Astrophysical implications of the Bumblebee model of ... · Astrophysical implications of the...

Astrophysical implications of the Bumblebee model of

Spontaneous Lorentz Symmetry Breaking

Gonçalo Dias Pereira Guiomar

Thesis to obtain the Master of Science Degree in

Engineering Physics

Supervisor(s): Prof. Dr. Vítor Manuel dos Santos Cardoso

Prof. Dr. Jorge Tiago Almeida Páramos

Examination Committee

Chairperson: Prof. Dr. Ana Maria Vergueiro Monteiro Cidade Mourão

Supervisor: Prof. Dr. Jorge Tiago Almeida Páramos

Members of the Committee: Prof. Dr. Amaro José Rica da Silva

November 2014

Acknowledgments

This thesis could not have been possible without the help and guidance of my supervisor, Professor

Jorge Páramos, with whom I rediscovered the joy of doing Physics. His immense patience when dealing

with my incompetence, along with his immense knowledge of unorthodox working places made this

work a fun and rewarding experience. Also, I would like to thank Professor Vítor Cardoso for his help

and availability in the process of realizing this thesis.

For my family, I am truly grateful for your continuous support and for providing me the opportunity

of realizing my goals, no matter how uncertain they might have seem in the past.

For my friends, who accompanied me throughout this journey, thank you for joining me in my

culinary digressions.

This last paragraph I dedicate to Geisa, for helping me collapse onto a better state of being.

i

Abstract

In this work the Bumblebee model for spontaneous Lorentz symmetry breaking is considered in the

context of spherically symmetric astrophysical bodies. A discussion of the modied equations of motion

is presented and constraints on the parameters of the model are perturbatively obtained. Along with

this, a detailed review of this model is given, ranging from the questioning of the basic assumptions of

General Relativity, to the role of symmetries in Physics and the Dark Matter problem.

Keywords

General Relativity, Bumblebee Model, Lorentz Symmetry Breaking, Stellar Equilibrium (English)

iii

Resumo

Neste trabalho, consideramos o modelo Bumblebee para a quebra espontânea da simetria de Lorentz

no contexto de corpos celestes com simetria esférica. Uma discussão das equações de movimento mod-

icadas é apresentada, juntamente com os constragimentos do modelo obtidos de modo perturbativo.

De modo a contextualizar o modelo e o problema em questão, uma revisão é apresentada, onde se

abordam temas tais como os fundamentos da Relatividade Geral, o papel das simetrias na física e o

problema da matéria escura.

Palavras Chave

Relatividade Geral, Modelo Bumblebee, Quebra da simetria de Lorentz, Equilíbrio Estelar

v

Contents

1 Introduction 2

1.1 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Essential concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 The limits of Einstein's Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Astrophysics 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Tolman-Openheimer-Volkov equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Polytropes and the Lane-Emden Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Symmetry Breaking 15

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Explicit and Spontaneous Breaking of Symmetries . . . . . . . . . . . . . . . . . . . . . 17

3.3 Observer and Particle LSB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Standard Model Extension and LSB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Vector Theories 25

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Aether Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 The Bumblebee Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Astrophysical constraints on the Bumblebee 37

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2 Static, spherically symmetric scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.3 Perturbative Eect of the Bumblebee Field . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.4 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Conclusions 45

Bibliography 47

vii

List of Figures

2.1 Lane-Emden solution for a spherical body in hydrostatic equilibrium. . . . . . . . . . . . 14

3.1 Snowakes generated through a Linenmayer rule set for the rst, second and third

iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 The Mexican Hat potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1 The Bullet Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.1 Prole of the relative perturbations pb/p0, pV /p0, ρb/ρ0 and ρV /ρ0 induced by the

Bumblebee. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 Allowed region (in grey) for a relative perturbation of less than 1% for pb and ρb. . . . . 43

5.3 Allowed region (in grey) for a relative perturbation of less than 1% for pV and ρV . . . . 44

ix

List of Tables

5.1 Selected stars that used as models for the numerical analysis of the Bumblebee pertur-

bation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Table of non-dimensional parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

xi

Abbreviations

GR - General Relativity

TOV - Tolman-Oppenheimer-Volko

LE - Lane-Emden

EP - Equivalence Principle

LLI - Local Lorentz Invariance

LPI - Local Position Invariance

EP - Equivalence Principle

WEP - Weak Equivalence Principle

SEP - Strong Equivalence Principle

LSB - Lorentz Symmetry Breaking

PPN - Parametrized Post Newtonian

CPT - Charge Parity Time Symmetry

GZK - Greisen-Zatsepin-Kuzmin

CMBR - Cosmic Microwave Background Radiation

CMBR - Cosmic Microwave Background Radiation

HI-RES - High Resolution Fly's Eye

SME - Standard Model Extension

SM - Standard Model

SSB - Spontaneous Symmetry Breaking

QED - Quantum Electrodynamics

MOND - Modied Newtonian Dynamics

FLRW - Friedmann-Lemaitre-Roberston-Walker

VEV - Vacuum Expectation Value

1

1Introduction

Contents

1.1 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Essential concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 The limits of Einstein's Relativity . . . . . . . . . . . . . . . . . . . . . . . 5

2

1.1 Thesis Outline

1.1 Thesis Outline

This work will begin by a small introduction of the relevant concepts needed from General Relativity

in the subsection 1.2 and the current observed limitations of the theory in subsection 1.4. The Tolman-

Oppenheimer-Volko equations are then introduced in Chapter 2 along with the Lane-Emden (LE)

model for spherical bodies, which will be needed in order to understand Chapter 5. In Chapter 3, the

concept of Lorentz Symmetry Breaking (LSB), in both its explicit and spontaneous form, is introduced

as a way to portray one of the fundamental properties of the model being used, the Bumblebee Model.

This will later be introduced in 4 as a particular case of the more general Aether Theories. The two

nal chapters (5,6) contain the kernel of this work: The application of the models presented as a way

of constraining their parameters using a spherical astrophysical body modeled by a polytrope. This

thesis closely follows the work done in [1].

1.2 Essential concepts

Presently, Einstein's general theory of relativity serves as a tool to understand a wide range of

phenomena. From the dynamics of compact astrophysical bodies, such as stars and black holes, to

cosmology, its striking predictions have sustained this theoretical framework and made it one of the

most relevant scientic achievements in the history of science [2].

The experimental conrmation of the existence of gravitational lensing, time dilation and gravi-

tational redshift have solidied this fact but this by no means imply that the theory is completely

correct; in fact, such consistence is a motivating factor for testing its limits even further.

Questioning the basic assumptions of General Relativity is thus a valid way to achieve this goal, as

they are the fundamental rule set from which it emerges. These basic assumptions can be expressed

in the following manner [3]:

• Weak Equivalence Principle - Bodies in free fall have the same acceleration independently of

their compositions.

• Local position invariance - The rate at which a clock ticks is independent of its position.

• Local Lorentz invariance - The rate at which a clock ticks is independent of its velocity.

This work shall focus on the second and third basic assumptions presented above. A more detailed

discussion will be presented in 3 in how this principle can be used to test the validity of physical

theories.

As a short introduction to the review that will be made in the following section, a small appetizer

is given in the context of special relativity, showing how one can test the limits of such theory.

The tests are based on testing the principles of relativity (physical laws are independent of the

inertial frame of reference used to describe them) and the constancy of the speed of light. In the case

of inertial frames of reference (which can always be found, at least in the vicinity of any given point in

spacetime), this translates into invariance under Lorentz transformations, a tenet of Special Relativity:

deviations from these transformations would also imply deviations from the underlying principles.

3

1. Introduction

This can be approached through the Robertson-Sexl-Mansouri formalism, which consists on the

known Lorentz transformations,

t′ =t− vx

c2√1− v2

c2

, x′ =x− vt√1− v2

c2

, y′ = y, z′ = z, (1.1)

altered in a way as to express a preferred frame of reference Σ(T, ~X). The transformations now take

the form,

T =t− ~ε.~xa

, ~X =~x

d−(

1

d− 1

b

)~v~x

v2~v + ~vT. (1.2)

With these transformations, it can be shown [3] that through an expansion of a, b, d and ~ε around

v/c2 one can obtain the following expression for the relative shift in the two way speed of light,

c2(θ, v)

c2(0, v)−1 = sin2 θ

[(δ−β)

(v

c

)2

+

(3δ2 − β2

4−β2−

β

2(1 + δ)− δ2 +

3

4(β− δ)2 cos 2θ

)(v

c

)4], (1.3)

with θ the angle between the velocity ~v of the frame of reference and the path of light, and αn, βn, δn, εn

the expansion terms of a(v), b(v), d(v) and ε respectively with n = 1, 2 the order of the expansion.

If any of the referred expansion terms are veried to be non-vanishing in some measurement, this

would imply the violation of Lorentz invariance.

1.3 General Relativity

Having introduced the general method, we now move onto dene the basic formalism of General

Relativity, along with the conventions that will be used in the remainder of this work.

General Relativity proceeds by claiming that the principle of relativity indeed applies to all frames

of reference, and not just inertial ones. Thus, instead of considering only invariance under Lorentz

transformations, it imposes general (dieomorphism) invariance, i.e. the laws of physics are invariant

under general (dierentiable) coordinate transformations.

The need for having a priori coordinate invariance implies that we need to use a scalar Lagrangian

density which, depends on the elds and their derivatives (up to second order in order to avoid Ostro-

gradsky's instabilities [6]). In Riemannian spacetime, this eld is the metric tensor and its derivatives

are embodied in the Ricci curvature scalar, leading to the standard Einstein-Hilbert action L = R. If,

however, we promote R to a fundamental variable, then more general forms are admissible

L =√|g| [Λ + bR+ c∇µ∇νRµν + ...+ f(R)] , (1.4)

with f(R) a possible function of the metric from which we can build modied versions of the basic

Einstein-Hilbert (which is obtained by selecting the linear terms alone).Variation of the Einstein-Hilbert

action leads to

δS =

∫dx4√−g

[− 1

16πG

(Rµν − 1

2gµνR

)+

1

2Tµν

]δgµν , (1.5)

with

Tµν = − 2√−gδ(√−gL)

δgµν. (1.6)

4

1.4 The limits of Einstein's Relativity

Imposing a null variation yields the Einstein's equations of motion,

Gµν ≡ Rµν − 1

2gµνR = 8πGTµν (1.7)

where G is the gravitational constant. This identity tells us one of the most important results of this

theory: The distribution of energy in spacetime dictates its curvature.

The doubly contracted Bianchi identities ∇νGµν = 0 then imply the conservation of the energy-

momentum tensor ∇νTµν = 0. In the absence of matter, one would obtain R = 0 for the Ricci scalar;

in a static and spherically symmetric spacetime, this leads to the Schwarzschild solution.

The following chapters will use these results to derive the Tolman-Oppenheimer-Volko equation

in chapter 2 or the results presented in the nal chapter.


Having introduced the basic formalism that will be used throughout this work, we shall now discuss

some of the known experimental limits that constrain General Relativity as well as the methods used

in obtaining them. These observational bounds can be used both to test the validity of the foundations

of GR and alternative theories of gravity [2].

Varying the action S =∫ √−gµνdxµdxν one then gets the geodesic equation of motion

d2xµ

dτ2= Γµαβ

dxα

dτ

dxβ

dτ. (1.8)

In the static weak-eld limit, the metric can be written as

g0i = gi0 = 0, g00 = −1− 2ΦNc2

, gij =

(1 +

2ΦNc2

)δij (1.9)

with ΦN the Newtonian gravitational potential. Inserting this metric into the geodesic equation we

get Newton's second law of motion for the (0, 0) component

d2xi

dt2= −Γi00 = −∂ΦN

dxi(1.10)

and, from Einstein's equations of motion 1.7, we get the Poisson equation

∇2ΦN = 4πGρ. (1.11)

This simply means that the metric plays the role of the Newtonian gravitational potential, with the

Christoel symbol behaving analogously to an acceleration.

Thus, by changing the metric one could explore how certain aspects of a theory cascade into the

equations of motion of the system.

1.4.1 Parametrized Post Newtonian Formalism

We now introduce the Parametrized Post Newtonian Formalism, or PPN for short. Behind every

metric theory of gravity lies the underlying principle that the eponymous tensor directly aects the way

in which the gravitational eld interacts with matter. As such, this formalism serves as a generalization

of the metric to include parameters which express certain symmetries and laws of invariance of the

5

1. Introduction

system, which then allows us to measure the deviation of these GR parameters in relation to Newtonian

gravity.

Assuming Local Lorentz and Position Invariance, along with conservation of momentum, it can be

shown that the metric tensor in PPN is given by,

g00 = −1 + 2U − 2βU2 − 2ξΦW + (2γ + 2α3 + ζ1 − 2ξ)Φ1 + 2(3γ − 2β + 1 + ζ2 + ξ)Φ2 (1.12)

+2(1 + ζ3)Φ3 + 2(3γ + 3ζ4 − 2ξ)Φ4 − (ζ1 − 2ξ)A− (α1 − α2 − α3)ω2U

−α2ωiωjUij + 2(2α3 − α1)ωiVi +O(ε3)

g0i = −1

2(4γ + 3 + α1 − α2 + ζ1 − 2ξ)Vi −

1

2(1 + α2 − ζ1 + 2ξ)Wi (1.13)

−1

2(α1 − 2α2)ωiU − α2ω

jUij +O(ε5/2)

gij = (1 + 2γU)δij +O(ε2), (1.14)

where β is a measure of the non-linearity of the law of superposition of gravitational elds, γ mea-

sures the curvature of the spacetime created per unit rest mass, α1, α2, α3 measure deviations from

Lorentz invariance, ζ the violation of Local Position invariance and α3, ζ1, ζ2, ζ3, ζ4 measure the pos-

sible violation of the conservation of momentum. The expressions for the gravitational potentials

U,Uij ,Φi,A, Vi,Wi are given in [3]. GR is characterized by β = γ = 1, and all other PPN parameters

vanish.

Current experimental tests on these parameters, particularly the couple γ, β, are the result of

measurements made by the Cassini 2003 spacecraft and helioseismology show that γ − 1 ≈ 2.3× 10−5

and β − 1 ≈ 3× 10−4, respectively [7].

1.4.2 Equivalence Principle, Lorentz and Position Invariance

We now move to the testing of the postulates presented in the previous section; the equivalence

principle and both the Local invariance principles (position and Lorentz), which are the fundamental

groundwork from which GR is built from. Tests for the Equivalence Principle can be divided into two

groups: those that test the weak version (WEP) and those that test the strong equivalence principle

(SEP).

The WEP states that the Equivalence Principle (all non-gravitational laws should behave in free-

falling frames as if there was no gravity) is satised by all interactions except that of gravity. One

could test its validity by simply measuring the dierence in the free-fall accelerations between two test

bodies a1 and a2 [3],

∆a

a=

2(a1 − a2)

a1 + a2=

(MG

MI

)

1

−(MG

MI

)

2

= ∆

(MG

MI

). (1.15)

where MG and MI represent the gravitational and inertial masses, or by directly measuring the ratio

MG/MI . For the latter, various experiments have been made where the most recent and strongest

constraint being given by Adelberger in Ref. [8] of |1−MG/MI | ≈ 1.4× 10−13.

Another consequence of the WEP is the existence of a gravitational Doppler eect in bodies which

travel trough a changing gravitational potential. One of the most historically relevant experiments

6


was made by Robert Pound and Glen Rebka [9], where photons where emitted by a moving source

and later absorbed by a stationary target located at the top of a tower. The absorption was only

possible as the relativistic doppler shift of the moving source cancelled the gravitational doppler eect

of the graviational eld of the earth. The measured change in frequency ∆ν/ν = 2.57± 0.26× 10−15

conrmed the existence of this eect, as predicted by GR.

The SEP, on the other hand, states that every measurement is independent of the velocity and

position of the laboratory, even accounting for the self-energy of massive bodies such as stars and

black holes.

Testing the SEP implies measuring the contributions of this gravitational energy which were not

considered in the WEP. In order to accomplish this we resort to the PPN formalism introduced above,

introducing the quantity

∆

(MG

MI

)= η

(Ω

Mc2

)(1.16)

where Mc2 is the total mass energy of the body and Ω its negative gravitational self-energy. Here

η = 4β − γ − 3 is a combination of PPN parameters. This combination implies that in GR one should

have η = 0, since γ = β = 1.

We now move on to the two nal assumptions presented in the previous section: Local Lorentz and

Position Invariance, or LLI and LPI respectively.

The phenomenological eect of moving in a reference frame relatively to a stationary one can be

probed by assuming a cosmological vector eld which collapses onto a non-vanishing minimum via

spontaneous symmetry breaking. These types of models have been proposed by Kostelecký [10] and

its impact on solar system observables was discussed in [11]. The mechanism underlying the breaking

of the symmetry is explained in chapter 3.

Experimental searches for the breaking of this symmetry have been made through diverse physical

phenomena. As a violation of this symmetry could imply the breaking of the CPT symmetry, numerous

proposals have been made for the possible testing of this possibility [12].

As referred above, the PPN formalism is also a good way in which to infer the departure of an

experimental phenomenon from what would be expected from GR. In what regards the parameters

that relate to preferred frame eects, the more relevant of these is α2, for which the observational limit

is |α2| < 4 × 10−7[13] and reects the existence of spin precession anomalies, along with α3, which

reects self-acceleration eects. For the latter, a measurement was made via pulsar statistics in order

to measure its deviation if it is non-vanishing, yielding the constraint α3 < 2.2× 1020 [14].

Another way to measure possible Lorentz violation is through the study of the Greisen and Zatsepin

& Kuzmin (GZK) cut-o. This stems from the interaction between protons with energies of the order

of 1020eV and Cosmic Microwave Background Radiation (CMBR) photons in nuclear reactions of the

type

p+ γCMB → p+ π0. (1.17)

Due to these reactions, the primary protons would have their energy decreased, suering a type of

7

1. Introduction

friction from the cosmic background radiation. The threshold for these reactions is given by

Ef =m2π + 2mπmp

4EγCMB≈ 1019eV, (1.18)

which implies that above this energy value, we should not observe cosmic rays with these energies on

Earth.

Conrmation of the GZK limit through experiments such as the Pierre Auger Observatory [15]

would impose strong constraints [16] on the possible observation of LV in the QED as discussed above.

As discussed in Ref. [17], the detection of those 1019eV photons would imply,

|a1| ≤ 10−25, |a2| = |a3| ≤ 10−7 (1.19)

which would indicate a very weak presence of LV at the high energy scale. However, recent results from

both HI-RES (High Resolution Fly's Eye) and the Pierre Auger collaborations have shown evidence

that the cut-o has not been statistically broken, which implies that Lorentz invariance has also been

maintained [18].

In what regards LPI, the already mentioned Pound-Rebka experiment can serve as a test for this

eect, with the measurement of the change in frequency given by

∆ν

ν=

(1 + µ)U

c2(1.20)

were µ = 0 in GR, would give an idea of how far the assumption holds. Measurements made with

hydrogen-maser frequencies on earth and on altitudes of ten thousand kilometres yield µ of |µ| <2× 10−4 [19].

Although LI has these strong constraints, the fact is that there is still room for obtaining deviations

in energy scales that are unreachable today [13, 20]. It is through this window of opportunity that we

shall peer through in the following sections.

8

2Astrophysics

Contents

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Tolman-Openheimer-Volkov equation . . . . . . . . . . . . . . . . . . . . . 10

2.3 Polytropes and the Lane-Emden Equation . . . . . . . . . . . . . . . . . . 12

9

2. Astrophysics

2.1 Introduction

In this section, the hydrostatic equilibrium equation is obtained via an approximation of the known

Tolmann-Oppenheimer-Volko equation which, is derived directly from Einstein's equations of motion.

Assuming a polytropic equation of state, these equations are the so called Lane-Emden dierential

equations, which have a solution depending on the polytropic index n alone. This model will be later

used in chapter 5 as a description of a non-perturbed star from which the subsequent analysis of the

Bumblebee model will follow.

2.2 Tolman-Openheimer-Volkov equation

The description of a star's interior when in hydrostatic equilibrium can be obtained in General

Relativity through the choice of an appropriate energy-momentum tensor along with a static spherically

symmetric metric.

Assuming that the uid inside the star behaves like a perfect uid, then the appropriate energy-

momentum tensor is given by,

Tµν = ρuµuν + p(gµν + uµuν), (2.1)

where the signature (−,+,+,+) is used both in this case and throughout the remainder of this work.

Given the static, spherically symmetric geometry, the Birkho metric is chosen, as it is described

by the line element,

ds2 = −e2ν(r)dt2 + e2λ(r)dr2 + r2dθ2 + r2 sin2(θ)dφ2 (2.2)

and so the energy-momentum tensor is given by

Tµν = diag(ρe2ν(r), pe2λ(r), pr2, pr2 sin2(θ)) (2.3)

and the trace by

T = gµνTµν = −ρ+ 3p. (2.4)

Using the trace-reversed form of the Einstein eld equations referred in Chapter 1

Rµν − 8π

(Tµν −

1

2gµνT

)= 0 (2.5)

the equations gttRtt,grrRrr and g

θθRθθ are then, respectively:

e2(ν−λ)

r

[r(λ′ − ν′)ν′ − rν′′ − 2ν′

]= −4πGe2ν(3p+ ρ) (2.6)

1

r

[2λ′ + r(λ′ − ν′)ν′ − rν′′

]= −4πGe2λ(p− ρ) (2.7)

e−2λ

[1 + e2λ − rλ′ + rν′

]= −4πGr2(p− ρ) (2.8)

Subtracting 2.6 and 2.7 then gives,

− 2e−2λ(λ′ + ν′)

r= 8πG(p+ ρ). (2.9)

10

2.2 Tolman-Openheimer-Volkov equation

For that we need to nd a relation between the density and the metric function λ(r). By using the

combination,

gttRtt + grrRrr − 2gθθRθθ = 0 (2.10)

we nd that

2[1− 8πGρr2 + e−2λ(−1 + 2λ′r)]

r2= 0 (2.11)

which gives us, through some manipulation, the equation

1− e−2λ + 2rλ′e−2λ = 8πGρr2 (2.12)

d

dr

(r − re−2λ

)= 8πGρr2. (2.13)

We can now obtain a relation pertaining λ

e−2λ = 1− 2Gm

r(2.14)

which can be obtained by calculating the mass through

m(r) =

∫ r

0

4πρ(r′)r′2dr′. (2.15)

After dierentiating λ one gets

− 2λ′e−2λ = G2m′r − 2m

r2= 8πGρr − 2Gm

r2, (2.16)

which, by substitution into equation 2.9, gives us a relation between the remainder metric function

ν(r) and the pressure,

ν′ =Gm+ 4πGpr3

r(r − 2Gm). (2.17)

We know from the twice contracted Bianchi identities that

∇νGµν = 0, (2.18)

which implies that the conservation of the energy-momentum tensor ∇νTµν = 0, from the Einstein

eld equations presented in Chapter 1.

By selecting the radial component µ = r, we arrive at the conservation equation for a perfect and

static uid,

p′ + ν′(p+ ρ) = 0, (2.19)

which allows us to reach a dierential equation for the pressure inside the star:

dp

dr=

(G(4πGpr3 +Gm)(p+ ρ)

r2

)(1− 2Gm

r

)−1

(2.20)

which is the so called Tolmann-Oppenheimer-Volko equation for hydrostatic equilibrium.

In order to solve the set of Eqs. 2.15 2.19 2.20 , which consists of a system of 4 unknown functions

λ, ν, p, ρ and three independent equations, we need another equation that relates the pressure and the

11

2. Astrophysics

density, i.e. an equation of state p = p(ρ). This motivates the next section, where we will be describing

the polytrope equation of state and the Lane-Emden dierential equation for a star in hydrostatic

equilibrium, as well as an introduction to the methodology applied in Chapter 5 for obtaining the

equations for the unperturbed star.

In order to proceed into the next section, we will assume the conditions of hydrostatic equilibrium,

i.e.:

p(r) << ρ(r), 4πp(r)r3 << m(r), 2Gm(r) << r, (2.21)

then the equation 2.20 can be approximated by its hydrostatic equilibrium version,

dp

dr= −Gm

r2ρ (2.22)

dm

dr= 4πr2ρ (2.23)

which when combined give us the Poisson equation ∇φ = 4πGρ in spherical coordinates

1

r2

d

dr

(r2

ρ

dp

dr

)= −4πGρ. (2.24)

2.3 Polytropes and the Lane-Emden Equation

With the previous conditions in mind, we can now assume that the pressure is given by the poly-

tropic relation in equation 2.25 for a certain polytropic index n

p = Kρ1+1/n, (2.25)

where K is the polytropic constant and ρ0 the baryonic mass density. The polytropic index describes

the basic thermodynamical processes. Taking n = 1 we get an isobaric sphere, n = 0 an isometric

one and n = ∞ gives the isothermal condition for that same sphere. The adiabatic processes are

related by n = 1/(γ−1) within this framework, where γ represents the adiabatic coecient γ = cp/cV .

These indexes can provide crude approximations to known astrophysical bodies [21]. Degenerate star

cores found in giant gaseous planets can be studied using a polytropic index of n = 3/2, boundless

systems which were rst use in the description of stellar systems by Arthur Schuster in 1883 with n = 5

and the rst solar models, which were rst proposed by Arthur Eddington (known as the Eddington

standard model of stellar structure), circa 1916 for a polytropic index of n = 3. Although these models

are rather simplistic, they are still useful as rst order approximations, as they allow for an easier

algebraic manipulation.

Dierentiation of 2.25 gives

dp

dr=

(n+ 1

n

)Kρ

1ndρ

dr(2.26)

which, when inserted into equation 2.24 changes into

1

r2

d

dr

[n+ 1

n

r2

ρKρ

1ndρ

dr

]= −4πGρ. (2.27)

12

2.3 Polytropes and the Lane-Emden Equation

We can now rewrite this dierential equation in its dimensionless form, by selecting the transfor-

mations

ρ = ρcθn(χ) , p = pcθ

n+1(χ), (2.28)

The dierential equation is now given by

(n+ 1)Kρ

1+ 1n

c

4πGρ2c

1

r2

d

dr

(r2 dθ

dr

)= −θn. (2.29)

where the underscript c refers to the central values of the quantities.

We can now dene χ = r/rn, where rn is dened through the star's radius and the terms present

in equation 2.29,

R = rnχf =

[(n+ 1)K

4πG

] 12

ρ1−n2nc χf . (2.30)

where χf signals the boundary of the spherical body (θ(χf ) = p(χf ) = ρ(χf ) = 0).

With all these components, we can now assemble the dimensionless Lane-Emden dierential equa-

tion:

1

χ

d

dχ

(χ2 dθ

dχ

)= −θn. (2.31)

The analytical solution for this equation was found by Chandrasekar [22] for a set of polytropic

indexes n = (0, 1, 5). These are given by, respectively

θ(χ) = 1− 1

6χ2, θ(χ) =

sin(χ)

χ, θ(χ) =

1√1 + χ2

3

(2.32)

where the nal solution is innite in radial extent.

As we want to calculate the LE solution with n values around the Eddington solution with n = 3,

these exact solutions will not be useful. We thus proceed with the analysis of the system via a numerical

method. We know that the boundary conditions for this system are given by

θ(0) = 1 (2.33)

θ′(0) = 0 (2.34)

and so by making a series expansion of our equation at the point χ = 0, we can then calculate the

initial conditions of our system for some point χ = χi, and solve numerically the equation for any n

up to 5, where the analytical solution is radially innite as referred above.

Throughout this work, three values for the polytropic index where chosen, with the intent of showing

how its variation aects the model. The values are n = 2.8, 3, 3.2 and so the need to obtain a general

expression for the expansion of θ(χ) around χ = 0 arises.

Assuming that the solution θ is symmetric under the transformation χ→ −χ and that the solution

θ is analytic we can expand θ as,

θ(χ) ≈ 1− χ2

6+nχ4

120+

(5n− 8n2)χ6

15120+ ... (2.35)

13

2. Astrophysics

where the terms of order between n = 6 and n = nf where not shown for simplicity. This is useful in

a numerical computation where it might be necessary the computation of a non-zero initial value close

to the real null initial condition.

Using these conditions one then obtains the following proles for the Lane-Emden function θ(χ),

as can be seen in Figure 2.1

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

χ

θ(χ)

n = 2.9n = 3n = 3.1

Figure 2.1: Lane-Emden solution for the a spherical body in hydrostatic equilibrium. The roots of the threefunctions are χ2.8

f = 6.191, χ3f = 6.896 and χ3.2

f = 6.768.

Besides obtaining the prole θ(χ), we can also relate the stars fundamental observed properties, its

mass and radius, and relate them with the central density through the LE solution.

In order to go from the set (M,R) to the set (ρc, pc) from which the LE solution takes its scaling

parameters, we start by integrating the relation ρ0(r) = m′/4πr2 in order to obtain

M =

∫ R

0

4πρ(r)dr = 4πr3nρc

∫ χf

0

χ2θndχ. (2.36)

By inserting the LE equation into the integral one then gets,

M = 4π

[(n+ 1)K

4πG

]3/2

ρ3−n2nc

[−χ2

fθ′(χf )

](2.37)

which allows us to obtain K in terms of the mass and radius of the star and the polytropic index n

K = G

(4πn

n+ 1[−χ2

fθ′(χf )]

1−nn χ

n−3n

f

)(MR−1

)n−1n . (2.38)

The central pressure of the system is then given by inverting rn,

ρc = (4π)nn−1

[R

χf

(K(n+ 1)

G

)−1/2]−2nn−1

(2.39)

completing the parameters needed to completely describe the star.

14

3Symmetry Breaking

Contents

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Explicit and Spontaneous Breaking of Symmetries . . . . . . . . . . . . . 17

3.3 Observer and Particle LSB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Standard Model Extension and LSB . . . . . . . . . . . . . . . . . . . . . . 21

15

3. Symmetry Breaking

3.1 Introduction

The concept of symmetry can be stated as an intrinsic property of a system which does not change

under certain transformations. Its usage throughout history can be traced back to both aesthetic and

technical means of describing apparent order and beauty, such as the apparent mirror symmetry of the

human body, or the more complex six-fold symmetry present in snow akes.

One of the most interesting aspects of this idea is the fact that when applied to physical systems,

the existence of symmetries restricts the possible outcomes that certain phenomena may have under

that system. This idea was stated by Pierre Curie in his Sur la symètrie dans les phénoménes physiques

[23], where he found that the thermal and electric properties of crystals varied with the underlying

crystal structure.

Figure 3.1: Snowakes generated through a Linenmayer rule set for the rst, second and third iterations.

A simple example can be given, as represented in gure 3.1. The snow ake was generated through

a formal rule set rst created by Aristid Lindenmayer, a Hungarian botanist whose work consisted in

trying to describe the evolution of the visual patterns present in the growth of yeast. The Linden-

mayer system takes an alphabet of symbols, an initial state for the system and a set of production

rules and applies the rules iteratively. In the case of our snowake, for each 60 angle a branch is

generated with two smaller branches, with each iteration adding onto the already growing pattern.

We have thus created a model that describes, up to a certain degree of certainty, a snowake which

one could nd in nature, only through the rules that describe its underlying symmetry. But what if

there were only two snowakes in the Universe, and one snowake was slightly dierent than the one

described by our model? Well, this would imply that one of the symmetries that described the "good"

snowake, would be broken by the "bad" one, and thus our model would have to be updated in order

to incorporate this fact.

This idea of looking for the underlying symmetries of a certain system and checking for their

existence in the Universe portrays the motivation behind the model used in this work, although it is

inserted in the larger context of both Einstein's Relativity and the Standard Model of Particle Physics.

The Principle of Relativity, as rst adopted by Einstein in his special theory of relativity, states

that the laws that govern the change of a physical system are independent of the chosen coordinate

systems, even when those are moving relatively to each other in uniform translational motion.

This principle, when coupled with the light postulate which guarantees a xed speed of light in all

16

3.2 Explicit and Spontaneous Breaking of Symmetries

inertial frames of reference, leads to the known Lorentz transformations between coordinate systems

which are moving relatively to each other.

This can be written succinctly, trough the matrix equation

x′ = Λ(v)x (3.1)

which, for a boost in the x direction for example, would be described by

ct′

x′

y′

z′

=

γ −βγ 0 0−βγ γ 0 0

0 0 1 00 0 0 1

c txyz

, (3.2)

representing thus the underlying symmetry of boosts in the x direction.

In General Relativity, this symmetry holds locally i.e. in distances small enough that the variations

in the gravitational eld are unnoticed. This is but a statement of the weak equivalence principle

which, as stated by Einstein, takes the form [24]

The outcome of any local non-gravitational experiment in a freely falling laboratory is in-

dependent of the velocity of the laboratory and its location in spacetime.

In the following sections, when speaking of Lorentz invariance, the concept behind the name will

follow closely what was described in the introduction. The goal is to provide a brief introduction to

the concept of Lorentz Symmetry Breaking (LSB) and its usage as a probe for studying fundamental

physical phenomena in both large and small scales.


The breaking of symmetries is a great source of richness in both physics and everyday life. Consider

then a hungry donkey who is placed between two stacks of hay and assume he always goes to the nearest

one. By the Principle of Sucient Reason put forward by Leibniz, the donkey cannot justify going

for either stack of hay and so it dies of hunger as it is unable to decide. Unless, of course, by some

unknown force of will, he breaks the symmetry of his dire situation and goes for one of those stacks

of hay! Thus it is justied that breaking a symmetry brings with it a lot more to life than simply

perturbing a perfect, albeit immutable situation.

In the case of physics, things are not so simple. We shall start by dening the two ways in which

these symmetries can be broken: explicitly and spontaneously, and provide some examples of these

kinds of phenomena.

For explicit symmetry breaking, the underlying symmetry that is broken appears in the description

of the physical laws themselves. This can take the form of certain terms in the Lagrangian density

that describes the system, as the dynamical equations are not invariant under that transformation.

One example of this is the parity violation in the weak interaction, which was initially proposed by

Yang & Lee circa 1950 [25] by studying the beta decay of cobalt-60. Up until this point the laws of

nature were assumed to be invariant under mirror reection transformations (as observed in gravity,

electromagnetism and the strong force), i.e. observing a certain experiment and its mirror reected

17


copy should yield the same results; however, in the case of beta decay, a preferred direction of the

emitted electrons was always observed.

In relativistic quantum mechanics, this symmetry portrays an invariance in the change between

particles and antiparticles within a system. Along with parity, charge conservation is also seems to be

broken in the weak interactions. The conjugation of both these symmetries (CP), was once thought

to be a valid symmetry of nature, but was later veried by Cronin & Fitch to be broken in the decay

of neutral kaons [26], which gave them the Nobel Prize in Physics in 1980.

Although the individual symmetries appear to not to be fundamental properties of our universe,

a combination of them does: a theorem proved by Schwinger, Pauli, Bell and Lüders around 1950

shows that the combination of charge conjugation, parity and time-reversal symmetries is conserved

in quantum eld theories which have Lorentz invariance, local causality and positive energy [27]. This

is called the CPT theorem and up to this day this symmetry seems to hold up experimentally [28].

Another example is the occurrence of symmetry breaking through non-renormalizable eects. Ef-

fective eld theories appear as low-energy approximations to a more overreaching theory, as they only

accurately describe the particles that fall within the energy range considered. Although the eects

of heavier particles do not appear on the low energy regimes, when moving on to higher energies,

symmetries which were assumed on the low end of the energy scale could be broken on the higher

energy theories.

As for the case of spontaneous breaking of symmetries, things are a bit dierent. Instead of the

asymmetry existing in the equations of motion themselves, it occurs instead in one of the possible

solutions to those equations, arising dynamically from the system.

A simple example can be given in order to illustrate this. Consider a cylindrical rod held hori-

zontally. If we let the rod fall, it will spontaneously choose a direction, breaking the initial rotational

symmetry. The state in which the symmetry of the system is broken is one of the innite solutions of

the dynamical equations governing the system.

This same phenomenon can be observed when a ferromagnet is cooled below its critical temperature

(Curie temperature Tc). Initially the system presents no magnetization, as it is has T > Tc, but a net

magnetization emerges as soon as T < Tc, where the spins align spontaneously in a given direction,

breaking the initial symmetry of the system.

In the context of particle physics, the occurrence of spontaneous symmetry breaking is fundamental

in explaining certain phenomena. One example of the breaking of a discrete symmetry is the Yukawa

interaction, which can be explained via the Lagrangian density

L =1

2(∂φ)2 − µ2φ2 − λφ4. (3.3)

where φ is a real scalar eld. This Lagrangian density represents a system consisting of a self-interacting

scalar eld φ, with the potential of the system consisting of the V (φ) = −µ2φ2 − λφ4 terms. It is

invariant under the global Z symmetry φ → −φ or, if we instead adopt a complex scalar eld, the

global U(I) symmetry φ→ φeiα, with α a constant.

If our potential has a minimum value at some point φ0 6= 0, then its symmetry would be eectively

18


broken. In this case, by taking µ2 = 0 the potential would have two possible minimum values φ0 =

±√

1/(2λ)µ, which when selected by the system would spontaneously break the underlined global

symmetry.

In the case of continuous SSB, the discovery of the Goldstone's Theorem [29] is fundamental in

explaining several of results in particle physics. As a result of the continuous symmetry being broken,

massless bosons appear (Goldstone bosons), with the number of bosons being equal to the number of

generators of the broken symmetry. These massless bosons are crucial in the understanding the Higgs

mechanism. In this case, the picture changes as the symmetry being broken is local, instead of the

global one described by the last example.

A similar model to the one given by equation 3.3 can be given in order to explain this mechanism.

Consider the Lagrangian of the complex scalar eld φ, φ∗ coupled with the electromagnetic eld

L = −(Dµφ)∗(Dµφ) + µ2φ∗φ− 1

4λ(φ∗φ)2 − 1

4FµνFµν (3.4)

where Dµφ = ∂φ − iqAµφ is the covariant derivative, and the potential can be identied similarly as

V (φ, φ∗) = µ2φ∗φ − 14λ(φ∗φ)2. This potential is what is called the "Mexican Hat Potential", due to

its shape, which can be seen on Figure 3.2.

Figure 3.2: The famous Mexican Hat potential which describes how a system spontaneously breaks its initialsymmetry by rolling onto the minimum of the potential, where a circle of innite possible solutions exist.

The minima is now given by a point in a circle on the complex plane, i.e. |φ|2 = µ2a2/2 with

a =√

4µ2/λ, and so the "rolling" of the eld to one of these values would eectively break the

symmetry.

If we dene the scalar eld φ as,

φ(x) =1√2

(α+ β(x) + iγ(x)) (3.5)

then our kinetic terms are changed into

− (Dµφ)∗(Dµφ) = −1

2(∂β)2 − 1

2(∂γ)2 − 1

2q2α2A2 + qαAµ∂

µβ + ... (3.6)

19


where we end up having a quadratic coupling between the gauge eld A and the scalar γ. With an

appropriate gauge change, given by Aµ(x) = Vµ(x) + (qα)−1∂µγ(x), this can be corrected, as now the

kinetic terms are give by

− (Dµφ)∗(Dµφ) = −1

2(∂β)2 − 1

2q2α2V 2 + .... (3.7)

This is the Higgs mechanism: The gauge change brought with it a new mass term with m = qα,

with the added advantage of absorbing the scalar eld γ and collapsing the potential onto a non-zero

minimum, thus breaking its symmetry.

The goal of this work is not to explain the Higgs mechanism in detail, but rather bring some

light into the relevance that the breaking of spontaneous symmetries has in explain certain physical

phenomena. The gravitational models that will be presented will have some aspects of these SSB in

some sense, which though not representing the same mechanism, are easier to understand having a

more complete conceptual baggage.

3.3 Observer and Particle LSB

Due to the extensive application of this symmetry in physical theories, discussing it without clarify-

ing which exact symmetry is broken brings with it a lot of confusion, as not all Lorentz Transformations

(LT's) are the same. We can group the dierent types into two groups: as Observer and Particle

Lorentz transformations. Following this labeling, we can characterize them in the following manner.

• Observer Lorentz transformations: These are the transformations one usually thinks of

when working in Special Relativity. They relate the observations of two inertial observers which

can have dierent velocities or can be rotated in some manner in relation to each other. Not

having this kind of invariance would mean that the choice of reference frame would alter the

results obtained when measuring particular phenomena, i.e. a particle's mass would vary from

one point to another. Due to this, theories must preserve this symmetry, something which is

accomplished by writing the laws of physics in terms of covariant equations. If one takes an

action S dened on a manifoldM with a metric g and which depends on some eld Φ(x), then

S[g,Φ(x)] =

∫

ML(g,Φ(x),∇Φ(x)) (3.8)

must be a Lorentz scalar in order to be invariant. The equations of motion derived from this

action will be invariant to boosts and rotations between reference frames of the form

xµ → x′µ = Λµνxν (3.9)

with the elds also being transformed accordingly.

• Particle Lorentz transformations: If in the rst case the dierence between dierent exper-

iments was only a change of coordinates, in this case two identical experiments can be boosted

or rotated relative to each other by the same observer. This happens because as we rotate our

20

3.4 Standard Model Extension and LSB

experiment in relation to a certain eld, the relations between them are also altered. In order to

clarify this fact, a couple of examples can be given.

Consider a system composed of a charge of massm and charge q being aected by a perpendicular

magnetic eld ~B. Its motion can be described by

md~v

dt= q~v × ~B, (3.10)

an equation that we know is valid in all reference frames. The particle in this situation would

move in a circular motion in a plane perpendicular to the eld. Suppose we made a particle LT

by means of a boost; the consequence of this would be a larger radius in the trajectory of the

particle, as its momentum would be increased. On the other hand, if we made an observer boost

along the particle's trajectory, the result would be dierent, as a drift in the particle would arise

due to now there being an electric eld.

The other example is also straightforward. Considering a mass on an inclined plane on the surface

of the earth [30], if one simply changes the system of coordinates that describe the system, the

acceleration that the mass obtains is the same in both reference frames, diering only from a

change of coordinates. However, if we rotate the inclined plane, that acceleration would change

perceptively as the direction of the gravitational eld in relation to the xed background would

be dierent (it would be as if we created a new ramp with a more drastic incline).

In conclusion, the type of Lorentz symmetry that is eectively broken in the following sections will

be of the second type, i.e. particle Lorentz transformations.


The Standard Model gives us an accurate description of the myriad of phenomena which occur

between the basic particles and forces at very small scales. On larger scales, this burden falls on the

classical description provided to us by General Relativity which, as far as experimental conrmation

goes, has proven itself within the class of physical eects it tries to eectively describe [10].

As these two eld theories are expected to merge at the Planck level, with energies around mP =

1019GeV, into a single unied and consistent description of nature, search for possible signals at this

scale is paramount to achieve a deeper understanding how this merger might theoretically occur. One

possible candidate for these signals is the breaking of Lorentz symmetry.

This work will be centered on a particular type of alternative gravitational theories which are

based on the introduction of vector elds in the Lagrangian density of our system, the so called Aether

theories.

The main motivation for this type of approach emerged from the work done initially by Alan

Kostelecký circa 1988, which consisted in the study of natural Lorentz symmetry breaking (LSB)

mechanisms in bosonic string theory, with the goal of trying to explain the relationship between the

26-dimensional spacetime needed for that theory, and the four at dimensions that we know of, with

21


the assumption that the breaking the symmetry could bring with it the compactication of extra

dimensions [31].

The raise in interest regarding the breaking of fundamental symmetries (such as Lorentz and

CPT violation) in eld theories then culminated on the creation of the Standard-Model extension. A

framework built from the core elements of the Standard Model and General Relativity, as Kostelecký

refers to in Ref. [10]

(...) suppressed eects emerging from the underlying unied quantum gravity theory might

be observable in sensitive experiments performed at our presently attainable low-energy

scales. (...) Any observable signals of Lorentz violation can be described using eective

eld theory. To ensure that known physics is reproduced, a realistic theory of this type must

contain both general relativity and the SM, perhaps together with suppressed higher-order

terms in the gravitational and SM sectors.

To give an idea, the Standard-Model Extension can be represented by an action consisting of a

partial sum of terms given by,

SSME = SSM + SLV + Sgravity + ... (3.11)

where SSM is the SM action (although with some gravitational couplings [10]) with a corresponding

Lagrangian density given by

LSM = Llepton + Lquark + LY ukawa + LHiggs + Lgauge (3.12)

SLV corresponds to the SM Lorentz and CPT-violating terms and Sgravity is the gravity sector of

the Lagrangian.

The Lorentz-violating terms on the SME take the form of Lorentz-violating operators coupled to

coecients which will be dened through Lorentz indexes. The existence of non-zero LV coecients

could appear through various mechanisms, one of which (and the one which will be crucial for the

models referred to herein) is spontaneous Lorentz violation (SLV).

The classication of the Lorentz-violating terms can be done through the observed properties under

CPT [10], as the breaking of this symmetry in Minkowski-spacetime implies Lorentz violation.

The pure gravity action can be written as,

Sgravity =1

2k

∫d4xLgravity (3.13)

with the Lagrangian density consisting of a Lorentz invariant and a Lorentz violating part Lgravity =

LLI + LLV + ... the latter being considered in the limit in which the torsion vanishes [10].

Although it is in this Lorentz violating Lagrangian density that we shall focus our attention, a

brief example follows of how the insertion of Lorentz breaking terms into our Lagrangian density can

change some known physical properties of a system.

Consider the following QED Lagrangian density with isotropic LV [17] which allows us to study LV

in the case of the photons and electron/positrons:

22


L = ψ(iγµDµ −m)ψ − 1

4FµνF

µν + ia1ψγiDiψ +

ia2

M2Djψγ

iDiDjψ +a3

4M2Fkj∂

2i F

kj . (3.14)

The Lorentz violating terms are coupled to parameters which allow to test the constraints on some

observable quantities. The rst two terms belong to the usual QED Lagrangian (the Dirac term

along with the electromagnetic energy). The remainder terms are the Lorentz violating ones, which

are coupled to the scale constant M and the dimensionless parameters a1, a2, a3. The u, j indexes

represent the spatial components of the respective terms. From this Lagrangian one can then derive

the dispersion relations, which will naturally depend on the LV parameters, as follows,

E2γ = k2 +

a3k4

M2(3.15)

E2e = m2 + p2(1 + 2a1) +

2a2p4

M2. (3.16)

which are clearly a deviation from the relativistic dispersion relations. The work done in [17] focused on

obtaining constraints for the ai parameters above by studying the interaction of high-energy photons

with the Earth's atmosphere and magnetic eld and how the detection of photon-induced showers with

energies above 1019eV would constrain those parameters to values of the order of |a1| ≤ 10−25, |a2| =|a3| ≤ 10−7, as referred to in chapter 1. The small values for these parameters are in agreement with

the fact that no compelling evidence exists for Lorentz violation.

In the following section a group of models which where created with the goal of trying to obtain

similar constraints will be presented. Focusing on the gravitational sector of the Standard Model

Extension, the aether models present themselves as phenomenological probes to test the existence of

LSB within astrophysical bodies and cosmologies.

23


24

4Vector Theories

Contents

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Aether Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 The Bumblebee Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

25

4. Vector Theories

4.1 Introduction

In the following section, a brief introduction to Einstein-Aether theories will be made. Starting

from the dening Lagrangian density of these theories, a brief historical review of how they came into

existence is made. As these theories are relevant in the context of dark matter, particularly as a result of

the emergence of the MOND theories (Modied Newtonian Dynamics) and later on the TeVeS models

which were used as an attempt to explain the phenomenon, the dark matter and energy problem will

also be described. Within the Aether models, two applications will be presented: one regarding the

change in the gravitational constant and the other regarding ination. Finally, the Bumblebee model

will be introduced. This model will later on be used in the work presented in chapter 5 as a way to

obtain astrophysical constraints on the existence of a vector eld with a non-vanishing expectation

value inside spherical star-like bodies.

4.2 Aether Theories

Although some of the more direct references [3235] mention the introduction of Lorentz breaking

symmetries in gravitational theories to the work of Kostelecký [10, 12, 31], the study of gravitationally

coupled vector eld theories dates back to a decade earlier, with the work of Will and Nordtvedt. Their

work, circa 1972, was based on using the Post-Newtonian Formalism (PPN) (see chapter 1 for more

details) to show the eects of a preferred frame of reference in the tides of the earth, the perihelion-shift

of the planets and the variation of the earth's rotation rate [36].

Aether models are based on the introduction of a vector eld in the Lagrangian density of the

system with a non-vanishing vacuum expectation value. Due to that property, the vector eld will

dynamically select a preferred frame at each point in space-time, spontaneously breaking Lorentz

invariance. This is a mechanism reminiscent of the breaking of local gauge symmetry in the Higgs

mechanism as explained in chapter 3 and in [37] and it serves as phenomenological representation of

the LSB terms in the gravitation sector of the SME, as discussed in Ref. [10].

Aether theories have been used, in parallel with the work of Kostelecký [10], as phenomenological

probes of LSB in quantum gravity and as models for ination and dark energy [32] and the corre-

sponding action consists in a 4-vector Aµ coupled to gravity, which can be written as,

S =

∫d4x√−g

[R

16πGN+ LAE(gµν , Aα)

]+ SM (gµν ,Φ) (4.1)

where SM stands for the matter action, gµν the metric and Φ the matter elds which couple only to

the metric [33, 35] and not to Aα.

For the Einstein-aether Lagrangean LAE we can start by writing it in the most general way given

by Ref. [33] (and the references found within),

LAE = Kαβµν − V (AµAµ) (4.2)

with V (AµAµ) being a general potential which depends on our vector Aµ.

The coecients Kαβµν can be given by

Kαβµν = K(1)αβµν +K(2)αβµν +K(3)αβµν (4.3)

26

4.2 Aether Theories

with

K(1)αβµν = (c1gαβgµν + c2δ

αµδ

βν + c3δ

αν δ

βµ + c4A

αAβgµν)(∇αAµ∇βAν) (4.4)

K(2)αβµν = (c5δανA

βAµ + c6gαβAµAν + c7δ

αµA

βAν + c8AαAβAµAν)(∇αAµ∇βAν) (4.5)

K(3)αβµν = c91

2FµνF

µν + c10δαµδ

βνRαβA

µAν + c11RδαµAαA

β (4.6)

in which the c4 and c8 gauge the relevance of the covariant derivatives along Aµ and Fµν = ∂µAν−∂νAµ.

The K(3) is the most relevant of the three, as it contains terms that represent the coupling be-

tween the vector eld and the geometry of the system given by the Riemann tensor (particularly the

c10δαµδ

βνRαβA

µAν and c11RδαµAαA

β terms) and the K(1) and K(2) are simplied to a single kinetic

term β(∇µAµ)2.

As stated previously these models are a subclass of Lagrangian densities where torsion is not

considered, in contrast to the more general case considered by Kostelecký in Ref. [10] and following

work starts with a Lagrangian density consisting of a terms mainly pertaining to the K(3) tensor, as

can be seen in Ref. [35]:

S =

∫d4x√−g

[R

16πG− β1

2FµνF

µν − β(∇µAµ)2 + c10RµνAµAν + c11RAµA

µ − V (AµAµ)

](4.7)

These models did not evolve from a vaccum, but rather from the complete opposite: the existence of

dark matter.

Hints for the existence of dark matter emerged in the early 1930s with the work of J. Oort and

F. Zwicky. Oort, by studying the Doppler shifts of stars in the Milky Way and thus obtaining their

velocities, observed that those stars had enough velocity to escape the gravitational pull provided by

the luminous mass of the galaxy i.e. the mass of the bodies that were directly visible [38].

The easiest way of studying stellar bodies is by measuring how much light they emit per unit of

time, i.e. their luminosity. Because we can accurately measure the mass of the Sun, we calculate its

mass-to-luminosity ratio ML and use it as a standard, which allows us to obtain estimates of the masses

of other astronomical bodies by comparison.

Zwicky used the M/L ratios of the nebulae in the Coma cluster in order to obtain their mass,

and discovered that this was only 2% of the average mass of the nebulae. This value was obtained by

dividing the total mass of the cluster by the number of observed nebulae (about 1000). [39]

Another way in which this mass discrepancy was observed was by assuming that galaxies in galaxy

clusters behaved like planets in the Solar System, with a velocity dispersion given by the Newtonian

expression

v(r) =

√GNm(r)

r. (4.8)

where m(r) is the mass enclosed within a sphere of radius r. Vera Rubin et al. [40] studied the rotation

curves of 60 galaxies and compared the results with what would be obtained by the expression above.

She found that the rotation curves of those galaxies where "at", meaning that for larger radii the

velocities of the galaxies increase until a threshold is reached and remain relatively constant for larger

27

4. Vector Theories

radii. This is unexpected because if one considers that only the luminous mass in the center of the

cluster as the only source of gravitational pull m(r) = M , the velocity should be higher closer to that

center and lower in the periphery, where the gravitational pull is smaller (as can be immediately seen

by the expression above). This then implied that that there is some sort of extended non-luminous

matter in the periphery of the galaxy that exerts a gravity pull large enough to keep the galaxies

rotating at those speeds.

Pieces of evidence from other sources also exist: X-ray radiation from the hot gas surrounding

galaxies was used by Vikhlinin et al. [41] to determine the mass distribution of the galaxies; Ratios of

baryonic to total mass in the order of 13% where found, again indicating the existence of extra mass

in those galaxies.

Gravitational lensing also provides some conrmation for these results, and some consider it as

a direct observation of dark matter, as it does not depend on the dynamics of the clusters [42].

Gravitational lensing consists in measuring masses through the deection of light presents when passing

through a gravitational eld, creating one of more images of the original object in a dierent location

where the object would be directly observed. Weak gravitational lensing (where a single image is

formed) has been used to infer the existence of dark matter in the Bullet Cluster [42], where the

matter distribution was found to be dislocated from the cluster's luminous center of mass. This

discrepancy can be seen in Figure 4.1, where the two sets of contour lines (blue and magenta as seen

in color) which represent the observed gravitational mass are located away from the larger physical

bodies (in black) that are part of the cluster.

Figure 4.1: F606W-band image of the Bullet Cluster. Image taken from [43]

As a nal example, Cosmic Microwave Background (CMB) radiation can also be used as a test for

inferring the existence of dark matter, through the measurement of the density parameters present in

the Friedman-Robertson-Walker model. Starting with the Friedman equation,

R2 − 8πG

3ρR2 = −kc2 (4.9)

28

4.2 Aether Theories

which describes a at, closed or open universe for k = 0,+1,−1, we can rewrite it by considering the

densitiy parameter given by

Ω =ρ

ρc=

8πGρ

3H2(4.10)

with H the Hubble parameter. This allows us to obtain the relation

H2(Ω− 1) =kc2

R2, (4.11)

which directly shows a relation between the matter density and the resulting curvature. The limits Ω =

1,Ω > 1,Ω < 1 make the correspondence between a at, closed and open Universe. CMB anisotropy

measurements allow us to obtain values for the total density parameter Ωtot = Ωm+Ωrad+ΩΛ and for

the isolated Ωm = ΩBaryonic+Ωnon−Baryonic parameter. Various measurements have been made, with

the most recent belonging to the Planck 2013 experiment [44]. Its results where Ωmh2 = 0.1423±0.0029,

Ωbaryonic = 0.02207 ± 0.00033 and for non-baryonic cold dark matter Ωch2 = 0.1196 ± 0.0031 (with

h = 6.2606957(29) × 10−34). These results are in agreement with previous experiments and show a

great disparity in the relative percentages of baryonic to non-baryonic matter, again indicating a strong

presence of dark matter in our Universe.

Numerous candidates have been proposed to ll the role of dark matter. Standard model neutrinos

along with sterile neutrinos [45] (same as Standard Model neutrinos but without weak interactions)

have been considered as candidates due to their weak interaction with baryonic matter. One of the

main problems in this assumption is that their abundance in the Universe does not allow them to be

the dominant component of dark matter. [45]

Axions, a result of CP violation physics postulated by Peccei and Quinn [46], have also been

considered, with searches currently being done by the Axion Dark Matter Experiment [47].

Another class of candidates encompasses the super-symmetric candidates emerging from SUSY.

Neutralinos have been widely studied as cold dark matter candidates due to being heavy stable particles

with coupling strengths in the order of the weak interaction [45]. Experiments such as the Cryogenic

Dark Matter Search [48] and more recently the Large Underground Xenon (LUX) [49] seek to detect

these kinds of weakly interacting massive particles (WIMPs) but the results don't seem to strengthen

the dark matter hypothesis.

For more comprehensive reviews of particle candidates to dark matter please check [45, 50, 51].

Besides particle candidates, other alternatives exist to the dark matter hypothesis. Modied New-

tonian Dynamics is one of such alternatives and it consists in a modication of Newton's laws in order

to explain the at rotation curves discussed earlier.

The main assumption in MOND theories is that Newton's law is modied with a dependence in the

acceleration of the system. For low accelerations we have, instead of the usual second law of motion:

~F = m~aµ(x) (4.12)

where µ(x) is any function exhibiting the asymptotic behaviors

µ (x) =

x if x 1

1 if x 1.(4.13)

29

4. Vector Theories

For small values of the ratio aa0

we can compare the acceleration to what we would normally have,

i.e.,

~F = m~g = m~aa

a0→ a =

√ga0. (4.14)

So, for the simple case of a body in an orbital motion around a central mass M, where g = GMr2 we

would have a centripetal acceleration given by a = v2

r so our velocity would be

v4 = GMa0, (4.15)

which would be dependent from the central mass M. The parameter a0 can now be tted to the results

obtained in the velocity curves yielding a0 = 1.2×10−10m/s2 [52]. One of the most interesting aspects

of this theory is that the value of a0 obtained from observations is in the same order of magnitude of

the Hubble constant, a0 ∼ cH0 [53].

Although this framework allows us to explain some of the velocity curves of individual galaxies,

some problems do exist. As Sanders and McGaugh explain [53],

There have also been several contributions attempting to formulate MOND either as a

covariant theory in the spirit of General Relativity, or as a modied particle action (modied

inertia). Whereas none of these attempts has, so far, led to anything like a satisfactory

or complete theory, they provide some insight into the required properties of generalized

theories of gravity and inertia.

One example of an attempted merger of both MOND and General Relativity is what is called the

Tensorial-Vector-Scalar theory, proposed by Bekenstein in 2004 [54]. The assumptions of the theory

are also its main source of criticism [55], as it assumes the existence, besides the metric gαβ and the

matter elds φi, one extra vector eld Uα and two extra scalar elds σ and φ, the rst of which lacks

of a convincing justication due to not having dynamic terms in the action. The action of this theory

also goes in the inverse direction of one of the main aspects of MOND that made it so appealing,

as its simplicity is completely lost, as one can see by the action S = SG + SV + SS + SM with the

corresponding terms given by

SG =

∫R(−g)1/2d4x (4.16)

SV = − K

32πG

∫ [gαβgµνU[α,µ]U[β,ν] −

2λ

K(gµνUµUν + 1)

](−g)1/2d4x (4.17)

SS =1

2

∫ [σ2hαβφ,αφ,β +

G

2l2σ4F (kGσ2)

](−g)1/2d4x (4.18)

SM =

∫L(φi)(−g)1/2d4x. (4.19)

Another point in which MOND theories fall short is the inability to solve some discrepancies in

particular matter dispositions of galaxy clusters, as discussed in [5658]. The matter distributions of

visible matter do not correlate exactly with the non-visible part, an eect which can be seen directly

in the Bullet Cluster, where two clusters of galaxies collided to form a matter distribution with two

distinct peaks of visible matter density, and two other zones where the gravitational lensing eect is

stronger, these being located both far away from each other and far relatively far away from the visible

density peaks.

30

4.2 Aether Theories

4.2.1 Eects on the Gravitational Constant

In this section the goal is to give an overview of some interesting aspects and results obtained by

applying these types of models to the study of cosmology. Starting by selecting the relevant terms for

our Lagrangian, we have

L = K(1)µναβ − λ(uµuµ +m2). (4.20)

where all but the c4 coecients are non-zero and our potential is a Lagrange multiplier eld associ-

ated with the variable λ, for which we will later on deduce its equations of motion. Following the

methodology presented by Ref. [59] and [60], we dene a new tensor given by,

Jµα = Kµναβ∇νuβ (4.21)

which allow us to obtain the equation of motion from the action above, with respect to the vector eld

uµ

∇µJµν = λuν . (4.22)

The constraint condition for this equation is

uµuµ = −m2 (4.23)

which basically says that we are solving our equations of motion in the vacuum expectation value of

our eld.

Given the adopted metric signature (−,+,+,+), we require that the vector uµ is timelike, which

implies that m2 > 0.

Multiplying the λ equation by uν on both sides gives

λ = − 1

m2uν∇µJµν . (4.24)

The metric that we will be using is the FLRW metric, which describes a spatially and isotropic

universe.

ds2 = −dt2 + a2(t)

(1

1− kr2dr2 + r2dΩ2

)(4.25)

The condition of spatial isotropy implies that our Aether vector eld must have only a temporal

component. Because of the condition gνµuµuν = −m2, the vector must be something like

uµ = (m, 0, 0, 0). (4.26)

Inserting this in equation 4.22 we get,

λ(t) = −3(c1 + c2 + c3)H2 + 3c2a

a(4.27)

which depends directly on the aether coecients dened in K(1) term of the previous section.

Besides this, another useful expression is that of the stress energy tensor which can be obtained

by varying the action with respect to the metric gµν . The calculations, albeit cumbersome, are spread

out in the literature related to these models [61], and its expression can be simplied to [59]:

31

4. Vector Theories

Tµν = 2c1(∇uβ∇νuβ −∇βuµ∇βuν)− [∇β(u(µJβν))+

∇β(uβJ(µν))−∇β(u(µJβν))]−

2

m2uα∇βJβαuµuν + gµνL.

(4.28)

The next step is to analyze how the introduction of the aether vector eld aects gravity. The rst

assumption is to consider that we have a system composed of both matter and the vector eld, which

can be expressed by the Einstein eld equations by the sum of the two stress-energy tensors

Rµν −1

2Rgµν = 8πGN (Tmatterµν + T aetherµν ). (4.29)

The gravitational constant GN is the one dened in the Aether Lagrangian from equation 4.1.

Matter is assumed to behave as a perfect uid, with a matter-energy tensor:

Tmatterµν = (ρm + pm)ηµην + pmgµν (4.30)

The Aether eld can be described in the same fashion, dening the corresponding pressure and

density

ρae = −3αH2 , pae = α

[H2 + 2

a

a

], (4.31)

with H = aa the Hubble parameter and α = (c1 + 3c2 + c3)m2 [59].

The Einstein equations lead, through the Bianchi identities, to the energy conservation equation

for a general density ρ and pressure p

ρ+ 3H(ρ+ p) = 0 (4.32)

If we use the denitions above for the aether density and pressure and insert it in equation (31),

we have,

˙ρae + 3H(ρae + pae) =

−3α2HdH

dt+ 3H(−3αH2 + α+ 2α

a

a) =

−3α2H(a

a−H2) + 3H(−2αH2 + 2α

a

a) =

−6αH(a

a−H2) + 6αH(

a

a−H2) = 0,

(4.33)

which proves that the selected terms are a good choice.

For the FLRW metric we have

R00 −1

2Rg00 =

3

8πGN

(H2 +

k

a2

)= T00 = ρae + ρm. (4.34)

The spatial components are given by

Rik −1

2Rgik =

1

8πGN

(H2 + 2

a

a+

k

a2

)= Tik = pae + pm (4.35)

Rewriting these equations in a form similar to the Friedmann equations, substituting the Aether

terms for pressure and density and assuming a at space k = 0, we get

H2 =8πGc

3ρm, (4.36)

a

a= −4πGc

3(ρm + 3pm) (4.37)

32

4.2 Aether Theories

with the denition of a new gravitational constant given by

Gc =GN

1 + 8πGNα. (4.38)

The aether energy density suggests that for a positive energy density, we must have α < 0; positivity

of H2 further implies that 1/(8πGN ) ∼M2p < α < 0 (where Mp ∼ 1019GeV is the Planck mass). This

would imply that our gravitational constant would increase in relation to the original G∗. Because the

acceleration equation (36) depends linearly on Gc, this implies that the net eect would be that the

acceleration rate of the expansion of the universe would become larger.

4.2.2 Ination

Based of the Lagrangian density used in the study above, one can also study the role that these

models play on ination. For this case the potential chosen by Kanno & Soda [62] has a Lagrange

multiplier potential similar to the one used above, where m2 = 1. The ci terms adopted are also

dierent, along with the expression for the timelike vector eld uµ

uµ = (1, 0, 0, 0) . (4.39)

The inital action is of a Scalar-Vector-Tensor theory, where besides the vector eld, we have a

scalar eld φ coupled to it. This new Lagrangian can be understood as a perturbation of the scalar

eld ination model, where a Lorentz violating vector eld is introduced. The inital action is given by

S =

∫d4x√−g

[1

16πGR− β1(φ)∇µuν∇µuν − β2(φ)∇µuν∇νuµ − β3(φ)(∇µuµ)2

−β4(φ)uµuν∇µuα∇νuα + λ(uµuµ + 1)− 1

2(∇φ)2 − V (φ)

] (4.40)

and one can immediately see that the terms pertaining solely to the scalar eld, are those used in the

most simple models that describe ination.

The metric used is

ds2 = −dt2 + a2(t)δijdxidxj . (4.41)

where a = eα.

Inserting the vector eld and the metric in the action gives[62],

S =

∫dt

1

Ne3α

[− 3

8πG

(1 + 8πGβ

)H2 +

1

2φ2 − V (φ)

]. (4.42)

where β = β1 + 3β2 + β3.

From this action the procedure is similar to what was done in the previous section, as we obtain

the relevant equations of motion for the variables of our system β, φ,H. They are, respectively

γH2 =1

3

[H2φ′

2β+V

β

](4.43)

γH ′

H+φ′2

2β+β′

β= 0 (4.44)

φ′′ +H ′

H+ 3φ′ +

V,φH2

+ 3β,φ = 0 (4.45)

33

4. Vector Theories

where Q = dQdη

dηdt with Q′ = dQ

dη and γ = 1 + 18πgβ .

The existence of Lorentz violation implies that the terms relating to it in our model are compar-

atively big to the scalar eld ones. This assumption, i.e. β >> 1&β >> f(φ, φ′′, β′) will imply that

γ → 1 and the rst and third equations will now be

H2 =V

3β(4.46)

φ′ +V,φ3H2

+ β,φ = 0 (4.47)

The choice of potential is that of a parabola, in order to nd the similarities between these conditions

and the slow roll conditions one forces when studying independent scalar eld ination. Besides that,

the authors suggest a quadratic coupling of the scalar eld to the Aether parameters β, so we have

β = ηφ2 ; V =1

2m2φ2. (4.48)

These conditions, along with equations (45) and (46) allow us to obtain the solutions

φ(α) =φ0

a4η(4.49)

H2 =m2

6η(4.50)

which represents an inationary model as a consequence of Lorentz violating parameters in our model.

This can be seen directly if we consider the way in which these parameters where choosen.

Because H = α, we dened a(t) = e2α(t) then α =˙a(t)

a(t) which is the denition of the Hubble

parameter. The deceleration parameter q can be related to H by

H

H2= −(1 + q)⇔ q = −

(1− H

H2

). (4.51)

Because H is a constant we have q = 1, which is the necessary condition for our theory to have

ination.

4.3 The Bumblebee Model

A model that contains a vector eld which dynamically break Lorentz symmetry is called a Bumble-

bee model. These models, although with a simpler form, contain interesting features such as rotations,

boosts and CPT violations.

This subclass of aether models posits the following action functional,

SB =

∫d4x

[1

16πG(R+ ξBµBνRµν) −1

4BµνBµν − V (BµBµ ± b2)

], (4.52)

where the Bµ is the same as the vector eld Aµ (but now following the KS convention), and Bµν being

the Bumblebee eld strenght given by

Bµν = ∇µBν −∇νBµ. (4.53)

The parameter ξ, with dimensions of M−2, represents the coupling between the Ricci tensor and

the Bumblebee eld Bµ and V the potential of the Bumblebee eld which, as in the case of the aether

34

4.3 The Bumblebee Model

theories, is the term that drives the breaking of the Lorentz symmetry of our Lagrangian by collapsing

onto a non-zero minimum at BµBµ = ∓b2. Here, Bµ is one of the Lorentz breaking coecients

referred in chapter 3. The presence of this coecient implies a preferred direction is selected at a

certain Lorentz frame, which implies that the equivalence-principle is locally broken for that particular

frame. Observations of Lorentz violation can emerge if particles or elds interact with the Bumblebee

eld [10].

It is worthy to repeat that within this local frame of reference, local particle Lorentz transformations

(referred to in chapter 3) can be performed without changing the local Bumblebee eld, as under these

transformations Bµ behaves as a set of four scalars. Rotations and boosts that change the local Lorentz

frame (observer Lorentz transformations) allow us to choose arbitrarily the local Lorentz frame being

observed, as under these transformations the Bumblebee eld behaves covariantly as a four-vector.

This allows us to maintain local Lorentz invariance despite having local particle Lorentz violation.

As referred in Ref. [34], a smooth quadratic potential of the form,

V = A(BµBµ ± b2)2 (4.54)

with A a dimensionless constant, is chosen. This potential allows both Nambu-Goldstone excitations

(massless bosons) besides the massive excitations for the cases of V = 0 and V 6= 0 respectively [10].

The other case mirrors one of the presented potentials in the aether model section above, i.e. a

linear Lagrange-multiplier potential which takes the form

V = λ(BµBµ ± b2). (4.55)

Both cases where studied from the particle physics point of view and, besides the spontaneous

lorentz breaking, these potentials present also the breaking of the U(1) gauge invariance and other

implications to the behavior of the matter sector, the photon and the graviton. A good review of

experimental proposals to test the result of Bumblebee models can also be found in [63].

Notice that the potential V is assumed to depend on BµBµ ± b2 with b 6= 0 the non-vanishing vev

signalling the spontaneous Lorentz symmetry breaking.

Variation of Eq. (4.52) with respect to the metric yield the modied equations of motion [10],

Rµν −1

2Rgµν = 8πG(TMµν + TBµν), (4.56)

where TMµν is the matter stress-energy tensor and TBµν the Bumblebee stress-energy tensor, dened

as

TBµν ≡ −BµαBαν −1

4BαβB

αβgµν − V gµν + 2V ′BµBν +ξ

8πG

[1

2BαBβRαβgµν −BµBαRαν

+1

2∇α∇µ(BαBν) +

1

2∇α∇ν(BαBµ)− 1

2∇2(BµBν)− 1

2gµνDα∇β(BαBβ)

].

(4.57)

The equations for the Bumblebee eld are

∇µBµν = 2V ′Bν − ξ

8πGBµR

µν , (4.58)

where the prime represents derivative in respect to the argument.

35

4. Vector Theories

No separate conservation laws are assumed for matter and the Bumblebee vector eld in the work

that will be presented in the following chapter. The covariant (non)conservation law (which is not

used) can be obtained directly from the Bianchi identities ∇µGµν applied to both sides of the modied

eld equations (5.7): this leads to ∇µTMµν = −∇µTBµν 6= 0, which may be interpreted as an energy

transfer between the Bumblebee and matter.

Studies using these models have recently emerged in the literature. The vacuum solutions for the

Bumblebee eld for purely radial, temporal/radial and temporal/axial Lorentz symmetry breaking

where obtained in [64]. For the rst case, a new black-hole solution was found where its Schwarzschild

radius presents itself with a slight perturbation. The second case was analyzed through the PPN

formalism where a set of PPN parameters was obtained. The nal case, due to a breaking of isotropy,

was not possible to analyze directly through the PPN formalism, although an estimation of the PPN

parameter γ was obtained.

Other work which directly deals with these models was done by Bluhm in Ref. [34], where the

possibility of a Higgs mechanism was analyzed. Studies referring to the electrodynamics of these elds

was done in Ref. [65], where the Bumblebee eld was interpreted as a photon eld and its propagation

velocity was studied, along with its implications on acelerator physics and cosmic ray observations.

The following chapter will feature original work which closely follows the work done in Ref. [1].

In it, these Bumblebee models are used in order to constrain the possibility of Lorentz violating elds

existing in astrophysical bodies such as the Sun.

36

5Astrophysical constraints on the

Bumblebee

Contents

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2 Static, spherically symmetric scenario . . . . . . . . . . . . . . . . . . . . . 38

5.3 Perturbative Eect of the Bumblebee Field . . . . . . . . . . . . . . . . . 39

5.4 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

37

5. Astrophysical constraints on the Bumblebee

5.1 Introduction

This chapter presents the results of a perturbation induced by the Bumblebee eld on a system

like the Sun. It follows the original work presented in Ref. [1].

5.2 Static, spherically symmetric scenario

Given that the relevant quantities such as the density, pressure and scalar curvature inside a

spherical symmetric body such as the Sun present a very strong radial variation in comparison with

the temporal component, the Bumblebee eld is chosen to be

Bµ = (0, B(r), 0, 0). (5.1)

Accordingly, a static Birkho metric is also selected,

gµν = diag

[−e2ν(r),

(1− 2Gm(r)

r

)−1

, r2, r2 sin2 θ

]. (5.2)

where m(r) is the mass prole in function of the radial coordinate, and it is assumed that the potential

takes a quadratic form, for simplicity,

V = A(BµBµ − b2)2, (5.3)

with the adopted sign reecting the spacelike nature of the Bumblebee eld.

For the radial case µ = r, the Ricci tensor is given by,

Rrr =G(m′r −m)(2 + rν′)

r2(r − 2Gm)− (ν′)2 − ν′′. (5.4)

The only non-vanishing component of Eq. (4.58) is for the component µ = r, 16πGV ′grr − ξRrr = 0

which yields,

2A(grrB2 − b2) =

ξ

16πGr3

[Gr2[m′ν′ + 2m(ν′′ + ν′2)] +Gr(2m′ −mν′)− 2mG− r3(ν′′ + ν′)

]. (5.5)

Through some algebraic manipulation we can the calculate the bumblebee eld strength,

B2 =

(1− 2Gm

r

)[b2 +

ξ

32πGAr3(Gr2[m′ν′ + 2m(ν′′ + ν′2)] +Gr(2m′ −mν′)− 2mG− r3[ν′′ + ν′])

].

(5.6)

In order to obtain the pressure and density equations, the trace-reversed eld equations are con-

sidered, giving,

Eµν ≡ Rµν − 8πG

[TMµν + TBµν −

1

2gµν(TM + TB)

]= 0, (5.7)

with TM and TB the traces of the stress energy tensors for normal matter and the Bumblebee eld,

respectively.

The stress-energy tensor for normal matter in the perfect uid hypothesis is given by,

TMµν = (ρ+ p)uµuν + pgµν , (5.8)

38

5.3 Perturbative Eect of the Bumblebee Field

where uµ is the four velocity; in the static scenario and given that uµuµ = −1, we have uµ = (e2ν(r),~0),

so that

TMµν = diag

(e2νρ,

p

1− 2Gmr

, r2p, r2p sin2(θ)

), (5.9)

with trace T = 3p− ρ.If we then consider the following combinations of the trace-reversed eld equations,

gttEtt − grrErr = 0,

gθθEθθ = 0.(5.10)

we can derive the equations that will allow us to obtain p(r), ρ(r) and ν(r). Without the Bumblebee

eld, these quantities (denoted with the subscript 0) are simply

p0(r) =r(r − 2Gm0)ν′0 −Gm0

4πGr3, (5.11)

ρ0(r) =m′0

4πr2, (5.12)

ν′0(r) = Gm0 + 4πp0r

3

r(r − 2Gm0), (5.13)

which, along with a state equation that relates p0 and ρ0, yields a closed set of four dierential equations

with four unknowns.

In the presence of the Bumblebee eld, the related eld equation (4.58) also has to be included;

solving Eq. (5.7), the pressure and density are then given by

p(r) =1

8πGr4

[rξB(r − 2Gm)2B′(2 + rν′) + r(8πGV r3 − 2Gm+ 2r(r − 2Gm)ν′)−

B2(r − 2Gm)(−2ξGm(−1 + r(ν′(−2 + rν′) + rν′′)) + r[16πGV ′r2

−ξ + rξ(ν′(−2 + rν′) + rν′′)])] ,

(5.14)

and

ρ(r) =1

8πGr4[−r2(8πGV r2 + ξ(r − 2Gm)2B′2 − 2Gm′) + rξB(r − 2Gm)(B′(−4r + 3Gm

+5Grm′)− r(r − 2Gm)B′′) + ξB2(3G2m2

−2G2rm(3m′ + rm′′) + r2[−1 +G(m′(4−Gm′) + rm′′)])].

. (5.15)

Although we have a complete set of equations that describe the behaviour of our system, the

solution of that set of equations implies very intensive numerical computations. This is expected as

we have up to second order derivatives on the ν(r) function inside the expression for the Bumblebee

eld, which itself appears inside both the pressure and density equations as a second derivative. The

combined expansion of all the terms in these expressions, combined with the fact that we would have

a fourth order system in our hands, brings a great deal of diculty to the problem. The linearisation

of the system of equations (not shown) does not simplify the problem signicantly, as the number of

terms in the equation would increase immensely.

5.3 Perturbative Eect of the Bumblebee Field

Since the stellar structure of the Sun is known to be well described by General Relativity, we

consider the perturbation to be of zero order, i.e. we replace the quantities on the r.h.s. of the

39


equations mentioned above by the unperturbed expressions for m0(r) and ν0(r) and the Bumblebee

eld. Regarding the latter, it is more straightforward to resort instead to Eq. (5.5), since at zeroth

order one has

Rrr = 8πG

(TMrr −

1

2grrT

M

)= 4πG(ρ0 − p0)grr, (5.16)

which leads to the following expression for the Bumblebee eld equation

B2(r) =

(1− 2Gm

r

)(b2 +

ρ0 − p0

8Agrr

). (5.17)

Since the unperturbed solutions ρ0(χ) and p0(χ) vanish at the boundary of the spherical body, the

above shows that the Bumblebee eld collapses onto its VEV as it crosses to its outer solution (where

TMµν = 0), BµBµ = b2. This is consistent with the approach followed in Ref. [64], where the latter

condition was also assumed.

Following the changes above, the expressions for the pressure and density can now be obtained,

giving

p(r) = p0 + [ξ(p− ρ)]2 +ξ2(ρ′ − p′)

2AGπr

(1− 2Gm

r

)3

(2 + ν′r) +ξ

Aπr3

(1− 2Gm

r

)2

[8Ab2

+ξ(ρ− p)]×[(2 + ν′r)(m−m′r) +

r

G

(1− 2Gm

r

)(1 + r(4Gπpr + 2ν′

−r[4Gπρ+ (ν′)2 + ν′′])− 2Gm

r[1 + r(ν′[2− ν′r]− rν′′)]

)],

(5.18)

ρ(r) = ρ0 −[ξ

8(ρ− p)

]2

+ξ2

128AGπr

(1− 2Gm

r

)2 [(4 +G

[mr− 9m′

])(p′ − ρ′)

+

(1− 2Gm

r

)3

(p′′ − ρ′′)r]

+ξ

64AGπr2

(1− 2Gm

r

)[8Ab2

+ξ(ρ− p)][2

(Gm

r

)2

+ 6Gm′(1−Gm′) + 2Gm′′r − 1− 2Gm

r(1 + 2Gm′′r)

],

(5.19)

The advantage of considering the simplistic model provided by the polytropic EOS Eq. (2.25) lies

in the possibility of rewriting the rather convoluted expressions above only in terms of the LE solution

θ(χ). For this, we now introduce the dimensionless parameters

α ≡ ξ2

R2G, β ≡ ξ3b2

R2G, γ ≡ Rs

R, (5.20)

where Rs ≡ 2GM is the Schwarzschild radius of the star, together with the form factor

φ ≡ 3M

4πρcR3, (5.21)

and the EOS parameter ωc ≡ pc/ρc.Using the relations (2.28) and the expression for ν′0(r) from Eq. (5.11), we can obtain

ν′0(χ) =3γ(χωcθ

1+n − θ′)2φχ2

f + 6γχθ′, (5.22)

and the form factor becomes φ = −3θ′(χf )/χf displaying the homology invariance of the LE Eq.

(2.31).

40

5.4 Numerical analysis

It is now possible to rewrite the above expressions for the pressure and density in terms of the

LE solution θ(χ) and its derivatives. This allows us to obtain a more manageable form: separating

the contributions to the pressure and density arising from the non-vanishing vev b and the potential

strength A as

p(χ) = p0(χ) + pb(χ) + pV (χ) + δ(χ), (5.23)

ρ(χ) = ρ0(χ) + ρb(χ) + ρV (χ)− δ(χ), (5.24)

we have

pb(χ) =β

16πφ3ξ2χ4fχ

(φχ2f + 3γχθ′)2 × [3γχ2θn(−1 + ωcθ)− (5.25)

2(−1 + χ[ν′(−2 + χν′) + χν′′])(φχ2f + 3γχθ′) + 3γχ(2 + χν′)(θ′ + χθ′′)],

ρb(χ) =β

16πφ3ξ2χ4fχ

(φχ2f + 3γχθ′)× [2φ2χ4

f + 3γχ(45γχθ′2 + (5.26)

χ[14φχ2fθ′′ + 9γχ2θ′′2 + 2φχ2

fχθ′′′] + 2θ′[7φχ2

f + 3γχ2(10θ′′ + χθ′′′)])],

pV (χ) =3γα2θn

1024π2Aφ4ξ2χ4fχ

2(φχ2

f + 3γχθ′)2 ×[(ωc − 1)(3γχ2θn[1− ωcθ] + (5.27)

2[−1 + χ(ν′[χν′ − 2] + χν′)](φχ2f + 3γχθ′))− χ

θ(2 + χν′)(θ′[θ([1 + n]φχ2

fωc + 3γ[ωcθ − 1]) +

3γχ([1 + n]ωcθ − n)θ′ − nφχ2f ] + 3γχθ[ωcθ − 1]θ′′)

],

ρV (χ) =3γα2θn

2048π2Aφ4ξ2χ4fχ

2(φχ2

f + 3γχθ′)×[χ

θ2(φχ2

f + 3γχθ′)(2[1− n]nχθ′2[φχ2f + 3γχθ′] + (5.28)

nθ[θ′(χθ′[−51γ + 2(1 + n)φχ2fωc + 6γ(1 + n)χωcθ

′]− 8φχ2f )−

χ(2φχ2f + 33γχθ′)θ′′] + [1 + n]ωcθ

2[θ′(8φχ2f + 51γχθ′) + χ(2φχ2

f + 33γχθ′)θ′′])−

2(2φ2χ4f + 3γχ[45γχθ′2 + χ(14φχ2

fθ′′ + 9γχ2θ′′2 + 2φχ2

fχθ′′′) +

2θ′(7φχ2f + 3γχ2[10θ′′ + χθ′′′])]) + 2ωcθ(2φ

2χ4f + 3γ + χ[45γχθ′2 +

χ(14φχ2fθ′′ + 9γχ2θ′′2 + 2φχ2

fχθ′′′) + 2θ′(7φχ2

f + 3γχ2[10θ′′ + χθ′′′])])

],

and

δ(χ) =9γ2α2θ2n(ωcθ − 1)2

4096π2φ2ξ2. (5.29)

The latter appears in both the pressure and density perturbations and, as shall be shown in the

following section, has a negligible impact when compared with the remaining contributions.


In Fig. 2, the prole of the contributions to Eq. (5.23) was shown for values between 0.1 and 10

solar masses. The radii varied between 0.1 and 1000 times those of the Sun. The motivation behind

having a much bigger variation in the radii in comparison to the masses stems from the fact that this

is what is observed in the set of chosen stars. Stars that have a big variance in radius show relatively

low variance in their mass. This can be seen by the chosen set of known stars in Table 5.1:

41


−28

−23

−18

−13

−8

−3

0.1R

R

1000R

logpbp0

ξb2 = 10−27, A = 10−11

n = 2.9

n = 3

n = 3.1

0 1 2 3 4 5 6−28

−23

−18

−13

−8

−3

0.1R

R1000R

χ

log

∣ ∣pVp0

∣ ∣

n = 2.9

n = 3

n = 3.1

−28

−23

−18

−13

−8

−3

0.1R

R

1000R

logρbρ0

ξb2 = 10−22, A = 10−6

n = 2.9

n = 3

n = 3.1

0 1 2 3 4 5 6−28

−23

−18

−13

−8

−3

0.1R

R

1000R

χ

log

∣ ∣ρVρ0

∣ ∣

n = 2.9

n = 3

n = 3.1

−28

−23

−18

−13

−8

−3

0.1MM

10M

logpbp0

ξb2 = 10−27, A = 10−11

n = 2.9

n = 3

n = 3.1

0 1 2 3 4 5 6−28

−23

−18

−13

−8

−3

0.1MM

10M

χ

log

∣ ∣pVp0

∣ ∣

n = 2.9

n = 3

n = 3.1

−28

−23

−18

−13

−8

−3

0.1MM

10M

logρbρ0

ξb2 = 10−22, A = 10−6

n = 2.9

n = 3

n = 3.1

0 1 2 3 4 5 6−28

−23

−18

−13

−8

−3

M

χ

log

∣ ∣ρVρ0

∣ ∣n = 2.9

n = 3

n = 3.1

Figure 5.1: Prole of the relative perturbations pb/p0, pV /p0, ρb/ρ0 and ρV /ρ0 induced by the Bumblebee.The parameters ξb2 and A were chosen so that the maximum of the perturbations reaches the adopted 1%limit, showed by the horizontal line in each plot.

42


Star Name Mass Radius

Wolf 359 0.09M 0.16RBetelgeuse 7.7− 20M 950− 1200RAntares 12.4M 883RVY Canis Majoris 17M 1420R

Table 5.1: Selected stars that used as models for the numerical analysis of the Bumblebee perturbation.

The abrupt variations in the perturbations for χ ≈ 1 and χ ≈ 2.7 are the result of the prole

changing between positive and negative values around those points and the adopted logarithmic scale.

The variation of the polytropic index n with the size of the star is also shown.

The values of the parameters (ξ, b, A) are chosen so that the maximum of the relative perturbations

is of the order of 1%, the order of magnitude of the current accuracy of the central temperature of the

Sun [6668]. For reference, the values of these parameters for ξ = 10−11, b = 10−8 are shown in table

5.2 for all the model stars considered in the numerical analysis.

−20 −10 0 10 20−30

−25

−20

−15

−10

−5

log(ξ[GeV 2])

log(

bGeV)

pbρb

Figure 5.2: Allowed region (in grey) for a relative perturbation of less than 1% for pb and ρb.

A small variation in the polytropic index does not cause signicant changes on the obtained bounds

for the system: in particular, n does not impact signicantly the value of ρb, as can be seen directly in

the equation for ρb. If we increase the radius (thus lowering γ and α) the impact of the non-vanishing

vev increases, while leading to a lower contribution from the potential term. If we increase the mass,

we end up having smaller eects on all quantities except ρV , which is rather insensitive to variations

of M .

If we x M = M and R = R, we can then nd values for the parameters of the model (ξ, b, A)

that lead to relative perturbations of less than 1%. The allowed parameter space can be obtained, as

depicted in Fig. 3. Notice that the allowed values for A are bounded from below, since this quantity

appears in the denominator of pV and ρV ; conversely, the region allowed for ξb2 is bounded from above.

Having said this, we can now obtain the bounds for the parameters of our model,

ξb2 . 10−23 ,ξ√A

GeV2 . 10−3 → ξ√AG

. 1034. (5.30)

43


−20 −18 −16 −14 −12 −10 −8 −6−30

−20

−10

0

log(ξ[GeV 2])

log(A

)

pVρV

Figure 5.3: Allowed region (in grey) for a relative perturbation of less than 1% for pV and ρV .

It is also worth noting that for the considered sets of parameters, the term δ(χ) is negligible in

comparison with the other terms in both the pressure and density equations, as mentioned after Eq.

(5.29): indeed, one can calculate numerically that δ . 10−34pV for the considered masses and radii.

M,R α× 10−33 β × 10−49 γ ωc0.1 M, R 1.18 1.18 4.07× 10−7 6.946× 10−7

M, R 1.18 1.18 4.07× 10−6 6.946× 10−6

10 M, R 1.18 1.18 4.07× 10−5 6.946× 10−5

M, 0.1 R 1.18 1.18 4.07× 10−5 6.946× 10−5

M, 1000 R 1.18 1.18 4.07× 10−9 6.946× 10−9

Table 5.2: Adimensional parameters used in the perturbative expressions for the pressure with ξ = 10−11,b = 10−8 and polytropic index n = 3.

And so we reach the end of this work. The equations, although quite complex in form, present

themselves in a more obvious way through this numerical analysis. Through the employment of the LE

equations and some dimensional analysis, we were able to separate the contributions of the Bumblebee

potential from the eld itself. This in turn allowed us to study their individual contribution on a set

of astrophysical bodies modeled through a polytropic equation of state. The obtained constraints are,

as will be discussed in the next section, a lot more stringent than the ones previously obtained in the

literature, thus giving relevance to the results.

44

6Conclusions

45

6. Conclusions

After a brief review of the relevant literature that brings context to the Bumblebee models, we

treated it as a zeroth order perturbation on a set of stars. We employed a set of stars which mimic,

in terms of order of magnitude, the radii and masses of some known stars. These stars where then

described by the Lane-Emden solution for a spherically symmetric body. Assuming that it follows

the underlying symmetry of the problem, we choose a Bumblebee eld with a radial component only.

Because the impact of the eld is considered as a perturbation, an attempt was made in order to

constrain the parameters of the model in such a way as to only cause a variation on the system of

roughly 1%, following the accuracy of our present modelling of the Sun.

The obtained constraint for the value of the potential driving the Bumblebee eld, ξb2 . 10−23, is

many orders of magnitude more stringent than the previously available bound ξb2 . 10−9, obtained

by resorting to tests of Kepler's law using the orbit of Venus [64]; by assuming that, in the presence

of matter, the Bumblebee eld is not relaxed at its vev, this study has also yielded a constraint on

the strength of the corresponding potential, ξ < 1034√AG, which is a new result for these models.

Although only a quadratic potential was considered in this study, the change of the power n showed

very little eect on the proles of the perturbation, as it s eect comes mostly as multiplicative factor

on the potential itself.

Future renements of this method could clearly include the use of a more accurate model for stellar

structure, as well as following a more thorough numerical analysis procedure, eectively solving the

(dierential) modied eld equations to rst order in the model's parameters. However, we must still

take into account that this should only rene the obtained bounds, having no eect on the order of

magnitude of the eect.

The application of this same methodology to the study of galaxies is also possible, in order to

gain further knowledge on the constraints to the parameters of our model, as well as the possibility of

describing galactic dark matter as a manifestation of the Bumblebee dynamics following analogue

eorts in both scalar eld [66] and vectorial Aether models [61, 69, 70]. In doing so, the addition of

a non-vanishing temporal component for a time evolving Bumblebee eld could also be considered, in

order to provide a results relevant at cosmological scales.

46

1

47

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