in relation with gravity theories, aresummarized in a comprehensive way by Wise [10] andCatren [8]. A detailed mathematical reference is givenby Sharpe [16], see also [17] for a summary and a com-parison with other mathematical approaches.

As in the case of usual gauge theories of particlephysics or of the Einstein-Cartan theory, the geometri-cal framework of Cartan geometry is a principal bun-dle P (H,M, π) with a connection ωC . However, thereare three important differences with respect to the usualcase:

1. The fiber is here a (topologically closed) subgroupH of a larger Lie group G,

2. The connection is a Cartan connection ωC whichtakes its values in the algebra g ⊃ h of G.

3. The connection ωC is, at each point p of P , a linearisomorphism between the tangent space TpP andthe Lie algebra g. This property requires that Ghas the same dimension as the tangent space TpP .

These properties are specific to the Cartan geometry, inparticular (2)-(3) distinguish the Cartan connection fromEhresmann’s, which by contrast takes its values in the Lie

8

algebra of the group H (the fiber) and does not satisfya property like (3). As a consequence of the above prop-erties, the tangent space of the base manifold M canbe locally identified with the tangent space g/h of thehomogeneous space9 G/H. Indeed, the third conditionprecisely states that the principal bundle P (H,M, π) issoldered to the base M .

For a (3+1)-dimensional manifold there are only threepossible homogeneous spaces: the (Anti)-de Sitter spacesand the Minkowski space. Each one being also a groupof symmetry: SO0(2, 3), SO0(1, 4) for Anti-de Sitter andde Sitter spaces respectively and the Poincare group forthe Minkowski space. The corresponding Cartan geome-tries have the property of being reductive; the definitionof this property can be found in [16, p197], along withits differences with the notion of reductive algebra. Fromour perspective it is sufficient to say that, for a reductiveCartan geometry, the Cartan connection takes the form

ωC = ω + θ, (8)

where ω is an Ehresmann connection h-valued one-form,on the principal fiber bundle P (H,M, π), and θ a g/h-valued one-form on P . Moreover, the definition of the re-ductive geometry, which relies on the existence of an iso-morphism between adjoint representations of the struc-ture group H (an Ad(H)-invariant decomposition of g,namely: g ' h ⊕ g/h), ensures that the two parts (ωand θ) of the Cartan connection ωC remain separatedunder a gauge transformation (a change of local sectionin P (H,M, π)).

For what may concern us, the main implication of the(Ad(H)-invariant) decomposition of the Lie algebra g isthat it allows us to split any g-valued form defined on P .In particular the curvature two-form ΩC of the Cartanconnection ωC which reads

ΩC := dωC + ωC ∧ ωC = Ωω + Θω, (9)

were Ωω and Θω stand respectively for the curvatureand the torsion of the Lorentz (Ehresmann) connectionω. An explicit calculation, using the fundamental repre-sentation of g, is given by Wise [10] for the three (tan-gent) homogeneous spaces: Anti-de Sitter, de Sitter andMinkowski. For the Minkowski case one obtains, bychoosing the Weitzenbock connection ω = ωW , the ex-pected result :

ΩC = ΘωW.

Introducing the reductive Cartan geometry thus solvesthe problem stated in the previous section: it accounts

9 Note that both G/H, with H a closed subgroup of G, being anhomogeneous space, and the fact that g/h can be identified withits tangent space are known results of differential geometry ofLie groups (see for instance [13] p. 294 for the former statement,and [16] p. 163, for the latter).

properly for both Lorentz and translational symmetriesthrough a connection whose curvature is the torsion.Moreover, the specific case of reductive Cartan geome-tries allows us to retrieve the framework of the orthonor-mal frame bundle: as shown in [16], for a reductive geom-etry the first part ω of the Cartan connection is preciselyan Erhesmann connection and the second part θ realizesthe soldering (see appendix A 3); in addition, the bundleP (H,M, π) is necessarily a reduction from the GL(R)frame bundle on M to the subgroup H leading thus tothe orhonormal frame bundle for H = SO0(1, 3).

VII. CONCLUSION

The results of the previous sections lead us to proposeto view TEGR in the context of reductive Cartan ge-ometries. The main reason is that, in our opinion, suchgeometries provide a more consistent framework than theusual translation-gauge theory. Precisely, we argued inSec. III and IV that

1. the conventional approach makes use of a transla-tional gauge field which does not appear, on math-ematical grounds, as a connection in a principal”translation-bundle”

2. moreover such a bundle could only be defined forspacetimes with trivial frame bundles, and finally

3. local Lorentz symmetry, taken into account in orderto implement covariance properties, appears as agauge field since it is a connection in the principalbundle of orthonormal frames.

Clearly, the Cartan connection corresponding to thePoincare symmetry does not share these drawbacks and,in the case of the Weitzenbock connection of TEGR, hastorsion for curvature. In addition, the frame bundle sol-dering to the base manifold, recognized as a source ofdifficulties in the gauge theoretical context, is a ”built-in” property for Cartan geometries.

On the other hand, the question about promotingTEGR to a legitimate gauge theory by building it witha Cartan connection is still open. Indeed, the structureof the reductive Cartan geometries differs from that ofthe usual gauge theories mainly because the connectiondoes not only relate to the symmetry group (here theLorentz group) of the principal bundle on which it isdefined (here the orthonormal bundle of frames): theconnection, in fact, takes its value in a larger lie alge-bra (here the Poincare algebra). This peculiarity couldbe considered as the ”non-standardness” inherited fromthe soldering property as mentioned in [3]. More im-portantly, although this Cartan version of TEGR cannotbe considered as a Poincare Gauge Theory (PGT)10, the

10 It could be part of a new class of ”Cartan-Poincare- Gauge The-ory” (CPGT) and termed ”Weitzenbock-CPGT”.

9

whole Poincare symmetry is required. This point shouldbe compared to the work of [18], which adopts a verydifferent approach, but still reaches the conclusion thatTEGR cannot be obtained without the whole Poincaresymmetry.

ACKNOWLEDGEMENTS

The authors wish to thank G. Catren and M. Lachieze-Rey for very useful discussions and references about dif-ferential geometry, and F. Helin for his help on somemathematical notions and discussions. MLeD acknowl-edges the financial support by Lanzhou University start-ing fund.

Appendix A: Definitions of, and comments on, somemathematical structures

1. Tetrads

We will take the definition of Nakahara’s Book [14](Sec. 7.8.1) that is : the tetrads (vierbein) are the coeffi-cients of the (field of) basis vectors ea non-coordinates,orthonormalized and preserving orientation. We havegµνe

µaeνb = ηab, gµν = eaµe

bνηab.

2. Comment on Ehresmann connection

An Ehresmann connection on a principal G-bundle ba-sically provides a G-invariant splitting of the tangentspace TpP into a vertical VerpP and a horizontal HorpPpart by defining (uniquely) the horizontal vectors asthose which belong to its kernel. Although always pos-sible, the tangent space TpP splitting into a vertical anda horizontal part is not unique. While the vertical partis always uniquely defined – the vertical vectors belongto the kernel of π∗, the horizontal part can be, in gen-eral, any complementary space of VerpP in TpP . In thissense, both verticality and horizontality are always de-fined. In addition there is a linear isomorphism betweenVerpP and the Lie algebra g (see for instance [13] p. 560)and between HorpP and Tπ(p)M through π∗ (see for in-stance [19] p. 255). The Ehresmann connection specifiesa unique horizontality which in turn is used to define theparallel transport of various tensorial objects.

Since a vector of TpP can be split in a unique way intoa horizontal and a vertical part, there is a map whichprojects the vector along HorpP to VerpP ' g, that is ag-valued one-form. Then, the definition of an Erhesmannconnection on a principalG-bundle is often made throughthat connection one-form ωE whose kernel specifies thehorizontal vectors of TP . This definition of horizontalityhas to be consistent with the group action. A formaldefinition, together with a comparison with the Cartanconnection, is detailed in [10].

3. Solder form

Let P (G,M,F, π) be a fiber bundle where M is thebase manifold, G the (Lie) structure group, F the fiber,P the total space, and π the projection from P onto M .The definition of the solder form can be found in theoriginal work of Kobayashi [6]. Using the definition ofthe vertical bundle – that is, the subbundle Ver(P ) ofTP defined as the disjoint union of the vertical spacesVerpP for each p in P – the definition given in [6] reads:

The bundle P (G,M,F, π) is soldered toM , if the following conditions are satisfied.

1. G is transitive on F .

2. dimF = dimM .

3. P (G,M,F, π) admits a section σ which will beidentified with M .

4. There exists a linear isomorphism of vectorbundles θ : TM −→ σ∗Ver(P ) from thetangent bundle of M to the pullback of thevertical bundle of P along the section σ.

This last condition can be interpreted as saying that θdetermines a linear isomorphism

θx : TxM −→ Vσ(x)P

from the tangent space of M at x to the (vertical) tan-gent space of the fiber at the point σ(x). This generaldefinition is pictured in Fig. 5.

θ

P (G,M,F, π)

TxM

Fx

x

f = σ(x)

M

u = (x, σ(x))

V erσ(x) (P )

FIG. 5. The solder form renders each tangent of the bundlebase isomorphic to the corresponding vertical tangent space

along the (global) section σ. In a rigorous way, θ, the solderform, is the bundle map between TM and σ∗Ver(P ).

The above general definition of soldering does not ap-ply as is to principal bundles. Its application to a princi-pal fiber bundle requires the use of an intermediate asso-ciated vector bundle discussed in appendix A 4, as shownin Fig.6 since a global section in a principal bundle wouldotherwise render it trivial. There, the soldering of theprincipal bundle is realised through the isomorphism ξbetween a horizontal and the associated vector bundle’s

10

P (G,M, π)

G

x

g = σP (x)

M

p = (x, g)

Horp(P )Vx

x

v = σ(x)

M

P ×ρ V

u = (x, σ(x))

V erσ(x) (P ×ρ V )

θ Θ

TxM

ξ = θ Θ

FIG. 6. The solder form renders a tangent of the principal bundle base isomorphic to the vertical of an associated vectorbundle. For a principal bundle, soldering is realised through isomorphism ξ between one principal bundle’s horizontal and theassociated bundle’s vertical at the corresponding section. This is made possible through the isomorphism Θ that always existsbetween the tangent to the base and any horizontal.

vertical tangent, effected thanks to the isomorphism Θbetween any horizontal of the principal bundle and thetangent to the base. In the case of the Frame bundle,as seen in Sec. II, the function of the isomorphism Θ isplayed by the canonical form θ.

In the case of Cartan reductive geometry, solderingtakes a specific form due to the various isomorphisms inwhich the g/h part of the splitting, related to the trans-lational part θ of the connection, g ' h ⊕ g/h of g asAd(H)-module, is involved (see Fig. 7). Indeed, thesoldering can further be particularized, through the pre-cising of these isomorphisms [see, in particular, 8].

4. Associated (vector) bundle.

We first follow here [15, p.54-55] and then make someremarks.

The idea is to build a fiber bundle in which the fiber Gof the original principal G-bundle is replaced by a mani-fold F on which the group G acts on the left. To this end,one defines a right action of the group G on the productspace P × F as follow: a g ∈ G maps (p, f) ∈ P × Fto (pg, g−1f) ∈ P × F . The set of the orbits (that isthe equivalence classes) corresponding to this action isdenoted P ×G F and is the total space of the associatedbundle. At first, P ×G F is just a set, the structure ofa fiber bundle is obtained as follow. One considers themapping: (p, f) ∈ P × F 7→ π(p) = x ∈ M . It induces aprojection π from P ×G F onto the base M . The fiber ofP×GF over x ∈M is π−1(x). Now, in a neighborhood Uof x, π−1(U) ∼ U ×G the action of G on π−1(U)×F is:(x, g′, f) 7→ (x, g′g, g−1f) with (x, g′, f) ∈ U ×G×F and

g ∈ G. The isomorphism π−1(U) ∼ U × G induces anisomorphism π−1(U) ∼ U × F . Then, one can introducea differentiable structure to ensure that π is a differen-tiable mapping from P ×GF to the base M . This in turnensures that P ×G F is a fiber bundle with base M , fiberF , and structure group G.

When F is a k dimensional vector space V on which Gacts on the left through a representation ρ, one obtainsan associated vector bundle denoted by its total spaceP×ρV , the right action on P×V is, in that case, (p, v) 7→(pg, ρ(g−1)v). We now restrict to this case.

The first remark is to recall that this rather compli-cated procedure is done to obtain a fiber bundle in whichthe fiber of the principal bundle is replaced by a vectorspace V , all other data (in particular transition func-tions) remain unchanged. It should be noted that theright action defined on P × V is different from the usualleft action ρ(g)v. One can be puzzled by the fact thatalthough the total space P ×ρ F is made of orbits, thefiber is the vector space V . Indeed, there is an identifi-cation there: following Fecko [13, Sec. 2.4.1] there is anon-canonical isomorphism between π−1(x) and V , givenby v 7→ [p, v] for an arbitrary but fixed p. Since the rightaction on the product space P × V moves both v and pthe fact that p is held fixed allows us to distinguish be-tween two elements u, v of V belonging to the same orbitthrough G, that is u = ρ(g−1)v (see Fig. 8). Accord-ingly, a linear structure on each fiber is provided throughπ−1(x) by [p, v]+λ[p, u] := [p, v+λu]. Different choices ofp do not change the result (they correspond to chooseinganother line parallel to the V axis on Fig. 8).

11

SO0(1, 3)

x

e

p = (x, e)

θ

LM

m

xM

LM×Ad(H)m

θ

M

ξ = θ θ = Id

V erp(LM) ' h

TxM ' Tx(G/H) ' m

T cxM ' G/H

Horp(LM) ' m

V er (LM×Ad(H) mm) '

FIG. 7. The solder form θ in the frame bundle LM within Cartan geometry is the identity between the base’s tangent planesand the vertical spaces of the associated vector bundle. Here the Cartan geometry is assumed to be reductive, so that one hasthe Ad(H)-invariant splitting of the Lie algebra g ≡ Lie(G) = h⊕m with h = Lie(H) = h and m = g/h.

p

v u = ρ(g−1)v

[p, v] [p, u]

pg[pg, ρ(g−1)v]

π−1(x)

V

FIG. 8. Linear structure in P ×ρ V .

5. On the affine connection

In the search for a correct geometrical description of atranslation gauge theory, the affine group presents a nat-ural way to combine the translation of diffeomorphismswith the linear group of all possible frame transforma-tions, which can be restricted to the Lorentz group. From[15, p136], the affine frame bundle AM can be built withan affine connection ω and the homomorphism

γ : GL(n;R)→ A(n;R)

a 7→

(a 0n0 1

)

from the frame bundle LM , and such that

γ∗ω = ω + φ

where ω is the connection on LM and φ is an Rn valuedconnection 1-form. Defining as usual the curvature ofsome connection ω with the Cartan structure equation:

Ωω := Dωω := dω + ω ∧ ω,

Dω being the covariant derivative associated to the con-nection ω, one obtains the relation

γ∗Ω = Ω +Dωφ. (A1)

Noticing the formal identity between the covariantderivative of φ and Eq. (2), we obtain precisely thesum of curvature and torsion required to obtain the fieldstrength for a TEGR translation-bundle

γ∗Ω = ΩL + Θ. (A2)

However, since it is not a curvature per se, the fieldstrength interpretation of the affine curvature pullbackis problematic.

12

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