Dimensional Reduction in Lovelock Gravitybjanssen/text/JavierMoreno-TFM.pdf · Dimensional Reduction in Lovelock Gravity F. Javier Moreno Trabajo Fin de M aster at the Instituto de

Dimensional Reduction in LovelockGravity

F. Javier Moreno

Madrid 2017

Dimensional Reduction in LovelockGravity

F. Javier Moreno

Trabajo Fin de Masterat the Instituto de Fısica Teorica UAM/CSIC

Madrid

presented byFrancisco Javier Moreno Gonzalez

under the supervision ofBert Janssen and Jose Alberto Orejuela

from the Universidad de Granadaand

the tutoring of Tomas Ortın Miguelfrom the Instituto de Fısica Teorica UAM/CSIC

Madrid, 30th August 2017

Examination Commitee:

Carlos Eiroa de San Francisco

Rosa Domınguez Tenreiro

Benjamin Montesinos Comino

Day of presentation:

8th September 2017

Contents

Abstract vii

1 Introduction 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives and Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Lovelock Gravity 32.1 Four-Dimensional Home Sweet Home . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Going Out a Bit to Explore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 You Came to the Wrong Neighborhood? . . . . . . . . . . . . . . . . . . . . . . . 5

3 Gravitation as a Gauge Theory 73.1 Poincare Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1.1 Poincare Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.2 Global Poincare Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.3 Local Poincare Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.4 Curvature Showing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 The Quest for the Cosmological Constant in (A)dS . . . . . . . . . . . . . . . . . 123.3 Expanding the Brain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Kaluza-Klein Dimensional Reduction 174.1 One-Dimensional Compactification . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Two-Dimensional Compactification . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 N -Dimensional Compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Conclusions 25

A Conventions 27A.1 The Levi-Civita Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

B Geometry from the Tangent Space 29B.1 Vielbein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29B.2 Vielbein Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31B.3 Curvature Tensors from the Tangent Space . . . . . . . . . . . . . . . . . . . . . 32

v

vi

Abstract

In this work, the compatibility of Lovelock gravity and Kaluza-Klein dimensional reduction isstudied. First, the Lovelock action is presented as a generalisation of the Einstein-Hilbert actionfor an arbitrary number of dimensions. This theory involves an action that consists of a sumof subsequently higher-order terms but it does not provide a numerical value for the couplingconstants. In order to obtain them, the gauge principle is applied giving the value of the constantsof the theory in D and D − 2 dimensions. With this two actions, Kaluza-Klein dimensionalreduction is applied over two dimensions and after comparing the coupling constants, they arefound to be different. The dimensional reduction is later applied to an arbitrary number N ofcompact dimensions in order to find a particular case of compatibility without any successfulresult.

vii

Chapter 1

Introduction

1.1 Background and Motivation

Einstein’s general relativity is a successful theory of gravitation that has explained a wide rangeof phenomena since it was published in 1915. In this theory, the Einstein-Hilbert action, whichis linear in terms of the curvature, yields second-order differential equations of motion of themetric. Although others terms of higher curvature can be added to the action, generally theyinvolve higher-order derivatives of the metric and therefore they are excluded.

However, in 1938, Lanczos proposed a quadratic term of curvature which leads to second-order equations of motion as well and therefore is not to be excluded when considering a validaction for gravity. This term is known as Gauss-Bonnet term. It turns out that in a four dimen-sional world with three spatial directions and one temporal direction, the Gauss-Bonnet term isa topological term and therefore it does not contribute to the equations of motion. On the otherhand, if the gravitational theory is made to live in a higher dimensional world, five or more, theGauss-Bonnet term becomes dynamical and then it does modify the equations of motion.

In 1971, Lovelock extended the result of Lanczos to an arbitrary number of dimensions. In histheory, the action was built as a sum of subsequently higher-order curvature terms in a genericnumber of dimensions, each of them with a coupling constant. He was also able to give a descrip-tion of which of the terms are dynamical in the equations of motion and which do not contributeto them relating the number of dimensions to the curvature order. A great success of this theoryis to include the Einstein-Hilbert action as a four-dimensional particular case. The fact of being asolving, inclusive theory concedes to Lovelock gravity an important argument to be studied withfurther detail: its implications, predictions, compatibility with other well-established theories,etc. Regrettably, Lovelock gravity suffers from a major problem, it does not provide the valuesof the coupling constants of each term. A complementary theory must be used in order to obtaintheir numerical values and this theory it is found in the gauge formulation of gravity.

Previously in 1956, Utiyama applied the gauge principle to the Poincare algebra and obtainedsuccessfully Einstein’s general relativity. This is an impressive result because if Einstein had notdiscover general relativity geometrically, Utiyama would have done algebraically in a lapse ofalmost forty years. With a modification of the Poincare algebra and extending the number ofdimensions, the theory can provide the undetermined coupling constants that the Lovelock the-

1

Dimensional Reduction in Lovelock Gravity 1. Introduction

ory alone was not able to yield.

However, an unresolved question when applying the gauge principle in Lovelock theory isthe uniqueness of the coupling constants for each term in different dimensions. Since their valuedepends on the dimension in which the theory lives, it is unclear whether it will coincide withanother higher-dimensional theory in which dimensional reduction has been applied. In otherwords, if the coupling constants of the (D−N)-dimensional Lovelock action are computed usingthe gauge principle, will they coincide with the coupling constants of a D-dimensional Lovelockaction in which N dimensions have been reduced?

The procedure of dimensional reduction is described in Kaluza-Klein theory. In 1919, Kaluzaconsidered a pure gravitational theory in five dimensions and decomposed it in a four dimen-sional gravitational field coupled to electromagnetic and to a massless scalar field. In 1926, Kleinsolved some issues present in the theory of Kaluza. His major contribution was its consider-ation of the fifth dimension to be a compact dimension. With this approach they unified thegravitational and electromagnetic interaction. Even though the theory was discarded because itsphenomenological implications could not meet the observations, it has inspired many unificationtheories due to its simplicity, elegance and mathematical consistence. This last attributes makethe dimensional reduction suitable to compare the coupling constants of the two Lagrangians.

When applying dimensional reduction from a higher to a lower dimensional theory a consistenttruncation can be made so that the theory keeps only the gravitational part allowing to comparetwo purely gravitational actions, namely, the Lovelock actions discussed above.

1.2 Objectives and Procedure

The objective of the present work is to provide an answer to the unresolved question posed inthe previous section.

In order to do so, the Lovelock action is to be presented as the D-dimensional generalisation ofthe Einstein-Hilbert and Gauss-Bonnet actions. When the problem with the coupling coefficientsis presented, the gauge formulation of the Einstein-Hilbert action is discussed from Poincare al-gebra. Since gauging this algebra the cosmological constant is not included a modification isnecessary. The solution is found gauging the (A)dS algebra, and doing so the zeroth-order andthe second-order curvature term, i.e. the cosmological constant and the Gauss-Bonnet term,are included in the theory. Once this action is reproduced, a higher dimensional extension isconstructed following the structure of the Lovelock gravity presented in the previous chapterusing the Levi-Civita tensor.

Since this extension has its coupling constants determined, a D-dimensional and (D − N)-dimensional actions are considered. After applying dimensional reduction to the former, the twoactions will have the same dimensions and then the coupling constants are compared. If theycoincide, the uniqueness of the coupling constants is granted and they do not depend on thedimensions in which they are built.

2

Chapter 2

Lovelock Gravity

In this section, the Lovelock action is to be presented as the D-dimensional generalisation of theEinstein-Hilbert and Gauss-Bonnet actions. The reason of choosing these two is the familiaritywith them so that the construction of Lovelock seems a natural extension.

2.1 Four-Dimensional Home Sweet Home

The action of general relativity is such that, through the principle of least action, yields theEinstein field equations of general relativity. Its expression is given by the sum of the Einstein-Hilbert Lagrangian LEH plus a term Lm describing any matter field that appears in the theory,

SGR =

∫d4x√|g| (LEH + Lm) =

∫d4x√|g|(R

2κ+ Lm

), (2.1)

where κ is the Newton’s constant, |g| is the absolute value of the determinant of the metric gµνand R is the Ricci scalar1. The Ricci scalar is defined as the trace of the Ricci curvature tensorwhich in turn is defined as the trace of the Riemann tensor,

R = R µµ = gµνRµν = gµνR ρ

µρν = gµνgρσRµρνσ. (2.2)

Besides, the Riemann tensor is written in terms of the Christoffel symbols as

Rµρνσ = gαµRαρνσ = gαµ

(∂νΓαρσ − ∂σΓαρν + ΓανλΓλρσ − ΓασλΓλρν

)(2.3)

and the Christoffel symbols are written in terms of the metric as

Γαρσ =1

2gαβ (∂ρgσβ + ∂σgρβ − ∂βgσρ) . (2.4)

Observe that the Ricci scalar, as (2.3) and (2.4) indicate, includes terms with second-orderderivatives of the metric

R = R(g, ∂g, ∂2g). (2.5)

This would imply that the equations of motion incorporate higher than second-order derivativesof the metric. However, instead of such scenario, the variation of the action (2.1) yields, as

1From now on the Ricci scalar will be also referred as the linear curvature term of a gravitational theory.

3

Dimensional Reduction in Lovelock Gravity 2. Lovelock Gravity

mentioned above, the Einstein field equations,

Rµν −R

2gµν = −κ 2√

|g|

∂(√|g|Lm

)∂gµν

, (2.6)

which are expressed only in terms of second-order derivatives of the metric. This is becausewhen computing the variation, the higher than second-order terms end up vanishing each other.Equations (2.6) are usually written as

Gµν = −κTµν , (2.7)

where the two introduced tensors Gµν and Tµν are respectively the Einstein and stress-energytensors. All these expressions look familiar, they are included in every lecture of general relativity.

The action (2.1) was modified shortly after published to include an additional term, thecosmological constant, making it to look like

SGRΛ =

∫d4x√|g|(R

2κ− Λ + Lm

), (2.8)

making the Einstein field equations to be

Gµν + Λgµν = −κTµν . (2.9)

Despite the modification, action (2.1) still conserves the interesting property mentioned aboveand keeps yielding second-order differential equations of motion.

2.2 Going Out a Bit to Explore

Although actions with higher-order curvature terms yield in general equations of motion withhigher than second-order derivatives, Lanczos discovered that the second-order curvature termαR2 +βRµνR

µν + γRµνρσRµνρσ can yield valid equations of motion when the constants take the

values (α, β, γ) = (1,−4, 1). This term is known as the Gauss-Bonnet term

LGB = R2 − 4RµνRµν +RµνρσR

µνρσ. (2.10)

When choosing these particular values for the constants α, β and γ, the higher than second-orderderivative terms cancel each other in the same way as the action of general relativity (2.8). If thisis the case, there is no reason to exclude the Gauss-Bonnet term from the action and thereforeit should be borne in mind when computing the equations of motion. The interesting fact isthat when the theory lives in four dimensions the Gauss-Bonnet term does not contribute to theequations of motion because it is a topological term. However, when considering a theory in fivedimensions or higher, this term does become dynamical and therefore must be included in theaction,

S5 =

∫d5x√|g|[R

2κ− Λ + LM + λ2

(R2 − 4RµνR

µν +RµνρσRµνρσ

)]. (2.11)

Notice that at this point the coupling constant λ2 of the Gauss-Bonnet term remains unknownbecause of the lack of means to determine it. Later, a discussion about its determination will be

4


done. For the time being, a quick look to expression (2.11) allows to separate the terms in theLagrangian for subsequently higher-order curvatures, from now, each term will be denoted Rd,where d indicates the order. Using this convention, the zero-curvature term can be identified asthe cosmological constant, the Ricci scalar as the first order and the Gauss-Bonnet term as thesecond one, respectively,

R0 = Λ, (2.12a)

R1 = R, (2.12b)

R2 = R2 − 4RµνRµν +RµνρσR

µνρσ. (2.12c)

Expressions (2.12b) and (2.12c) can also be written in terms of only Riemann tensors and theantisymmetric product of Kronecker deltas as

R1 = δα[µ δβν]R

µναβ = R, (2.13a)

R2 = 6δα[µ δβν δ

γρδδσ]R

µναβ R ρσ

γδ = R2 − 4RµνRµν +RµνρσR

µνρσ, (2.13b)

allowing to infer how the third order curvature term R3 is to be constructed as well as otherhigher curvature terms. It would be very convenient for the future if these relations are written interms of contractions of Levi-Civita tensors instead of antisymmetric products of Kronecker. Inappendix A.1, the relation between these two objects is discussed and using it, the two curvatureterms yield

R1 =|g|4εµνρσε

αβρσR µναβ = R, (2.14a)

R2 = −|g|4εµνρσε

αβγδR µναβ R ρσ

γδ = R2 − 4RµνRµν +RµνρσR

µνρσ. (2.14b)

Notice that the absolute value of the determinant in these two expressions comes from thecontraction of Levi-Civita tensor.

2.3 You Came to the Wrong Neighborhood?

As commented before, it is very natural to try to extend the curvature terms when looking(2.14a) and (2.14b). The Lovelock Lagrangian is constructed taking into account this propertyconsidering an arbitrary number terms of curvature,

LL =

bD/2c∑d=0

λdRd, (2.15)

where bD/2c is the integer part of D/2 and λd is the coupling constant2 of the curvature termRd, which according to the previous discussion takes the form

Rd =(−1)d−1 |g|

2dεµ1ν1...µD/2νD/2

εα1β1...αD/2βD/2

d∏i=1

R µiνiαiβi

(2.16)

2Notice that now makes sense to identify λ2 as the coupling constant of the Gauss-Bonnet term since it is theterm with second-order curvature (d = 2).

5


for d ≥ 1 and identifying the cosmological constant term of zero curvature,

R0 = Λ. (2.17)

In this Lagrangian, along with the zero-curvature term already discussed, the linear curvatureterm d = 1 which is completely identified with the Einstein-Hilbert term (2.1) as well as thesecond-order curvature term d = 2, which is the Gauss-Bonnet are present. Remember that forD = 4 Gauss-Bonnet term is only a topological term and therefore it does not contribute tothe equations of motion. The higher-order curvature terms are identically zero and this can bechecked from expression (2.16). If a term of order three is considered, the Levi-Civita must havesix indices whereas there are only four different. Thus, any higher-order curvature vanishes.

The whole reasoning of the previous chapter can be extended as well for an arbitrary numberof dimensions yielding a really interesting result. The Lagrangian proposed by Lovelock provedthat if D dimensions are considered, there are exactly bD/2c terms constructed with respectivelyhigher-curvature tensors such that, when applying the minimal action principle, only equationsof motion with second order derivatives are obtained. This means that each term of order d isunique and vanishes if D < 2d, is a topological term if D = 2d and if D > 2d it contributes tothe equations of motion.

This is an outstanding result because the Lovelock action is completely able to include theEinstein-Hilbert action plus the Gauss-Bonnet term explaining its properties. However, there isa problem with this theory. It is insufficient to construct the whole Lagrangian, since it doesnot provide the numerical values of the coefficients λd, which in (2.15) represent the couplingconstant of each term of the series.

In order to compute the value of the coefficients, a complementary theory must be used. Sincea gravitational theory can be constructed as a gauge theory, Lagrangian (2.15) is to be reproducedfollowing its principle fixing the numerical value of the undetermined coupling constants.

6

Chapter 3

Gravitation as a Gauge Theory

Once Lovelock gravity has been presented as the generalisation of Einstein-Hilbert Lagrangian,it is time to compute the coupling constants. In order to do so, the Lovelock Lagrangian is tobe constructed as a gauge theory, but first it would be convenient to show that this is possible.In the present chapter, the Einstein-Hilbert action (2.1) is to be constructed applying the gaugeprinciple to the Poincare algebra in D = 4 dimensions. Afterwards, since gauging the Poincarealgebra does not include the cosmological constant term, it will be modified to include so, thismodification is the (anti-)de Sitter algebra. When the Einstein-Hilbert action plus a cosmologicalconstant is reproduced, the generalisation for D dimensions is to be done following the discussionon the previous chapter.

3.1 Poincare Gauge Theory

When a given Lagrangian is invariant under certain group of constants transformations, thetheory is said to show a global symmetry. When the transformations are made to be local,the Lagrangian is no longer invariant. However, if new fields are introduced in the Lagrangianto cancel the extra terms, the invariance is restored. These fields introduce interactions tothe original theory. If this is the case, from a matter field Lagrangian and a global symmetry,general relativity could be constructed with the procedure described. It turns out that it actuallyis because the matter fields living in a manifold are invariant under the Poincare group.

3.1.1 Poincare Algebra

First, a notion of the Poincare transformations is to be presented. The Poincare group ISO(3,1)is the isometry group of the Minkowski space-time ηab and it consists on the semidirect productof the translations group T1,3 and the Lorentz group SO(3, 1),

ISO(3,1) = T1,3 o SO(3, 1). (3.1)

An generic element of this group can be written as

xb 7→ x′b = Λbcxc + ab, (3.2)

where Λbc is a Lorentz transformation and ab is an arbitrary constant vector. The infinitesimalversion of this transformation is given by

x′b = xb + ξb(xc) = xb + θbcxc + εb, (3.3)

7

Dimensional Reduction in Lovelock Gravity 3. Gravitation as a Gauge Theory

where εb are the infinitesimal translations, and the infinitesimal Lorentz transformations θbc areantisymmetric θbc = −θcb. This relation is found when substituting the infinitesimal Lorentztransformation Λbc = δbc + θbc + . . . in the condition of leaving invariant the Minkowskian space-time,

ηab = ΛcaΛdbηcd. (3.4)

Because of this antisymmetry relation, in a four-dimensional space-time there are only six in-dependent components (trace components are zero). Three of them are the rotations aroundthe axis and the other three are boosts. Together with the four translations, the infinitesimaltransformations are ten constant parameters.

The generators of this group are the operators that yield the transformations (3.3). For therotations part, the generator Mab must verify

x′c = ei2θabMabxc ≈

(1 +

i

2θabMab

)xc = θcax

a. (3.5)

This happens whenMab = i (xa∂b − xb∂a) . (3.6)

Now that the generators of the rotations are found, it is turn for the generators Pa of thetranslations. They must verify

x′b = eiεcPcxb ≈ (1 + iεcPc)xb = xb + εb (3.7)

and this holds whenPa = −i∂a. (3.8)

The commutation relations between the generators of the Poincare group form the Poincarealgebra

[Mab,Mcd] = iηadMbc − iηbdMac − iηacMbd + iηbcMad, (3.9a)

[Pa,Mbc] = iηabPc − iηacPb, (3.9b)

[Pa, Pb] = 0, (3.9c)

where, in order to obtain them, the relation [∂a, xb] = ηab has been used.

3.1.2 Global Poincare Symmetry

Once the Poincare algebra has been defined, it is time to see if a matter Lagrangian is, indeed,invariant under the Poincare group. Consider a Lagrangian depending on the coordinates xµ ofa manifold M, the field χ(xµ) and its first derivative term ∂µχ(xµ), this is

Lm = Lm [χ(xµ), ∂µχ(xµ), xµ] . (3.10)

The interest now is to see how does the Lagrangian change under infinitesimal variations of theseterms.

A Poincare transformation acts on the coordinates yb of a tangent space TP (M) of themanifold exactly as (3.2), thus, infinitesimally

y′b = yb + ξb(yc) = yb + θbcyc + εb. (3.11)

8


Notice that the notation has been changed from xb to yb in order to distinguish from the coor-dinates xµ of the manifold. Regarding the field χ (xµ), it is infinitesimally transformed like

χ(xµ)→ χ′(x′µ) = χ(xµ) + δχ (xµ) , (3.12)

where

δχ (xµ) =

(− i

2θabMab + iεaPa

)χ (xµ) ≡ Pχ(xµ), (3.13)

with Mab supposed to be expressed in its corresponding representation. Notice that in expression(3.13) the term Paχ(xµ) vanishes because the operator is differentiating with respect to thetangent space coordinates whereas the field depends on the manifold coordinates, it is keptthough, for future convenience. Regarding the variation of the terms which include a partialderivative ∂µχ(xν), they verify

δ [∂µχ(xν)] = ∂µ [δχ(xν)] = P∂µχ(xν). (3.14)

The total variation of Lagrangian (3.10) is given by

δLm = L′m(x′)− Lm(x) = δLm + ξµ∂µLm + ∂µξµLm = 0, (3.15)

where

δLm =∂Lm

∂χδχ+

∂Lm

∂ (∂µχ)δ (∂µχ) = 0, (3.16)

are the Euler-Lagrange equations, which are assumed to hold. The second term of (3.15) is atotal derivative term which does not modify the equations of motion and the last term vanishesas well when recalling the expression of ξµ from (3.3). This obtained result means that thePoincare transformations are a global symmetry of the constructed Lagrangian. However, if nowthe parameters of the transformation are made to be local, this is no longer the case.

3.1.3 Local Poincare Symmetry

When the Poincare transformations are functions of the coordinates of the manifold,

yb 7→ y′b = Λbc (xµ) yc + ab (xµ) , (3.17)

they are said to be local. Their local character changes the infinitesimal transformation ofcoordinates (3.11) to

y′b = yb + θbc (xµ) yc + εb (xµ) . (3.18)

When this is the case, invariance (3.15) is violated. Even though the transformation rule pre-sented in (3.13) of the field χ(xµ) remains unchanged,

δχ (xµ) =

[− i

2θab (xµ)Mab + iεa (xµ)Pa

]χ (xµ) = P (xµ)χ(xµ), (3.19)

the transformation of the derivative of the field ∂µχ (xν) is different to the previous one (3.14)arising a new term,

δ [∂µχ (xν)] = ∂µ [δχ (xν)] = ∂µP (xν)χ (xν) + P (xν) ∂µχ (xν) . (3.20)

9


This term ∂µP (xν) breaks the symmetry that the theory enjoyed when the parameters wereglobal. However, if a new derivative, called gauge covariant derivative, is defined as

Dµχ (xν) = (∂µ −Aµ)χ (xν) , (3.21)

where Aµ are some introduced gauge fields of the form

Aµ (xν) =i

2ω abµ (xν)Mab − ieaµ (xν)Pa, (3.22)

the transformation rule (3.20) recovers the expression of the global symmetry case,

δ [Dµχ (xν)] = P (xν)Dµχ (xν) . (3.23)

Therefore, a Lagrangian which depends on this covariant derivative instead of the previous regularderivative,

Lm = Lm [χ(xµ),Dµχ(xµ), xµ] , (3.24)

is again invariant under the localised Poincare transformations. Regarding the variation of thefields Aµ, they are found to be

δAµ = − i

2DµθabMab + i

(Dµεb + θbce

cµPb

), (3.25)

where

Dµθab = ∂µθab + ω a

µc θcb + ω a

µc θac, (3.26a)

Dµεb = ∂µεb + ω a

µc εa. (3.26b)

It is important to notice that the defined gauge fields ω bµa and eaµ must be taken into account

in the total Lagrangian LT of the theory as they will be part of an additional kinetic term,

LT = Lk + Lm. (3.27)

The term Lk is given by the field strengths which are computed with the commutator of twocovariant derivatives, this is

[Dµ,Dν ]χ (xρ) =

[i

2R abµν Mab − iR a

µν Pa

]χ(xρ), (3.28)

where the quantities R abµν and R a

µν have been defined as

R abµν = ∂µω

abν − ∂νω ab

µ + ω acµ ω b

νc − ω acν ω b

µc , (3.29a)

R aµν = Dµeaν −Dνeaµ. (3.29b)

3.1.4 Curvature Showing Up

The choice of naming the introduced gauge fields as ω bµa and eaµ is not casual. In the present

section they are to be identified as the spin connection and the Vielbein respectively. The defi-nition and properties of such fields are found in appendix B. After identifying the gauge fields,the obtained field strength R ab

µν from (3.29a) turns out to be the curvature tensor and therefore

10


proportional to the Einstein-Hilbert Lagrangian.

In order to check that the gauge fields are indeed the geometric fields mentioned, the trans-formation rule of the introduced gauge fields ω ab

µ is computed. It is particularly easily to do sosince it coincides with the covariant derivative of the infinitesimal Lorentz transformation (3.26),this is,

δω abµ = ∂µθ

ab + ω aµc θ

cb + ω bµc θ

ac. (3.30)

This expression can be compared with the transformation of the spin connection from (B.16). Itturns out that they are exactly the same and therefore they can be treated as the same tensor.If this is the case, there are strong indications to think that the gauge field eaµ is actuallythe Vielbein and assuming this equivalence (3.29b) vanishes when it is written in terms of thecovariant derivative Dµ, this is,

R aµν = Dµeaν −Dνeaµ = 0. (3.31)

In order to prove this condition, the existence of inverse gauge fields eµa are required and theymust verify

eaµeµb = δba, (3.32a)

eµaeaν = δνµ, (3.32b)

similarly to the Vielbein. When this condition is fulfilled, the relation

Ω cab + ω c

ab − ω cba = 0 (3.33)

is found, where ω cab = eµaω c

µb and the new coefficients have been defined as

Ω cab = eµae

νb

(∂µe

cν − ∂νecµ

). (3.34)

Combining (3.33) and (3.34), the spin connection can be expressed in terms of the fields eaµ inthe same manner as (B.28), where the Ω c

ab are the coefficients of anholonomy. In the light ofthe results, it is natural to identify the field eaµ as the Vielbein.

This is a remarkable result that allows to identify the nature of the field strength R abµν as

the curvature tensor. This means that the total Lagrangian LT from (3.27) is actually

LT = λ µνab R ab

µν + Lm, (3.35)

where the factors λ µνab represent the coupling constant of the Lagrangian and a tensorial quantity

so that the product with the field strength is a scalar. When choosing

λ µνab =

1

2κeµae

νb, (3.36)

the first term of Lagrangian (3.35) is completely identified to the Einstein-Hilbert Lagrangianplus the matter field term (2.1),

LT ≡ LGR, (3.37)

yielding the correct action from the previous chapter,

SGR =

∫d4x√|g|(

1

2κeµae

νbR

abµν + Lm

). (3.38)

11


With this derivation, it has been shown that general relativity can be successfully reproducedusing the gauge principle. However, as it may be noticed in action (3.38), the cosmologicalconstant is absent. In order to include it another algebra different from (3.9) must be considered,this is the (anti-)de Sitter algebra.

3.2 The Quest for the Cosmological Constant in (A)dS

The (anti-)de Sitter (also written (A)dS) algebra is quite similar to the Poincare algebra. Thefirst two commutators [Mab,Mcd] and [Mab, Pc] remain exactly the same whereas the last one(3.9c) is modified to be

[Pa, Pb] =iε

L2Mab. (3.39)

The parameter ε in this expression can take the values ε = ±1 differentiating between (A)dS andtherefore the signature of the cosmological constant. Regarding the other introduced parameterL, it is the radius of the (A)dS space. Notice that in the limit when the radius tends to infinity,

limx→∞

[Pa, Pb] = 0, (3.40)

the whole algebra of Poincare (3.9) is recovered.

The (A)dS space is invariant under the symmetry group SO(1,4) (or SO(2,3) respectively).These groups are really similar to the Lorentz group SO(1,3) with the only difference of an extraspatial (or temporal) direction. Because of this similarity, in analogy to the Lorentz algebra the(A)dS algebra can be written as[

Mab, Mcd

]= iηadMbc − iηbdMac − iηacMbd + iηbcMad. (3.41)

In this expression, the metric ηab = diag(1,−ε,−1,−1,−1) has the extra direction of the spacegroup, with the value of ε determining its spatial or temporal character. Using the expression(3.41) the generators are to be expressed as

Mab= Mab, (3.42a)

Ma?= LPa, (3.42b)

and along with η?? = −ε, algebra (3.41) reduces to (3.9). With the only difference of the lastcommutator being different from zero, the procedure from section 3.1 can be re-written taken itinto account. This time the variation of the generic field has the form

δχ = − i

2θbaMabχ = − i

2θabMab − iθa?Ma? = − i

2θabMab − iaaPa, (3.43)

where the identification

θab= θab, (3.44a)

θa?= −aa

L, (3.44b)

holds. The gauge fields ω abµ , can also be decomposed into

ω abµ = ω ab

µ , (3.45a)

ω a?µ = −

eaµL, (3.45b)

12


which yields the same gauge covariant derivative as in the Poincare case

Dµχ = ∂µχ+i

2ω abµ Mabχ = ∂µχ+

i

2ω abµ Mabχ− ieaµPaχ. (3.46)

Notice that now the variation of the fields changes with respect to Poincare because the trans-lations do not commute, the variation of the gauge field,

δω abµ = Dµθ

ab, (3.47)

now is decomposed into

δω abµ = Dµθ

ab − ε

L2

(eaµa

b − ebµaa), (3.48a)

δeaµ = Dµaa + θac e

cµ. (3.48b)

Likewise, the curvature tensors are different to the Poincare case,

R abµν = ∂µω

abν − ∂ν ω ab

µ − ω acµ ω b

νc + ω acν ω b

µc , (3.49)

yielding the decomposition

R abµν = R ab

µν − ε

L2

(eaµe

bν − eaνebµ

), (3.50a)

R a?µν = −

R aµν

L, (3.50b)

where R abµν and R a

µν are the same expressions from (3.29).

In order to construct the Einstein-Hilbert Lagrangian plus the cosmological constant term,the discussion from section 2.2 is to be recalled. The Levi-Civita tensor can be used to constructthe different terms of the Lagrangian when contracted with a quadratic term of curvature

L = N εµνρσ εabcdeRab

µν R cdρσ V c. (3.51)

In this expression, V c = δe? is a vector that breaks the symmetry SO(1,4) (SO(2,3)). Thenormalisation constant N must be chosen in such a way that the term with first-order curvaturehas the same value for the coupling constant as the Einstein-Hilbert-Lagrangian. If the four-and five-dimensional Levi-Civita tensors are related as

εabcd? = εabcd, (3.52)

using the contractions between them specified in appendix A.1, Lagrangian (3.51) yields

L = |g|−1

[R

2κ+

3ε

L2κ+εL2

8κ

(R2 − 4RµνR

µν +RµνρσRµνρσ

)], (3.53)

when the normalisation constant has been chosen as

N =εL2

32κ. (3.54)

13


When doing so, the coupling constants of the different terms are fixed both for the zero-curvatureterm,

Λ = − 3ε

L2κ, (3.55)

which is actually the cosmological constant, and for the Gauss-Bonnet term. Notice that in spiteof fixing the coupling constant of the Gauss-Bonnet term, since a D = 4 space-time is beingconsider it is a topological invariant as previously discussed.

Now that the Einstein-Hilbert Lagrangian with cosmological constant and the Gauss-Bonnetterm has been successfully constructed, following the reasoning of Lovelock a D-dimensionalLagrangian could be constructed as well.

3.3 Expanding the Brain

The D-dimensional extension of Lagrangian (3.53) is such that resembles its structure and in-cludes D/2 curvature terms contracted with two D-dimensional Levi-Civita tensors, accordingto the discussion of the previous chapter this looks like

LD = NDεµ1ν1...µD/2νD/2 εa1b1...aD/2bD/2c

D/2∏i=1

R aibiµiνi V c, (3.56)

where ND is the normalisation constant of the Lagrangian for D dimensions, εµ1ν1...µD/2νD/2 andεa1b1...aD/2bD/2c

are the D- and (D + 1)-dimensional Levi-Civita tensors, related as

εa1b1...aD/2bD/2= εa1b1...aD/2bD/2?, (3.57)

analogously to the lower dimensional case (3.52). Notice that if a generic number of dimensionsis considered, the commutator (3.39) also changes to

[Pa, Pb] = iεL−2D Mab, (3.58)

where LD is the (A)dS space radius in the new dimensions but the structure of the algebraremains the same so the description of the previous section still holds.

Lagrangian (3.56) can only be constructed when the number of dimensions D is an evennumber because for each curvature tensor introduced, two more indices are added to the Levi-Civita tensor. This is a key feature whose implications will be commented afterwards. For thetime being, when the Lagrangian is expanded using the term (3.50a), the following expression isobtained,

LD = NDD/2∑d=0

(D/2

d

)(−2ε

L2D

)D/2−dεµ1ν1...µdνdλ1...λD/2−dεa1b1...adbdλ1...λD/2−d

d∏i=1

R aibiµiνi . (3.59)

This expression can be shortened using the definition of curvature term (2.16) along with thecontraction of Levi-Civita tensors to look like

LD = − |gD|−1NDD/2∑d=0

(D/2

d

)2D/2(−ε)D/2−d(D − 2d)!

L2(D/2−d)D

Rd. (3.60)

14


Analogously to the previous section, the normalisation ND is determined with the condition ofhaving the same coupling constant for the first order curvature term (this is d = 1) regardless ofthe dimension considered. This imposes the relation

−ND(D/2

1

)2D/2(−ε)D/2−1(D − 2)!

LD−2D

R =R

2κD, (3.61)

which re-arranging the terms yields

ND =(−1)D/2εD/2−1LD−2

D

2D/2D(D − 2)!κD. (3.62)

Notice that the Newton constant κD in expression (3.61) is also different from the previousconstant in four dimensions and it will be taken into account. Introducing the obtained value ofthe normalisation constant in the previous expression (3.59), the obtained Lagrangian is

LD = |gD|−1D/2∑d=0

(D/2

d

)(−ε)d+1(D − 2d)!L

2(d−1)D

D(D − 2)!κDRd, (3.63)

where the coupling constants are completely identified using again expression (2.15) as

λd =

(D/2

d

)(−ε)d+1(D − 2d)!L

2(d−1)D

D(D − 2)!κD. (3.64)

As usual, the term d = 0 yields the cosmological constant, which obviously changes its valuedepending on the dimension considered

ΛD ≡ λ0 = −(D − 1)ε

L2DκD

. (3.65)

Now that the undetermined coupling constant from the previous chapter have been computedusing the gauge principle, it is important to remark a difference between Lagrangian (3.63) andthe one indicated in (2.15). Expression (2.15) is able to produce a Lagrangian for even and odddimensions whereas (3.63) is constrained to an even dimension. Since the question of this workis whether the coupling constants match when applying dimensional reduction, the procedure ofthe next chapter is conditioned by this property. The only possibility to compare the constants

is to fix the values of λDd and λ(D−N)d (with N an even number) of a D-dimensional Lagrangian

and (D −N)-dimensional Lagrangian respectively.

15


16

Chapter 4

Kaluza-Klein Dimensional Reduction

In this chapter the uniqueness of the found coupling constants using the gauge principle is tobe analysed through Kaluza-Klein dimensional reduction. The Kaluza-Klein theory describes aprocedure to obtain a four dimensional gravitation theory coupled to electromagnetic field whenapplying dimensional reduction to a five dimensional gravitational theory. The existence of thisfifth dimension was justified with the concept of compact, periodic dimension, an imperceptiblecoordinate with similar length to the Planck scale. In the present work this theory is to beused as a tool to check the uniqueness of the coupling constants for each term of the LovelockLagrangian. Instead of reducing from five dimensions to four dimensions, the D-dimensionalLovelock action is to be reduced to D −N dimensions.

The structure of the present chapter is the following: First, one dimensional reduction isapplied and the results are discussed and interpreted focussing on the gravitational part of thedecomposition. Afterwards, dimensional reduction is again applied over another dimension sothat the total number of reductions is even to fulfil the requirement mentioned in the previouschapter. Once the action has been reduced, it would be straight-forward to reduce a highereven number of dimensions, this is N times. Finally, the constants of the (D −N)-dimensionalLagrangian are compared with the ones fixed using the gauge theory.

A final comment is necessary. Since the radius of the compact dimension is similar to thePlanck length, when working at scales much greater than it, all the fields lose their dependencyon the compact coordinate ωn. Because of this, the partial derivative of any field with respectto this coordinate is identically zero,

∂ωn = 0. (4.1)

This limit is often referred to as the low-energy limit.

Regarding the notation of this chapter, from now on, D-dimensional objects will be writtenwith a hat

xµ, gµν , Rσ

µνρ , . . .

whereas (D −N)-dimensional objects without it

xµ, gµν , Rσ

µνρ , . . .

with µ = 0, 1, . . . , D and µ = 0, 1, . . . , D−N . As mentioned above, the compact dimensions arenoted as ωn = xD−nn with n = 1, . . . , N . Every periodic dimension may have a different radius

17

Dimensional Reduction in Lovelock Gravity 4. Kaluza-Klein Dimensional Reduction

to each other, forming a N-torus, this is

M1,D−1 =M1,D−N−1 × S1 × · · · × S1︸︷︷︸N times

.

4.1 One-Dimensional Compactification

The aim is to compactify the Lovelock action that results from Lagrangian (3.63), this is

SD =

∫dDx

√|gD|

D/2∑d=0

λdRd. (4.2)

The D-dimensional metric gµν from which the square root determinant√|gD| appears in the

previous expression can be decomposed in a (D− 1)-dimensional metric gµν , a vector gµω and ascalar gωω. The usual ansatz for it is given by

gµν =

(e2αφ+2Ω0gµν − γ2e2βφAµAν −γe2βφ

−γe2βφ −e2βφ

)(4.3)

with α, β, γ and Ω0 arbitrary constants that are usually determined conveniently1. Regarding thefields, the scalar field φ is called the dilaton and the electromagnetic potential Aµ. In the originalwork of Kaluza and Klein, the electromagnetic potential would yield the electromagnetic fieldwhen compactifying. However, as previously mentioned, the interest is to compare two purelygravitational theories, therefore Aµ is set to zero. Under this assumption, metric (4.3) is diagonalwith the line element being

ds2 = e2αφ+2Ω0gµνdxµdxν − e2βφdω2. (4.4)

In order to simplify the calculations, the Vielbein of the metric are to be computed usingexpression (B.4), yielding

eaµ = eαφ+Ω0eaµ, (4.5a)

eaω = ezµ = 0, (4.5b)

ezω = eβφ, (4.5c)

where the same notation for the compact coordinate applies this time for the flat index xa =(xa, z). If the components of the inverse Vielbein are computed, they are found to be

eµa = e−αφ−Ω0eµa, (4.6a)

eµz = eωa = 0, (4.6b)

eωz = e−βφ, (4.6c)

where eaµ are the Vielbein of metric gµν and eµa are, respectively, the Vielbein of the inversemetric gµν . Now, with the Vielbein perfectly computed, it is really easy to express the volumeelement from (4.2) in terms of the (D − 1)- dimensional metric as√

|g| =∣∣∣eaµ∣∣∣ = eβφ

∣∣∣eαφ+Ω0eaµ

∣∣∣ = eβφe(D−1)(αφ+Ω0)∣∣eaµ∣∣ = e[(D−1)α+β]φ+(D−1)Ω0

√|g|. (4.7)

1Notice that, in this section, the compact dimension is noted as ω instead of ω1 for simplicity.

18


The next step is to reduce the curvature term Rd. Recalling that this term is built up withRiemann tensors (and its contractions) and since the Vielbein are already computed, it is to bedone in terms of the spin connection ω c

ab, whose expression without the hat notation is given in

(B.28), in this case it is

ω cab

=1

2

(Ω eadηcdηbe + Ω e

bdηcdηae − Ω c

ab

). (4.8)

In order to compute them, the components of anholonomy coefficients are needed. According toexpression (B.7), they are given by

Ω cab

= eµaeνb

(∂µe

cν − ∂νecµ

), (4.9)

whose non-trivial components are

Ω cab = e−αφ−Ω0 (Ω c

ab + α∂aφδcb − α∂bφδca) , (4.10a)

Ω zaz = βe−αφ−Ω0∂aφ. (4.10b)

Notice that when computing the components of the partial derivative,

∂a = eµa∂µ, (4.11)

the expression (4.1) must be recalled from the discussion of the low energy limit and thereforethey are given by

∂z = eµz∂µ + eωz∂ω = 0, (4.12a)

∂a = eµa∂µ + eωa∂ω = e−αφ−Ω0eµa∂µ = e−αφ−Ω0∂a. (4.12b)

With all these ingredients, the components of the spin connection are completely determined,

ω cab = e−αφ−Ω0 (ω c

ab + α∂bφδca − α∂cφηab) , (4.13a)

ω zzb = βe−αφ−Ω0∂bφ, (4.13b)

ω czz = βe−αφ−Ω0∂cφ, (4.13c)

and therefore the Riemann tensor, given by expression (B.33) is computed with the notation ofthis chapter

R dabc

= ∂aωd

bc− ∂bω

dac + ω e

bcω dae − ω e

ac ωd

be+ Ω e

abω dec , (4.14)

yielding the components

R dabc = e−2αφ−2Ω0

(R dabc − α2∂aφ∂cφδ

db + α2∂aφ∂

dφηbc + αDaDcφδdb

−αDaDdφηbc + α2∂bφ∂cφδ

da − α2∂bφ∂

dφηac − αDbDcφδda + αDbD

dφηac

−α2 (∂φ)2 ηbcδda + α2 (∂φ)2 ηacδ

db

), (4.15a)

R zazc = βe−2αφ−2Ω0

[∂aφ∂cφ (β − 2α) + α (∂φ)2 ηac +DaDcφ

]. (4.15b)

Regarding the contractions of the Riemann tensor, namely, the Ricci tensor and the Ricci scalarare given by

Rac = R babc

, (4.16a)

R = ηacRac, (4.16b)

19


which means

Rab= e−2αφ−2Ω0Rab + αD2φηab + [(N − 3)α+ β]DaDbφ

+[β2 − (N − 3)α2 − 2αβ

]∂aφ∂bφ+

[(N − 3)α2 + αβ

]φ (∂φ)2

, (4.17a)

Rzz= −e−2αφ−2Ω0

βD2φ+

[(N − 3)αβ + β2

](∂φ)2

, (4.17b)

R = e−2αφ−2Ω0R+ 2 [(N − 2)α+ β]D2φ

+[(N − 2) (N − 3)α2 + 2 (N − 3)αβ + 2β2

](∂φ)2

. (4.17c)

Remember that these expressions are the building blocks of the curvature Rd. The length of theexpression of each of them means a great complexity when reducing a higher curvature term froma dimension to a lower one. In order to illustrate the complete phenomena, the Einstein-Hilbertterm is to be reduced and a solution to the complexity is to be found analysing the result. TheEinstein-Hilbert term, after plugging the obtained expression of the reduced Ricci scalar (4.17c)and the metric (4.7), yields

SD

∣∣∣∣d=1

=2π`12κD

∫dD−1x

√|gD|e[(D−3)α+β]φ+(D−3)Ω0

R+ 2 [(N − 2)α+ β]D2φ

+[(N − 2) (N − 3)α2 + 2 (N − 3)αβ + 2β2

](∂φ)2

. (4.18)

Notice that the compact dimension ω has been integrated over the circumference S1 of radius `because the independence with respect to any field as commented before. The second derivativeterm would yield third-order equations of motion. In order to avoid this situation, it is integratedby parts, thus expression (4.18) is now

SD

∣∣∣∣d=1

=2π`1e(D−3)Ω0

2κD

∫dD−1x

√|gD−1|e[(D−3)α+β]φ

R

−α (D − 2) [(D − 3)α+ 2β] (∂φ)2. (4.19)

The next step is to pay attention to the conformal factor present in the action. Generallyspeaking, when a conformal factor ef(x) is present in any Lagrangian, for instance

L =√|gD−1|ef(x)R, (4.20)

it is said to be expressed in the Jordan frame. Since the usual actions are written in terms ofthe Einstein frame, this is

L =√|gD−1|R, (4.21)

it is enough to conveniently choose the value of the arbitrary constant

α = − β

D − 3, (4.22)

making the conformal factor to disappear allowing to work in the usual Einstein frame. Usingthis, expression (4.19) yields

SD

∣∣∣∣d=1

=2π`1e(D−3)Ω0

2κD

∫dD−1x

√|gD−1|

[R− D − 2

D − 3β2 (∂φ)2

]. (4.23)

20


All this procedure has been done for the Einstein-Hilbert term but the validity of it is limited.Consider the Gauss-Bonnet term, this is

LGB = λ2

(R2 − 4RabR

ab + RabcdRabcd). (4.24)

When plugging the expressions of the Riemann tensors and its contractions (4.15) and (4.17),the conformal factor that previously was found to be ef(x) = e[(D−3)α+β]φ(x) in (4.19) is now

ef(x) = e[(D−5)α+β]φ(x), (4.25)

making useless the relation (4.22) imposed to keep working in the Einstein frame. This alsohappens for higher-curvature terms d = 3, d = 4 and subsequently. The only possibility to stayalways in the Einstein frame is to do a consistent truncation of the dilaton field φ(x) keepingonly the gravitational part of the theory, which is actually the one important for the results ofthe work. Applying these changes to expression (4.19), it yields a simpler expression,

SD

∣∣∣∣d=1

=2π`1e(D−3)Ω0

2κD

∫dD−1x

√|gD−1|R, (4.26)

in which the first order curvature term R1 has been compactified from (4.17c) in a extraordinarilyuncomplicated expression

R1 = e−2Ω0R1, (4.27)

that allows to easily extend to other curvature terms d in the future. Finally, in order to get ridof the extra factors of the action (4.26), if the Newton constant is chosen to be

κD = 2π`1e(D−3)Ω0κD−1, (4.28)

the coupling constant of the one-dimensional reduced Einstein-Hilbert action is found to havethe same structure in D dimensions and in D − 1 dimensions.

Taking advantage of the knowledge achieved with this particular case of the Lovelock action,it is time to come back to the general case (4.2). The general curvature term after applyingdimensional reduction yields

Rd = e−2dΩ0Rd. (4.29)

Using this expression, the action (4.2) yields

SD = 2π`1e(D−1−2d)Ω0

∫dD−1x

√|gD−1|

D/2−1∑d=0

(D/2

d

)(−ε)d+1(D − 2d)!L

2(d−1)D

D(D − 2)!κDRd, (4.30)

where the explicit expression of the coupling constants λd has been recalled from (3.64). Therelation between the Newton constant in D dimensions and in D − 1 dimension was previouslyfound and must hold whatever the term, therefore, the action is

SD = e−2dΩ0

∫dD−1x

√|gD−1|

(D−1)/2∑d=0

(D/2

d

)(−ε)d+1(D − 2d)!L

2(d−1)D

D(D − 2)!κD−1Rd. (4.31)

At this point, the following step is to relate the radius of the (anti-)de Sitter space of a higherdimension LD with the radius of a lower dimension. Since the construction of the Lovelock actionof the previous chapter requires an even dimension, the next radius that can be determined isLD−2. This means that compactifying over two dimensions is necessary.

21


4.2 Two-Dimensional Compactification

The necessity of compactifying over two dimensions has been showed. The fact of having appliedthe consistent truncation and considered only the gravitational part has simplified greatly theprocedure and allows to retake the computations from the previous section and extend themin a quite straight-forward way. The difference is that now the compactification is over twodimensions, because of this, the reduction of the metric is now√

|gD| = e(D−1)Ω0e(D−2)Ω0√|gD−2| = e(2D−3)Ω0

√|gD−2|. (4.32)

The geometry of these two coordinates is a 2-torus S1 × S1 with two different radii `1 and `2.Regarding the curvature terms Rd, it can be checked that their reduction is now

Rd = e−4dΩ0Rd. (4.33)

Putting altogether the Lovelock action is this time

SD = 4π2`1`2e(2D−3−4d)Ω0

∫dD−2x

√|gD−2|

D/2−1∑d=0

(D/2

d

)(−ε)d+1(D − 2d)!L

2(d−1)D

D(D − 2)!κDRd. (4.34)

As previously, if the Newton constant in D-dimensions is chosen so that the term d = 1 isequivalent to the Einstein-Hilbert term, a condition to relate it with the constant of two lowerdimensions is found,

κD = 4π2`1`2e(2D−7)Ω0κD−2. (4.35)

Moreover, unlike before, another condition to relate the value of the (anti-)de Sitter LD withLD−2 in two lower dimensions is available. The simplest term that allows to relate them is thed = 0 one, this is, the cosmological constant. The cosmological constant in D dimensions is givenby

ΛD ≡(D − 1)ε

L2DκD

, (4.36)

while the cosmological constant in D − 2 dimensions is

ΛD−2 ≡(D − 3)ε

L2D−2κD−2

. (4.37)

When comparing the first term of the two actions in D and D−2 dimensions, a relation betweenthe two (anti-)de Sitter radii is given by

LDLD−2

=D − 1

D − 3e4Ω0 , (4.38)

and plugging this expression in action (4.34), it yields

SD =

∫dD−2x

√|gD−2|

D/2−1∑d=0

(D/2

d

)(−ε)d+1(D − 2d)!

D(D − 2)!κD−2

(√D − 1

D − 3LD−2

)2(d−1)

Rd. (4.39)

Now that finally the action has been reduced from D to D−2 dimensions, it can be comparedwith the action

SD−2 =

∫MD−2

dD−2x√|gD−2|

D/2−1∑d=0

(D/2− 1

d

)(−ε)d+1[D − 2(d+ 1)]!L

2(d−1)D−2

(D − 2)(D − 4)!κD−2Rd, (4.40)

22


obtained gauging the (A)dS algebra in D− 2 dimensions. The two expressions (4.39) and (4.40)match when d = 0 and d = 1 (expected because imposed with the two conditions for κD andLD), but the terms with higher curvature d ≥ 2 do not, independently of the dimension D of thetheory. A natural question arises now about the choice of compactifying two dimensions. It isuncertain whether compactifying other even number of dimensions the coupling constant wouldcoincide, some of them or all of them. Since the assumptions made allow to extend easily thereduction to a generic number of dimensions, it is to be done in next section.

4.3 N-Dimensional Compactification

In order to study the compatibility of the coupling constants after N dimensional reductions,the first task is to compactify the geometric objects involved in the action. Previously done forthe two-dimensional case, this time the Riemann tensor, Ricci tensor and Ricci scalar take theform

R dabc = e−2NΩ0R d

abc , (4.41a)

Rab = e−2NΩ0Rab, (4.41b)

R = e−2NΩ0R. (4.41c)

This results can be used to generalise the introduced scalar Rd, which happens to be reduced as

Rd = e−dNΩ0Rd. (4.42)

Likewise, for this case the volume element yields

√|gD| =

N∏n=1

e(D−n)√|gD−N | = e

(ND −

N∑n=1

n

)Ω0√

|gD−N | = eN(D−N+12 )Ω0

√|gD−N |. (4.43)

Taking into account these increase in the number of compact dimensions the relation betweenthe Einstein constants κD and κD−N , which for the case N = 2 was computed in (4.35), is nowrequired to be

κDκD−N

= (2π)NeN(D−N+12−2)Ω0

N∏n=1

`n. (4.44)

Respectively, the second condition (4.38) can be revisited to compute the relation between theradii of (A)dS,

L2D

L2D−N

=D − 1

D −N − 1e2NΩ0 . (4.45)

Introducing these new relations in the Lovelock action, it yields

SD =

∫dD−Nx

√|gD−N |

D−N2∑

d=0

(D/2

d

)(−ε)d+1(D − 2d)!

D(D − 2)!κD−N

(√D − 1

D −N − 1LD−N

)2(d−1)

Rd,

(4.46)

23


and this is to be compared with

SD−N =

∫dD−Nx

√|gD−N |

D−N2∑

d=0

(D−N2

d

)(−ε)d+1(D −N − 2d)!L

2(d−1)D−N

(D −N)(D −N − 2)!κD−NRd. (4.47)

A careful look in these expressions reveals that there are not any new combinations of thethree-tuple (D, d,N) that match the coupling constants of the two actions computed using bothformalisms. This result means that regardless the dimension and regardless the curvature order,the coupling constants are not related using this two theories and therefore the uniqueness ofthe Lovelock Lagrangian is not guaranteed fixing the values through the gauge principle.

24

Chapter 5

Conclusions

The Lovelock Lagrangian has been presented, discussed the motivations and the reasons to con-sider this theory as a candidate to describe gravitation in higher dimensions. The problem ofindetermination of coupling constants has been addressed.

In order to fix the value of the coupling constants, the gauge principle has been used. First,it has been showed that the Einstein-Hilbert Lagrangian can be reproduced when applying thegauge principle to the Poincare group of transformations. This procedure does not include thecosmological constant so, in order to obtain it, a modification to the algebra of the group hasbeen considered, the (anti-)de Sitter algebra. After obtaining the Einstein-Hilbert Lagrangianplus the cosmological constant, a natural extension of the theory has been constructed to includea generic number of dimensions and therefore, the Lovelock Lagrangian.

Once the values of the coupling constants have been determined, Kaluza-Klein dimensionalreduction is applied to it with the goal of studying the uniqueness of the determination of theconstants when applying the gauge principle.

First, a reduction over two dimensions, with the geometry of a two-torus, is performed andonly the cosmological term and the Ricci scalar term, which coincides with the Einstein-Hilbertaction, are compatible. This is rather natural because by construction, in order to find the rela-tions between the Einstein’s constant and respectively the radii of the (A)dS, they were imposedto do so. However, for terms of higher curvature the coupling constants are different showingthat uniqueness is not guaranteed when obtaining them using the gauge principle.

In an attempt to find a particular N dimensional reduction that shows compatibility for anyhigher curvature term among the coupling constant using the geometry of a n-torus, the proce-dure has showed again non-matching constants, reinforcing the conclusion of arbitrariness whendetermining the coupling constant of the Lovelock series using the gauge principle.

Further questions arise at this point, because even though the dimensional reduction overtorus does not agree the terms of the two actions, a reduction over different topologies may doso. This is left for future research.

25

Appendix A

Conventions

A.1 The Levi-Civita Symbol

The D-dimensional Levi-Civita symbol is a completely antisymmetric object, defined as

εµ1...µD =

1 if (µ1 . . . µD) is an even permutation of (0 1 . . . D − 1) ,

−1 if (µ1 . . . µD) is an odd permutation of (0 1 . . . D − 1) ,

0 otherwise,

(A.1)

which is equivalent toεµ1...µD = δ1

[µ1· · · δDµD]. (A.2)

The Levi-Civita symbol is a fundamental symbol: it has these values in all coordinate systems.In order for this to be true, the Levi-Civita symbol has to transform as a pseudo-tensor densityof weight w = +1 under general coordinate transformations

εα1...αD = sgn

(∂y

∂x

) ∣∣∣∣∂y∂x∣∣∣∣ D∏i=1

∂xµi

∂yαiεµ1...µD . (A.3)

Using these transformation rules the invariant volume element is given by√|g|dDx =

√|g|εµ1...µD

D∏i=1

dxµi . (A.4)

An alternating symbol with upper indices can also be defined as

εν1...νD =

(D∏i=1

gµiνi

)εµ1...µD , (A.5)

such thatε01...D−1 = (−1)D−1 |g|−1 . (A.6)

The contraction of two Levi-Civita symbols is given by

εµ1...µdν1...νD−dεµ1...µdρ1...ρD−d = (−1)D−1 |g|−1 d! (D − d)!δρ1[ν1

· · · δρD−d

νD−d], (A.7)

from which yields two interesting cases, namely

εµ1...µDεµ1...µD = (−1)D−1D! |g|−1 , (A.8a)

εµ1...µDεν1...νD = (−1)D−1D! |g|−1 δν1[µ1

· · · δνDµD]. (A.8b)

27

Dimensional Reduction in Lovelock Gravity A. Conventions

28

Appendix B

Geometry from the Tangent Space

B.1 Vielbein

LetMD be a D-dimensional manifold. A property of the differential manifold is that each pointP can be locally described by its tangent TP (MD). The coordinates xµ of MD define at everyP a basis |eµ〉 = ∂µ ∈ TP (MD). This basis is called coordinate basis and it is related to anarbitrary basis |ea〉 through

|eµ〉 = eaµ |ea〉 , (B.1a)

|ea〉 = eµa |eµ〉 . (B.1b)

The coefficients eaµ are the components of the vector |eµ〉 in the new basis |ea〉. The matrixeµa is called Vielbein1 whereas eaµ is the inverse Vielbein, verifying

eµaeaν = δνµ, (B.2a)

eµaebµ = δba. (B.2b)

In a particular tangent space TP (MD), the Vielbein allows to transform the components of atensor of type (1, 1) from a basis to another as

Tµν = eµaebνT

ab , (B.3a)

T ab = eaµeνbT

µν , (B.3b)

leaving a trivial generalisation to the tensors of type (p, q). Using this transformation prop-erty, the metric tensor gµν , which is curved in general, can be locally expressed with Cartesiancoordinates,

gµν = eaµebνηab, (B.4a)

ηab = eµaeνbgµν . (B.4b)

This procedure can be generalised to the whole tangent bundle of the manifold by making theVielbein functions depending on the coordinates xν of the manifold,

eaµ = eaµ(xν). (B.5)

1Vierbein if D = 4, Funfbein when D = 5 and subsequently.

29

Dimensional Reduction in Lovelock Gravity B. Geometry from the Tangent Space

Due to the dependency of the Vielbein with the coordinates of the manifold, a feature arises, thenon-zero commutativity of partial derivatives acting on an arbitrary field χ, this is

[∂a, ∂b]χ = −Ω cab ∂cχ, (B.6)

where Ω cab are called the anholonomy coefficients,

Ω cab = eµae

νb

(∂µe

cν − ∂νecµ

). (B.7)

If another base |e′a〉 is considered in TP(MD

), it will be related to the coordinate basis through

|eµ〉 = e′aµ |e′a〉 , (B.8a)

|e′a〉 = e′µa |eµ〉 . (B.8b)

Since the relation between the basis |ea〉 and |e′a〉 are the Lorentz transformations(Λ−1

)ab, then

the Vielbein verifies that

e′aµ = Λabebµ, (B.9a)

e′µb =(Λ−1

)abeµa. (B.9b)

From this, a straight-forward relation between any tensor of type (1, 1) is found using expression(B.3),

T ′ab = e′aµe′νbT′µν = ecµe

νdΛ

ac

(Λ−1

)dbTµν = Λac

(Λ−1

)dbT cd . (B.10)

Although the Lorentz transformations are global in a tangent flat TP(MD

), they are local in the

tangent bundle T(MD

), namely Λab = Λab(x). This is the reason of being called local Lorentz

transformation. Once again this means troubles with the partial derivative acting on a tensor,the result shall not be a tensor,

∂µTab = ∂µ

[Λaici

(Λ−1

)djbj

]T cd + Λac

(Λ−1

)db∂µT

cd . (B.11)

In order to solve this situation, and following the same procedure of Riemannian geometry, aderivative that acts properly under the local Lorentz transformations is defined,

DµTab = ∂µT

ab + ω a

µc Tcb − ω c

µb Tac , (B.12)

where ω bµa is called the spin connection and transforms as

ω′ bµa =

(Λ−1

)caΛbdω

dµc + Λbc∂µ

(Λ−1

)ca. (B.13)

Notice that, although the spin connection does not transform like a tensor due to the last term,the difference of two spin connections actually does. An aside before continuing is that theinfinitesimal variation of this connection requires the infinitesimal Lorentz transformation,

Λab ≈ δab −i

2θcd

(Scd)ab, (B.14)

where the set of matrix operators(Scd)ab

is a representation of the Lorentz algebra defined as(Scd)ab

= iηcaδdb − iηdaδcb . (B.15)

30


Thus, the variation of the spin connection yields

δωµab=i

2θcd

(Scd)eaωµeb +

i

2θcd

(Scd)ebωµea + ∂µθcd

(Scd)eaδeb

= −θeaωµeb − θebωµea + ∂µθab. (B.16)

Back with the topic, the constructed covariant derivatives Dµ do not commute when acting onvectors in the same way as covariant derivatives ∇µ of Riemannian geometry,

[Dµ, Dν ]V b = R bµνa V

a, (B.17a)

[Dµ, Dν ]Va = R bµνa Vb, (B.17b)

where R bµνa are the Riemann curvature tensors, defined as

R bµνa = ∂µω

bνa − ∂νω b

µa − ω cµa ω

bνc + ω c

νa ωb

µc . (B.18)

At this point, it turns out that the manifoldMD can be described in terms of the curved metricgµν and the affine connection Γρµν as well as in terms of the Vielbein and the spin connection ω b

µa .A priori, both descriptions are independent in spite of being related through (B.4), however, theycan be related using the Vielbein postulates.

B.2 Vielbein Postulates

In general, the constructed derivatives Dµ are not covariant with respect to general changes ofcoordinate systems, therefore a totally covariant derivative Dµ (respect to curved indices andflat indices) must be defined. For a (2, 2) type tensor with two covariant and contra-variant flatindices and with the other two curved indices, the aforementioned covariant derivative is definedas

DµTaαbβ = ∂µT

aαbβ + ω a

µc Tcαbβ − ω c

µb Taαcβ + ΓαµγT

aγbβ − ΓγµβT

aαbγ . (B.19)

This derivative is finally covariant with respect to general changes of coordinate systems as wellas with respect to Lorentz transformations in the tangent space. It is used in the formulation ofthe first Vielbein postulate, which reads

Dµeaβ = 0. (B.20)

According to the previous expression, this means that

Dµeaβ = ∂µe

aβ + ω a

µc ecβ − Γγµβe

aγ = 0, (B.21)

which yields a relation between the spin connection and the affine connection,

ω aµc = Γγµβe

aγeβc − eβc∂µeaβ, (B.22)

and this allows to establish the relation between curvature tensors with mixed curved and flatindices,

R σµνρ = eaρe

σbR

bµνa . (B.23)

It is clear that both tensors carry the same information about the geometrical properties of themanifold and therefore both formalisms are completely equivalent.

31


The second Vielbein postulate is the connection compatibility condition with the metric∇ρgµν = 0, using expression (B.4) it can be written in terms of the Vielbein as

∇ρ(eaµe

bνηab

)= 0, (B.24)

which implies together with the first postulate (B.20) the compatibility of the spin connectionω bµa with the Minkowskian metric,

Dµηab = 0. (B.25)

This compatibility condition implies a relation of antisymmetry on the last two indices of thecurvature tensor R ab

µν = −R baµν showing up when computing the commutator,

[Dµ, Dν ] ηab = R aµνc η

cb +R bµνc η

ca = R abµν +R ba

µν = 0. (B.26)

Likewise, the compatibility (B.25) forces an antisymmetry relation on the two last indices of thespin connection, ωµab = −ωµba, because

Dµηab = ∂µηab − ω cµa ηcb − ω c

µb ηac = −ωµab − ωµba = 0. (B.27)

Since in the present work only manifolds equipped with the Levi-Civita connection are consid-ered2, expression (B.22) can be written with all its indices flat as

ω cab = eµaω

cµb =

1

2

(Ω ead η

cdηbe + Ω ebd η

cdηae − Ω cab

), (B.28)

where the definition of anholonomy coefficient (B.7) has been used to shorten the expression.Furthermore, an interesting relation when subtracting two spin connections with its first pair ofindices exchanged is found,

ω cab − ω c

ba = Ω cab , (B.29)

the operation yields the anholonomy coefficients, this fact will be used afterwards to avoid sometedious calculations.

B.3 Curvature Tensors from the Tangent Space

Previously, in expression (B.23), a relation between the Riemann tensor and the curvature tensorwritten in terms of the spin connection is found. Now, the objective is to find the relationsbetween the contractions of R b

µνa and its relation to the Ricci tensor Rµν and the Ricci scalarR. The first contraction is

Rµa = eνbRb

µνa = ∂aωc

cb − ∂cω bca − ω c

µa ωb

bc + eνbΓρµνω

bρa , (B.30)

which transforms properly under general change of coordinate systems and Lorentz transforma-tions. The contraction with two flat indices can also be constructed as

Rab = eµaRµb = ∂aωc

cb − ∂cω cab − ω c

ab ωd

dc + ω dca ω

cdb , (B.31)

and this transforms as a rank two tensor under Lorentz transformation and as a scalar undergeneral change of coordinate systems. This tensor is the flat version of the Ricci tensor,

Rab = eµaeνbRµν , (B.32)

2Torsion equal to zero.

32


which is also obtained from the completely flat curvature tensor

R dabc = eµae

νbR

dµνc

= ∂aωd

bc − ∂bω dac − ω e

ac ωd

be + ω ebc ω

dae − ω e

ab ωd

ec + ω eba ω

dec

= ∂aωd

bc − ∂bω dac − ω e

ac ωd

be + ω ebc ω

dae − ω e

ab ωd

ae − Ω eab ω

dec , (B.33)

contracting the second and fourth indices

Rac = R babc = ηbdRabcd. (B.34)

Notice that in equation (B.33), the relation (B.29) has been used. The contraction of the Riccitensor gives the Ricci scalar,

R = ηabRab = gµνRµν , (B.35)

and it coincides when using the flat indices and the curved indices. It is obvious that the sym-metry and antisymmetry relations hold in both formalisms.

With all the geometric objects expressed with curved and flat indices, the gravitational actionscan be constructed with any of the both formalisms carrying the same information.

33

Acknowledgements

I would like to thank above all my supervisors, Bert and Jose Alberto because in spite of requiringguidance and assistance they were always available to help me through this thesis every singletime I needed.

34

Bibliography

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[3] David Lovelock. The Einstein Tensor and its Generalizations. Journal of MathematicalPhysics, 12(3):498–501, 1971.

[4] Ryoyu Utiyama. Invariant theoretical interpretation of interaction. Physical Review,101(5):1597, 1956.

[5] Theodor Kaluza. Zum unitatsproblem der physik. Sitzungsber. Preuss. Akad. Wiss.Berlin.(Math. Phys.), 1921(966972):45, 1921.

[6] Oskar Klein. Quantentheorie und funfdimensionale relativitatstheorie. Zeitschrift fur Physik,37(12):895–906, 1926.

[7] Jose Alberto Orejuela. Teorıa de Kaluza Klein. 2015.http://www.ugr.es/~bjanssen/text/TFM_JoseAlbertoOrejuelaGarcia.pdf.

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[10] Evelyn Karina Rodrıguez Duran et al. Teorıa de Einstein-Lovelock e invariancia de gaugetipo Maxwell. PhD thesis, Universidad de Concepcion. Facultad de Ciencias Fısica yMatematicas. Departamento de Fisica, 2013.

[11] Christos Charmousis. Higher order gravity theories and their black hole solutions. Physicsof Black Holes, pages 299–346, 2009.

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