An Analysis of Theories of Multiple Spin-2 Fields

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An Analysis of Theories of Multiple Spin-2 Fields

Joakim Flinckman

Supervisor: Fawad Hassan Stockholm University Department of Physics Master’s Degree

Physics, Master Project, 60 ECTS 2019-2020

Stockholms Universitet/Stockholm University SE-106 91 Stockholm

Phone: 08 - 16 20 00 www.su.se

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Abstract

Until recently, no consistent theory of more than two interacting spin-2 fields was known, except for a collection of pairwise bimetric couplings. In 2018, Hassan and Schmidt-May presented a theory with couplings beyond pairwise for an arbitrary number of fields in terms of vielbeins [1]. This theory was proved to propagate the correct number of modes and is hence free of the Boulware-Deser ghost.

The spin-2 fields are represented by vielbein in this theory, but vielbeins are not the minimal covariant description of spin-2 fields, and they contain non-physical Lorentz degrees of freedom. Metrics provide a more familiar and natural representation of spin-2 fields, and in this thesis, such a formulation of the theory is presented. Though a formal proof of the absence of ghost modes in the metric formulations remains to be formulated, it is argued to be the first consistent multimetric theory beyond pairwise coupling.

In this thesis, the known consistent theories of spin-2 fields are reviewed, in both the metric and vielbein formulation. The revision is preceded by the construction of a metric formulation of the new multivielbein theory. Multiple methods for expressing the theory in terms of metric are presented, and both the vielbein and metric field equations are derived and analysed. The equations of motion of the non-physical degrees of freedom are shown to give rise to constraints on the physical fields. These constraints are interpreted and solved, both numerically and analytically to conclude that non-trivial metric configurations exist. The constraints of both the new and previously known pairwise metric coupling are geometrically analysed in terms of overlapping null cones. The pairwise coupling is shown to allow overlaps that might give rise to undesired causal properties which do not seem to occur in the more general multimetric theory.

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List of Abbreviations

dRGT de Rham, Gabadadze and Tolly DoF Degrees of Freedom

EoM Equation(s) of Motion

GCT General Coordinate Transformations GR General Relativity

Λ-CDM Λ Cold Dark Matter LC Lorentz Constraints QFT Quantum Field Theory

Notation and Conventions

• In this thesis, the notion of degrees of freedom is considered, and this will refer to different concepts depending on the context. While it should be evident from the context, it might refer to one of three different notions:

– The number of independent variables in an equation.

– The number of variables in phase space that is needed to represent the classical system in question.

– The number of propagating modes, where one mode is a pair of conjugate variables in phase space.

• The set of smooth (p, q)-tensor fields will be denoted Tq^p(M ) := T^∗M^×p× T M^×q → C^∞(M ) where T M/T^∗M denotes the set of all smooth vector/one-form fields on the spacetime manifold M, C^∞(M ) the set of all smooth functions from M to R and T M/T^∗M the tangent/cotangent bundle. A short introduction to abstract tensors is presented in Appendix A.3.

• A symmetric non-degenerate (0,2)-tensor gI will be referred to as a metric. The (2,0)-tensor field g_I⁻¹ will be referred to as the "inverse metric" and will be assumed to exist everywhere. The inverses of both the metric and other objects will often be indicated only by the placement of the indices, e.g. the components of g⁻¹ will be written g^αβ:= (g⁻¹)^αβ and the inverse vielbein e⁻¹ as e^αA:= (e⁻¹)^α_A.

• All metric considered are Lorentzian and the "mostly positive" convention is used, e.g. the Minkowski metric would be

η = ηµνdx^µ⊗dx^ν= −dt ⊗ dt + dx ⊗ dx + dy ⊗ dy + dz ⊗ dz.

Often the tensor product ⊗ will be neglected to lighten up notation but is still implicit in an expression such as dxdy := dx ⊗ dy.

• Indices can be raised and lowered by the musical isomorphism between vectors and one-forms. Since both spacetime indices and Lorentz indices are dealt with, the corresponding isomorphism must be used. See Appendix A.3 for a definition of the musical isomorphism.

• In this report, it is assumed that spacetime is endowed with a linear torsion-free connection (covariant derivative), such that ∇α(gIµν) = 0, referred to as the Levi-Civita connection. In the case of multiple metrics, every metric has it is own connection. This is important for the existence of the Einstein- Hilbert action, but will not be used extensively elsewhere.

• The flat spatial metric with components δij will be denoted as ˆδ, while the identity operator, with components δjⁱ will be written as 1. Note that these two operators have the same matrix but are two different mathematical objects. One is a (0,2)-tensor and the other a (1,1)-tensor or endomorphism.

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• Einstein’s summation convention is used for both Greek, lower and upper case Latin indices. Greek indices, α, β, ... ∈ {0, 1, ..., d}, enumerate components of a spacetime object in a coordinate basis, i.e.

ωαX^α := Pd

α=0ωαX^α. Capital Latin indices A, B, C, ... enumerate components of a Lorentz space, e.g. ηAB = Λ^C_Aη_CDΛ^D_B. Some objects, such as vielbeins, have mixed indices and these are summed individually, e.g. eÂµ, as in gµν = eÂ_µη_ABe^B_ν. The conventions should be self-evident but some sub- or superscripts will not be summed, these will, in general, be indexed by I, J, K, L..., e.g. in the case of multiple metrics gI = g₁, ..., g_N, I will not be used for summation. These types of indices can be raised or lowered to simplify notation and do not correspond to any covariant och contravariant structure, i.e. gI ≡ gÎ. In some places indices such as ν and µ are used without referring to spacetime indices, then they are placed in parenthesis, e.g. Ci^(ν)has only one index i that can be used for contraction.

• In this thesis, two types of basis are used, coordinate basis (also referred to as chart induced or holonomic basis), and orthonormal basis. The coordinate basis is defined by the chart (U, x), where U is an open subset of the spacetime manifold, x : U → R^d is the coordinate map, and x^α : U → R is referred to as the components of x. The set of partial derivatives ∂α := _∂x^∂_α form a basis of T U e.g a vector Y is written Y = Y^α∂_α:= Y^α_∂x^∂_α. The spatial components are written Xⁱ starting from the middle of the alphabet, (i, j, k, ... ∈ {1, 2, ..., d}). This basis is generically not orthonormal since g(∂α, ∂β) = gαβ is not necessarily equal to the Minkowski metric. The indices of a coordinate frame are referred to as spacetime indices.

A non-coordinate basis, or orthonormal frame¹, is a set {eA}, eA∈ T M, such that g(eA, eB) = ηAB. The indices of an orthonormal basis, A, B, C, ... ∈ {0, 1, ..., d}, are referred to as Lorentz indices and the spatial indices in a orthonormal frame are indicated by a, b, c, ... ∈ {1, 2, ..., d} starting from the top of the alphabet.

• Matrix notation for rank-2 tensors such as metrics, vielbeins and transformations will often be used in this report to simplify equations, and the same symbol will be used for the abstract tensor, and its matrix, e.g. the symbol g can represent both the abstract metric and its components in a basis (gµν := g(∂µ, ∂ν)) arranged in a matrix in the standard way. The notation should be self-evident and clear from the context, but below are some examples to elucidate possible confusions:

– Summed indices should be adjacent and are moved by the transpose operation g_µν= e^A_µη_ABe^B_ν = (e^T)_µ^Aη_ABe^B_ν ⇐⇒: g = e^Tηe.

– Sometimes both matrix and index notation is used at the same time: S^µν =p

g⁻¹f^µ

ν where the product under the square root is the matrix of the composition of g⁻¹and f, g⁻¹◦for (AB)^αβ

is short for the A^ασB^σ_β.

– Since (A^T)⁻¹= A⁻¹T the notation A⁻^T can be used without ambiguity.

It is worth pointing out that the notation h will sometimes also be used to denote the trace of a metric perturbation, i.e. h = Tr(h) = η^αβhαβ but this should be clear from the mathematical context.

• Throughout the thesis√

−g :=p−det(g) iff g is a metric; usually g, f, h or gI, fI are used for metrics.

This notation will also be used for the Riemannian metrics such as the spatial metric,√

γ :=pdet(γ). The determinant is here defined to be the "naive" determinant of the matrix representation of the bilinear form g and transforms as a scalar density, while the measure √

−gd⁴x is invariant under general coordinate transformations. For more details, see Appendix A.6.

• Proper homogeneous orthochronous Lorentz transformations will simply be referred to as Lorentz transformations in this report, i.e. Λ is a Lorentz transformation iff: Λ^TηΛ = η, Λ⁰0> 0and det(Λ) = 1 and is denoted Λ ∈ SO⁽1, 3) =: O(η). Other orthogonal groups will also be considered, such that similarly L^TgIL = gI and will be denoted LI ∈O(gI), i.e. the orthogonal group with respect to gI.

1There are other non-coordinate basis, but only orthonormal frames will be considered in this thesis.

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• The word isometry will be used in the pure mathematical meaning and should not be confused with the Killing symmetries of spacetime, i.e. an isometry is a bijective map X between metric spaces, such that for the metrics f and g: f = X^∗g (the pullback of g under X, see Appendix A for definition) or in matrix notation, f = X^TgX.

• Some general remarks:

– In this thesis, only the Boulware-Deser ghost will be considered, and any reference to ghost modes will hence refer to the Boulware-Deser ghost.

– All fields considered in this thesis are continuous classical fields, and quantisation will not be considered.

– The dimensionality of spacetime will be assumed to be 3+1 unless otherwise stated, but most of the analysis is valid for higher dimensions.

– The spacetime manifold will be considered to be nice enough for most standard approaches in GR.

– Fields with spin less than two will be referred to as lower spin fields.

– All equations are written in natural units unless otherwise specified, i.e. c = ~ = G = 1.

Acknowledgements

I would like to thank my supervisor Fawad Hassan for introducing me to this exciting and deep subject and for his patience and knowledgeable explanations. I would like to thank Francesco Torsello and Mikica Kocic for exciting and stimulating discussions and their forbearance with my many questions. Thanks to Anders Lundkvist for taking part in teaching me general relativity in earlier semesters and answering my many questions about the bimetric theory. Thanks to my office mate Tobias Wipf for our many discussions, particularly during the start-up of both our projects and mental support during the Covid-19 pandemic.

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Introduction

Two of the most successful models of modern physics are the standard model of particle physics (SM) and General Relativity (GR). The SM describes the electromagnetic force and, the weak and strong nuclear interactions, and GR describes gravity. While the SM is constructed in the language of quantum field theory (QFT), GR is a classical theory. GR is the foundation of the currently best cosmological model Λ-CDM and can be used to explain phenomena on a vast range of scales, ranging from black holes and neutron stars, local phenomenon, to large scale structures in the universe. Both the SM and GR are relativistic field theories where the field equations are obtained from an action by Hamilton’s principle of stationary action.

The fields can be classified using the Wigner classification in terms of mass and spin. The mass and spin determines the mathematical representation of the field and affects the form of the field equations.

The SM contains both massive and massless fields of different spins interacting with each other. The fields are arranged in so-called multiplets which provide the theory with an intricate structure. For instance, the SM variant of Maxwell’s theory of electromagnetism is embedded in the more general structure of the electroweak theory, SU(2)W×U(1)Y together with three other massive spin-1 fields. The multiplet construction is crucial for the structure of the theory, its phenomenological validity and theoretical consistency.

On the other hand, GR contains one single massless spin-2 field which interacts with the lower spins fields through the curvature of spacetime, so-called minimal coupling. This is, in a sense, the simplest non-linear spin-2 theory and naturally leads to the question if there are other consistent spin-2 theories. Using the experience from the SM, one might ask, if there are consistent theories of multiple spin-2 fields and if they have a multiplet structure similar to the SM? Since spin-2 fields are intrinsically linked to gravity, these theories might result in modified theories of gravity, and one can consider if GR could be embedded in a more general multiplet structure like electromagnetism is contained in electroweak theory?

GR is a well-established and tested theory, and any modification must consequently explain the same phenomena that GR successfully describes, e.g. solar system dynamics, cosmological structures, and gravitational waves. Nevertheless, GR does have some observational weaknesses. The universe appears to have an accelerating expansion and something, referred to as dark energy, is needed to explain this self-acceleration.

This phenomenon can be dealt with by introducing a cosmological constant in Einstein’s field equations, and while this remedies the expansion of spacetime, its origin and value remain a mystery. One proposed origin would be the energy of the vacuum itself, but calculating this from the SM leads to one of the most significant discrepancies between theory and observation in physics and is referred to as the cosmological constant problem.

Another observational shortcoming of GR and the SM is the problem of dark matter, i.e. most of the matter content of the universe seems to interact only through gravity and has no known counterpart in the SM. This problem is often hypothesised to be solved by a new massive particle that interacts only very weakly with the other particles but does contribute to the mass content of the universe and consequently affects gravity.

While these problems are both intrinsically linked to gravity and might justify modifications of GR; there are additionally theoretical reasons to investigate other theories of gravity and more general theories of spin-2 fields. While lower spin theories are understood both at a classical and a quantum level, only the simplest spin-2 theory, i.e. GR, is adequately studied and its quantum theory is still not known. Consequently, theories with multiple spin-2 fields might be of interest for a quantum theory of gravity.

Apart from Maxwell’s equations, all the simplest field equations contained in the SM were first discovered theoretically and only later shown to have physical significance. This provides a strong motivation to investigate more general theories of spin-2 fields, and a natural progression is to consider a massive spin-2 field. In 1939, Fierz and Pauli showed [2] that it is possible to formulate a consistent theory at a linear level. The mass term turned out to be very restrictive for the theory to propagate the correct number of modes. A massive spin-s field should have 2s + 1 propagating modes or degrees of freedom (DoF). Hence a massive spin-2 field should have five DoF, while the general linear massive spin-2 theory of Fierz and Pauli propagates six. The sixth mode has a kinetic term with the wrong sign in the Hamiltonian, making it a so-called ghost, rendering the theory nonphysical.

It turns out that most naive modifications of the simplest spin-2 theories have such ghost instabilities,

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and this is one of the reasons why research into the subject had little progress between the 1940s and 2010s.

In 2010 de Rham, Gabadadze and Tolley (dRGT) found a non-linear completion of the Fierz and Pauli theory, which was proved to be ghost-free by Hassan and Rosen [3]. This was the first non-linear theory of a massive spin-2 on a flat spacetime [4, 5] and the action was later generalised to an arbitrary background by Hassan and Rosen [6].

In order to create non-trivial scalar terms, the mass term contained an extra metric, which was chosen by hand. The additional metric served as a predetermined geometric background. This extra tensor field was later promoted to a dynamical physical field by Hassan and Rosen and resulted in the first ghost-free multi-spin-2 theory, bimetric theory or bimetric relativity [7]. This theory had seven propagating DoF, and at a linear level, the fields could be diagonalised into mass eigenstates, i.e. a massive (DoF=5), and a massless (DoF=2) mode could be identified.

While bimetric relativity is an interesting theory in itself, the bimetric interaction provides the possibility to create theories with more than two metrics. Using multiple copies of the bimetric interaction, one could create ghost-free theories with pairwise interacting spin-2 fields. These multimetric theories are based on pairwise interactions, i.e. two metrics in one interaction term, but no interaction beyond pairwise was known.

In 2013, Rosen and Hinterbichler [8] formulated a theory of multiple interacting spin-2 fields in terms of vielbeins (also known as vierbein or tetrads if d = 4). The interaction term contained up to d(=dimensionality of spacetime) vielbeins, but it was later shown that the most general form was not necessarily ghost-free [9].

The only subset which was known to be ghost-free was equivalent to the previously known bimetric pairwise coupling.

In 2018, Hassan and Schmidt-May [1] successfully constructed a non-pairwise multivielbein interaction and proved it to be ghost-free. The interaction was a subset of the theory presented by Rosen and Hinter- bichler, but not of the previously known pairwise coupling. How to formulate the theory in terms of the more familiar and natural metric formulation has remained unresolved, and this is where this current work starts.

Structure of the thesis: This thesis is divided into three main parts: Part I serves as an introduction to the subject of spin-2 fields and presents only previously known results. This part can be skipped by a reader familiar with the bimetric theory, theories of multiple spin-2 fields, and the 3+1 and vielbein formalisms.

The formalism and methods introduced in Part I are used in Part II to present the previously unknown work and new multimetric theories. The Appendix is heavily used for detailed calculations and technical matters that provides a more rigorous framework for some of the methods used.

Part I starts with a background on relativistic field theories, theories of spin-2 fields and modifications of gravity. Later, GR is expressed in both the Lagrangian and the Hamiltonian formalism to identify the dynamic DoF and the constraint. In Section 3 GR is formulated in terms of a constrained vielbein system.

Later dRGT massive gravity and bimetric theory are introduced, and a method to create pairwise multimetric theories is described. Part I is concluded with a brief introduction to the Hassan-Schmidt-May multivielbein theory [1]. Part II starts with Section 3, where a method of expressing a vielbein action in terms of metrics is presented. This method is then applied to the Hassan-Schmidt-May theory, and the constraint and field equations are derived. They are shown to be equivalent to ones obtained in the vielbein theory under mild restrictions, and the equations of motion are analysed at a linear level. The constraints are geometrically analysed in terms of overlapping null cones, and some specific solutions are presented. Part II is concluded with a summary with conclusions and an outlook of the subject of multimetric theories and proposed further research.

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Part I

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1 Research Background and Motivation

Relativistic fields can be classified by the Wigner classification where the spin and mass correspond to the Casimir invariants of the fields irreducible representation of the Poincaré group [10], i.e. the spacetime symmetries of Minkowski space. The representation of the field determines the spin of the field, e.g. a scalar field has spin-zero, while a vector field has spin-one. The spin and mass determine the dynamical DoF of the field where a massive spin-s field has 2s + 1 physical DoF, and a massless field has two DoF². In a covariant formulation of such theories, the representation of the field typically has more variables than physical DoF, i.e.

there is a redundancy in the formulation. This redundancy is typically removed by constraints or symmetries that reduces the number of non-physical phase space variables to the correct number. The simplest classical equations of motion and the number of propagating modes of the free fields from QFT and GR are listed in Table 1. These can all be obtained from an action and Hamilton’s principle of stationary action. For spins less than 2, there is a clear pattern, where the massive equation is obtained from the massless equation by merely adding a mass term. Considering the theory of a spin-1 field, it is evident from the table that the number of propagating modes changes between the massless and massive case. As we will see, this provides some complications, and in Section 2.2 a similar problem will appear for the theory of a massive spin-2 field.

It is, therefore, worth considering some details of the spin-1 case before progressing.

In the covariant formulation of Maxwell’s theory, the electromagnetic field is described by the vector potential A^µ. A^µ has four components, while the physical field should have two modes or polarisations, hence there is a redundancy in the description. The field equations and the action can be described in terms of the field-strength tensor Fµν := ∂µAν− ∂νAµ, from which the physical electric and magnetic fields E and B can be obtained³. The field strength is invariant under the transformation Aµ7→ A⁰_µ= Aµ+ ∂µϕfor any differentiable scalar function ϕ

F_µν⁰ = ∂µA⁰_ν− ∂νA⁰_µ= ∂µAν− ∂νAµ= Fµν.

Hence Aµand A⁰µ give the same field-strength tensor and therefore the same EoM, and electric and magnetic fields; therefore, this invariance or gauge symmetry represents a non-physical redundancy in the representation Aµ. ϕ(x) can be chosen to eliminate a component of Aµ while Fµν remains unchanged, e.g. if one chooses ∂^µ∂µϕ(x) = −∂^µAµ(x), one parameter if fixed in terms of the others and the EoM reduces to a wave equation

∂_µF^µν= ∂_µ∂^µA^ν− ∂_µ∂^νA^µ= ∂_µ∂^µA^ν = 0.

Spin Massless DoF Massive DoF

Spin-0 Klein-Gordon

∂µ∂^µφ = 0 1 Klein-Gordon

(∂µ∂^µ− m²)φ = 0 1 Spin-¹⁄2 Dirac

i /∂ψ = 0 2 Dirac

(i /∂ − m)ψ = 0 2

Spin-1 Maxwell

∂_µF^µν = 0 2 Proca

∂_µF^µν− m²A^ν= 0 3 Spin-³⁄2

Rarita-Schwinger

i /∂ψ^µ= 0 2 Rarita-Schwinger

(i /∂ − m)ψ^µ= 0 4

Spin-2 Einstein

R_µν−¹₂g_µνR + Λg_µν = 0 2

Table 1: Table of the classical vacuum equations of motion for fields of different spins and masses. The DoF of each field is denoted and corresponds to half the number of phase space variables for each dynamic field.

2Except for the scalar field, which has 1 DoF.

3The field-strength is a 2-form defined as F := B + E ∧ dt where E = Exdx + Eydy + Ezdz, B := Bxdy ∧ dz + Bydz ∧ dx + Bzdx ∧ dy and F = Fµνdx^µdx^ν. See A.5 for definition of forms and the wedge product.

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There is a residual gauge symmetry, where adding a ψ that solves the wave equation, i.e. ∂µ∂^µψ = 0 to A^µ, again can be used to fix a component of A^µ. Hence this, so-called U(1) gauge symmetry eliminates two components, leaving only two physical polarisation modes of the photon.

Now turning to the Proca theory of a massive vector field, the mass term m²A^νbreaks the U(1) symmetry, and the EoM are no longer invariant under the shift of the field by ∂µϕ. Hence the process from above can not be used to eliminate variables, and naively there seem to be four propagating DoF in the Proca equation, but a massive spin-1 field should have 3 DoF. However, contracting the EoM with ∂ν it becomes

∂ν∂µ(∂^µA^ν− ∂^νA^µ) − m²∂νA^ν= 0.

Since ∂^µA^ν− ∂^νA^µ is antisymmetric, and ∂µ∂ν is symmetric, the equation now requires ∂νA^ν = 0. This equation can be used to fix one variable, eliminating one DoF, resulting in three physical DoF of a massive vector field. Imposing this constraint, the EoM takes a simpler form

∂µF^µν− m²A^ν = ∂µ∂^µA^ν− ∂^ν∂µA^µ− m²A^ν= (∂µ∂^µ− m²)A^ν= 0.

This is precisely the EoM of Maxwell theory in the Lorentz Gauge (∂νA^ν = 0) with an added mass-term.

The difference here is that the relation ∂µA^µ was forced upon the theory by a constraint and there is no residual gauge symmetry. Similar constraints will be essential in theories of higher spins since the covariant formulation provides higher redundancy as the spin increases, e.g. GR is represented by a metric with ten DoF but should have only two.

Apart from the correct number of DoF of a field, other complications can appear. For a classical field theory to be consistent, one must require the absence of fields with the wrong signs of their kinetic or mass terms in the action. Modes with a negative kinetic term in the Lagrangian are referred to as ghosts, while modes with a positive mass term are called tachyons

L = −(∂tϕ)²+ ... Ghost mode L = +m²ϕ²+ ... Tachyonic mode.

The absence of ghost fields is necessary to keep the Hamiltonian bounded from below. A Hamiltonian which is not, could in principle transfer an arbitrary amount of energy to other fields, making the theory unstable and inherently non-physical. These types of ghost modes often arise in modifications of spin-2 theories from the extra DoF and are therefore of great interest for the rest of this study. Ghost fields are also problematic from a quantum perspective as these result in states with a non-positive norm and hence produce negative probabilities and non-unitary evolution [11]. The elimination of ghosts is required in a consistent field theory and was one of the main obstacles in formulating a consistent theory of a massive spin-2 field.

It has been claimed that one, in principle, could ignore ghost modes if it decouples from the other fields and mghost> Λ_cutoff[12], which implies that the ghost is never exited. In this case, the mode might not be problematic, but since this might lead to an unstable quantum theory and a classical theory without a ghost is more natural, only ghost-free theories will be considered to be consistent in this thesis.

The spin-s ≤ 1 fields play a crucial role in the SM and are all well-understood both at a classical level and quantum level. Even spin-³/2[13] have been of great interest for supergravity and supersymmetry, e.g.

as the supersymmetric partner of the graviton. Fields with higher spin than 2 are highly constrained [14], e.g. no massless interacting such fields exist on Minkowski space [15], and no general local, interacting theories are believed to exist. Non-local theories such as string theory do allow for such modes, and they arise in AdS/CFT [16] in terms of infinite spin ladders, where all spins must exist, 1,³/2, 2,⁵/2, 3, .... However, restricting the analysis to local interacting theories, the only still unexplored area is the spin-2 sector where only the massless case is well-understood. In the next section, some of the existing consistent theories of spin-2 fields are introduced, and in Part II a new such theory is presented.

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2 Existing Spin-2 Theories

This chapter starts with a short introduction of the mathematical framework of general relativity and the conventions used for its formalism. Later, using the Hamiltonian formalism and 3+1 decomposition the Einstein-Hilbert action is shown to propagate the correct number of modes for a massless spin-2 field. GR is formulated in terms of vielbeins, and modifications of GR are discussed. Linear gravity and the Fierz- Pauli action are presented together with the dRGT massive gravity theory. This theory is argued to be unsatisfactory and is generalised to a bimetric theory, which is presented in both the metric and vielbein formulation. The chapter is concluded with an introduction to theories of multiple spin-2 fields in terms of vielbeins.

2.1 General Relativity

2.1.1 Introduction to General Relativity

Before considering more general spin-2 theories, let us introduce the simplest non-linear such theory, general relativity. GR is a relativistic field theory of a metric field on a smooth, Lorentzian manifold M, referred to as spacetime. The metric can be used to define a unique Levi-Civita connection⁴∇which, in turn, provides a geometry on M. The connection or covariant derivative is described in a chart by the Christoffel symbol as follows

∇X(Y ) = X^α∇α(Y^β∂β) = X^α∇α(Y^β)∂β+ X^αY^β∇α(∂β) =: X^α∂αY^β∂β+ X^αY^βΓ^γ_βα∂γ.

Where X and Y are spacetime vectors. The connection is defined by its action on a vector field, but it can equally well act on any (p, q)-tensor by the use of the musical isomorphism, producing p + q Christoffel symbols⁵. Using the connection, it is possible to define parallel transport on the manifold and the Riemann curvature tensor, such that for a set of vectors X, Y, Z and a one-form ω

R(ω, Z, X, Y ) : = ω ∇_X∇_Y − ∇_Y∇_X− ∇_{[X,Y ]} Z

= ωαR(dx^α, ∂β, ∂γ, ∂δ)Z^αX^βY^γ = ωαR^α_βγδZ^βX^γY^δ.

It should be stressed that the convention for the sign of the Riemann tensor can differ, but here the sign convention of Wald [17] is used. The Riemann tensor components in the coordinate basis are entirely determined by the Christoffel symbols and therefore only depends upon the metric and its derivatives

R^α_βγδ= ∂_γΓ^α_βδ− ∂δΓ^α_βγ+ Γ^α_γεΓ^ε_βδ− Γ^α_δεΓ^ε_βγ. (2.1) Using the Riemann tensor and the metric, it is possible to construct the Ricci tensor Rµν := R^α_µαν and the Ricci scalar R := g^µνR_µν. In GR, the spacetime metric is dynamic, and the EoM affects both the notion of length, through the metric and curvature through the Levi-Civita connection. The EoM are given by the Einstein field equations, which can be obtained by varying the Einstein-Hilbert action

SEH= M_Pl² Z

M

d⁴x√

−g (R − 2Λ).

Where MPl² = _16πG^~c is the Planck mass, and Λ a cosmological constant. Adding a minimally coupled matter action Sm =Rd⁴x√

−g Lm to SEH and varying the action with respect to g^µν the Einstein field equations take the form

Rµν−1

2Rgµν+ Λgµν = Gµν+ Λgµν = 1

2M_Pl² Tµν. (2.2)

4i.e. the unique connection given by ∇αg_βγ= 0.

5The Christoffel symbols are determined entirely in term of the metric and its derivatives, gαβΓ^β_γδ =

1

2 ∂γgαδ+ ∂δgαγ− ∂αgγδ.

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Here Tµν:= −^√²

−g (δ√

−g Lm)

δg^µν is the energy-momentum tensor, which specific form is provided by the matter Lagrangian Lm. The field equations of the matter fields are affected by the curvature provided by the metric, and the EoM of the metric are in turn dependent on the matter fields through Tµν. The theory, therefore, has a natural geometric interpretation and the quote by Wheeler, "Spacetime tells matter how to move;

matter tells spacetime how to curve" [18] is an intuitive one.

One crucial difference between the Einstein field equations, Eq. 2.2 and the rest of the field equations in Table 1 is that it is not linear. This is a manifestation of the equivalence principle, i.e. gravity couples the same to all kinds of energy, including its own. This gives rise to the interactions of the metric with itself, resulting in a non-linear theory.

2.1.2 General Relativity in 3+1 Formalism

GR is often presented in the Lagrangian formalism and the Einstein field equations. It is not apparent from the action or the EoM what the physical field content is, and how many propagating modes the theory has.

This will be clear when performing a Hamiltonian analysis by transforming the action to the Hamiltonian formalism using the 3+1 decomposition. It will allow us to identify the number of propagating modes with canonical kinetic terms and show that the theory indeed has the correct number of DoF for a massless spin-2 field. Before doing this, let us do the following observation; the contracted Bianchi identity give

∇µG^µν = ∂0G^0ν+ ∂kG^kν+ Γ^µ_µσG^σν+ Γ^ν_µσG^µσ ≡ 0.

Since Gµν contains at most second-order time derivatives, so does ∂0G^0ν and therefore G^0ν can only contain first-order time derivatives. Gµν is symmetric; therefore there are in total ten equations and twenty phase space variables, the metric components and their time derivatives. The evolution is given by Eq. 2.2, a second-order PDE, hence the equation G0ν∝ T0ν provides constraints on the initial conditions. Hence there are only six dynamical equations and twelve phase space variables⁶. In the Hamiltonian analysis, it will be apparent that there are even fewer dynamical DoF.

While the Lagrangian density is naturally covariant, the Hamiltonian and specifically the canonical momentum requires a departure from the usual covariant formalism because it refers to a notion of time.

In a non-covariant system, only the time derivatives ˙q of the variables q are replaced by p = ^∂L_{∂ ˙}_q during the Legendre transform, H = piqⁱ−L. For a Hamiltonian analysis of a covariant system, one needs to decompose the spacetime manifold to separate notions of space and time. This seems to contradict the idea of spacetime and GR, but the notion of a spacetime manifold is in-fact crucial for a geometrical interpretation of gravity, see Appendix A.2. The 3+1 decomposition does exactly this without losing the spacetime structure as a four- dimensional manifold. While the equations will no longer be manifestly 4d covariant, the general covariance will be evident in the form of constraints, that will be of importance for the number of propagating modes.

In Appendix A.3, the 3+1 decomposition is described through a proper construction and below follows a brief introduction to the idea.

Even though they were not the first to do so, Arnowitt, Deser and Misner (ADM) [19] formulated GR in the 3+1 language and analysed the Hamiltonian. The main idea of 3+1 decomposition is to consider a set of spacelike hypersurfaces Σt ⊂ M, together with an orthogonal timelike direction and consider the induced⁷spatial metric on Σtand consider how the hypersurfaces evolve in time, i.e. when moving through the manifold in the timelike direction. The total four-metric can be decomposed using the 3+1 variables

ds²= gµνdx^µdx^ν= −N²dt²+ (dxⁱ+ νⁱdt)γij(dx^j+ ν^jdt) Or in matrix notation: g =−N²+ ν^Tγν ν^Tγ

γν γ

. (2.3)

Where N > 0 is termed the lapse, ν is a spatial 3-vector tangent to Σ named the shift, and γ is the induced positive-definite 3-metric on Σ.

6Where four DoF are fixed by a General coordinate transformation (GCT).

7by the embedding of Σtin M .

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Introducing the shift operator

χ(ν) :=1 0 ν 1

then χ(ν)χ(µ) = χ(ν + µ)

allows the metric to be written using matrix notation as g = χ^T(ν)diag(−N², γ)χ(ν), which gives the form of the inverse metric

g⁻¹ = χ(−ν)diag(−N⁻², γ⁻¹)χ^T(−ν) = 1 N²

−1 ν^T

ν N²γ⁻¹− νν^T

. (2.4)

Note that in a basis γij = g_ij but γ^ij 6= g^ij and that γ canonically provides a musical isomorphism, such that one can raise and lower indices on spatial vectors/one-forms, e.g. νi= γ_ijν^j.

The lapse, N, and shift, ν, have useful interpretations of both the foliation and of the shape and orientation of the null cones generated by g. These details will become more important as multiple metrics are introduced on the spacetime manifold, and an overview of how the 3+1 variables affect the null cone can be found in Appendix A.4.

Now there is a clear notion of spatial variables, and it is possible to define the canonical conjugate momentum to the spatial metric

π^ij = ∂LEH

∂ ˙γij

= Mpl

√γ

N γîkγ^jl− γîjγ^kl ˙γkl− ∇ⁱν^j− ∇^jνⁱ+ 2γîj∇kν^k .

It can be expressed in terms of the extrinsic curvature K^ij, defined in Appendix A.3 which endows a geometric interpretation of the canonical momentum as well.

π^ij = Mpl

√γ (K^ij− γ^ijK).

Where K = γijK^ij is the trace of the extrinsic curvature. The Einstein-Hilbert action can now be brought to the form

SEH= Z

d⁴x π^ij˙γij+ N C + νⁱCi . (2.5) The explicit form⁸ of the Hamiltonian constraint C and the momentum constraint Ci are not of importance for what follows, but only that they are functions of only γ and π. Since N and ν do not have any time derivatives, they are non-dynamical fields. They also appear linear in the action, meaning that they are Lagrange multipliers and their EoM are constraints on the dynamic variables, C(γ, π) = 0 and Ci(γ, π) = 0. The decomposition allows the identification of the dynamical variables and the constraints of the theory and provides a means of calculating the number of free phase space variables. Since γ and π are symmetric, they have six DoF each, N has one and ν three, giving a total of sixteen DoF. Four DoF can be removed by imposing the constraints C = 0, Ci = 0, leaving twelve DoF. Since the action is invariant under general coordinate transformations (GCT), a transformation can be used to fix some of the dynamical variables γij

and π^ij, leaving 8 DoF. The EoM of the GCT fixed variables are now instead constraints, fixing N and ν in terms of dynamical variables, leaving four DoF. In principle, the time derivative of the constraints could give rise to secondary constraints [20], but these are all trivially zero for the Einstein-Hilbert action. Hence there are four DoF in phase space corresponding to two propagating modes, as should be the case for a massless spin-2 field and the theory propagates the correct number of modes for a massless spin-2 field.

2.1.3 Vielbein Formalism

As seen in the previous sections, the natural mathematical object to covariantly describe a spin-2 field, is a metric. There are, however, different ways to represent the DoF of a spin-2 field; one way is using a vielbein.

8C = M_pl²√

γ R(γ) + (M_Pl²√ γ )⁻¹

π²

2 − π^ijπij

, Ci= 2√

γ ∂^j(πij).

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Vielbeins are also referred to as orthonormal basis, tetrads or vierbeins in 4 dimensions. One of the reasons for the many names is that they can be interpreted in multiple equivalent ways. Vielbeins play an essential role in the generalisation to theories of multiple spin-2 fields and will here be introduced in the familiar framework of GR.

So far, all equations have been formulated in the coordinate induced bases, {∂µ} / {dx^µ}. Even though these basis are dual to each other, i.e. dx^α(∂_β) = δ^α_β the basis vectors are not in general orthonormal. This is evident from the fact that

g(∂_α, ∂_β) = g_αβ6= η_αβ g⁻¹(dx^α,dx^β) = g^αβ6= η^αβ.

One could instead consider a set of vectors {eA}and one-forms eÂ , A = 1, 2, 3, 4 which are dual eÂ(eB) = δÂ_B and where the inner product fulfil

g(eA, eB) = ηAB g⁻¹(e^A, e^B) = η^AB.

These sets of vectors and one-forms form an orthonormal basis of the tangent and cotangent space in a similar way as the coordinate basis, and they can be expressed in terms of each other

g(eA, eB) = g(e^α_A∂α, e^β_B∂β) = e^α_Agαβe^β_B= ηAB

g⁻¹(eÂ, e^B) = g⁻¹(eÂ_αdx^α, e^B_βdx^β) = eÂ_αg^αβe^B_β = ηÂB.

For eA = e^α_A∂α and eÂ = eÂ_αdx^α to be a compatible basis, e^αA and eÂα must be invertible change of basis operators. From the duality property of the basis it follows that

eÂ(e_B) = eÂ_αe^β_Bdx^α(∂_β) = eÂ_αe^α_B= δ^! Â_B.

Hence e^αA and eÂα are "inverses" of each other in the sense that eÂα = (e^α_A)⁻¹. So the coordinate basis vectors/one-forms can be expressed as linear combinations of the orthonormal vectors, ∂α = eÂ_αeA and dx^α= e^α_AeÂ. This gives the natural relation between the spacetime and Minkowski metric in both component and matrix form

gαβ= e^A_αηABe^B_β ⇐⇒ g = e^Tηe (2.6)

g^αβ= e^α_Aη^ABe^β_B ⇐⇒ g⁻¹= e⁻¹η⁻¹e⁻^T.

e^A or their components e^Aα are called vielbeins, and when represented as a matrix e = (e⁰, e¹, e², e³)is a general invertible matrix with sixteen components. Technically, the vectors eA define a smooth section of the frame bundle over spacetime, i.e. a coordinate or frame at each spacetime point. Hence these are also referred to as an orthonormal frame. It is worth noting that the vielbeins, just like a coordinate basis, can in general only be chosen locally and not necessarily even in a neighbourhood.

A vielbein can be interpreted as a map between the tangent space of spacetime and Minkowski space in the sense that the pull-back map is such that g = e^∗η ⇐⇒ g(· , ·) = η(e(·), e(·)), which in component or matrix form becomes exactly Eq. 2.6. Since e is invertible, this gives the notion of a local bijective isometry between each tangent space and Minkowski space (TpM, g)

∼e

= (R^1,3, η), and represents the fact that every point in spacetime is locally isomorphic to R^1,3. From special relativity its known that (R^1,3, η)is invariant under the Lorentz group SO(1, 3) and this implies that the map L = e⁻¹◦ Λ ◦ eis a map from T M to itself, see Eq. 2.7. This can be denoted L^TgL = g and L ∈ O(g), i.e. L is an orthogonal transformation of g.

Later, these orthogonal transformations will play an important role in the new metric formulation of the multi-spin-2 theory.

(T M, g) (T M, g)

(R^1,3, η) (R^1,3, η)

e L

e Λ

(2.7)

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Due to this isometry, the local Lorentz invariance of spacetime is apparent in the sense that for any Λ ∈ SO(1, 3)

g = e^Tηe =η = Λ^TηΛ = (Λe)^TηΛe.

Hence the two vielbeins e and Λe represent the same physical spacetime metric. This relates the ten metric components to the sixteen components of the vielbein, by complementing the metric components with the three Lorentz boosts and three spatial rotations of SO(1,3). Since the physical metric is blind to the Lorentz transformation, this results in an extra redundancy of variables when describing spin-2 fields in terms of vielbeins.

It should be mentioned that even though a vielbein has two indices and is C^∞-linear in each slot, it is not a tensor. The two different indices of e^Aα relate to two different vector spaces. This means that vielbein components do not transform like a (1, 1)-tensor; instead, it is a map between T M and R^1,3.

2.1.4 Vielbein Formulation of General Relativity

In this section, the vielbein formulation of GR is introduced using a 3+1 decomposition of spacetime. It should be noted that the vielbein formalism can be used without a 3+1 decomposition.

The boosts of a Lorentz transformation can always be used to transform a vielbein into a lower triangular form, such that

e = Λ N 0 Eν E

, e⁻¹= 1 N

1 0

−ν N E⁻¹

Λ⁻¹. (2.8)

Here E are spatial vielbeins, i.e. the vielbeins of the spatial metric γ = E^TδE ⇐⇒ gˆ ij = γij = E^a_iδabE^b_j⁹. From this; it can quickly be confirmed that e^Tηe gives the metric in the form of Eq. 2.3 and that any Lorentz parameter dependence vanishes. The Einstein-Hilbert action, in terms of the metric, can easily be transformed into the vielbein theory by substituting g with e^Tηe

SEH= M_Pl² Z

d⁴x√

−g R(g) = M_Pl² Z

d⁴x det(e)R(e).

Where Proposition 7 have been used, R(g) and R(e) denote the Ricci scalar in terms of the metric and vielbein, respectively. Using the 3+1 variables, the action can be brought to the form

SEH= M_Pl² Z

dx⁴det(E)NR(E) − K²+ K^ijKij .

R(E)is the Ricci scalar of the spatial vielbein and Kij is the extrinsic curvature in terms of vielbeins given by

Kij= 1 2N

˙E^a_iδabE^b_j+ ˙E^a_jδabE^b_i− ∇iνj− ∇jνi

.

This Lagrangian can be transformed into the Hamiltonian formalism, and the details can be found in [8,21]

but are not of importance here. The final form is, however, interesting

SEH= Z

d⁴x

π(E)ⁱ_aE˙^a_i− N C − νⁱCi−1 2λ^abPab

. (2.9)

The form of C and Ci is again not important, only that they are independent of N and ν, resulting in four constraints on the dynamical fields. There are more constraints than in the metric formulation, specifically Pab = E^c_iδ_c[aπⁱ_b]. These constraints are related to the spatial part of the Lorentz transformations and are

9Note that the a, b indices are here the spatial part of the Lorentz index while i, j are the spatial coordinate indices.

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crucial for the metric and vielbein formulation to have the same number of physical DoF. Using these extra constraints, it is possible to calculate the number of propagating modes.

The spatial vielbein and its canonical momentum have nine DoF each, resulting in eighteen dynamical variables. C = 0 and Ci = 0 together with invariance under GCT fixes eight DoF, leaving ten. Pab = 0 is antisymmetric and are three first-class constraints¹⁰. This removes six DoF, leaving four phase space variables or two propagating modes.

The critical observation here is that there are additional constraints in the vielbein formulation that are not present in the metric action. Constraints like these are important in multivielbein theories and must be dealt with when formulating a vielbein action in terms of metrics.

Note that transitioning from a metric to vielbein formulation is in general only a matter of substituting g with e^Tηe. In the case of GR, the action simplifies, and the dynamical variables E^aiare singled out. However, it is now much harder to transform back to a metric formulation if the process before was unknown.

2.1.5 Spin-2 Fields and Modifications of Gravity

General Relativity is a theory of a massless spin-2 field, and therefore spin-2 theories are inherently linked to the notion of spacetime curvature and gravity. The theories considered in Part II could be interpreted as modifications to GR, and it is worth considering GR and some of the restrictions that exist to its modification.

In this section, the phenomenological success of GR and restrictions on modifications will briefly be reviewed.

For a more comprehensive review, see [22].

GR has been extensively tested since Einstein’s publication in 1916 [23]. It provides a framework for extreme gravitational objects such as neutron stars and black holes and reproduces Newtonian dynamics in the weak gravity limit. GR describes gravity on both solar system and cosmological scales and exten- sive experiments have been conducted verifying its predictions on a vast collection of phenomena such as gravitational redshifts [24], gravitational lensing [25], binary pulsars [26] and gravitational waves [27]. The phenomenological success of GR, therefore, puts strict restrictions on any modification thereof, as any new theory must also be in line with these experiments.

Apart from the observational constraints, Lovelock’s theorem provides theoretical limitations of how a theory of a massless spin-2 field can be modified. Lovelock’s theorem states that the only local, one metric, 2^nd-order action in 4 dimensions is the Einstein-Hilbert action [28, 29, 22]. Hence to alter the theory of gravity, one of these premises must be discarded and while the subject of modified gravity is vast, only additional spin-2 fields will be of interest in this thesis. It turns out that multiple massless spin-2 fields can not interact [30]¹¹; therefore, the theory of a massive spin-2 field has been of interest in the construction of general spin-2 field theories.

Even though GR has been hugely successful, some observations are not satisfactorily explained by GR.

In 1933, Zwicky [31] observed velocity dispersion in galaxy clusters that implied that these are filled with a large amount of invisible matter which only seem to interact gravitationally. Rubin [32] later concluded the same thing from observations of rotation curves of galaxies, and this has later been confirmed by multiple measurements of independent phenomena, e.g. measurements of the cosmic microwave background [33] and observations of the bullet cluster [34]. The cause of these observations is referred to as dark matter, and in the standard model of cosmology, Λ−CDM it is assumed to be in the form of a still unknown massive particle. There is, however, no known counterpart with the correct properties of dark matter in the SM.

Since the only known interaction of dark matter is gravitational, the phenomenon might be explained by a modified theory of gravity.

Another still unexplained phenomenon that justifies inquiries into modified theories of gravity is the cosmological constant problem [35]. From observations, the universe seems to have an accelerating expansion [36,37] on a large scale. One can include a cosmological constant in the Einstein-Hilbert action, which can be chosen, such that spacetime has an accelerated expansion. The constant can be interpreted as an energy density of the vacuum, and if this vacuum energy is assumed to come from the known SM fields, it yields

10In the language of Dirac [20].

11i.e. there is no ghost-free theory with a second-order action that has only interacting massless spin-2 fields.

An Analysis of Theories of Multiple Spin-2 Fields