RG - Aldrovandi - Teleparallel Gravity

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An introduction to

TELEPARALLEL GRAVITY

R. Aldrovandi and J. G. Pereira

Instituto de Fsica Teorica, UNESP

Sao Paulo, Brazil


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Foreword

These notes began to emerge in 2001, as part of the notes of a two-month courseon general aspects of gravitation, delivered at the Instituto de Fsica Teorica, UNESP,Sao Paulo. Improved versions were then written for a series of lectures on teleparallelgravity, delivered by one of the authors at the Universidad de Concepcion, Concepcion,Chile (2005), at the National Central University, Chung-Li, Taiwan (2006), and at theUniversidad Tecnologica de Pereira, Pereira, Colombia (2007). They contain actuallymore material than presented in the lectures, allowing those interested in exploringfurther the subject to do so. The reader should be warned that the ideas presented

here are strongly biased by the authors point of view on the subject, and are essentiallybased on the research developed by them in the recent years. They are actually anexpanded version of the relevant publications, which includes also many correctionsto the original texts. An important point to be considered is that they presuppose areader familiar with general relativity, as no review of this topic is provided. Althoughthe authors are the solely responsible for the contents of the notes, they owe much totheir former and present collaborators: V. C. de Andrade, H. I. Arcos, T. V. Aucalla,A. L. Barbosa, L. C. T. Guillen, M. Cal cada, R. A. Mosna, Yu. N. Obukhov, D. J.Rezende, G. Rubilar, K. H. Vu and C. M. Zhang.

Sao Paulo, May 2007

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Contents

1 Introduction 1

1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Linear Frames and Tetrads . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Fundamental Fields of Teleparallel Gravity 8

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 The Gauge Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Translational Covariant Derivative . . . . . . . . . . . . . . . . . . . . 102.4 Field Strength: Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 The Weitzenbock Connection . . . . . . . . . . . . . . . . . . . . . . . 13

3 Gravitational Coupling Prescription 15

3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Translational Coupling Prescription . . . . . . . . . . . . . . . . . . . . 163.3 Lorentz Coupling Prescription . . . . . . . . . . . . . . . . . . . . . . . 17

3.3.1 General Covariance Principle . . . . . . . . . . . . . . . . . . . 173.3.2 Lorentz Covariant Derivative . . . . . . . . . . . . . . . . . . . . 18

3.4 Gravitational Coupling Prescription . . . . . . . . . . . . . . . . . . . . 193.4.1 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.2 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . 203.4.3 Teleparallel Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Particle Mechanics 244.1 Free Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1.2 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Gravitationally Coupled Particles . . . . . . . . . . . . . . . . . . . . . 264.2.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.2 Strong Equivalence Principle . . . . . . . . . . . . . . . . . . . . 294.2.3 Newtonian Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 A Global Formulation for Gravity 32

5.1 Phase Factor Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2 Colella-Overhauser-Werner Experiment . . . . . . . . . . . . . . . . . . 33

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5.3 Gravitational Aharonov-Bohm Effect . . . . . . . . . . . . . . . . . . . 355.4 Quantum Versus Classical Approaches . . . . . . . . . . . . . . . . . . 375.5 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Lagrangian and Field Equations 41

6.1 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.2 Matter Energy-Momentum Density . . . . . . . . . . . . . . . . . . . . 436.3 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.4 Gravitation Energy-Momentum Density . . . . . . . . . . . . . . . . . . 466.5 Bianchi Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7 Teleparallel Equivalent of Some Known Solutions 50

7.1 Teleparallel de Sitter Solution . . . . . . . . . . . . . . . . . . . . . . . 50

7.1.1 Description of the Solution . . . . . . . . . . . . . . . . . . . . . 507.1.2 The de Sitter Force Equation . . . . . . . . . . . . . . . . . . . 517.2 The Teleparallel Schwarzschild Solution . . . . . . . . . . . . . . . . . . 527.3 The Teleparallel KerrNewman Solution . . . . . . . . . . . . . . . . . 55

8 Gravitational Interaction of the Fundamental Fields 58

8.1 Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588.2 Dirac Spinor Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

8.2.1 The Coupled Dirac Equation . . . . . . . . . . . . . . . . . . . . 598.2.2 Torsion Decomposition and Spinors . . . . . . . . . . . . . . . . 60

8.3 Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

8.4 Spin-2 Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638.4.1 The Fierz Formulation . . . . . . . . . . . . . . . . . . . . . . . 638.4.2 Spin-2 Field in the Presence of Gravitation . . . . . . . . . . . . 668.4.3 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 67

9 Duality Symmetry 68

9.1 Dual Operation for Soldered Bundles . . . . . . . . . . . . . . . . . . . 689.2 Duality Symmetry in Gravitation . . . . . . . . . . . . . . . . . . . . . 709.3 In Search of Duality Symmetry . . . . . . . . . . . . . . . . . . . . . . 71

9.3.1 Torsion Decomposition and Spinors . . . . . . . . . . . . . . . . 71

9.3.2 Self Dual and Anti-Self Dual Fields . . . . . . . . . . . . . . . . 729.4 A Self Dual Gravitational Theory . . . . . . . . . . . . . . . . . . . . . 739.5 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

10 Torsion and Gravitation 77

10.1 The Torsion Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7710.2 Gravitational Coupling Prescription Revisited . . . . . . . . . . . . . . 7810.3 The Connection Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

10.3.1 Translations in the Connection Space . . . . . . . . . . . . . . . 7910.3.2 Equivalence under Connection Translations . . . . . . . . . . . . 80

10.4 Example: the Spinning Particle . . . . . . . . . . . . . . . . . . . . . . 82

10.5 The Role of Torsion in Gravitation . . . . . . . . . . . . . . . . . . . . 84

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11 Doing without the Weak Equivalence Principle 86

11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8611.2 Separating Inertia from Gravitation . . . . . . . . . . . . . . . . . . . . 87

11.2.1 Passage to a General Frame . . . . . . . . . . . . . . . . . . . . 8711.2.2 Inertial Versus Gravitational Effects . . . . . . . . . . . . . . . . 88

11.3 On the Lack of Universality . . . . . . . . . . . . . . . . . . . . . . . . 8911.4 The Electromagnetic Case as an Example . . . . . . . . . . . . . . . . 9011.5 Non-Universal Coupling Prescription . . . . . . . . . . . . . . . . . . . 9111.6 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9211.7 Universality and General Relativity . . . . . . . . . . . . . . . . . . . . 94

12 Epilogue 97

12.1 Gravity and the Quantum . . . . . . . . . . . . . . . . . . . . . . . . . 97

12.1.1 General Relativity Versus Universality . . . . . . . . . . . . . . 9712.1.2 Equivalence Versus Uncertainty Principles . . . . . . . . . . . . 9812.2 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

References 101

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Chapter 1

Introduction

1.1 Preliminaries

The first attempt to unify gravitation and electromagnetism was made by H. Weyl in1918.1 Despite not successful, this beautiful proposal introduced for the first time thenotions of gauge transformations and gauge invariance, and can be considered as theseed that has grown to what is known today as gauge theories. A second attempt inthe same direction was made by A. Einstein about ten years later. It was based on themathematical structure of distant parallelism, also referred to as absolute or telepar-allelism. The crucial idea was the introduction of a tetrad field, a field of orthonormalbases of the tangent spaces at each point of the four-dimensional spacetime. However,

the specification of the tetrad involves the specification of sixteen components, whereasthe gravitational field, represented by the spacetime the metric, requires only ten com-ponents. The six additional degrees of freedom ensued by the tetrad was then supposedby Einstein to represent the electromagnetic field.2 This attempt of unification did notsucceed either, but like Weyls work, it introduced concepts that remain important upto the present day.

It is interesting to observe that Einstein did not agree with Weyls unification the-ory [3]: Although your idea is so beautiful, I have to declare frankly that, in my opinion,it is impossible that the theory corresponds to nature. Weyl did not agree with Ein-steins theory either [3]: I prefer not to believe in distant parallelism for a number ofreasons. First, my mathematical intuition objects to accepting such an artificial geome-try; I find it difficult to understand the force that would keep the local tetrads at differentpoints and in rotated positions in a rigid relationship. This difference of opinions ledto an intense exchange of letters between Einstein and Weyl, which involved also otherphysicists, like W. Pauli, who wrote to Weyl ... let me emphasize that side of matterconcerning which I am in full agreement with you: your incorporation of spinor theoryinto gravitational theory. I am as dissatisfied as you are with distant parallelism andyour proposal to let the tetrads rotate independently at different space-points is a truesolution.

After this initial period of intense activities, which included also contributions from

1

An account of the history underlying the birth and evolution of gauge theories can be found inRef. [1]2A description of the teleparallelbased Einsteins unification theory can be found in Ref. [2].

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E. Cartan and R. Weitzenbock, owing basically to the failure in unifying gravitationand electromagnetism, the notion of teleparallelism experienced no new advances in thefollowing three decades. In the sixties, Mller [4] rescued the idea of teleparallelism, but

with the purpose of describing gravitation only. In the sequel, Pellegrini & Plebanski [5]found a Lagrangian formulation for teleparallel gravity, a problem that was reconsideredlater by Mller [6]. In 1967, Hayashi & Nakano [7] formulated a gauge theory for thetranslation group, which was further developed by Hayashi [8]. A few years later,Hayashi [9] pointed out the connection between this theory and teleparallelism, and anattempt to unify these two developments was made by Hayashi & Shirafuji [10] in 1979.According to this approach, general relativity a theory that involves only curvature was supplemented with teleparallel gravity a theory that involves only torsion,and presents three free parameters which should be determined by experiment. Thistheory, called new general relativity, represented a new way of including torsion into

general relativity, an alternative to the scheme provided by the usual EinsteinCartanSciamaKibble approach [11]. The fundamental difference between these two theorieswas that, whereas in the former torsion is a propagating field, in the latter it is not, apoint which can be considered a drawback of this model.

Teleparallel gravity, understood as a three-parameter theory, is usually consideredas a particular case of more general gravity theories, like for example the Poincare andthe metric-affine gauge theory [12]. According to this point of view, teleparallel gravitywould represent only the contribution of torsion to the gravitational interaction, whichin the general case is supposed to involve also contributions from other geometricalmagnitudes, like for example curvature and nonmetricity. However, as is well known,

for a specific choice of the free parameters, teleparallel gravity shows up as a theorycompletely equivalent to Einsteins general relativity, in which case it is usually referredto as the teleparallel equivalent of general relativity. This equivalence can be consideredas a compelling evidence that curvature and torsion does not represent independentdegrees of freedom, but alternative ways of representing the same field. From thispoint of view, because of the difference in the dynamical role played by torsion, theteleparallel equivalent of general relativity cannot be considered a particular case ofmore general theories.

Although usually reserved for the three-parameter theory, the name teleparallelgravity will be used in the present text as a synonimous for the teleparallel equivalentof general relativity. As in the study of any gravitational theory, it is fundamentalto have a clear idea of the concepts of coordinate systems, linear frames, tetrads, andmetrics. It is also crucial the notion of connection, as well as its main properties:curvature and torsion. For this reason, we begin by reviewing these concepts.

1.2 Linear Frames and Tetrads

The geometrical setting of any theory for gravitation is the tangent bundle, a naturalconstruction always present in spacetime. In fact, at each point of spacetime thebase space of the bundle there is always a tangent space attached to it the

fiber of the bundle on which the gauge group acts. We use the Greek alphabet( , , , . . . = 0, 1, 2, 3) to denote the indices related to spacetime, and the first half of

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the Latin alphabet (a , b , c, . . . = 0, 1, 2, 3) to denote the indices related to the tangentspace, assumed to be a Minkowski spacetime with the metric

ab = diag(+1, 1, 1, 1). (1.1)The second half of the Latin alphabet (i , j , k , . . . = 1, 2, 3) will be reserved for spaceindices. The spacetime coordinates, therefore, will be denoted by {x}, whereas thetangent space coordinates will be denoted by {xa}. Such coordinate systems define, ontheir domains of definition, local bases for vector fields, formed by the sets of gradients

{} {/x} and {a} {/xa}, (1.2)

as well as bases {dx} and {dxa} for covector fields, or differentials. These bases aredual, in the sense that

dx = and dx

a b = ab. (1.3)

On the respective domains of definition, any vector or covector can be expressed interms of these bases, which can furthermore be extended by direct product to constitutebases for general tensor fields.

We are going to use the notation {ea, ea} for general linear frames. A holonomicbase like {a}, related to coordinates, is a very particular case of linear base. Any set offour linearly independent fields {ea} will form another base, and will have a dual {ea}whose members are such that ea(eb) =

ab. These frame fields are the general linear

bases on the spacetime differentiable manifold whose set, under conditions making ofit also a differentiable manifold, constitutes the bundle of linear frames. Of course, onthe common domains they are defined, the members of a base can be written in termsof the members of the other, that is,

ea = ea and e

a = ea dx, (1.4)

and conversely. We can consider general transformations taking any base {ea} into anyother set {ea} of four linearly independent fields. These transformations constitute thelinear group GL(4,R) of all real 4 4 invertible matrices. Notice that these frames,with their bundles, are constitutive parts of spacetime. They are automatically present

as soon as spacetime is taken to be a differentiable manifold [13].Analogously, we are going to use the notation {ha, ha} for a generic tetrad field, a

linear frame connected with the presence of a gravitational field. Consider the space-time metric g, with components g, in some dual holonomic base {dx}:

g = gdx dx = gdxdx. (1.5)

A tetrad field ha = ha will be a linear base which relates g to the tangentspace

metric = abdxadxb by

ab = g(ha, hb) = ghahb

. (1.6)

This means that a tetrad field is a linear frame whose members ha are (pseudo) or-thogonal by the metric g. We shall see later how two of such bases are related by

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the Lorentz subgroup of the linear group GL(4,R). The components of the dual basemembers ha = hadx

satisfy

ha

ha

=

and ha

hb

= a

b, (1.7)so that Eq. (1.6) has the converse

g = ab hahb. (1.8)

Anholonomy the property of a differential form which is not the differentialof anything, or of a vector field which is not a gradient is commonplace in manychapters of Physics. Heat and work, for instance, are typical anholonomic coordinateson the space of thermodynamic variables, and the angular velocity of a generic rigidbody is a classical example of anholonomic velocity. In the context of gravitation,anholonomy is related, through the equivalence principle, to the very existence of a

gravitational field. Given a Riemannian metric as in (1.8), the presence or absence ofa gravitational field is fixed by the anholonomic or holonomic character of the formsha = hadx

. We can think of a change of coordinates {xa} {x} represented bydxa = (x

a) dx and dx = (ax) dxa. (1.9)

The 1-form dxa is holonomic, just the differential of the coordinate xa, and the objectsx

a are the components of the holonomic form dxa written in the base {dx}, withax

its inverse. Thus, such a coordinate change is just a change of holonomic basesof 1-forms. For the dual base we have the relations

= (xa) a and a = (ax

) . (1.10)

Take now a dual base ha such that dha = 0, that is, not formed by differentials.Apply the anholonomic 1-forms ha to /. The result, h

a = h

a , give the compo-nents of each ha = hadx

along dx. The procedure can be inverted when the hasare linearly independent, and defines vector fields ha = ha

which are not gradi-ents. Because closed forms are locally exact, holonomy/anholonomy can be given atrivial criterion: a form is holonomic iff its exterior derivative vanishes. A holonomictetrad will always be of the form ha = dxa for some coordinate set {xa}. For such atetrad, the metric tensor (1.8) would be simply the components of the Lorentz metric transformed to the coordinate system {x}.

An anholonomic basis

{ha

}satisfies the commutation table

[ha, hb] = fcab hc, (1.11)

with fcab the so called structure coefficients, or coefficient of anholonomy. The frame{} has been presented above as holonomic precisely because its members commutewith each other. The dual expression of the commutation table above is the Cartanstructure equation

dhc = 12

fcab ha hb = 1

2(h

c hc) dx dx. (1.12)

The structure coefficients represent the curls of the base members:

fcab = hc([ha, hb]) = ha

hb(h

c hc) = hc[ha(hb) hb(ha)]. (1.13)

If fcab = 0, then dha = 0 implies the local existence of functions (coordinates) xa suchthat ha = dxa. The tetrads are gradients when the curls vanish.

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1.3 Connections

In order to define derivatives with a well-defined tensor behavior (that is, which are

covariant), it is essential to introduce connections , which are vectors in the lastindex but whose non-tensorial behavior in the first two indices compensates the non-tensoriality of the oordinary derivatives. Linear connections have a great degree ofintimacy with spacetime because they are defined on the bundle of linear frames, whichis a constitutive part of its manifold structure. That bundle has some properties notfound in the bundles related to internal gauge theories. Mainly, it exhibits soldering,which leads to the existence of torsion for every connection [13]. Linear connectionsinparticular, Lorentz connectionsalways have torsion, while internal gauge potentialshave not.

It is important to remark at this point that, from a formal point of view, curva-

ture and torsion are properties of a connection [13]. Strictly speaking, in the contextof gauge interactions, there is no such a thing as curvature or torsion of spacetime,but only curvature or torsion of connections. This becomes evident if we notice thatmany different connections are allowed to exist in the very same spacetime [14]. Ofcourse, when restricted to the specific case of general relativity, where the only con-nection present is the LeviCivita connection, universality of gravitation allows it tobe interpreted as part of the spacetime definition. However, in the presence of differ-ent connections with different curvature and torsion, it seems far wiser and convenientto take spacetime simply as a manifold, and connections (with their curvatures andtorsions) as additional structures.

A spin connection A is a connection assuming values in the Lie algebra of theLorentz group,

A =12

Aab Sab, (1.14)

with Sab a given representation of the Lorentz generators. On the other hand, a tetradfield relates internal with external tensors. For example, if Va is a Lorentz vector,

V = ha Va (1.15)

will be a spacetime vector. However, in the specific case of connections, an additionalvacuum term appears when transforming internal to external indices, and vice versa.In fact, a general linear connection is related to the corresponding spin connection

Aab through

= hah

a+ ha

Aabhb. (1.16)

The inverse relation is, consequently,

Aab = hahb

+ hahb

. (1.17)

Equations (1.16) and (1.17) are simply different ways of expressing the property thatthe total that is, acting on both indices covariant derivative of the tetrad vanishesidentically:

ha ha + Aabhb = 0. (1.18)

A connection

is said to be metric compatible ifg g g = 0. (1.19)

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From the tetrad point of view, by using Eqs. (1.16) and (1.17), this equation can berewritten in the form

hc(ab)

Ada db

Adb ad = 0, (1.20)

or equivalentlyAba = Aab. (1.21)

The underlying content of the metric preserving property, therefore, is that the spinconnection is Lorentzian. Conversely, in the presence of nonmetricity, the spin connec-tion will not assume values in the Lie algebra of the Lorentz group.

The curvature and the torsion tensors of the connection Aab are defined respectivelyby

Rab = Aab Aab+ AaeAeb AaeAeb (1.22)

and

Ta = ha ha+ Aaehe Aaehe. (1.23)Using the relation (1.17), they can be expressed in a purely spacetime form:

R ha hb Rab = + (1.24)and

T ha Ta = . (1.25)The connection coefficients can be conveniently decomposed according to3

=

+ K, (1.26)

where

=12

g (g+ g g) (1.27)is the torsionless LeviCivita connection of general relativity, and

K =12

(T + T

T) (1.28)

is the contortion tensor. In terms of the spin connection, the decomposition (1.26)assumes the form

Aca =

Aca+ Kca, (1.29)

where

Aca is the Ricci coefficient of rotation, the spin connection of general relativity.Now, since the spin connection is a tensor in the last index, we can write

Aabc = Aab hc

. (1.30)

It can thus be easily verified that, in the anholonomic basis ha, the curvature andtorsion components are given respectively by [15]

Rabcd = hcAabd hdAabc + AaecAabd AaedAebc + fecdAabe (1.31)

and

Tabc = f

abc + A

acb A

abc. (1.32)

3The magnitudes related with general relativity will be denoted with an over .

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Seen from this frame, therefore, torsion includes the anholonomy. Use of (1.32) forthree combinations of the indices gives

Aa

bc = 1

2(fa

bc + Ta

bc + fbca

+ Tbca

+ fcba

+ Tcba

). (1.33)When torsion vanishes, as in general relativity, we obtain the usual expression of theRicci coefficient of rotation in terms of the anholonomy:

Aabc = 12(fabc + fbca + fcba). (1.34)

1.4 Lorentz Transformations

The base {ha} is far from being unique. There exists actually a six-fold infinity oftetrad fields

{ha = ha

}

, each one relating g to the Lorentz metric by Eqs. (1.6)

and (1.8). This comes from the fact that, at each point of the Riemannian spacetime,Eq. (1.8) only determines the tetrad field up to transformations of the six-parameterLorentz group in the tangent space indices. Suppose in effect another tetrad {ha} suchthat

g = ab hahb = cd h

ch

d. (1.35)

Contracting both sides with hehf

, we arrive at

ab = cd (hcha

)(hdhb

). (1.36)

This equation says that the matrix with entries

a

b = h

a

hb

, (1.37)which gives the transformation

ha =

ab hb, (1.38)

satisfiescd

ca

db = ab. (1.39)

This is just the condition that a matrix must satisfy in order to belong to (the vectorrepresentation of) the Lorentz group.

Under a local Lorentz transformation ab ab(x), the tetrad changes accordingto (1.38), whereas the spin connection undergoes the transformation

Aab = acAcd b

d + ac bc. (1.40)

It should be noted that, due to its non-covariant transformation law, if a connectionvanishes in a given frame, it will not vanish in any other frame related to the first by alocal Lorentz transformation. In the same way, it is easy to verify that Ta and R

ab

transform covariantly:

Ta = ab Tb and R

ab =

ac b

dRcd. (1.41)

Due to the covariant behavior of the tetrad, this means that , T and R

are

all invariant under a local Lorentz transformation:

= , T

= T

, R

= R

. (1.42)

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Chapter 2

Fundamental Fields of Teleparallel

Gravity

2.1 Preliminaries

Teleparallel gravity corresponds to a gauge theory for the translation group. From theto the point of view of this theory, curvature and torsion are able to provide, each one,equivalent descriptions of the gravitational interaction. According to general relativity,a theory fundamentally based on the weak equivalence principle, curvature is used togeometrize the gravitational interaction. The gravitational interaction, in this case, isachieved by letting (spinless) particles to follow the geodesics of the curved spacetime.

In general relativity, therefore, geometry replaces the concept of gravitational force, andthe trajectories are determined, not by force equations, but by geodesics. Teleparallelgravity, on the other hand, attributes gravitation to torsion, but in this case torsionaccounts for gravitation not by geometrizing the interaction, but by acting as a force.As a consequence, there are no geodesics in teleparallel gravity, but only force equationsquite analogous to the Lorentz force equation of electrodynamics [16].

What is the reason for gravitation to present two equivalent descriptions? Theanswer to this question is related to a quite peculiar property of gravitation: univer-sality. Like the other fundamental interactions of nature, gravitation can be describedin terms of a gauge theory just teleparallel gravity. Universality of free fall, on theother hand, makes it possible a second, geometrized description, based on the weakequivalence principle just general relativity. As the sole universal interaction, itis the only one to allow also a geometrical interpretation, and hence two alternativedescriptions. From this point of view, curvature and torsion are simply alternativeways of describing the gravitational field [17], and consequently related to the samedegrees of freedom of gravity. It is important to remark that, in the eventual lack ofuniversality, the geometrical description of general relativity would break down [18]. Itis equally important to mention that, although the existing experimental data seem toindicate that gravitation is in fact universal at the classical level, there are no evidencesthat gravitation remains universal at the quantum level. A more detailed discussion ofthis point will be presented in Chapter 11.

One may wonder why a gauge theory for the translation group, and not for otherspacetime group. The reason for this is related to the source of gravitation, that is,

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energy and momentum. As is well known from Noethers theorem [19], these quantitiesare conserved provided the physical system is invariant under spacetime translations.It is then natural to expect that the gravitational field be represented by a gauge

theory for the translation group. This is similar to the electromagnetic field, whosesource the electric four-current is conserved due to invariance of the theoryunder transformations of the unitary group U(1), the gauge group of Maxwells theory.Observe that the invariance of a physical system under Lorentz transformations has todo only with changes of frames. In fact, whereas a localLorentz transformation relatesdifferent classes of frames, a global Lorentz transformation relates frames inside one ofthese classes.

2.2 The Gauge Potential

The geometrical setting of teleparallel gravity a gauge theory for the translationgroup is the tangent bundle: at each point x of spacetime there is attached aMinkowski tangent-space on which the gauge transformations take place. A gaugetransformation is defined as a local (point-dependent) translation of the tangent-spacecoordinates,

xa

= xa + a, (2.1)

with a a(x) the transformation parameters (see Fig. 2.1). In terms of the genera-

xa

xa

T x

R

x

R

Figure 2.1: Spacetime with the Minkowski tangent space at x.

tors of infinitesimal translations

Pa = /xa a, (2.2)

the corresponding infinitesimal transformation can be written in the form

xa = bPb xa. (2.3)

Of course, as an Abelian group, the translation generators satisfy the commutation

relation[Pa, Pb] = 0. (2.4)

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It should be remarked that, due to the peculiar character of translations, any gaugetheory including these transformations will differ from the usual internalthat is,Yang-Mills typegauge models in many ways, the most significant being the presence

of a tetrad field. This means essentially that the gauge bundle will always presentthe soldering property [13], and consequently the internal and external sectors of thetheory will be connected.

As a gauge theory for the translation group, the fundamental field of teleparallelgravity is the gauge potential B = B(x

), a field assuming values in the Lie algebraof the translation group:

B = Ba Pa. (2.5)

It is important to remark that the gauge potential B, as well as the transformationparameter a, does not depend on xa, but only on x. This means that

cBa = 0 (2.6)

andc

a = 0. (2.7)

Consequently, like in any gauge theory, the basis Pa is gauge invariant. This is acrucial point for the consistency of the theory. In fact, if we write an element of thetranslational gauge group in the form

U = exp[a(x) Pa], (2.8)

condition (2.7) guarantees that, even for a point-dependent parameter, the translationgroup remains Abelian, that is,

[U, U] = 0.

2.3 Translational Covariant Derivative

Let us consider now a general source field . IfV is an open interval in spacetime, isrepresented by a local section V of the fiber bundle, which is given by a differentiableapplication of the form

V : V

1(V), (2.9)

where is a projection from the fiber into spacetime [14]. Now, a fiber bundle isalways locally trivial, that is, 1(V) is always diffeomorphic to V G, with G thegauge group:

1(V) V G. (2.10)Such diffeomorphism, usually called a local trivialization, is given by

fV : 1(V) V G. (2.11)

Denoting the group dimension by d, we see that a local section V is an application ofthe form

x f1V (x, A), (2.12)

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where x defines a point in spacetime, and the group coordinate A(x) A(x)defines a point in the fiber over x, with A = 1, . . . , d. Therefore, a general source field, defined as a section of the bundle, must depend on both coordinates x and A:

= (x, A). (2.13)

In the case of the internal, or YangMills gauge theories for compact groups, thedependence of any source field on the internal coordinates A is through a phase,

(x, A) = exp[iA(x) TA] (x), (2.14)

with TA an element of the corresponding Lie algebra. For the case of the non-compacttranslation group, each fiber is a Minkowski spacetime, and the group coordinates arerepresented by xa(x), a = 1, . . . , 4. In this case, the dependence of on xa(x) is not

through a phase, and we write simply

= (x, xa). (2.15)

As a consequence, under an infinitesimal translation of the tangent space coordinates,it transforms according to

= aa. (2.16)In this expressioin, stands for the functional change at the same xa, which is therelevant transformation for gauge theories. Of course, the above transformation is alsoat the same spacetime point x.

Using now the general definition of covariant derivative [14]

D = Ba a

, (2.17)

the translationalcovariant derivative of is found to be

D = + Ba a. (2.18)

Equivalently, we can write [16]

D = h

a

a, (2.19)where

ha = xa + Ba Dxa (2.20)

is a nontrivial that is, anholonomic tetrad field.

Comment 2.1 It is important to observe that, when we write the translational covariant derivativein the form (2.19-2.20), we have already chosen a specific class of frames to describe the theory. Thisclass of frame is characterized by the fact that, in the absence of gravitation, that is, when Ba =

a,it reduces to the class of inertial (or holonomous) frames. For this reason, it can be considered as apreferred class of frames. Of course, we could equally have chosen to describe the theory in anotherclass of frames. In this case, the tetrad would have the form

ha = Dxa + Ba. (2.21)

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In this expression,Dxa = xa + Aab xb,

with Aab a vacuum connection related to the inertial properties of the frame only. The anholonomy

of the tetrad, in this case, will include two parts: one related to the gravitational field Ba, andanother related the non-inertiality property of the frame. This class of frames, therefore, does notreduce to the class of inertial frames in the absence of gravitation. Although in the preferred class offrames (2.28) teleparallel gravity assumes a simpler form, it can of course be described in any class offrames. A more detailed discussion on this point will be presented in Chapter 11.

The gauge transformation ofBa can be obtained from the covariance ofD. Thetransformed covariant derivative is

D

=

+ Ba a . (2.22)

Substituting = + cc, (2.23)

we see that, in order to be covariant, that is, in order to satisfy

(D) aa(D) = aa[ + Baa], (2.24)

the translational gauge potential must transform according to

Ba = Ba a. (2.25)

2.4 Field Strength: TorsionAs usual in gauge theories, the field strength of teleparallel gravity is given by thecovariant derivative of the gauge potential B = B

aPa. From the general definition

(2.17) of covariant derivative, and using the transformation (2.25), it is found to be1

Ta = B

a Ba. (2.26)

Adding and subtracting the derivative of the trivial tetrad ea = xa, that is, adding

the vanishing term

e

a

ea

xa

xa

= 0,we get

Ta = (xa + Ba) (xa + Ba). (2.27)

Sincex

a + Ba = Dxa ha (2.28)

is the tetrad field, we obtain the usual expression

Ta = ha ha. (2.29)

1The magnitudes related with teleparallel gravity will be denoted with an over

.

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Like in any gauge theory, the field strength can also be obtained from the com-mutation relation of covariant derivatives. Using the translational covariant derivative(2.18), one can easily verify that

[D,D] =

TaPa, (2.30)

from where we see that the field strength

T =

TaPa is also a field assuming valuesin the Lie algebra of the translation group. By using the transformations (2.1) and(2.25), the tetrad is found to be gauge invariant:

ha

= ha. (2.31)

Consequently, as expected for an Abelian gauge theory,

Ta is also invariant under a

gauge transformation.

2.5 The Weitzenbock Connection

Multiplying Eq. (2.29) by ha yields

ha

Ta = ha h

a ha ha. (2.32)

The piece

hah

a =

(2.33)

is the so called Weitzenbock connection.

Comment 2.2 It should be remarked that R. Weitzenbock has never written such a connection.In spite of this, the name Weitzenbock connection is commonly used to denote the particular case ofa general Cartan connection.

Since torsion is defined by

T =

, (2.34)

we see that the field strength of teleparallel gravity is nothing but torsion written inthe tetrad basis:

T

a = h

a

T

. (2.35)

The Weitzenbock connection is a connection presenting a non-vanishing torsion, butvanishing curvature. In fact, as an explicit calculation shows, its curvature tensorvanishes identically:

R =

+

0. (2.36)

In terms of

T, the commutation relation (2.30) becomes

[D,D] =

T D. (2.37)

We see from this equation that torsion coincides with the anholonomy of the transla-tional gauge covariant derivative.

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From the definition (2.33) of the Weitzenbock connection, it follows naturally that

ha

h

a

ha = 0. (2.38)

This is the so called distant, or absolute parallelism condition, from which teleparallelgravity got its name. As a consequence of this condition, we see from the relation(1.17) that the corresponding Weitzenbock spin connection vanishes identically:

Aab hahb+ ha

hb = 0. (2.39)

This is one of the most important properties of teleparallel gravity. In this case, therelation (1.29) assumes then the form

Aca = 0

Kca. (2.40)

Of course, relations (2.39) and (2.40) are true only in one specific class of frames. In

fact, since

Aab is a connection, if it vanishes in a given frame, it will be different fromzero in any other frame related to the first by a local Lorentz transformation.

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Chapter 3

Gravitational Coupling Prescription

3.1 Preliminaries

The coupling of matter fields to gravitation encompasses two distinct parts. The firstis a consequence of the replacement of the Minkowski metric ab by a general pseudo-Riemannian metric g representing a true gravitational field. This part of the couplingis universal in the sense that it affects equally all matter fields. As we are going to see,this coupling follows naturally from the gaugefication of translations.

The second part of the gravitational coupling prescription comes from the couplingof the spins of the matter fields to gravitation. Of course, this part is not universalas it depends on the spin content of the field. As we are going to see, this part of

the coupling follows from the principle of general covariance [20]. This principle statesthat a special relativity equation can be made to hold in the presence of gravitation ifit is generally covariant, that is, it preserves its form under a general transformation ofthe spacetime coordinates. The process runs as follows. In order to make an equationgenerally covariant, a connection is always necessary, which is in principle concernedonly with the inertial properties of the coordinate system under consideration. Then,by using the equivalence between inertial and gravitational effects, instead of repre-senting inertial properties, this connection can equivalently be assumed to representa true gravitational field. In this way, equations valid in the presence of gravitationare obtained from the corresponding special relativity equations. Of course, in a lo-cally inertial coordinate system, they must go back to the corresponding equations ofspecial relativity. The principle of general covariance, therefore, can be seen as anactive version of the equivalence principle in the sense that, by making a special rela-tivity equation covariant, and by using the strong equivalence principle, it is possibleto obtain its form in the presence of gravitation. The usual form of the equivalenceprinciple, on the other hand, can be interpreted as its passive version in the sense thatthe special relativity equation must be recovered in a locally inertial coordinate sys-tem, where the effects of gravitation are absent. The crucial point is that, when thepurely inertial connection is replaced by a connection representing a true gravitationalfield, the principle of general covariance naturally defines a covariant derivative, andconsequently a gravitational coupling prescription.

Comment 3.1 It should be emphasized that general covariance by itself is empty of physical contentas any equation can be made generally covariant. Only when use is made of the strong equivalence

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principle, and the inertial compensating term is interpreted as representing a true gravitational field,the principle of general covariance can be seen as an alternative version of the strong equivalenceprinciple [21].

3.2 Translational Coupling Prescription

As we have already seen, the translational gauge-covariant derivative of a general mat-ter field is given by

D = + Baa. (3.1)

The corresponding gravitational coupling prescription, therefore, is achieved by replac-ing

D = + Baa. (3.2)

Now, due to the presence of the tetrad fieldha Dxa = xa + Ba, (3.3)

the covariant derivative (3.1) can be rewritten in the form

D = Dxa a. (3.4)

Despite mathematically equivalent to (3.1), there are conceptual differences betweenthem. In fact, since in the absence of gravitation

Dxa = x

a ea,we see that, in the case of the covariant derivative (3.4), the gravitational coupling

prescription is transferred to the covariant derivative of the coordinate xa. In fact, theminimal coupling prescription in this case is achieved by replacing

xa Dxa = xa + Ba, (3.5)

that is,ea ha. (3.6)

Equivalently, it amounts to

ab eaeb g ab hahb. (3.7)Since the generators Pa = a are derivatives which act on matter fields through their

arguments, every source field in nature will respond equally to their action, and conse-quently will couple equally to the translational gauge potentials. All of them, therefore,will feel gravitation the same. This is the origin of the concept ofuniversalityaccordingto teleparallel gravity.

Comment 3.2 In spite of the different structure, the derivative (3.4) keeps the translational co-variance property. To see it, we write

D = haa

= haa( + cc), (3.8)

where we have used the fact that ha is gauge invariant. We can then write

D = D + h

aa(

cc). (3.9)

Considering that ac = 0 and cha = 0, we get

(D) = cc(D). (3.10)

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3.3 Lorentz Coupling Prescription

3.3.1 General Covariance Principle

The description of the general covariance principle presented in Section 3.1 refers to itsusual holonomic version. An alternative, more general version of the principle can beobtained in the context of nonholonomic frames. The basic difference between thesetwo versions is that, instead of requiring that an equation be covariant under a generalcoordinate transformation, in the nonholonomic-frame version the equation is requiredto transform covariantly under a localLorentz rotation of the frame. Of course, in spiteof the different nature of the involved transformations, the physical content of bothapproaches are the same [22]. The frame version, however, is more general in the sensethat, contrary to the coordinate version, it holds for integer as well as for half-integer

spin fields.Like in the holonomic version, when the purely inertial connection is replaced by aconnection representing a true gravitational field, the non-holonomic principle of gen-eral covariance naturally defines a Lorentz-covariant derivative, and consequently also agravitational coupling prescription. The process of obtaining this coupling prescriptioncomprises then two steps. The first is to pass to a general nonholonomic frame, whereinertial effects which appear in the form of a compensating term are present.Then, by using the strong equivalence principle, instead of inertial effects, the compen-sating term can be replaced by a connection representing a true gravitational field. Inthis way, a gravitational coupling prescription is naturally obtained.

Comment 3.3It should be remarked that, like general coordinate covariance, Lorentz covarianceby itself is empty of dynamical content. This means essentially that the local Lorentz symmetry is not

a gauge, but a kinematic symmetry related to changes of frames only. In fact, the Lorentz connectionthat shows up from the requirement of local Lorentz covariance, even when considered as a truegravitational field, is not an independent field: it is completely determined by the translational gaugepotential or equivalently, by the tetrad field. One should not expect, therefore, any dynamicaleffect coming from a gaugefication of the Lorentz group.

Passage to a nonholonomic frame

Let us consider the Minkowski spacetime of special relativity. In this spacetime one cantake the frame a = a

as being a trivial (holonomous) tetrad, with components a.

Consider now a local, that is, point-dependent Lorentz transformation ab = ab(x).It yields the new frame ha = ha

, with components ha ha(x) given by

ha = a

b b. (3.11)

Notice that, on account of the locality of the Lorentz transformation, the new frameha is nonholonomous, [hb, hc] = f

abc ha, with the coefficient of nonholonomy given by

fabc = hb hc

(ha ha). (3.12)

Making use of the orthogonality property of the tetrads, we see from Eq. (3.11) that theLorentz group element can be written in the form b

d = hbd. From this expression,

it follows thatcd (hab

d) = 12

(fbca + fa

cb fcba) . (3.13)

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Let us consider now a vector field vc in the Minkowski spacetime. Its ordinaryderivative in the frame a is

avc = a

vc. (3.14)

Under a local Lorentz transformation, the vector field transforms according to Vd =dc v

c, and it is easy to see that avc = ba d

cDbVd, where

DaVc = haVc + cd (habd) Vb. (3.15)

In the frame ha, therefore, using the identity (3.13), the ordinary derivative (3.14)acquires the form

DaVc = haVc + 12 (fbca + facb fcba) Vb. (3.16)The freedom to choose any tetrad {ha} as a moving frame in the Minkowski spacetimeintroduces the compensating term 1

2(fb

c

a+ f

a

c

b fcba

) in the derivative of the vectorfield. This term, of course, is concerned only with the inertial properties of the frame.

Replacing inertia with gravitation

According to the general covariance principle, the derivative valid in the presence ofgravitation can be obtained from the corresponding Minkowski covariant derivative(3.16) by replacing the inertial compensating term by a connection Acab representinga true gravitational field. Considering a general Lorentz-valued connection presentingboth curvature and torsion, it follows from the relation (1.32) that [15]

Ac

ba Ac

ab = Tc

ab + fc

ab, (3.17)

with Tcba the torsion of the connection Acab. Use of this equation for three different

combination of indices gives

12

(fbca + fa

cb fcba) = Acba Kcba, (3.18)

whereKcba =

12

(Tbca + Ta

cb Tcba) (3.19)

is the contortion tensor. Equation (3.18) is completely general, and is the crucialpoint of the approach. It is actually an expression of the equivalence principle in thesense that, whereas its left-hand side involves only inertialproperties of the frames, itsright-hand side contains purely gravitationalquantities.

3.3.2 Lorentz Covariant Derivative

Substituting this expression in the covariant derivative (3.16), it becomes

DaVc = haVc + (Acba Kcba) Vb. (3.20)

In terms of the vector representation [24]

(Seb)cd = i (

ce bd cb ed) (3.21)

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of the Lorentz generators, the covariant derivative (3.20) can be rewritten in the form

DaVc = haV

c + i2 A

eba

Keba (Seb)

cd V

d. (3.22)

Although obtained in the case of a Lorentz vector field, the compensating term(3.13) can be easily verified to be the same for any field. In fact, denoting by

U U() = exp i2

cdJcd

the element of the Lorentz group in an arbitrary representation Jbc, it is easy to verifythat [22]

(haU)U1 = i

4(fbca + facb fcba) Jbc. (3.23)

In the case of a general field carrying an arbitrary representation of the Lorentz group,therefore, the covariant derivative (3.22) acquires the form

Da = ha + i2

Aeba Keba

Jeb. (3.24)

In the presence of curvature and torsion, therefore, the coupling prescription impliedby the general covariance principle amounts to replace

a Da. (3.25)

3.4 Gravitational Coupling Prescription

3.4.1 General CaseUsing the relations

Aeb = ha A

eba and K

eb = h

aK

eba, (3.26)

the covariant derivative (3.24) can be rewritten in the form

Da = haD, (3.27)where

D =

i2 Aab

KabJab (3.28)is a generalized FockIvanenko derivative. This means that the gravitational couplingprescription can be split in two parts: one, corresponding to the (universal) transla-tional coupling prescription, which is given by

ea ha, (3.29)

and another, corresponding to the (non-universal) spin coupling prescription, which isgiven by

D. (3.30)Put together, they yield the full gravitational coupling prescription in the presence of

curvature and torsion:ea haD. (3.31)

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3.4.2 General Relativity

The relevant spin connection of general relativity is the so called Ricci coefficient of

rotation

A, a torsionless connection assuming values in the Lie algebra of the Lorentzgroup:

A=12

Aab Sab. (3.32)

In this case, the compensating term (3.18) is written as

12

(fbca + fa

cb fcab) =

Acba, (3.33)

and consequently the coupling prescription in general relativity amounts to replace

ea

Da = ha

D, (3.34)

where

D = i2

Aab Sab (3.35)

is the usual Fock-Ivanenko covariant derivative [25]. Of course, in this case ha is a

general tetrad field, and not the tetrad of teleparallel gravity, which is written in termsof the translational gauge potential Ba.

It is important to remark that the spin connection

Aab is not an independent field.In fact, in terms of the Levi-Civita connection, it is written as [ 26]

A

a

b = h

a

hb

+ h

a

hb

ha

hb

, (3.36)

where

is the Levi-Civita covariant derivative of general relativity. The inverserelation is

= ha

Aab hb+ ha

ha ha

Dha. (3.37)We see from these expressions that the connection is completely determined by thetetrad or equivalently, by the metric. This means that, as already emphasized, thelocal Lorentz symmetry is not dynamical (gauge), but essentially a kinematic symme-try. In other words, like in special relativity, it is related to changes of frames only.In contrast to the global Lorentz transformations, which relate frames inside one spe-

cific class (inertial frames, for example), local Lorentz transformations relate framesbelonging to different classes (inertial and non-inertial frames, for example).

Comment 3.4 Since the spin connection is not an independent field, one cannot expect any dy-namics to be obtained from a gaugefication of the Lorentz symmetry.

A tetrad field can be used to transform Lorentz into spacetime tensors, and vice-versa. For example, a Lorentz vector field Va is related to the corresponding spacetimevector V through

Va = haV. (3.38)

As a consequence, the FockIvanenko covariant derivative

D of a general Lorentz

tensor field can also be related to the Levi-Civita covariant derivative

of the cor-responding spacetime tensor. For example, take the vector field Va for which the

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FockIvanenko derivative is written with the Lorentz generator (3.21). It is then aneasy task to verify that

DVa = ha

V, (3.39)

where

V = V +

V. (3.40)

However, in the case of half-integer spin fields, as is well known, there exists no space-time representation for spinor fields [27]. This means that no Levi-Civita covariantderivative can be defined for these fields. Thus, the only possible form for the co-variant derivative of a Dirac spinor , for example, is that given in terms of the spinconnection,

D = i2

Aab Sab , (3.41)

whereSab 12ab = i4 [a, b] (3.42)

is the Lorentz spin-1/2 generator, with a the Dirac matrices. We may say, therefore,

that the Fock-Ivanenko derivative

D is more fundamental than the Levi-Civita co-variant derivative

in the sense that it can be defined for both tensor and spinorfields.

3.4.3 Teleparallel Gravity

Since in teleparallel gravity the Weitzenbock spin connection

A vanishes, the compen-sating term (3.18) in this case assumes the form

12

(fbca + fa

cb fcba) = 0

Kcba. (3.43)

This means that the teleparallel spin connection, which will be denoted by

cab, isgiven by a zero connection minus the contortion tensor [28]:

ca = 0

Kca. (3.44)

Notice that the zero connection appearing in Eq. (3.44) is crucial in the sense that itis the responsible for making the right hand-side a true connection.

Like any connection assuming values in the Lie algebra of the Lorentz group, theconnection

=12

ab Sab (3.45)

is anti-symmetric in the first two indices. Furthermore, under an infinitesimal localLorentz transformation with parameters ab ab(x), it changes according to

ab =

Dab, (3.46)

where Dab = ab +

ac cb

cb ac (3.47)

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is the covariant derivative with

ab as the connection.The teleparallel coupling prescription, therefore, can be written in the form

ea Da = ha D, (3.48)

with

D = i2

ab Sab + i2

Kab Sab (3.49)

the teleparallel FockIvanenko derivative operator.For the specific case of a Lorentz vector field Vc, the teleparallel FockIvanenko

derivative is

DVc = Vc +

ca Va Vc

Kca Va. (3.50)

Using the relation (3.38), the corresponding teleparallel covariant derivative of the

spacetime vector V, which we denote by

D, is

DV = V

+

K

V. (3.51)

Analogously to the general relativity case, these derivatives satisfy the relation

DV = hc

DVc (3.52)

Since

K =

, (3.53)

we see that the teleparallel covariant derivative

D is nothing but the LeviCivita

covariant derivative

of general relativity rephrased in terms of the Weitzenbockconnection. The same is true of the FockIvanenko derivative

D.To fix the idea, let us consider the total covariant derivative of the tetrad, which in

the context of general relativity is given by

hb +

hb Aabha = 0. (3.54)

The teleparallel version of this expression can be obtained by substituting

and

Aab by their teleparallel counterparts. Using Eqs. (3.44) and (3.53), one gets

hb +

K

hb

0 Kab

ha = 0. (3.55)

Since the contortion terms cancel out, this relation is the same as the absolute paral-lelism condition

hb +

hb = 0. (3.56)

This shows the consistency of identifying the teleparallel spin connection as minus thecontortion tensor plus the zero connection.

The teleparallel FockIvanenko derivative (3.49) presents all necessary propertiesto define a consistent coupling prescription for teleparallel gravity:

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It is consistent with the strong equivalence principle. Actually, as we have seen,it is just the covariant derivative implied by the general covariance principle, anactive version of the strong equivalence principle.

It is completely completely equivalent with the coupling prescription of generalrelativity, even in the presence of spinor fields. Actually, it is just what is neededso that it turns out to be the minimal coupling prescription of general relativityrephrased in terms of magnitudes of the teleparallel structure.

For any matter field, it results completely equivalent to apply the minimal cou-pling prescription in the Lagrangian or in the field equation.

When used to describe the interaction of the electromagnetic field with gravita-tion, it does not violate the U(1) gauge invariance of Maxwells theory.

Comment 3.5 Sometimes, the Weitzenbock spin connection

A is assumed as the spin connectionof teleparallel gravity. In this case, it is said that, for spinor fields , the ensuing Fock-Ivanenkoderivative

D = i2

Aab Sa

b (3.57)

reduces to an ordinary derivative [10]:

D = . (3.58)For this reason, it is usually asserted that spinor fields break the equivalence between general relativityand teleparallel gravity. However, there are several problems associated with this coupling prescription.For example, it results different to apply this coupling prescription in the Lagrangian or in the field

equation, which is a rather strange result. When applied to the problem of a massive particle inthe presence of curvature and torsion, the ensuing equation of motion is found to be given by theauto-parallel equation. Although this equation reduces to the geodesic equation for vanishing torsion,it present two important drawbacks: it violates the equivalence principle, and it cannot be derivedfrom an action principle. Furthermore, when used to describe the interaction of the electromagneticfield with gravitation, the above coupling prescription violates the U(1) gauge invariance of Maxwells

theory. One can then say that there is no compelling arguments supporting the choice of

A as thespin connection of teleparallel gravity.

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Chapter 4

Particle Mechanics

4.1 Free Particles

The purpose of this chapter is to study the equation of motion of massive particles inthe presence of a gravitational field, in the context of teleparallel gravity. To fix thenotation, we begin by reviewing the case of a free particle.

4.1.1 Basic Notions

Let us consider the Minkowski spacetime, whose quadratic interval is

d2 = ab dxa dxb. (4.1)

Since the metric ab is Lorentz invariant,

ab = acb

d cd, (4.2)

the squared interval d2 is clearly invariant under global Lorentz transformations. Wedefine now the trivial tetrad

ea = xa. (4.3)

By trivial we mean a tetrad that can always be reduced to a holonomic form whenwritten in a specific class of frames. The spacetime metric

g0 = ab ea eb (4.4)

will also represent a Minkowski spacetime, whose quadratic interval is

ds20

= g0dx dx. (4.5)

In fact, from expressions (4.3) and (4.4), we see that

ds20

= ab dxadxb d2. (4.6)

The spacetime four-velocity of a free particle is defined by

u0

=dx

ds0. (4.7)

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Hence, along the trajectory of the particle, we can write

ds0 = u0 dx. (4.8)

This four-velocity can be written in terms of the tangent-space indices as

ua0

= ea u0

, (4.9)

where

ua0

=dxa

ds0. (4.10)

The spacetime interval along the particles trajectory can, therefore, be also written inthe form

ds0 = u0a dxa. (4.11)

4.1.2 Equation of Motion

A free particle of mass m is represented by the action integral

S= mcba

ds0. (4.12)

Using the relations obtained in the previous section, it can be rewritten in the forms

S =

mc

b

a

u0 dx =

mc

b

a

u0a dxa. (4.13)

Now, under a spacetime variation

x x + x, (4.14)the action (4.12) undergoes a variation

S= mcba

ds0. (4.15)

From the expression (4.6), and using the definition (4.10), one can verify that

ds0 = u0a dxa. (4.16)

Using the property [, d] = 0, the invariance of the action yields

S= mcba

u0a dxa = 0. (4.17)

Integrating by parts and neglecting the surface term,1 we obtain

mcb

a

du0a xa = 0, (4.18)

1The surface terms can be neglected because x = ax xa = 0 at the extremes of the integral.

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or equivalently,

mc

ba

eadu0ads0

ds0 x = 0. (4.19)

Due to the arbitrariness of x, we can equate to zero its coefficient, which yields

eadu0ads0

= 0. (4.20)

This is the equation of motion of a free particle.

Comment 4.1 Using the relation (4.9), the above equation of motion can be rewritten in the form

du0

ds0 ea u0 u0a = 0. (4.21)

Defining the inertial connection

= ea ea, (4.22)

it becomesdu0

ds0 u0 u0 = 0. (4.23)

This is the equation of motion in a general coordinate system {x}. In a Euclidian coordinate system,the connection vanishes, yielding the usual form of the equation of motion of a free particle:

du0

ds0= 0. (4.24)

4.2 Gravitationally Coupled Particles

4.2.1 Equation of Motion

Since in classical mechanics particles are not represented by fields, the gravitationalcoupling prescription cannot be achieved through the replacement (3.2). Instead, it hasto be represented by the replacememnt (3.5) or equivalently by (3.6). In consonancewith these replacements, the spacetime metric becomes

g0 = ab ea eb g = ab hahb, (4.25)

and consequently

ds20 = g0dx dx ds2 = gdx dx = ab ha hb. (4.26)Of course, when the tetrad is non-trivial, ds2 will no longer represent the quadraticMinkowski interval. Concomitantly, the four-velocity of a particle will change accordingto

u0

=dx

ds0 u = dx

ds(4.27)

and

ua0

= xa dx

ds0 ua = ha dx

ds= ha u

. (4.28)

Using the above expressions, the spacetime interval along the particles trajectory canbe rewritten in the form

ds = g u dx = ab u

a hb. (4.29)

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Now, as we have already seen, the action representing a free particle is

S=

m c

b

a

u0a

xadx. (4.30)

The corresponding action in the presence of gravitation, therefore, can be obtained byapplying the coupling prescription

u0a ua and xa Dxa, (4.31)with D given by (3.5). The result is

S= m cba

[ua dxa + Ba ua dx

] . (4.32)

This is the teleparallel version of the action. The first term of Eq. (4.32) represents theaction of a free particle in the gravitationally modified spacetime, and the secondthe coupling of the particles mass with the gravitational field. In fact, a variation ofthe first term of the action (4.32) yields the equation of motion

duads

= 0,

which represents a free particle in the gravitationally modified spacetime.Observe that, due to the gauge structure of teleparallel gravity, the action assumes

a form similar to the action of a charged particle in an electromagnetic field. In

fact, if the particle has additionally an electric charge e and is in the presence of anelectromagnetic field A, the total action has the form

S= ba

pa dx

a + paBa dx

+e

cA dx

, (4.33)

with pa = mcua the particle four-momentum. Observe also that, by using the identity

ua dxa + Ba ua dx

= ua(dxa + Ba) = ua h

a ds, (4.34)with ds the spacetime line element, the action (4.32) reduces to

S= m cba

ds, (4.35)

which is the usual general relativity form of the action. We see in this way that (4.32)is equivalent to the corresponding general relativity action. When written in the form(4.35), however, the interaction of the particle with the gravitational field is describedby the metric tensor g, and not by the coupling of the gauge potential B

a with the

particle four-momentum pa.

Comment 4.2 It is important to remark that, although the spacetime metric is not the field variable

of teleparallel gravity, it is changed by the presence of a gravitational field. As a consequence, the partof the action representing a free particle is written in this spacetime, which is no more a Minkowskispacetime. It does not represent, therefore, a true free particle, whose motion would take place in a

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Minkowski spacetime. Only when Ba = 0, the first term of the action (4.32) will reduce to the actionof a free particle, that is, the action of a particle in Minkowski spacetime. This is similar to whathappens in the ordinary application of the equivalence principle in general relativity. To see that, let

us consider the geodesic equation du

ds+

u

u = 0. (4.36)

In a locally inertial coordinate system,

= 0, and the geodesic equation is said to reduce to theequation of motion of a free particle:

du

ds= 0. (4.37)

However, this is not completely true because, even when the connection vanishes at a point, theinterval ds still refers to the gravitationally modified spacetime remember that curvature can neverbe made to vanish. Only when the gravitational field (that is, curvature) is assumed to vanish, dswill represent the interval of a Minkowski spacetime, and ( 4.37) will, in fact, represent the equationof motion of a free particle.

The invariance of the action (4.32) under the spacetime variation (4.14) yields

ba

[ua dxa + ua dx

a + Ba ua dx + Ba ua dx

+ Ba ua dx] = 0.

Using the identity ha = dxa + Ba, it becomes

ba

[ua dxa + Ba ua dx

+ ha ua + Ba ua dx

] = 0. (4.38)

Comment 4.3 Let us consider the quadratic spacetime interval

ds2 = ab ha hb.

As an explicit calculation shows,

(ds) = ab ha hb

1ds

.

Using the definition (4.28), it becomes

(ds) = ab ha ub + ab u

aub (ds). (4.39)

Since ab uaub = 1, we get

ab ha ub ha ua = 0. (4.40)

Using (4.40), the variation (4.38) becomes

ba

[ua dxa + Ba ua dx

+ Ba ua dx] = 0. (4.41)

Integrating by parts the first and the last terms and neglecting the surface terms, weget b

a

[dua xa Ba ua dx + dBa ua x + Ba dua x] = 0. (4.42)

Substitutingxa = x

a x, Ba = Ba x

, dBa = Ba dx

, (4.43)

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and using that

Ta = Ba Ba, (4.44)

we obtain ba

ha

duads

Ta ua u

x ds = 0. (4.45)

From the arbitrariness of x, we get finally

haduads

=

Ta ua u, (4.46)

or alternatively,duads

=

Tbac ub uc. (4.47)

This is the force equation governing the motion of the particle, in which the teleparallelfield strength

Tbac plays the role of gravitational force.Since the tetrad ha and the tangent space metric ab are both gauge invariant, the

spacetime metricg = ab h

a ha, (4.48)

as well as the corresponding interval

ds2 = gdx dx (4.49)

will also be gauge invariant. As a consequence, the four-velocity

ua = ha

1ds

= ha u

(4.50)

is also found to be gauge invariant. This is consistent with the fact that both ha andu are gauge invariant. Now, under a gauge transformation, we have that

dxa = dxa + da (4.51)

andBa = Ba da. (4.52)

Therefore, the action integral (4.32), as well as the corresponding equation of mo-

tion (4.47), are both gauge invariant.

4.2.2 Strong Equivalence Principle

Let us begin by rewriting the force equation (4.46) in a purely spacetime form. Inorder to do it, we use the relation

haduads

=

duds

u u, (4.53)

where

is the spacetime particle four-acceleration. We then get

duds

u u =

T u u. (4.54)

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The lefthand side of this equation is the Weitzenbock covariant derivative of the four-velocity u along the world-line of the particle. The presence of the torsion tensor onits righthand side, as already stressed, shows that in teleparallel gravity torsion plays

the role of gravitational force. By using the identity

Tu u = Ku u, (4.55)

this equation can be rewritten in the form

duds

K

u u

= 0. (4.56)

Using then the identity (6.11), it becomes

duds

u u = 0. (4.57)

This is precisely the geodesic equation of general relativity, which means that the trajec-tories followed by spinless particles are geodesics of the underlying Riemann spacetime.In a locally inertial coordinate system, the first derivative of the metric tensor vanishes,and the LeviCivita connection

=12

g (g+ g g) (4.58)

vanishes as well. Consequently, the geodesic equation (4.57) becomes the equation ofmotion of a free particle. This is the usual version of the (strong) equivalence principleas formulated in general relativity [20].

It is important to notice that, by using the torsion definition (2.34), the forceequation (4.54) can be written in the alternative form

duds

u u = 0. (4.59)

Since

is not symmetric in the last two indices, this is not a geodesic equation.This means that the trajectories followed by spinless particles are not geodesicsorautoparallelsof the underlying Weitzenbock spacetime. In a locally inertial coordi-

nate system, the first derivative of the metric tensor vanishes, and if we remember thatg = ha ha, we see that

g

+

= 0. (4.60)

This means that, in this coordinate system, the Weitzenbock connection

becomesskewsymmetric in the first two indices. In this coordinate system, therefore, owing tothe symmetry ofu u, the force equation (4.59) reduces to the equation of motion of afree particle. This is the teleparallel version of the (strong) equivalence principle [16].

4.2.3 Newtonian Limit

Let us consider now the Newtonian limit, which is obtained by assuming that the grav-itational field is stationary and weak. This means respectively that the time derivative

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of Ba vanishes, and that |Ba| 1. Accordingly, all particles will move with a suf-ficient small velocity so that2 ui can be neglected in relation to u0. In this case, theequation of motion (4.59) can be written as

d2xds2

00 c2

dt

ds

2= 0. (4.61)

Now, taking into account that the field is stationary, up to first order in Ba, we obtainfrom the definition (2.33) of the Weitzenbock connection that

00 = B00. (4.62)

The equation of motion (4.61) is then equivalent to the two equations

d2xds2

= c2 B00

dtds

2

, (4.63)

andd2t

ds2= 0, (4.64)

where x0 = x0 = ct, the components ofx are given by xi = xi, and stands for the

gradient. The solution of the second equation is that dt/ds equals a constant. Dividingby (dt/ds)2 puts the first equation in the form

d2

xdt2 = c2 B00. (4.65)

The corresponding Newtonian result is

d2x

dt2= , (4.66)

where

= GM|x| (4.67)

is the gravitational potential. Comparing (4.65) with (4.66) we see that

c2 B00 = . (4.68)

2We use i ,j ,k, . . . = 1, 2, 3 to denote the space components of the spacetime indices.

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Chapter 5

A Global Formulation for Gravity

5.1 Phase Factor Approach

The fundamental field of teleparallel gravity is the gauge potential Ba. In this for-mulation, gravitation becomes quite analogous to electromagnetism. Based on thisanalogy, and relying on the phase-factor approach to Maxwells theory, a teleparal-lel nonintegrable phase-factor approach to gravitation can be developed [29], whichrepresents the quantum mechanical version of the classical gravitational Lorentz force.

As is well known, in addition to the usual differentialformalism, electromagnetismpresents also a global formulation in terms of a nonintegrable phase factor [30]. Ac-cording to this approach, electromagnetism can be considered as the gauge invariant

action of a nonintegrable (path-dependent) phase factor. For a particle with electriccharge q traveling from an initial point P to a final point Q, the phase factor is givenby

e(P|Q) = exp

iq

c

QP

A dx

, (5.1)

where A is the electromagnetic gauge potential. In the classical (non-quantum) limit,the nonintegrable phase factor approach yields the same results as those obtained fromthe Lorentz force equation

dua

ds=

q

mc2Fab u

b. (5.2)

In this sense, the phase-factor approach can be considered as the quantum generaliza-tion of the classicalLorentz force equation. It is actually more general, as it can be usedboth on simply-connected and on multiply-connected domains. Its use is mandatory,for example, to describe the Bohm-Aharonov effect, a quantum phenomenon takingplace in a multiply-connected space.

In analogy with electromagnetism, Ba can be used to construct a global formula-tion for gravitation. To start with, let us notice that the electromagnetic phase factorhas the form

e(P|Q) = exp i

Se

, (5.3)

where Se is the action integral describing the interaction of the charged particle with

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the electromagnetic field. Similarly, we can write the gravitational phase factor as

g(P|Q) = exp 1

Sg , (5.4)

where Sg is the action integral describing the interaction of a particle of mass m withgravitation. According to Eq. (4.32), this action is given by

Sg = QP

Bapa dx, (5.5)

with pa the particle four-momentum. However, since the phase factor is essentially aquantum object, the momentum now is to be written as

pa = i a. (5.6)Furthermore, because a coincides with the translational generators Pa, the actionassumes the form

Sg = iQP

B dx, (5.7)

with B = BaPa. The gravitational phase factor is then written as [29]

g(P|Q) = exp

i

QP

Ba a dx

. (5.8)

Similarly to the electromagnetic phase factor, it represents the quantum mechanicallaw that replaces the classicalgravitational Lorentz force equation (4.46). Also similarto the electromagnetic case, the integral in the phase factor changes by a surface termunder gauge transformations.

Apparently, the gravitational phase factor (5.8) seems not to depend on Plancksconstant . However, note that, when acting on a wave function representing a massiveparticle, the translation generators are to be replaced by

a i

pa, (5.9)

which depends on . In other words, is hidden in the translation generators whichenters the momentum definition in quantum mechanics.

5.2 Colella-Overhauser-Werner Experiment

As a first application of the gravitational nonintegrable phase factor (5.8), we considerthe Colella-Overhauser-Werner (COW) experiment [31]. It consists in using a neutroninterferometer to observe the quantum mechanical phase shift of neutrons caused bytheir interaction with Earths gravitational field, which is usually assumed to be New-tonian. Now, as we have already seen, a Newtonian gravitational field is characterizedby the condition that only B00 = 0. Furthermore, as the experience is performed withthermal neutrons, it is possible to use the small velocity approximation. In this case,

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the gravitational phase factor (5.8), when acting in the wave function of such particles,can be written as

g(P|Q

) = exp im

Q

P c

2

B00 dt

, (5.10)

where we have used (5.9), as well as the fact that u0 = 1 for thermal neutrons. Inthe Newtonian approximation, c2 B00 can be identified with the (homogeneous) EarthNewtonian potential [see Eq. (4.68)], which is given by

c2 B00 = g z. (5.11)

In this expression, g is the gravitational acceleration, assumed not to change signifi-cantly in the region of the experience, and z is the distance from Earth taken fromsome reference point. Consequently, the phase factor can be rewritten in the form

g(P|Q) = exp

img

Q

P

z(t) dt

exp i. (5.12)

AB

C

D

EF

r

s

2

Figure 5.1: Schematic illustration of the COW neutron interferometer.

Let us now compute the phase through the two trajectories of Fig. 5.1. First, weconsider the trajectory BDE. Assuming that the segment BD is at z = 0, we obtain

BDE =mg

ED

z(t) dt. (5.13)

For the trajectory BCE, we have

BCE =mg

CB

z(t) dt +mgr

EC

dt. (5.14)

As the phase contribution along the segments DE and BC are equal, we get

BCE BDE = mgr

EC

dt. (5.15)

Since the neutron velocity v is constant along the segment CE, we have

E

C

dt sv

=sm

h, (5.16)

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where s is the length of the segment CE, and = h/(mv) is the de Broglie wavelengthassociated with the neutron. We thus obtain

= s 2grm2

h2, (5.17)

which is exactly the gravitationally induced phase difference predicted for the COWexperience [31].

5.3 Gravitational Aharonov-Bohm Effect

As a second application we use the phase factor (5.8) to study the gravitational analogof the Aharonov-Bohm effect [32]. The usual (electromagnetic) Aharonov-Bohm effectconsists in a shift, by a constant amount, of the electron interferometry wave pattern,in a region where there is no magnetic field, but there is a nontrivial gauge potential Ai.Analogously, the gravitational Aharonov-Bohm effect will consist in a similar shift ofthe same wave pattern, but produced by the presence of a gravitational gauge potentialB0i.

Comment 5.1 As the phase shift of the COW esperiment is produced by the coupling of theneutron mass with the component B00 of the translational gauge potential, it can be considered as agravitoelectric Aharonov-Bohm effect. A similar denomination is used in the electromagnetic case [33].

Phenomenologically, this kind of effect might be present near a massive rapidly rotatingsource, like a neutron star, for example. Of course, differently from an ideal apparatus,

in a real situation the gravitational field cannot be eliminated, and consequently thegravitational Aharonov-Bohm effect should be added to the other effects also causinga phase change.

Let us consider first the case in which there is no external field at all. If the electronsare emitted with a characteristic momentum p, then its wavefunction has the de Brogliewavelength = h/p. Denoting by L the distance between slit and screen (see Fig. 5.2),and by d the distance between the two holes, when the conditions L , L x andL d are satisfied, the phase difference at a distance x from the central point of thescreen is given by

0(x) =2xd

L. (5.18)

This expression defines the wave pattern on the screen.We consider now the case in which a kind of infinite gravitational solenoid pro-

duces a purely static gravitomagnetic field flux concentrated in its interior. In the idealsituation, the gravitational field outside the solenoid vanishes completely, but there is anontrivial gauge potential B0i. When we let the electrons to move outside the solenoid,phase factors corresponding to paths lying on one side of the solenoid will interfere withphase factors corresponding to paths lying on the other side, which will produce anadditional phase shift at the screen. Let us then calculate this additional phase shift.The gravitational phase factor (5.8) for the physical situation described above is

g(P|Q) = exp imc

Q

P

u0 B0 dr

, (5.19)

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SOLENOID

(2)

x

(1) z

d

L

Figure 5.2: Schematic illustration of the Aharonov-Bohm electron interferometer.

where B0 is the vector with components B0i = B0i. Sinceu0 = [1 (v2/c2)]1/2,

and considering that the electron velocity v is constant, we can write

g(P|Q) = expimc

QP

B0 dr

. (5.20)

Now, denoting by 1 the phase corresponding to a path lying on one side of thesolenoid, and by 2 the phase corresponding to a path lying on the other side, the

phase difference at the screen will be

2 1 = mc

B0 dr. (5.21)

It can be rewritten in the form

=E c

, (5.22)

where E= mc2 is the electron kinetic energy, and

=

B0 dr =

( B0) d

H d (5.23)

is the H flux inside the solenoid. In components, the gravitational field H is writtenas

Hi =1

2ijk (jB0k kB0j) = 1

2ijk

T0jk , (5.24)

with

T0jk the torsion of the Weitzenbock connection that is, the teleparallel fieldstrength. Remembering that the axial torsion is defined by

a = (1/6)

T , (5.25)

it is then an easy task to verify that Hi =

ai, with

ai = 12

0ijk

T0jk (5.26)

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the space components of the axial torsion. This shows that Hi coincides with the axialtorsion, which in turn represents the gravitomagnetic component of the gravitationalfield [34].

Expression (5.22) yields the phase difference produced by the interaction of theparticles kinetic energy with a gauge potential, which gives rise to the gravitationalAharonov-Bohm effect. As this phase difference depends on the energy, it appliesequally to massive and massless particles. There is a difference, though: whereas formassive particles it is a genuine quantum effect, for massless particles, due to the theirintrinsic wave character, it can be considered as a classical effect. In fact, for E= h,Eq. (5.22) becomes

=

c, (5.27)

and we see that, in this case, the phase difference does not depend on the Plancks

constant.In contrast with the electromagnetic Aharonov-Bohm effect, the phase difference

in the gravitational case depends on the particle kinetic energy, which in turn dependson the particles mass and velocity. Like the electromagnetic case, however, the phasedifference is independent of the position x on the screen, and consequently the wavepattern defined by (5.22) will be whole shifted by a constant amount. It is also im-portant to mention that the requirement of invariance of the gravitational phase factorunder the translational gauge transformation (2.25) implies that

E= nhc, (5.28)

with n an integer number. Differently from the elec

RG - Aldrovandi - Teleparallel Gravity

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