FOUNDATIONS OF MASSIVE GRAVITY
ANDREW MATAS
Submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
Dissertation Advisor: Prof. Claudia de Rham
Department of Physics
CASE WESTERN RESERVE UNIVERSITY
August 2016 CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the dissertation of
Andrew Matas
candidate for the degree of Doctor of Philosophy∗
Committee Chair
Claudia de Rham
Committee Member
Stacy McGaugh
Committee Member
Glenn Starkman
Committee Member
Andrew J. Tolley
Date of Defense
May 19th, 2016
∗We also certify that written approval has been obtained for any proprietary
material contained therein.
i Contents
List of Figures vii
1 Introduction 1
1.1 Notions of gravity ...... 1
1.1.1 Gravity is an attractive force ...... 2
1.1.2 Gravity is curved space-time ...... 4
1.1.3 Gravity is a fundamental interaction ...... 5
1.2 A massive graviton and its implications ...... 9
1.3 Building models of massive gravity ...... 16
1.3.1 Stability ...... 17
1.3.2 Strong-coupling and continuity ...... 19
1.3.3 Extra dimensions and massive gravity ...... 22
1.4 Overview of dissertation ...... 24
2 Basics of Massive Gravity 29
2.1 Setting the stage ...... 31
2.1.1 Consistent m → 0 limit of a massive spin-1 ...... 31
ii 2.1.2 Ghost modes ...... 33
2.1.3 Massless spin-2 ...... 35
2.2 Toward a theory of massive gravity ...... 37
2.2.1 Non-interacting massive spin-2 field ...... 37
2.3 Non-linear formulations of massive gravity ...... 41
2.3.1 Ghost-free potential ...... 43
2.3.2 Vielbein formulation ...... 44
2.3.3 St¨uckelberg for gravity ...... 45
2.4 Absence of the Boulware-Deser mode ...... 47
2.4.1 ADM form ...... 48
2.4.2 Counting degrees of freedom ...... 49
2.4.3 Mini-superspace ...... 51
2.5 Minkowski decoupling limit ...... 52
2.5.1 Interaction scales ...... 54
2.5.2 Explicit form of decoupling limit ...... 56
2.5.3 Galileon interactions ...... 58
2.6 Realization of the Vainshtein mechanism ...... 59
2.6.1 Vainshtein mechanism ...... 59
2.6.2 Boulware-Deser ghost ...... 61
2.6.3 Vainshtein mechanism for stars ...... 62
2.7 Bi-gravity and multi-gravity ...... 65
2.8 Summary ...... 66
iii 3 Dimensional deconstruction 68
3.1 Deconstruction of a scalar ...... 71
3.2 Deconstruction for gravity ...... 74
3.3 Deconstruction of the metric ...... 76
3.4 Consistent deconstruction procedure ...... 79
3.4.1 Different discretization of derivative ...... 84
3.4.2 Multi-gravity ...... 85
3.5 Strong-coupling from deconstruction ...... 86
3.5.1 Strong-coupling scale at large N ...... 87
3.5.2 The 5D origin of the strong-coupling scale ...... 91
3.5.3 Gauge fixed continuum theory ...... 91
3.5.4 Generic gauge for continuum theory ...... 92
3.6 Outlook ...... 93
4 Kinetic interactions in massive gravity 95
4.1 Kinetic term for massive spin 1 ...... 98
4.2 Candidate kinetic interactions ...... 100
4.2.1 Leading order interactions ...... 101
4.2.2 Non-linear St¨uckelberg decomposition ...... 103
4.2.3 Non-linear completions ...... 104
4.3 Deconstruction-inspired ansatz ...... 106
4.4 Systematic argument ...... 109
4.4.1 St¨uckelberg criteria ...... 110
iv 4.4.2 Decoupling limits ...... 112
4.4.3 Algorithm ...... 114
4.4.4 Quadratic order ...... 116
4.4.5 Cubic order ...... 116
4.4.6 Quartic order ...... 119
4.4.7 Extension to higher order ...... 122
4.5 Relationship to the coupling to matter ...... 124
4.5.1 Effective vielbein coupling ...... 124
4.5.2 Effective metric ...... 126
4.6 Outlook ...... 128
5 Galileon radiation from binary systems 130
5.1 Galileons with different scales ...... 133
5.2 Perturbations around a spherically symmetric background ...... 134
5.2.1 Keplerian source ...... 134
5.2.2 Background Galileon field ...... 137
5.2.3 Perturbations for Galileon ...... 138
5.3 Formula for the power emission ...... 141
5.4 Power emission for the cubic Galileon ...... 144
5.5 Power emission for the general Galileon ...... 147
5.5.1 Mode functions ...... 148
5.5.2 General form for the power ...... 149
5.5.3 Power in quadrupole ...... 151
v 5.5.4 Breakdown of perturbation theory ...... 152
5.5.5 Hierarchy of masses ...... 157
5.6 Hierarchy between two strong coupling scales ...... 159
5.6.1 Power emission ...... 160
5.6.2 Quadrupole radiation ...... 162
5.7 Outlook ...... 162
6 Conclusions 165
6.1 Summary ...... 165
6.2 Outlook ...... 167
A Lovelock interactions 169
vi List of Tables
4.1 Coefficients for ghost-free kinetic term at quartic order ...... 120
vii List of Figures
1.1 Newtonian T-shirt...... 3
1.2 Fundamental Forces Comic...... 5
1.3 Scattering processes...... 7
1.4 Gravitational wave polarizations...... 11
1.5 Hulse-Taylor pulsar...... 12
1.6 Ghost instabilities...... 17
1.7 Vainshtein radius...... 20
1.8 Kaluza-Klein compactification...... 22
viii Foundations of Massive Gravity
ANDREW MATAS
General Relativity (GR) is a relativistic theory of gravity which has a large number
of theoretical and observational successes. From the perspective of quantum field
theory, GR can be thought of as a theory of a massless spin-2 particle called the
graviton. It is a fundamental question to ask how the graviton behaves if it has a
small but non-zero mass. In this dissertation I shall study the recently constructed
theory of ghost-free massive gravity, which avoids the pernicious Boulware-Deser ghost that had thwarted previous attempts to study massive gravity.
In Chapter 1 I will give a broad overview of the context, motivation, and model- building issues underlying massive gravity. Then in Chapter 2 I will give a detailed pedagogical introduction to ghost-free massive gravity focusing on concepts that will be used throughout the remainder of the work.
In Chapter 3 I will derive the ghost-free structure of massive gravity from an ex- tra dimensional perspective through a process known as Dimensional Deconstruction.
Before my work it had been an open question whether Deconstruction could be con- sistently applied to gravity. The key insight relies on using the elegant formulation of General Relativity in terms of the vielbein.
Inspired by Deconstruction, in Chapter 4 I will discuss the possibility of non- standard kinetic interactions in massive gravity. I will show that the only consistent derivative interactions for a massive spin-2 particle must be the same as in GR. This is remarkable because there is no known symmetry reason for this to be the case,
ix since massive gravity breaks diffeomorphism invariance.
Finally, in Chapter 5, in order to connect with observations I shall consider the radiation emitted from binary systems in Galileon theories, which are scalar theories that can mimic the behavior of a massive graviton. This work extends the under- standing of the Vainshtein screening mechanism into a time-dependent situation.
x Chapter 1
Introduction
In this work I will be concerned with an extension of General Relativity (GR) known as massive gravity. Before diving into the details of this theory, it is useful to recall the basic ideas underlying our modern understanding of gravity.
1.1 Notions of gravity
On the one hand, gravity is a familiar phenomenon. Gravity makes apples fall from trees and is responsible for the tides.
On the other hand, gravity is mysterious. Gravity connects us to the universe.
Newton taught us that the same force that pulls us to the Earth also holds the Earth to the Sun, the Sun to the Galaxy, the Galaxy to the Local Group, the Local Group to the
Local Supercluster, and that holds galaxy clusters together in an enormous network of filaments mapped by large scale structure. Gravity collapses stars into black holes, regions so dense that light cannot escape. Under the influence of gravity, the universe
1 expands. The expansion rate has recently been observed to be accelerating, and if this acceleration continues forever then ultimately all matter and energy will be diluted away to almost nothing.
In this opening section I will give three different perspectives on the nature of gravity. These different pictures look quite different, but all of these pictures are useful and relevant in different regimes, and whenever two or more are valid they agree in their quantitative predictions.
1.1.1 Gravity is an attractive force
The theory of gravity published by Isaac Newton in 1687 states that gravity is an at- tractive force between massive objects. Between any two objects is an attractive force proportional to the masses of the objects, and inversely proportional to the square of the distance between them. The proportionality constant is Newton’s gravitational constant G, which sets the strength of the gravitational force. These statements can be summarized in an equation so simple and profound that it can be found on T-shirts
(see Figure 1.1).
From this one (apparently) simple equation, one can make an enormous array of precise predictions: planets orbit the sun in elliptical orbits and a bowling ball and a basketball dropped simultaneously from the top of the Tower of Pisa will hit the ground at the same time.
Despite its success and continued utility, Newton’s theory of gravity is unsatisfying in at least one respect. In Newton’s theory the gravitational force is instantaneous:
2 Figure 1.1: This T-shirt illustrates the Newtonian view of gravity as a force. Image from http://www.nerdytshirt.com/physics-tshirts.html. the force between two objects at a given time depends on the distance between the objects at that same time. This raises puzzles in extreme situations with fast moving objects. For example, if the sun were to suddenly disappear1, would this information be transmitted to earth instantaneously, or would the information take a finite time to propagate? Intuitively we might say that there should be a finite time. This problem is even more compelling from the perspective of special relativity. In special relativity there is no notion of absolute time, and the need to have a finite time of propagation goes from being intuitive to being mandatory.
Our modern understanding of gravitation is given by the theory of GR, developed by Einstein, to which we now turn.
1For readers rightly uncomfortable with the non-conservation of mass in this example, we can imagine that a mischievous alien race has very suddenly and quickly moved the sun to another place.
3 1.1.2 Gravity is curved space-time
Within the framework of GR, gravity is interpreted as the curvature of space-time.
Freely falling masses follow straight paths through curved space-time. When you
stand on a scale, the scale is not supporting you against the force of gravity, the scale
is preventing you from following the natural straight line path through space-time.
The curvature of space-time is affected by matter and energy through Einstein’s
Equations
1 8πG R − Rg = T . (1.1) µν 2 µν c2 µν
On the left hand side are quantities describing the curvature, and on the right hand
side the quantity Tµν describes the mass (and energy) of matter living in space-time.
We can also see that this equation relates gravity (through the presence of Newton’s constant G) and relativity (through the speed of light c).
Whenever space-time is approximately flat, GR approaches Newton’s theory. How-
ever, there is a key difference, even in this limit. The gravitational force is no longer
instantaneous. Einstein’s equations allow for propagating wave solutions. In the
case when space-time is approximately flat, Einstein’s equations reduce to a linear
wave equation describing propagating disturbances in the metric. Solutions to this
linearized equation describe gravitational waves2, which travel at the speed of light
c. The speed can be seen directly from the linearized wave equation. The existence
2The notion of a gravitational wave is not limited to linearized GR, however linearized GR provides a very useful description of freely propagating gravitational waves.
4 Figure 1.2: ”Of these four forces, there’s one we don’t really understand.” ”Is it the weak force or the strong–” ”It’s gravity.” From https://xkcd.com/1489/. of gravitational waves propagating at c provides an answer to the question posed about the sun’s disappearance: the earth would orbit for eight minutes before careen- ing off into space. Shortly before the time of this writing, gravitational waves have been directly detected by the Laser Interferometer Gravitational-Wave Observatory
(LIGO) [1].
GR has been summarized by Wheeler, “Space-time tells matter how to move; matter tells space-time how to curve” [2]
1.1.3 Gravity is a fundamental interaction
A different picture of forces has developed through the development of particle physics and the Standard Model (see Figure 1.2). While gravity governs the motion in the heavens and keeps us fixed on earth, the forces responsible for keeping the earth solid and warm apparently behave quite differently. There are three (known) such forces, the electromagnetic force and the strong and weak nuclear forces. Our modern understanding of these forces is that they are described by the exchange of force-
5 carrying particles, associated with quantum mechanical fields.
It is possible, and fruitful, to formulate GR in similar terms, at least in special
situations such as when space-time is asymptotically flat and the curvature is not too
large (for example see [3], for a more modern effective field theory approach see [4]).
This allows us to think of gravity on similar footing as other known forces, and allows
us to use techniques from other areas in physics to understand gravity. We emphasize
that while this gives us a different picture of gravity, it agrees quantitatively in terms
of physical observables.
A familiar example of a fundamental interaction is electromagnetism. Like charges
repel and opposites attract because charges interact with the electromagnetic field.
An electric charge generates an electric field, and this field spreads through space
until it pushes or pulls on another charge. Associated with the electromagnetic field
spreading is a propagating wave, which we experience (at least for a range of wave-
lengths) as light. The color of light is related to the wavelength of this wave. Light
travels at the speed of light3 c.
Associated with the electromagnetic field is a quantum mechanical particle (or
indivisible packet of energy) called the photon. Quantum mechanically, the photon
is the basic force-carrying entity of the electromagnetic force (see Figure 1.3). Two
negatively charged electrons repel as they exchange photons. The fact that electro-
magnetic waves travel at the speed of light translates into the fact that the photon is
3It may sound tautological to say light travels at the speed of light, but strictly speaking calling c the ‘speed of light’ in special relativity is misleading. In special relativity, the parameter c describes the space-time geometry and is not directly related to any particle. In practice this does not usually matter since the photon is massless (or at least has a very tiny mass) and so it travels at (or at least very near) c.
6 Figure 1.3: To illustrate the picture of a force as being due to an exchange of particles, in the left figure we see electrons e− scattering by exchanging a photon γ. In the right figure, the earth and moon interact by exchanging a graviton hµν. Time runs vertically upward in these diagrams.
a particle with zero mass.
We can make similar statements about the gravitational field. At the classical
level, as we have discussed, masses interact with the gravitational field. Gravitational
waves propagate through this field. Quantum mechanically, there is expected to be
a particle of the gravitational field that is called the graviton. One can think of
(and calculate) the force between the earth and the moon as being mediated by an
exchange of gravitons (see Figure 1.3).
This framework allows us to incorporate quantum effects into gravity in a calcula-
ble way, when the energies involved are not too large. Scattering processes involving
exchanges of gravitons, with typical energy E, can be computed as a perturbative series in E/MPl, where MPl is an energy scale known as the reduced Planck scale
r c M ≡ ~ = 2.4 × 1018 GeV/c2. (1.2) Pl 8πG
This is a remarkable quantity which combines quantum mechanics (through Planck’s
7 constant ~), relativity (c), and gravity (G). The Planck scale is very large (the
−15 Large Hadron Collider (LHC) probes energies that are a ‘measly’ 10 MPl) and therefore the gravitational interaction is very weak (the electrical repulsion between two electrons is about 1042 times larger than the gravitational attraction).
When fundamental particles collide at energies E & MPl, the perturbative picture we are describing of graviton exchange breaks down and a more complete description is needed (indeed the very notion of a particle may cease to make sense in this regime).
We will not discuss this issue further in this work.
In macroscopic situations relevant for astrophysics, gravity is of course an impor- tant force even though quantum effects are not relevant. This can occur because the systems involved (such as the sun) are macroscopic and contain many particles. As a result, even though the mass of the sun is enormously larger than MPl (by a factor
38 of around 10 ), E/MPl is negligibly small for each individual particle in the system.
We emphasize that this picture underlies much of modern cosmology. Future ob- servations may detect primordial gravitational waves from the early universe. Within the framework of cosmic inflation, these originate as quantum fluctuations in the grav- itational field in an inflating background [5]. Primordial gravitational waves may be observed by looking at B-mode polarization in the cosmic microwave background [6,7], or may contribute to a stochastic gravitational wave background [8].
8 1.2 A massive graviton and its implications
In this work we will be interested in extending GR. The graviton is normally taken to be a massless particle with spin-2. However, intuitively, since mass is a continuous parameter, one may expect that there should only be a small difference in observable quantities in a theory of a massless graviton, and a theory of a graviton with a small mass.
In this section I will discuss some of the phenomenological and theoretical implica- tions of massive gravity (the theory of a massive graviton). Massive gravity introduces one new dimensionful parameter, the mass of the graviton m (which I will often quote in units of eV/c2). While I report some observational bounds, this is not meant to be a comprehensive review of different bounds on the graviton mass (see [9,10] for some discussions), but rather to give a sense of how different observations fit together.
Gravitational waves. As we have discussed, gravitational waves are propagating disturbances in the gravitational field. The effects of the mass can be seen very directly in terms of these waves.
The mass of the graviton affects gravitational waves in two ways.
First, a mass modifies the speed of propagation. The speed of the waves is de- termined by its dispersion relation, which relates the frequency and wavelength of a gravitational wave. The dispersion relation (on a flat background geometry) is
9 determined by the mass of the particle associated with that wave by4
m2c4 ω2 = c2k2 + , (1.3) ~2
where ω is the angular frequency and k is the wavenumber5. A packet of gravitational
waves travels at the group velocity
dω m2c5 vg ≡ = c − + ··· < c, (1.4) dk 2~2ω2
where ··· refers to terms that are negligibly small when the mass is small mc2 ~ω.
When m 6= 0, waves of different frequency travel at different speeds. This can lead to an experimental test of the graviton mass, [9,12].
Recently gravitational waves have been directly detected by LIGO from the in- spiral of a binary black hole system. When m is nonzero, low frequency gravitational waves travel more slowly than high frequency waves. This modifies the predicted gravitational waveform by inducing a frequency dependent phase delay relative to the prediction for GR. Performing a search for this kind of modification to the wave- form, LIGO has placed a bound on the graviton mass m < 1.2 × 10−22 eV/c2 at 90%
confidence [13].
Second, a mass introduces new polarization states (see Figure 1.4). While the
waves associated with a massless spin-2 particle have 2 polarizations, the waves asso-
4“This is the sophisticate’s version of the layperson’s E = mc2.”—Zee, Quantum field theory in a nutshell [11]. 5Note that even though ~ appears, here we are talking about classical gravitational waves. Quan- tum mechanics comes in through interpreting the parameter m in the dispersion relationship as the mass of a particle.
10 Figure 1.4: This figure shows the effect of different gravitational wave polarizations acting on a ring of test particles. The direction of motion of the gravitational wave is shown in each figure. The top row displays the tensor modes (+, ×). The second row shows the two vector modes (note that the x and y axes are marked on the figure). The last row shows the two possible scalar modes, the left is the longitudinal mode and the right is the conformal mode. GR has the two tensor modes only. A healthy theory of massive gravity has two tensor, two vector, and one scalar polarization. The sixth scalar mode appears as a ghost if it is present in massive gravity. This figure is based on Figure 8 of [9].
11 Figure 1.5: The decay of the orbital period of the binary pulsar system PSR B1913+16, compared with the prediction from General Relativity. Figure from [14] ciated with a healthy theory of a massive spin-2 particle have five. As we will review in greater detail in Chapter 2, understanding the physics of these additional states
(in particular the scalar mode) is crucial to understanding massive gravity.
Gravitational waves have been indirectly detected by studying binary pulsars.
In 1974 Hulse and Taylor discovered the first binary pulsar system PSR B1913+16
[14,15]. This system consists of two neutron stars in orbit around a common center.
GR predicts that as the neutron stars orbit, the system emits gravitational waves and the orbit slowly decays. Subsequent observations of PSR B1913+16 (and other systems discovered later, for a review see [16]) confirm that the rate of the spin- down is consistent with the prediction from the emission of gravitational waves (see
Figure 1.5).
In massive gravity we expect the spin-down rate of pulsars to be (ever so slightly)
12 larger because the system can radiate into the additional polarization states. I will discuss this further in Chapter 5 where we discuss calculating the gravitational wave emission from binary pulsar systems.
Gravitational force. Newton’s law of gravitation can be modified by adding a mass. Within the Newtonian approximation we expect the mass will effectively cut off the gravitational force at distances larger than the Compton wavelength λg ≡ ~/mc.
The smaller the mass, the larger the range of the gravitational force. We can express this in terms of the gravitational potential sourced by a star of mass M
e−r/λg V (r) = GM . (1.5) r
For distances that are short compared with the Compton wavelength r λg, this looks like the potential ∼ 1/r of Newtonian gravity. However for large distances r λg, the force is exponentially suppressed. This effect is known as the Yukawa suppression. Solar system tests of planetary motion bound the Compton wavelength
12 −22 2 to be λg < 2.8 × 10 km or m < 4.4 × 10 eV/c [9].
In addition, the new polarization states mediate additional forces, often called
fifth forces since it is often said that there are four forces in fundamental physics
(recall Figure 1.2). Physically this occurs because there are more polarization states that can be exchanged contributing to an attractive force in a Feynman diagram such as Figure 1.3. These polarization states couple to matter differently than the tensor modes present in GR and their effect cannot simply be absorbed into a rescaling of
13 Newton’s constant, as we will describe in more detail in Section 1.3.2. Since gravity
is well tested in the solar system, these forces need to be screened using a screening
mechanism. Screening mechanisms allow the gravitational forces to depend on the
local environment in a way that recovers GR in regions where it is successful [17].
Despite the existence of screening mechanisms, solar system tests are stringent
enough to put tight bounds on the graviton mass from this additional contribution
to the Newtonian force. In particular these forces predict an anomalous precession
of the moon’s orbit around the earth, which is tightly constrained by Lunar Laser
Ranging experiments [18]. This bound is model dependent, but can be as strong as
m < 10−32 eV/c2 in some models [19,20].
Cosmic acceleration. Over the past 15 years a wide array of observations have confirmed that the universe’s rate of expansion is accelerating [21]. In GR, in the absence of spatial curvature the rate of expansion of the universe, called the Hubble parameter H, is related to the energy density in the universe ρ by the Friedman equation
8πG H2 = ρ. (1.6) c2
Within GR, an explanation for the acceleration of the universe can be given by the
existence of a cosmological constant. The cosmological constant contributes a con-
stant amount of energy density to the Friedman equation, which leads to a constant
rate of expansion and therefore an exponential growth in the scale factor of the uni-
14 verse. Physically, we can interpret this constant energy density ρvac as energy density associated with the vacuum itself. The data suggests that the vacuum energy has a
obs −3 4 value ρvac ∼ (10 eV) [21]. However, there is still room for alternative explanations, and ultimately experiment should decide between different possibilities.
The possibility of alternative explanations of cosmic acceleration is especially tan- talizing in light of the cosmological constant problem (for reviews see [22,23]). Quan- tum mechanically, every field has a zero point motion. The field will contribute some energy even in vacuum. The vacuum energy we observe is a combination of the ‘bare’
(intrinsic) vacuum energy, plus contributions from the vacuum energy of each field6
obs bare 4 4 ρvac = ρvac + α1m1 + α2m2 + ··· , (1.7)
bare where ρvac is the bare value (which is not computable although may in principle be determined in a fundamental theory) α1,2 are constants, and m1,2 are the masses of particles associated with the fields present in nature, and the ··· refers to a sum over all fields. The heaviest fundamental particle that has been observed is the top quark,
2 4 mtop ∼ 170 GeV/c , leading to a contribution to the vacuum energy ∼ (170 GeV) ∼
57 obs 10 ρvac. In order to produce the small observed value, the bare value of the vacuum
bare energy ρvac must cancel the contribution to the top quark, as well as the corrections from other particles, to enormous precision. This is logically possible, but extremely uncomfortable, and such a high degree of fine-tuning of the theory suggests there may
6More precisely there is an equation of this form when one performs a matching between a low energy effective field theory valid at energies below the mass of (say) m1 and a UV theory valid above the mass of m1. In this setup the coefficients α1, α2, ··· are calculable in terms of parameters of the UV theory. See [23] for more discussion.
15 be more to the story.
As a result there has been a push to find alternative explanations for cosmic acceleration. Massive gravity can address cosmic acceleration in various ways. It may be possible for a massive graviton to degravitate a large cosmological constant [24,25].
Roughly speaking we expect gravity to be cutoff on large distances because of the
Yukawa suppression. As a result massive gravity might not see the full strength of a cosmological constant. An alternative scenario is called self-acceleration where the graviton forms a condensate whose energy density drives the cosmic acceleration [26].
In order to be relevant for cosmic acceleration, the graviton should have a mass
m ∼ H ~ ∼ 10−33 eV/c2, (1.8) 0 c2
where H0 is the Hubble parameter today. We will often take this as our fiducial value for m.
1.3 Building models of massive gravity
Having described some of the potential applications of a massive graviton, I will now discuss issues in constructing realistic models of a massive graviton. Finding viable theories is not a trivial task. Indeed, a consistent interacting theory of a massive graviton was not discovered until recently [27, 28]. Here we describe some of the major themes that we will discuss throughout the rest of this thesis.
16 Figure 1.6: Illustrating different ways to think about a ghost instability. The left figure shows how a ghost manifests itself classically. The right figure illustrates an example quantum process where the vacuum decays into ghosts and ordinary matter, in this case a photon. As an example of a specific theory which would contain this 1 2 −2 2 2 vertex, we may consider the lagrangian L = + 2 (∂φ) + Λ φ Fµν, where Λ is a scale needed for dimensional reasons and where Fµν = ∂µAν − ∂νAµ is the electromagnetic field strength tensor. The real scalar field φ is a ghost field because it has the wrong sign kinetic term.
1.3.1 Stability
The first challenge in constructing a healthy theory of massive gravity is related to the fact that there are six possible polarizations for a symmetric two index tensor hij
(see Figure 1.4), but a healthy theory of a massive spin-2 field should only have five
polarizations, corresponding to the 2S + 1 = 2(2) + 1 = 5 spin states [29].
When the sixth mode appears, it appears as a ghost instability. This problem was
recognized already for non-interacting massive spin-2 particles in [30]. As recognized
in that paper, there is a specific choice of mass term in the Lagrangian which removes
the instability. However, a realistic model of massive gravity necessarily involves
an interacting massive spin-2 particle. Early attempts to build models of massive
gravity found that the ghost that was vanquished at the linear level returned when
interactions was included, this instability is known as the Boulware-Deser (BD) ghost
[31].
17 A ghost is a particle with negative kinetic energy. This is extremely problematic
as can be seen from multiple points of view (see for example [32]). At the classical
level, we can see how a ghost behaves by considering a particle released from height
on a potential well (see Figure 1.6). Since the total energy E = K + V (where K
is the kinetic energy and V is the potential energy) must be conserved, an ordinary
particle with K > 0 will roll in the potential without ever rising above the height from which it was dropped. Energy conservation binds the particle to a finite region of the potential. However for a ghost particle with K < 0, the particle can rise up
the potential, increasing V , while simultaneously moving faster and faster to make K
more and more negative, keeping the combination E = K + V constant. This strange behavior is clearly unphysical. Of course, analogous results hold in field theory when ghost-like and ordinary fields interact.
The most serious version of the problem is that if the ghost interacts with ordi- nary matter, then a process is possible where the vacuum spontaneously decays into combinations of ghosts and ordinary particles. This can happen because the posi- tive energy of the ordinary particle(s) is compensated by the negative energy of the ghost(s). Furthermore, if we trust the theory to arbitrarily high energies, then there is an infinite amount of available phase space for this process and so the decay rate is infinite. Even if we consider a more realistic scenario where we do not trust the theory to arbitrarily high energies and we regulate the divergence by cutting off the allowed energy, one still finds that a ghost is a very fast instability [33].
The BD ghost can be avoided by carefully choosing the potential interactions
[27, 28]. I will make use of these stability requirements in Chapter 4 when I will use
18 these requirements to constrain extensions to the kinetic structure of massive gravity.
While this discussion can become technical, there is also something beautiful about
how the internal consistency of a theory is so constraining.
1.3.2 Strong-coupling and continuity
Intuitively, as long as the mass of the graviton is sufficiently small, all predictions for
physically observable quantities in massive gravity should be close to those in GR.
While this is widely thought to be the case, showing how this occurs in detail is not
straightforward.
In GR, a standard approach is to treat the gravitational field as weak, or more
precisely to take the metric to be approximately flat. Various observable quantities,
such as the orbit of the moon around the earth or the bending of light by the sun,
can be computed within this regime.
However, when one applies the same weak field approximation to massive gravity,
one finds there is a discrete difference between the predictions of observable quantities
in massive gravity, and the predictions of observable quantities in GR, even when the
mass m is arbitrarily small. For example, fixing Newton’s constant G from non- relativistic terrestrial probes, one finds that the predictions for the bending of light by the sun are different in the two theories. This is known as the van Dam-Veltman-
Zakharov (vDVZ) discontinuity [34, 35]. Naively, it implies that the graviton mass must be identically zero, which is an extremely strange conclusion.
A possible resolution to this puzzle was proposed by Vainshtein [36] (see also
19 Figure 1.7: An illustration of the Vainshtein mechanism. For distances less than the Vainshtein radius r?, strong-coupling effects suppress the interactions of extra graviton modes and leads to the recovery of GR. Outside of this radius, the fifth force effects are unsuppressed and there are in principle large deviations from GR. The boundary is smeared to represent the fact that there is a continuous transition between these different regions.
20 [37,38] for a more modern take).
Vainshtein found that around a matter distribution with mass M, it is only consis- tent to treat the gravitational field as weak at large distances. In ghost-free theories of massive gravity larger than a characteristic scale now known as the Vainshtein radius
GM 1/3 r ∼ . (1.9) ? m2
−33 2 7 For the sun, taking m ∼ 10 eV/c , the Vainshtein radius is r? ∼ 100 pc.
It may seem strange that the gravitational fields can be treated as weak in GR, but are very strong in massive gravity. However, while the calculation of an observable quantity may be different in the two theories, the theories should agree on physically observable quantities. In fact, we will see that the strong-coupling is responsible for decoupling the helicity-0 polarization state and the usual helicity-2 modes present in
GR remain weakly coupled.
While the Vainshtein mechanism is well understood in static, spherically symmet- ric situations (for example see [39,40]), it is an ongoing research effort to understand how the Vainshtein mechanism operators in more general settings. In Chapter 5 I will study the Vainshtein mechanism in a time dependent situation, binary pulsars.
21 Figure 1.8: On the left hand side we see how extra dimensions can be wrapped up so that they are hard to see at large distances. On the right hand side we see the associated Kaluza-Klein tower of gravitons. The mass scale is set by the size of the extra dimension ∼ a.
1.3.3 Extra dimensions and massive gravity
Massive gravity is intimately connected to GR when formulated with extra dimen- sions, where extra dimensions refer to spatial dimensions beyond the three we observe.
Since GR in more spatial dimensions is a consistent theory, we can use extra dimen- sions as a tool to understand consistent theories of massive gravity. Indeed ghost-free massive gravity theory was discovered by using extra dimensional models [41–43].
Historically, an important extra dimensional model giving rise to a theory of mas- sive gravity is the Dvali-Gabadadze-Porrati (DGP) model [44]. The DGP model is a brane-world model consisting of a brane (or membrane) living in a bulk space with four spatial dimensions of infinite extent. Ordinary standard model fields are confined to the brane, but gravity can explore the bulk.
DGP exhibits many features of massive gravity. At sufficiently short distances
7In fact this is likely an underestimate as it assumes that the sun is in a vacuum with no other matter present. As one considers spheres of larger size around the real sun, the spheres will enclose additional sources of matter which will in turn contribute to the Vainshtein screening, and so in reality the Vainshtein regime may be even larger.
22 (smaller than the Hubble scale), an observer on the brane sees a graviton that has a soft (resonance) mass [45, 46]. It avoids the Boulware-Deser ghost, and the theory exhibits a Vainshtein mechanism. Studying DGP led to many insights that have been relevant for massive gravity. For example, Galileon scalar field theories [39] (see [47] for an introduction), which are a class of scalar theories which exhibit the Vainshtein mechanism and play a large role in massive gravity, were discovered by studying
DGP [48,49].
Another much older approach to extra dimensions, that does not use branes, is to have compact (finite size) extra dimensions (see Figure 1.8). This is known as a Kaluza-Klein (KK) compactification [50, 51]. The KK model consists of GR in a
4 + 1 dimensional space-time, with one dimension compactified on a circle. From the perspective of a 3+1 dimensional observer, this theory looks like a theory of a massless graviton (as well as additional massless states8, and an infinite tower of massive gravitons, with masses given by the inverse size of the compact dimension, in appropriate units.
It is tantalizing to use this infinite tower to find a consistent theory of a massive graviton. However, in the simplest KK compactification, the tower is evenly spaced in mass so there is never a regime where only one graviton is relevant. At energies much below the mass of the lightest graviton in the tower, none of the massive states are relevant. When we work at energies large enough to produce the first massive graviton, however, there are many other states with a similar mass that cannot be neglected.
8In particular these massless states are a massless spin-1 (which could be associated with elec- tromagnetism) and a massless spin-0 (which is not associated with any observed particle).
23 A ‘holy grail’ of massive gravity would be to find an explicit compactification which produced a single massive graviton widely separated from the other states in the tower.
I will apply some of these to studying massive gravity using an approach known as Dimensional Deconstruction. Deconstruction consists of taking a theory defined in extra dimensions, and discretizing the extra dimension to produce a theory of interacting fields in one dimension lower. Dimensional Deconstruction has been suc- cessfully applied to Yang-Mills theories [52, 53]. However an outstanding issue has been to apply Deconstruction to gravity in a way that removes the BD ghost [54–59] .
In Chapter 3 I will resolve this by providing a procedure that shows how a consistent theory of massive gravity emerges from Deconstruction, using the elegant vielbein formulation of GR.
1.4 Overview of dissertation
Publications during dissertation work
During my dissertation work I have been involved in multiple research projects which have led to several publications and developments. Some of these publications form the basis of chapters in this work. I have marked these with a dagger (†).
• Extra dimensions
I have shown how to derive ghost-free massive gravity using Dimensional De-
construction, as I describe in Chapter 3. Previously it had not been known how
24 to apply Deconstruction to gravity to produce a ghost-free theory. This work
formed the basis of follow-up work on extensions of massive gravity that I will
also describe.
– † “Deconstructing Dimensions and Massive Gravity.” C. de Rham, A. Matas and A. J. Tolley, Class. Quant. Grav. 31, 025004 (2014) arXiv:1308.4136 [hep-th].
In addition, I have studied the motion of classical point particles in extra dimen- sions. This work is completely separate from massive gravity. Compactifications can break symmetries globally, for example a standard Kaluza-Klein compact- ification breaks 5D Lorentz invariance, and if the extra dimensions have the topology of a Klein-Bottle, the compactification breaks translation invariance as well. This breaking of symmetries can lead to additional forces on point par- ticles moving through a compact space which are not present in flat space, which we describe. These results may have implications for brane-world scenarios.
– “Point particle motion in topologically nontrivial space-times” A. Matas, D. Muller and G. Starkman, Phys. Rev. D 92, no. 2, 026005 (2015) arXiv:1505.01937 [hep-th].
• Extensions of massive gravity
Given the construction of ghost-free massive gravity, there has been a lot of interest in extending the theory and exploring the full space of consistent re- lated theories. I have been particularly interested in extensions to the kinetic structure (interactions involving derivatives), as I will describe in Chapter 4. My work on this has implications for other models in the literature, such as to generalized matter couplings and to charged spin-2 fields.
– † “New Kinetic Interactions for Massive Gravity?” C. de Rham, A. Matas and A. J. Tolley, Class. Quant. Grav. 31, 165004 (2014) arXiv:1311.6485 [hep-th].
25 – “Interactions of Charged Spin-2 Fields” C. de Rham, A. Matas, N. Ondo, and A. J. Tolley, Class. Quant. Grav. 32, 175008 (2015) arXiv:1410.5422 [hep-th]. – “New Kinetic Terms for Massive Gravity and Multi-gravity: A No-Go in Vielbein Form” C. de Rham, A. Matas and A. J. Tolley, Class. Quant. Grav. 32, 215027 (2015) arXiv:1505.00831 [hep-th]. – † “Cutoff for Extensions of Massive Gravity and Bi-Gravity,” A. Matas, Class. Quant. Grav. 33, no. 7, 075004 (2016) arXiv:1506.00666 [hep-th].
• Galileons and phenomenology
I have studied Galileon scalar field theories, which emerge in the decoupling limit of massive gravity as I describe in Chapter 2. These theories are useful toy models which capture many non-trivial features of the full theory of mas- sive gravity. As a result they are useful for studying phenomenology, such as radiation from binary pulsars which I describe in Chapter 5.
– † “Galileon Radiation from Binary Systems” C. de Rham, A. Matas and A. J. Tolley, Phys. Rev. D 87, no. 6, 064024 (2013) arXiv:1212.5212 [hep- th]. – “Superluminality in the Bi- and Multi- Galileon” P. de Fromont, C. de Rham, L. Heisenberg and A. Matas, JHEP 1307, 067 (2013) arXiv:1303.0274 [hep-th].
This work has led to an invited talk at a workshop which led to a published review
– “Testing General Relativity with Present and Future Astrophysical Obser- vations” E. Berti et al, Class. Quant. Grav. 32, 243001 (2015) doi:10.1088/0264- 9381/32/24/243001 arXiv:1501.07274 [gr-qc].
Organization of dissertation
The rest of this dissertation is organized as follows.
• In Chapter 2, I will give a more technical introduction to massive gravity, con-
centrating on the issues and techniques that will arise in later chapters.
26 • In Chapter 3, I will discuss Dimensional Deconstruction, which is a way to
derive the ghost-free structure of massive gravity from GR in extra dimensions.
This is based on [60].
• In Chapter 4, I will discuss kinetic interactions in massive gravity. I will show
that the kinetic structure must be the same as in GR in order to avoid a ghost
instability, and I will also discuss implications of these results for the coupling
to matter fields. This is largely based on [61] and parts of [62].
• In Chapter 5, I will discuss binary pulsar tests. This is based [63], as well as
the closely related work [64].
• I will summarize my results and provide an outlook in Chapter 6.
Conventions
I will use units where Planck’s constant ~ and the speed of light c are set to unity,
~ = c = 1 (which to paraphrase Sidney Coleman is fine so long as you remember that
when LeBron James shoots a free throw the ball has a tiny velocity but an enormous
angular momentum).
I use Greek indices µ, ν, ··· to refer to space-time indices, and Latin letters a, b, ···
to refer to local Lorentz indices, so the metric has the form gµν and the vielbein
a has the form eµ. Unless otherwise specified I will assume that space-time has four
dimensions. I use the mostly plus metric convention so that ηµν = diag(−1, 1, 1, 1). I p use MPl to refer to the reduced Planck mass which is given by MPl = ~c/8πGN =
27 2.4 × 1018 GeV. I use coordinates xµ = (x0, x1, x2, x3) where x0 = t, and wavenumber
µ 0 1 2 3 0 µ ~ k = (k , k , k , k ) where k = ω is the angular frequency, so kµx = −ωt + k · ~x.
I will symmetrize and antisymmetrize tensor indices with unit weight, so for ex- ample T(µν) ≡ (Tµν + Tνµ)/2 and T[µν] ≡ (Tµν − Tνµ)/2. I use square brackets to refer
µ to the trace of a matrix, [M] ≡ Mµ . The Levi-Civita symbol is normalized so that
ε0123 = +1.
28 Chapter 2
Basics of Massive Gravity
In this section I will review the basic ideas that underlie the construction of ghost-free massive gravity. This serves as an introduction to the theory, as well as allowing me to introduce many different kinds of arguments and issues that will be important in the remaining chapters.
The central idea of massive gravity as an extension of General Relativity (GR) is that the limit in which the graviton mass m vanishes,
m → 0, (2.1)
is a smooth limit, and the physical predictions approach those of GR. This is why we expect massive gravity to be a ‘good’ modification of GR: there are small corrections under control and we can put bounds on m. As we will see, understanding this in detail is actually very subtle and not fully understood. Indeed as written, the limit is not well defined because we have not specified what we are holding fixed. As we will
29 see, while observable quantities may approach GR, the theory itself does not approach
GR but rather GR plus a set of decoupled degrees of freedom. Nevertheless, the continuity of this limit is the basic physical idea in the back of our minds. It informs how we will formulate the theory since we will want to write the theory in a form where the m → 0 limit can be taken in a clear way.
The rest of this Chapter is organized as follows. First in Section 2.1 I will set the stage by providing a few useful examples to introduce terminology and to highlight issues that will be important later. In particular I will show how to carefully take the limit m → 0 for a spin-1 field, I will give an explicit example discussing ghost instabilities, and I will briefly recall some facts and terminology about massless spin-2
fields (General Relativity).
After setting the stage I will work toward building an interacting gravitational theory of a massive spin-2 field. In Section 2.2 I will discuss the free theory, and discuss the issues that arise in particular the vDVZ discontinuity and the possibility of ghost degrees of freedom. I will also describe why self-interactions are a necessary part of a gravitational theory. In Section 2.3 I will extend this discussion to the non- linear level, explaining the generic form of a massive gravity theory, as well as giving explicit forms for the mass term in different languages. In Section 2.4 I will briefly describe the Hamiltonian (ADM) argument for the absence of a ghost, focusing on the mini-superspace which I will make use of in later chapters.
After having built up the non-linear theory, I will then describe how to approxi- mate the theory at low energies on a Minkowski background. In Section 2.6, I will give an explicit realization of the Vainshtein mechanism in a static spherically symmet-
30 ric setup. Lastly in Section 2.7 I will describe certain extensions to massive gravity
known as bi-gravity and multi-gravity, which will play a role in some later chapters.
I conclude this chapter in Section 2.8 with a summary describing the most important
points.
2.1 Setting the stage
This section is meant to be introductory. We review several well-known results in
order to introduce some working examples of a scalar and vector field that we will
return to several times, for example when we discuss the kinetic terms in Chapter 4,
and to define terms and ideas that we will use.
2.1.1 Consistent m → 0 limit of a massive spin-1
The free theory of a massive spin-1 particle Aµ is given by the Proca Lagrangian
1 1 L = − F F µν − m2A Aµ + A J µ, (2.2) 4 µν 2 µ µ
where Fµν ≡ ∂µAν − ∂νAµ and indices are raised and lowered with respect to ηµν.
J µ is an external source. An external source allows us to probe the theory, without explicitly introducing new dynamical fields that obey their own equations of motion.
We are interested in the m → 0 limit of this theory. The most direct approach is to simple set m = 0 in the action given in Eq. 2.2. However, this is not a consistent limit.
The reason is that if we set m = 0, the number of degrees of freedom discontinuously
31 changes from 3 to 2. Formally this occurs because when we set m = 0 in Eq. 2.2, the action is invariant under a U(1) gauge symmetry Aµ → Aµ + ∂µλ (provided that the
µ source is conserved ∂µJ = 0). We can use the freedom to choose λ to remove one degree of freedom in Aµ.
Instead, we want to take a decoupling limit where the third degree of freedom
(which we can identify as the helicity-0 mode of the photon) decouples from the system. To implement the decoupling limit, we first re-introduce the broken U(1) gauge invariance of the massless theory by introducing a new field via the so-called
St¨uckelberg procedure
∂µπ Aµ −−−−→ Aµ + , (2.3) replace m where the factor of m is chosen for later convenience. We now have a gauge invariance under transformations
Aµ → Aµ + ∂µλ, π → π − mλ. (2.4)
The Proca Lagrangian given in Eq.2.2 becomes (up to total derivative terms)
1 1 π∂ J µ L = − F F µν − (∂π)2 − mπ∂ Aµ + m2A Aµ + A J µ − µ . (2.5) 4 µν 2 µ µ µ m
Note that π appears with a canonically normalized kinetic term, meaning that the
2 1 coefficient of (∂π) is equal to − 2 . In that case as we send m → 0 the kinetic term
32 for π remains fixed. We may now consistently take the limit m → 0, resulting in1
1 1 L = − F F µν − (∂π)2 + A J µ. (2.6) m→0 4 µν 2 µ
Here we nicely see that we have taken a limit where the number of degrees of freedom
has not changed, and where π has completely decoupled from the system.
A question that might arise is, why should the kinetic term be built out of the gauge invariant field strength tensor Fµν even though gauge invariance is broken. I will return to this issue in Chapter 4. The real reason is that any other kinetic term would result in a kinetic instability. Since the notion of a ghost is at the core of this work, in the next section we will introduce this class of pathologies.
2.1.2 Ghost modes
A ghost is a field with a wrong sign kinetic term. In order to understand the problem, it is useful to consider a simpler model with two scalar fields φ and χ where χ has
the wrong sign kinetic term
1 1 1 L = − (∂φ)2 + (∂χ)2 + m2 χ2 − V (φ, χ). (2.7) 2 2 2 ghost
The Hamiltonian (or energy) density of this system is
1 1 H = φ˙2 + (∂ φ)2 − (χ ˙ 2 + (∂ χ)2 + m2 χ2 + V (φ, χ), (2.8) 2 i 2 i ghost
1 −1 µ Assuming m ∂µJ → 0, which is needed for the limit to be consistent. In other words, matter should couple to Aµ in a U(1) covariant way at least up to O(m).
33 2 i where (∂iφ) ≡ ∂iφ∂ φ. The main thing to note is that even if V (φ, χ) is positive
definite, the energy is unbounded below because of the kinetic energy of χ. The fact
that φ and χ interact means that the system can explore regions with arbitrarily large
gradients consistent with energy conservation. This is the field theory version of the
discussion in Figure 1.6.
Quantum mechanically, the interactions in V (φ, χ) allow for processes where the
vacuum spontaneously produces φ and χ particles. The only constraint from energy
conservation is that the energy of the particles be ≥ mghost. If we trust the theory to be a valid description at all energies, then there is an infinite phase space volume allowed for this process and the decay rate is infinite. Alternatively if we only trust the theory to be a good description up to some cutoff in energy Ec.o., then the rate will be finite and depend on Ec.o.. If Ec.o. < mghost (that is, we do not trust this theory to be a good description of physics for modes with energy E > mghost), then we may consistently treat the decay process as simply an artifact of using the Lagrangian beyond its regime of validity. The full theory will include some new physics at Ec.o.,
and it may be that the new physics can remove the ghost.
Lastly we give the example of an Ostrogradsky ghost, which is a special case
associated with higher derivatives. As we will see in more detail in Chapter 4, an
Ostrogradsky ghost on the helicity zero mode occurs where the kinetic term was taken
2 to be different from Fµν. A simple case of an Ostrogradsky ghost is given by the two
field model
1 1 2 L = − (∂φ)2 − φ φ − V (φ). (2.9) 2 2 Λ2
34 We can rewrite this action in a completely equivalent form by introducing a new field
χ
1 1 L = − (∂φ)2 + φ χ + Λ2χ2 − V (φ). (2.10) 2 2
Integrating out χ (that is, eliminating χ using its own equations of motion2) repro- duces Eq. 2.9. However we can now write the Lagrangian in an equivalent form by doing a field redefinition φ = φ˜ − χ which yields
1 1 1 L = − (∂φ˜)2 + (∂χ)2 + Λ2χ2 − V (φ˜ − χ), (2.11) 2 2 2
Now the ghost mode is manifest since the action has the same form as Eq. 2.7, with mghost = Λ.
2.1.3 Massless spin-2
Finally before turning to a massive spin-2 it is useful to state several facts about a massless spin-2 to fix conventions.
The Lagrangian for a massless spin-2 can be written as
µν 1 µν αβ hµνT L = − hµνE αβh + , (2.12) 4 2MPl
where Tµν is an external source (representing the energy and momentum tensor,
2The language comes from the path integral formulation of quantum field theory. For fields that appear in the Lagrangian quadratically and whose equations of motion are algebraic, eliminating a field using its equation of motion is equivalent to doing a path integral over that field.
35 analogous to the external current in Eq. 2.2), and where the Lichnerowicz operator is
defined by its action on a symmetric tensor field hµν as
1 E αβ h = − h − 2∂ ∂ hλ + ∂ ∂ h + η ∂ ∂ hαβ − h µν αβ 2 µν λ (µ ν) µ ν µν α β 1 = − ε αγσε ∂ ∂λhβ . (2.13) 2 µ νβλσ γ α
The Lichnerowicz operator is normalized to agree with the linearized Einstein tensor,
2 Gµν = (Eh)µν + O(h ) when gµν = ηµν + hµν. The Lichnerowicz operator has the following properties
µν νµ µν µν E ρσ = E ρσ = E σρ = Eρσ
µραβ µναβ ∂µE = E ∂αχβ = 0, (2.14)
where χµ is an arbitrary one-form. Note these properties are easier to see directly
from the εε structure.
The action is invariant under the linear diffeomorphisms (linear diffs), a gauge
symmetry
hµν → hµν + ∂µξν + ∂νξµ. (2.15)
At the non-linear level, gravity is described by the Einstein-Hilbert action, which
36 is the action for General Relativity
M 2 Z √ S = Pl d4x −gR, (2.16) EH 2
−1 where R is the Ricci curvature scalar and where gµν = ηµν + MPl hµν. The linear gauge transformations given in Eq. 2.15 become invariance under general coordinate transformations, or diffeomorphisms (diffs), under which the metric transforms as
∂xρ ∂xσ g (x) → g˜ (y) = g (x(y)), x → y = y(x). (2.17) µν µν ∂yµ ∂yν ρσ
Much like in the spin-1 example, we expect a mass term to break the invariance of the theory under these transformations.
2.2 Toward a theory of massive gravity
2.2.1 Non-interacting massive spin-2 field
I will discuss the Fierz-Pauli theory in the St¨uckelberg language in some depth, be- cause I will use the logic presented here when I discuss kinetic interactions in Chap- ter 4.
The starting point is the free theory (which is described by a Lagrangian quadratic in the fields). The simplest tensor capable of describing spin-2 degrees of freedom is
37 a rank-2 tensor hµν. The Fierz-Pauli theory of a massive spin-2 field is given by
µν 1 µν αβ 1 2 µν 2 hµνT L = − hµνE αβh − m hµνh − h + , (2.18) 4 8 2MPl
µν where h ≡ η hµν. The normalization of the mass term is chosen so that the propa- gating degrees of freedom (mass eigenstates) have a mass m.
Following the logic in Section 2.1.1, we now introduce a St¨uckelberg field to re- introduce linear diffs
1 hµν −−−−→ hµν − (∂µBν + ∂νBµ) , (2.19) replace m where the factor of m−1 is chosen for later convenience.
Then the action to quadratic order becomes
1 1 L = − h (Eh)µν − F F µν 4 µν 8 µν µν m µν ν 1 2 µν 2 hµν Bν∂µT + Bν (∂µh − ∂ h) − m hµνh − h + − , (2.20) 4 8 2MPl mMPl
where Fµν ≡ ∂µBν − ∂νBµ. Note that the scalings in Eq. 2.19 were chosen precisely so that these fields are canonically normalized.
We still cannot take the m → 0 limit, because the helicity-0 mode is still contained inside the massive vector Bµ. We therefore introduce another St¨uckelberg field
∂µπ Bµ −−−−→ Bµ + , (2.21) replace m
38 µ µν where ∂ π = η ∂νπ.
In this form we can take the limit m → 0 in a safe way, that is without losing any
degrees of freedom.
To summarize, starting from Eq. 2.19 we make the St¨uckelberg replacements
2∂(µBν) 2∂µ∂νπ hµν −−−−→ hµν − − , (2.22) replace m m2 and take the limit m → 0. In the future, when doing St¨uckelberg replacements we will now simply use an arrow instead of explicitly writing replace, but it is impor- tant to keep in mind the distinction between a St¨uckelberg replacement and a gauge transformation. We can write the entire decoupling limit m → 0 as3
µν 1 µν αβ 1 µν 1 µ ν µν hµνT L = − hµνE αβh − FµνF − hµν (∂ ∂ π − πη ) + . (2.23) 4 8 4 2MPl
Note that π only gets a kinetic term from mixing with hµν. We can make this explicit
by doing a field redefinition
˜ hµν = hµν + πηµν, (2.24)
leading to
˜ µν 1˜ µν ˜αβ 1 µν 3 2 hµνT πT L = − hµνE αβh − FµνF − (∂π) + + . (2.25) 4 8 4 2MPl 2MPl
3 −2 µν Provided that the matter source satisfies m ∂µT → 0 in this limit.
39 vDVZ Discontinuity Note the coupling
πT (2.26) 2MPl survives even in the limit m → 0, even for a conserved source. As we send m →
0, π remains coupled to T . This leads to an apparent finite difference in physical predictions no matter how small the value of the mass. This is the van Dam-Veltman-
Zakharov (vDVZ) discontinuity.
At first sight, this seems like a disaster. No matter how small m, there is no way we can have continuity with GR. In fact, the resolution proposed by Vainshtein is that the free theory is a bad approximation in a regime that becomes larger and larger as we send m → 0.
2 µν Ghost mode In the Fierz-Pauli mass term, the terms h and hµνh appear with a relative coefficient of −1. This is no accident, as any other combination would not give rise to a consistent theory of massive gravity. In order to see this more precisely, we can allow for a more general mass term
µν 1 µν αβ 1 2 2 2 hµνT L = − hµνE αβh − m αhµν − βh + . (2.27) 4 8 2MPl
Introducing St¨uckelberg fields as in Eq. 2.22 and focusing on the scalar-tensor sector
(which is the most relevant for this discussion), and focusing on the part of the action
40 that is nonvanishing as m → 0 we find,
1 α − β L = − h E µν hαβ − h (∂µ∂νπ + (α − β) ηµν π) − ( π)2 .(2.28) scalar−tensor 4 µν αβ µν 2m2
From here we see the presence of a dangerous higher derivative operator. I will use
similar techniques to identify the presence of a ghost in Chapter 4.
Comparing with Eq. 2.9, we see that the mass of the ghost
3 m2 m2 = , (2.29) ghost 2 α − β where the 3/2 comes from accounting for the normalization of π in Eq. 2.25. The special Fierz-Pauli potential occurs for α = β, in which case mghost is infinite.
So far we have described the linear theory of a massive graviton in some detail.
When the spin-2 field couples to matter, the spin-2 field necessarily develops addi- tional interactions. We now turn to a more realistic theory of an interacting massive graviton.
2.3 Non-linear formulations of massive gravity
A generic interacting theory of Lorentz invariant massive gravity has the form
M 2 Z √ m2 S[g ] = Pl d4x −g R − U(gµν, η ) + S [g , ψ]. (2.30) µν 2 2 µν matt µν
41 The action depends on a dynamical metric gµν. The potential U contains contractions
of the dynamical metric with the Minkwoski metric ηµν.
When perturbed to leading order around Minkowski, the potential should have the form of the Fierz-Pauli choice. At higher order it is less clear what the mass term should be. A natural first guess for the potential is to covariantize the Fierz-Pauli mass term
1 U = gµνgρσ (H H − H H ) , (2.31) 2 µρ νσ µν ρσ
where
Hµν ≡ gµν − ηµν, (2.32)
and we use a capital Hµν to recognize that we are not assuming that H is a small
quantity. However, it was shown by Boulware and Deser that this combination, and
any function of this combination, is plagued by a return of the ghost instability [31]. It
has recently been shown that there is a choice of potential which avoids this problem,
to which we now turn.
42 2.3.1 Ghost-free potential
The general form of the potential which avoids the Boulware-Deser ghost was given
in [28]
4 X µν Ug.f. = − αnLn[K(g , ηρσ)], (2.33) n=0
where
√ µ µ µα K(g, η) ν ≡ δν − g ηαν, (2.34)
and where the mass terms are given by
L0[K] = 4!
L1[K] = 3![K]