The Square-Root Isometry of Coupled Quadratic Spaces

Home , Null vector, Square root of a matrix

On the relation between vielbein and metric formulations of spin-2 interactions

Mikica B Kocic

S (or S2)

X X X S S (V, f2) (V, h2) (V, g2)

X X S2 S2 X X S2 S2 X X X X Ξ(µ) S S Ξ(ν) (V, fˆ) (V, f) (V, h) (V, g) (V, gˆ) X Lˆ Eˆ m m L S (or S2) E et es t ⊕ s ⊕

Λ2 Λ Λ1 (Rm,n, η) (Rm,n, η) (Rm,n, η) (Rm,n, η) 2 ] 2 S = g ◦ f[ = S2

Supervisor: S. F. Hassan

Department of Physics Stockholm University Master’s Thesis, 45 HE credits Autumn term 2014

Abstract

Bimetric theory is an extension to general relativity that introduces a secondary symmetric rank-two tensor field. This secondary spin-2 field is also dynamical, and to avoid the Boulware-Deser ghost issue, the interaction between the two fields is obtained through a potential that involes the matrix square-root of the tensors. This square-root “quantity” is a linear transformation, herein referred to as the square-root isometry. In this work we explore the conditions for the existence of the square-root isometry and its group properties. Morever we study the conditions for the simultaneous 3+1 decomposition of two fields, and then, in terms of null-cones, give the (local) causal relations between fields coupled by the square-root isometry. Finally, we show the algebraic equivalency of bimetric theory and its vielbein formulation up to a one-to-one map relating the respective parameter spaces over the real numbers.

Sammanfattning

Den bimetriska teorin är en utökning av den allmänna relativitetsteorin som introduc- erar ett sekundärt symmetriskt tensorfält av rang-två. Det här sekundära spin-2 fältet är också dynamiskt, och för att undvika Boulware-Deser spöke, erhålls vaxelverkan mellan de två fältena genom en potential som er baserad på kvadratrotsmatris av två tensorfält. Den “kvadratroten” är en linjär avbildning som kallas kvadratrotsisometri. I detta arbete utforskas förutsättningar för existensen av kvadratrotsisometrin och ges dess egenskaper i termer av gruppteori. Därutöver utforskas förutsättningarna för den samtidiga 3+1 dekom- positionen av två tensorfält och sedan, i termer av ljuskoner, ges de (lokala) kausala relation- erna för tensorfält kopplade genom kvadratrotsisometrin. Slutligen bevisas den algebraiska ekvivalensen mellan den bimetriska teorin och dess vielbein formulering upp till en bijektiv relation mellan respektive parameterutrymmen över de reella talen.

Contents

Introduction iii Notation...... vii

1 Quadratic Spaces, Fundamentals1 1.1 Deﬁnitions...... 1 1.1.1 Bilinear Forms...... 1 1.1.2 Quadratic Spaces...... 3 1.2 Matrix and Tensor Notation...... 4 1.3 Regular Spaces...... 6 1.3.1 Canonical Dualities...... 6 1.3.2 Inner Product and Norm...... 7 1.4 Transpose and Adjoint Transformations...... 8 1.4.1 Transpose...... 8 1.4.2 Adjoint...... 8 1.5 Isometries and Conformal Maps...... 10 1.5.1 Isometries and Pullbacks...... 10 1.5.2 Conformal Maps...... 11 1.6 Orthogonality and Orthogonal Splitting...... 12 1.6.1 Orthogonal Group...... 13 1.7 Isotropy and Hyperbolic Splitting...... 14 1.8 Orthogonal Decomposition...... 15 1.9 Classiﬁcation of Real Quadratic Spaces...... 17

2 The Square-Root Isometry 19 2.1 Coupled Quadratic Spaces...... 19 2.2 The Square-Root Isometry...... 20 2.3 The Square-Root Group Actions and Orbits...... 23 2.4 Coupled Symmetrization...... 27

3 Geometry of Decompositions 29 3.1 Structure of O(m, n) ...... 29 3.1.1 Polar Decomposition of O(n) ...... 30 3.1.2 Polar Decomposition of O(m, n) ...... 31 3.2 Partitioned Matrices...... 34 3.3 m+n Decomposition of Quadratic Forms...... 36 3.4 Vielbeins in Diﬀerential Geometry...... 39

Mikica Kocic, The Square-Root Isometry, MSc thesis, SU 2014 Printed: 2015-03-11 ii CONTENTS

3.5 Decomposition of Vielbeins...... 42 3.6 Quadrics...... 44

4 Symmetrization Condition 47 4.1 Decomposition of the Symmetrization Condition...... 47 4.2 The Symmetrizing Quadratic Space...... 50 4.3 Topology of the Symmetrization Condition...... 50

5 Simultaneous Proper Decomposition 53 5.1 Alternative Square-Roots...... 54 5.2 Reﬂection Group of the Square-Root Isometry...... 56 5.3 Symmetrizing Space Covered by Null-Cones...... 59 5.4 Intersecting Null-Cones...... 60 5.5 Simultaneous Diagonalization by Congruence...... 61

6 Application 65 6.1 Review of Bimetric Theory...... 65 6.2 Algebraic Relation to Vielbein Formulation...... 69

7 Summary and Discussion 73

Appendix A Tedious Derivations 75

References 81

Index 87 Introduction

For almost a century it has been known that the theory of general relativity (GR) describes the gravitational force with immaculate agreement with observations. Likewise, as a pillar of the standard model of particle physics (SM), since around the 1950s it has been widely accepted that physical laws are shaped in the form of a quantum theory of ﬁelds (QFT). The signiﬁcance of these theories, upon which all modern physics rests, is well pointed out by Steven Weinberg [Wei95]:

“If it turned out that some physical system could not be described by a quantum ﬁeld theory, it would be a sensation; if it turned out that the system did not obey the rules of quantum mechanics and relativity, it would be a cataclysm.”

Nonetheless, it is evident that we need the theoretical developments which are beyond the standard model and general relativity. The origins of dark energy and dark matter are just to name some of the unresolved topics in present-day physics. One of the approaches towards new physics is to build novel theoretical models from a top-down perspective, like string theory, which might eventually lead us to the ultimate theory of nature. From this viewpoint, GR and the contemporary quantum field theories are mere low-energy approximations and ‘effective field theory’ limits of a more fundamental theory. However, although our current field theories are certainly not ‘fundamental,’ we know that any relativistic quantum theory will take the form of a field theory when describing particles at sufficiently low energies [Wei95]. An equally viable approach is therefore to reuse the existing tools of field theories, and then from a bottom-up perspective search for a consistent theory of interacting spin-2 fields which will extend general relativity. One such classical field theory is a ghost-free bimetric theory, which is an extension to GR that introduces a dynamical secondary symmetric rank-two tensor field. The meaning of ‘ghost-free’ and ‘spin-2 field’ will be highlighted by reviewing the methodology used for constructing a quantum field theory. In general, fields are physical quantities that have a value at each point in spacetime. Subject to whether these values are characterized by numbers or quantum operators, we can distinguish between classical and quantum fields. The energy of the system, which is represented by the Hamiltonian, has a crucial role for the consistency of a field theory. Provided that we have a consistent classical field for which the Hamiltonian has a lower bound (that is, for which the energy of the system is bounded from below), the process of quantization is ‘fairly’ simple either using the canonical quantization or the path integrals formalism; this process will consequently convert classical fields into operators acting on quantum states of QFT.1 However, if the Hamiltonian is not bounded from below, a classical

1Of course, one must be careful not to introduce inconsistencies and violate the unitarity during quantization (which is not always possible).

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Mikica Kocic, The Square-Root Isometry, MSc thesis, SU 2014 Printed: 2015-03-11 iv INTRODUCTION

field will be inherently unstable allowing the transfer of an infinite amount of energy to coupled fields, which is considered unphysical. In such case, the negative kinetic term will give rise to the negative probability states, i.e., to the loss of unitarity in the resulting quantum theory. Therefore, a consistent classical field with a Hamiltonian having the ‘healthy’ kinetic term is a necessary condition for a consistent QFT. A field with the inconsistent kinetic term bearing the wrong sign is called a ghost field.2 After constructing a QFT, we can formally define particles as “representations of the inhomogeneous Lorentz group” [Wig39, Wei95]. This rather cumbersome statement is equivalent to saying that particles are represented by the degrees of freedom in the flat spacetime characterized by mass and spin (or helicity), and carried by fields as quanta. What is important is that the origin of the classification by mass and spin are the spacetime symmetries of a field relative to inhomogeneous Lorentz transformations. In particular, depending on whether the value of a specific field at each point of spacetime is given by (or, transforms as) a scalar, a vector, a spinor or an n-rank tensor, the spin of the field is given by 0, 1, 1/2 or n, respectively.3 A typical example is the electromagnetic (EM) field, which is a massless spin-1 vector field that, upon quantization, yields photons with two possible circular polarization states corresponding to the two helicity states. Additionally, the standard model of particle physics comprises quantum fields with spin-0 (Higgs boson), spin-1 (gauge bosons) and spin-1/2 (fermions). From a field theory perspective, GR can be interpreted as the classical field theory of a massless spin-2 field with nonlinear self-interactions. In GR, the field is given by a symmetric nondegenerate rank-two tensor gµν(x) which functions as a metric. Although GR is consistent and ghost-free, it cannot be quantized because its Lagrangian is nonrenormalizable [Ham08]. General relativity is the unique theory of a nonlinear massless spin-2 interactions and the search for extensions and alternatives has been a challenging endeavor since its formulation. A theory of a massive spin-2 field is not simple to derive, and a number of problems arise if we want this field to describe the ‘graviton’ interacting with other particles. The first attempt to give a mass to the spin-2 field dates back to 1939 when Fierz and Pauli added a mass term to the free spin-2 action [FP39]. The theory was not interacting and linear, and the construction of a model to describe nonlinear interactions between massive spin-2 fields remained without apparent success for decades because of the infamous Boulware-Deser (BD) ghost instability [BD72a, BD72b]. Later developments of approx- imation methods in [AHGS03, CNPT05] enabled the proposal of an admissible action for nonlinear massive gravity that is free of the BD ghost [dRG10, dRGT11]; this is the so-called dRGT model, which is a theory of a single massive spin-2 field. The proof that the dRGT model is indeed ghost-free was given in [HR12a] completing this model as the first consistent description of nonlinear spin-2 interactions.

2A field with the wrong sign in front of its mass term, i.e., with the source of instability in potential energy, is called a tachyonic field. Tachyonic fields are not so bad; in fact, they play a significant role in physics—like in spontaneous symmetry breaking, wherein the Higgs field of the standard model is a tachyonic field in its uncondensed phase that is associated with the field vacuum residing at a local maximum of the potential. After reaching the local minimum (by ‘tachyon condensation’), the tachyonic field becomes benign. 3Note that, in addition to spacetime symmetries, fields may also have other types of symmetries, such as electric charge U(1) in quantum electrodynamics, or color charge SU(3) in quantum chromodynamics; being independent on space or time, these symmetries are called internal symmetries. Moreover, some symmetries can be purely accidental, as, for instance, the global lepton number U(1) in the standard model of particle physics. INTRODUCTION v

To formulate a nonlinear massive gravity for a spin-2 field gµν, one also needs a second symmetric rank-two tensor fµν (for the reasons explained later) that is usually denoted as the “background metric,” which was taken to be the Minkowski metric in the dRGT model (fµν ≡ ηµν). A generalization to any background metric was presented in [HR11], for which the absence of the BD ghost was proven in [HRSM12, HR12c, HSMvS12a]. A further generalization was in providing dynamics to this secondary symmetric rank-two tensor fµν without destroying the consistency of the theory, which was done in [HR12b, HR12c]. This model, with the dynamical secondary metric fµν, is called the ghost-free bimetric theory, 4 which is a theory of two interacting spin-2 fields gµν and fµν. The main feature of this theory is that, to avoid the BD ghost, the interactions are necessarily written in terms of the p µσ µ square-root of the two fields g fσν ν. The mass spectrum around the Einstein solutions of bimetric theory was furtherJ analyzedK in [HSMvS13] with the conclusion that the mass spectrum describes nonlinear interactions between the massive and massless spin-2 fields where both the fields bear a superposition of the massless and the massive modes. Finally, a reformulation of bimetric theory in terms of vielbeins has been proposed in [HR12d].5 However, there are still some open problems which are in the scope of this work. Firstly, the vielbein formulation [HR12d] relied on the ghost-proof from bimetric theory and also asked for a special “symmetry” condition on the vielbeins to ensure the existence of the square-root isometry. This raised the question about the equivalency of these two formulations [HSMvS12b, DMZ12, BDP13] (the vielbein formulation avoids the usage of this square-root isometry, otherwise). As mentioned before, to avoid the BD ghost issue in bimetric theory, the interaction between two spin-2 fields, given by tensors g and f, is obtained as a function of the matrix square-root S defined through S2 = g−1f. This square- root “quantity” is a linear transformation, referred to as the square-root isometry in this work (the formal definition is given in Chapter2). Given two arbitrary tensors, this square-root may not exist in the domain of real numbers. Secondly, the ghost analysis of bimetric theory is done in the Hamiltonian framework using ADM formalism [ADM08] which further relies on the 3+1 decomposition of spacetime. The 3+1 formalism is based on the slicing of the four-dimensional spacetime by three- dimensional spacelike hypersurfaces, effectively amounting to a decomposition of spacetime into “space”+“time” [Gou12]. In bimetric theory when using ADM formalism, it was/is silently assumed that the 3+1 decomposition can be simultaneously employed for both tensor fields. In GR, with a single metric tensor, this slicing is always locally possible, and even globally under some reasonable conditions on the causal structure of a spacetime (the so-called global hyperbolicity) [Ler55, Ger67, Ger70] (for the summary see [Wal84]). In the case of several tensor fields, however, the simultaneous 3+1 decomposition is quite limiting presupposition. It is rather a very special condition put by hand and not coming out from the theory. Finally, in view of the initial value formulation of bimetric theory, inability to ensure a common spacelike hypersurface for both metrics becomes an obstacle for well-posedness of the Cauchy problem and a development of the initial data, and moreover, the missing common notion of sustained timelikeness raises the question about causality violations.

4A detailed review of bimetric theory is given in Section 6.1 on page 65. 5Shortly, vielbeins are a way to establish local orthonormal frames on a manifold equipped with an arbitrary metric. The formal deﬁnition is, in the context of quadratic spaces, given in Section 1.9 on page 18 (in the context of diﬀerential manifolds, vielbeins are discussed in Section 3.4). vi INTRODUCTION

Therefore, the aim of the work is to explore the following questions: • What are conditions for the existence of the square-root isometry? • In terms of groups and group actions, what are properties of the square-root isometry? • Given two arbitrary metric tensors coupled by the square-root isometry, what are conditions for the simultaneous 3+1 decomposition? • In terms of null-cones, what are (local) causal relations of metric tensors coupled by the square-root isometry? • What are constraints on the parameter space for the equivalency of bimetric theory and its vielbein formulation? It is noteworthy that all of the herein presented results are the outcome of explorations that originally started with the last question: on the connection between bimetric theory and its reformulation in terms of vielbeins. Nonetheless, the obtained results are more general and applicable to quadratic forms of any signature, so the scope of this work has shifted to a study of the geometric and algebraic properties of the square-root isometry in the language of coupled quadratic spaces. With this reason, the acquired assertions are stated in deductive fashion ascending from the most general properties of the symmetric bilinear forms. This work is organized as follows. In Chapter1, we set up the notation and highlight the basic concepts of symmetric bilinear and quadratic spaces. This chapter brings nothing new, it just exposes the common algebra needed for the rest of the work. Readers familiar with the notions of bilinear and quadratic forms may prefer to proceed directly with Section 1.4 Transpose and Adjoint Transformations, and then with Chapter2 The Square-Root Isometry. The key propositions of Chapter1 are the orthogonal decomposition of quadratic spaces, Sylvester’s law of inertia and the invariance of the classification of quadratic forms. In Chapter2, we introduce the square-root isometry as an isometry which is also a self-adjoint transformation. We then briefly investigate the square-root isometry group actions on quadratic spaces and the resulting orbits. The key proposition of this chapter is the so-called symmetrization condition which asserts the correspondence between the self-adjoint isometries and the underlying orthogonal transformations on coupled quadratic spaces. In Chapter3, we develop the orthogonal decompositions of quadratic spaces in addition to the decompositions of their isometries. This is done both in matrix notation as well as in the component form. At the end, this will give rise to the so-called m+n decomposition, which will later be used to study the square-root symmetrization condition in more detail. In Chapter4, we derive explicit expression for the symmetrization condition of the square-root isometry in terms of vielbeins. We also give the geometrical and topological interpretation of coupled quadratic spaces in terms of their null-cones. Finally, in Chapter5, we show that, if the square-root isometry exists between two quadratic spaces, then we may always find the basis where these coupled quadratic spaces can be simultaneously decomposed. In Chapter6, we narrow our considerations to 3+1 decompositions and apply the earlier obtained propositions to the relation between vielbein and metric formulations of the interacting classical spin-2 fields. This chapter begins with review of bimetric theory and ends with the redefinition of shift variables showing the algebraic relation between bimetric theory and its vielbein formulation. In Chapter7, we summarize the key results and give directions for further research, in particular a connection to the (global) causality issues. Notation

Bil(V,W ; K) a vector space of bilinear forms on V × W over K, p.1 Sym(V,K) a vector space of symmetric bilinear forms on V × V over K Quad(V,K) a vector space of quadratic forms on V over K Mat(m × n, K) m×n matrices with elements in K Mat(n, K) square n×n matrices with elements in K GL(n, K) the general linear group, { S ∈ Mat(n, K) | det S 6= 0 }

T O(m, n) the orthogonal group { S ∈ GL(m + n, R) | S ηm,nS = ηm,n } O(V, b) the orthogonal group of the quadratic space (V, b), p. 13

Aij a matrix with elements Aij J K ∗ b[ a map W → V deﬁned by a bilinear form b : V × W → K, p.6 ] b an inverse to b[, for a non-degenerate bilinear form b f = S∗b a pullback f(·, ·) = b(S(·),S(·)), p. 11 B a bilinear space associated with a matrix B L M λ1, ··· , λn a bilinear space associated with a diagonal matrix δijλj L m,n m,nM m,n m J n K R , (R , η) a regular real quadratic space R ≡ −1 ⊥ +1 , p. 18 m,n L M L M ηm,n, η the bilinear form of R with the associated matrix η = (−δm) ⊕ δn ˆ ˆ n δn, δ the standard bilinear form of R with the associated matrix δij × × J K K K ≡ K \{0K }, a multiplicative group of K K×2 K×2 ≡ { α2 | α ∈ K× }, a group of squares d(V, b) the discriminant, d(V, b) ≡ (−1)n(n−1)/2 det(V, b), n = dim(V ) √ 2 S :(V, f) ,→(V, g) a square-root isometry from f to g, where f[ = g[ ◦ S , p. 22 X1/2 the principal square-root of a matrix X (see Remark I.1) √ X an arbitrary matrix square-root sign(X) a matrix sign function, sign(X) = X(X2)−1/2 = X−1(X2)1/2 ] ∂µ|P a coordinate basis that spans a tangent space TP (M), p. 40 J µ K ∗ dx[ |P a coordinate basis that spans a cotangent space TP (M) J ] K v ∈ TP (M) a vector, an element of a tangent space TP (M) ∗ ∗ ω[ ∈ TP (M) a covector, an element of a cotangent space TP (M)

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Mikica Kocic, The Square-Root Isometry, MSc thesis, SU 2014 Printed: 2015-03-11 viii NOTATION

Remark I.1. As in [Hig08, HJ94], we make distinction between several classes of matrix square-roots. The square-roots of a nonsingular complex matrix X can be classified relative to the Jordan canonical form of X (see Theorem 1.26 in [Hig08]). The first group contains a finite number of primary square-roots each having a nonzero sum of any two of their eigenvalues. The second group of nonprimary square-roots (this class may be empty) comprises finitely many parametrized families of matrices where each family contains infinitely many square-roots having the same spectrum. Furthermore, for a nonsingular complex matrix X − 1/2 without eigenvalues on R , we have the principal square-root, denoted by X , which is unique (Theorem 1.29 ibid.) and whose all of the eigenvalues are in the open right half-plane so that X1/2 is a primary matrix function of X. In particular, if X is real then its principal √ square-root is also real. We shall use X to denote an arbitrary square-root.

Remark I.2. The matrix sign function has a fundamental connnection with the matrix square- root and the polar decomposition. The properties of the sign function are given by Theorem 5.1 in [Hig08].

Remark I.3. In the ﬁrst two chapters, we do not use any special notation to discriminate vectors and covectors. Later on, ]/[ will be used as super/subscripts to label vectors/covectors, ] µ ] µ e.g., as in v = v ∂µ and ω[ = ωµdx[ . ˆ a i ˆ Remark I.4. δn = δab , In = δ b and Is = δ j are diﬀerent objects: δn is a bilinear form n ˆ J K J K n J n K of (R , δn), In is the identity map id : R → R and Is is the identity map id : Vs → Vs.

Remark I.5. In the case of quadratic spaces with the signature (1, n), the Greek indices µ, ν, . . . are used to denote the components relative to the coordinate bases. These indices are often called the coordinate (or world or spacetime) indices. We use index 0 for a temporal part and i, j, . . . to denote the indices reduced to a spatial part. On the other hand, the roman indices A, B, . . . are used to denote the components relative to the orthonormal bases. These indices are often called the Lorentz indices. In this case, we use a, b, . . . to denote the spatial part indices. For more details, see Section 3.4 on page 39.

Terminology

The terminology is standard and further explanations appear in [Lan02, Bou98, CBDMDB77, Nak03, Sch85, O’M73]. As usual, we limit the use of the term operator for those linear transformations whose domain coincides with codomain. More speciﬁcally, the following terms are employed:

homomorphism a linear transformation Hom(V,W ) or L(V,W ) endomorphism a linear operator End(V ) ≡ Hom(V,V ) monomorphism an injective linear transformation V,→ W epimorphism a surjective linear transformation isomorphism a bijective linear transformation automorphism a bijective linear operator Aut(V )

Monomorphisms are sometimes called the embeddings. Notice also that the set of all linear transformations from V to W , i.e., L(V,W ), forms a vectors space. Chapter 1

Quadratic Spaces, Fundamentals

This chapter highlights the important parts of algebra needed for the rest of the work. The subject is purely mathematical and yields nothing new. Here we set up the notation and expose the basic concepts of symmetric bilinear and quadratic spaces where most of the propositions are adopted from [Lan02, Bou98, Sch85, O’M73, HK71, EKM08, Art88]. We begin with the deﬁnitions of the bilinear and quadratic forms, and then introduce the matrix of a bilinear form needed to establish the isomorphism between the space of forms and the space of square matrices. Section 1.4 emphasizes the transpose and adjoint transformations. Finally, we study the concept of an isometry and the group preserving quadratic forms. The key statements of this chapter are the orthogonal decomposition of quadratic spaces, Sylvester’s law of inertia and the invariance of the classiﬁcation of real quadratic forms.

1.1 Deﬁnitions

We assume that the terms vector space, dual vector space and field are familiar. At first, K will denote an arbitrary field of characteristic not two,1 but towards the end we will take K to be specialized either to the real R or the complex numbers C. We consider only finite-dimensional vectors spaces over K.

1.1.1 Bilinear Forms

Deﬁnition 1.1. Let V and W be vector spaces over the same ﬁeld K.A bilinear form on the space V × W is a map

b : V × W → K, b :(v, w) 7→ b(v, w), (1.1) which is linear in each slot (argument), that is

1◦ b(v + v0, w) = b(v, w) + b(v0, w), b(αv, w) = αb(v, w), (1.2) 2◦ b(v, w + w0) = b(v, w) + b(v, w0), b(v, αw) = αb(v, w), (1.3)

1 The characteristic of a field K with the additive identity 0K ∈ K is the smallest positive number n of times a so that a + a + .... + a = 0K , if such a number n exists for all a ∈ K, and zero otherwise (if this sum never reaches 0K , the field has the characteristic zero). Over the fields of characteristic two, symmetric bilinear forms are essentially different and require a separate treatment [Sch85].

Mikica Kocic, The Square-Root Isometry, MSc thesis, SU 2014 Printed: 2015-03-11 2 Chapter 1. Quadratic Spaces, Fundamentals for all v, v0 ∈ V , w, w0 ∈ W and α ∈ K. The ordered triple (V,W ; b) is called a bilinear space over K. When W = V , we simply say that b is a bilinear form on V . 2 Let further X be a subspace of V . Then the restriction of b to X, b|X×X , is a bilinear form on X and is called a subform of b, which we shortly denote by bX := b|X×X . Consequently, (X, bX ) is a bilinear space, a subspace of (V, b).

The set of all such bilinear forms constitutes a subspace of KV ×W [Kal07], that is, the set of bilinear forms on V is a vector space over K, denoted by Bil(V,K) (or L(V,W ; K)).

Remark 1.1. A bilinear form is a special case of a more general multilinear map. Let

V1,...,Vn and U be vector spaces over a common ﬁeld K; then a multilinear map is a function of several variables

f : V1 × · · · × Vn → U (1.4) that is linear separately in each variable, i.e., such that for each slot i = 1, ..., n, if all of the variables but vi are held constant, then f(v1, . . . , vn) is a linear function of vi. A multilinear map of two variables is called the bilinear map. If the codomain U of a multilinear map is the ﬁeld itself, U = K1, the map is called the multilinear form [Sze04].

Deﬁnition 1.2. If b is a bilinear form over V × W , then bT denotes the transposed bilinear form bT : W × V → K deﬁned by

(v, w) 7→ bT(v, w) := b(w, v), ∀v ∈ V, ∀w ∈ W. (1.5)

When W = V , the bilinear form b on V × V is called symmetric, iﬀ it satisﬁes

b(v, w) = b(w, v), ∀v, w ∈ V. (1.6)

Hence, b is symmetric iﬀ b = bT. In such case, the ordered pair (V, b) is called a symmetric bilinear space over K.

As we earlier noted, the set Bil(V,W ; K) of bilinear forms on V × W is a vector space over K. The space Bil(V ; K) contains the subspace Sym(V ; K) of symmetric bilinear forms on V .

Remark 1.2. The bilinear form b is called skew-symmetric, iﬀ it satisﬁes

b(v, w) = −b(w, v), ∀v, w ∈ V. (1.7)

Our attention will be turned to symmetric bilinear forms and to the orthogonal geometry that is shaped by such forms. On the other hand, if the bilinear form is skew-symmetric, the geometry is called symplectic. If the form is rather sesquilinear and also hermitian (or symmetric sesquilinear), the geometry is called unitary.

2 If f : A → B is a function from A to B, then the restriction of f to X ⊆ A is the function f|X : X → B that is only deﬁned on X ∩ dom f. 1.1 Deﬁnitions 3

1.1.2 Quadratic Spaces

Deﬁnition 1.3. Let V be a vector space over K. A map q : V → K is called a quadratic form on V , iﬀ

1◦ q(αv) = α2q(v), ∀α ∈ K, ∀v ∈ V ; (1.8) ◦ 1 2 bq(v, w) := 2 [q(v + w) − q(v) − q(w)] , ∀v, w ∈ V, such that bq is a bilinear form on V. (1.9)

◦ (Note that bq is necessarily symmetric.) The axiom 2 which introduces a map Pol : q 7→ bq is often denoted as the polarization identity; consequently, bq is called the bilinear form associated with q (or the polar form of q). The ordered pair (V, q) is called a quadratic space

(over K). Conversely, if b is a bilinear on V , then the map qb : V → K deﬁned by

qb : v 7→ qb(v) := b(v, v) (1.10) is a quadratic form, and qb is called the quadratic form associated with b. Clearly, the set Quad(V,K) of quadratic forms on V is a vector space over K.

The quadratic space (V, b) is said to represent a scalar α ∈ K× iff there exists a vector × v ∈ V such that qb(v) = α, i.e., iff α ∈ qb(V ). We say that V is universal iff qb(V ) = K . Remark 1.3. The polarization identity can take many forms

1 1 bq(v, w) = 2 [q(v) + q(w) − q(v − w)] = 4 [q(v + w) − q(v − w)] . (1.11)

Lemma 1.1. Let b be a bilinear form and q a quadratic form on V ; then,

1 T bqb = 2 (b + b ) and qbq = q. (1.12)

Moreover, bqb = b iﬀ b is symmetric.

Proof. Assuming the ﬁeld with characteristic not two, the proof directly follows from the polarization identity, therefore the polarization is bijective [Sch85].

The previous lemma enables us to identify quadratic spaces Quad(V,K) with symmetric bilinear spaces Sym(V,K) by means of the inverse correspondences

−1 Pol : q 7→ bq, Pol : b 7→ qb, (1.13) which implies that all concepts for symmetric bilinear spaces (like non-degeneracy or the notion of isometries and orthogonality) can be directly transferred to quadratic spaces and vice versa.

Remark 1.4. In the case of a ﬁeld K with char = 2, it is possible that the quadratic form q : V → K may not have the polarization, that is, it is possible that there is no symmetric bilinear form b such that q(v) = b(v, v). However, from now on, of our primary concern are the ﬁelds K = R or C both having the characteristic zero, so we shall not dwell on this issue anymore. 4 Chapter 1. Quadratic Spaces, Fundamentals

1.2 Matrix and Tensor Notation

To perform actual computations we shall employ the components of vectors and bilinear forms in a specific ordered basis; moreover, these components will be organized as indexed one- and two-dimensional arrays which will lead us to matrix notation. Most of the pure treatments of bilinear forms in mathematics and abstract algebra do not impose a notational difference between the components of vectors and covectors to be based on the placement of indices. It will be seen, however, that such distinction is beneficial. Therefore, as it is done in tensor notation, we shall use the upper indices to denote the components of vectors belonging to an ordinary vector space V , and the lower indices to denote the components of covectors (i.e., of linear forms) belonging to a dual vector space V ∗. To be more specific, ∗ i let E = ei i∈I be an ordered basis of the vector space V and E = e i∈I be an ordered basis of VJ ∗ Kwhere I is a finite index set, and let further v and ω be someJ vectorK and covector ∗ i i in V and V , respectively; then we write v = eiv and ω = ωie implying the Einstein summation convention with the dummy index i ranging over I. Furthermore, for finite dimensional vector spaces V and W , it can be shown that the bilinear forms on V × W are in one-to-one correspondence with linear maps V ⊗ W → K [Sze04, Lee03]; since these linear maps constitute the dual space of V ⊗ W , we can treat bilinear forms as the elements of (V ⊗ W )∗ =∼ V ∗ ⊗ W ∗ consistently imposing the usage of lower indices for denoting their components.

Deﬁnition 1.4. Let E = ei i∈I and F = fj j∈J be ordered bases of the vector spaces V and W , ranging over ﬁniteJ indexK sets I andJJ ,K respectively. Let further b a bilinear form on V × W ; then

[b]EF = bij , bij = b(ei, fj), i ∈ I, j ∈ J , (1.14) J K is called the matrix of b with respect to the bases E, F. In the special case W = V , we write,

[b]E = bij , bij = b(ei, ej), i, j ∈ I. (1.15) J K T T T It is clear that the matrix [b ]E of a bilinear form b is the transposed matrix [b]E and that b is symmetric iﬀ [b]E is symmetric. Remark 1.5. We will often utilize the same symbol for a bilinear form b and its associated matrix [b]E , unless confusion might arise. i i Example 1.1. Let v = eiv and w = eiw be two vectors in V . Then we can write

i j i j i j T b(v, w) = b(eiv , ejw ) = v b(ei, ej)w = v bijw = [v]E [b]E [w]E , (1.16) or for short b(v, w) = “ vTbw ”. (1.17)

Hence, in matrix notation, we identify a vector v from V with the matrix column vector vi . This is consistent with the abuse of notation that, if we write the ordered basis E as J K the matrix row vector E = ei ∈ Mat(1 × n, V ), then the vector v ∈ V can be written as i J K T v = E[v]E = eiv where [v]E ∈ Mat(n × 1,K). Note that v is a row vector, but with the indices still up. Example 1.2. In matrix notation, a pullback S∗b is given by the matrix STbS (see Deﬁni- tion 1.13). 1.2 Matrix and Tensor Notation 5

Remark 1.6. For linear transformations, the inner matrix product (the ordinary matrix multiplication) correspond to the composition of maps. Consider two linear transformations i A : U → W and B : V → U where ei is a basis of U and e the canonical basis of its dual ∗ j j J K J K U such that ei(e ) = δi . In the component form, the composition A ◦ B(·) = A(B(·)) is i j l i j l i j l done by contracting two consecutive indices as in A je (elB k) = A je (el)B k = A jδl B k = i j A jB k where a down-index on the left is consistently contracted with the oncoming up- index to reﬂect the composition. Introducing the concept of a matrix of a bilinear form requires a change in this index-down/up policy since both indices of a bilinear form are down. The transpose on the left side of a bilinear form restores the consistency as it changes the character of U to U ∗ and also the order of the composition (see Deﬁnition 1.10). Of course, the components are just numbers so they can freely be placed in any order when using i k l j l k i j component notation; for example, v Bi Ak bljw is the same as bljA kB iv w . However, the former expression is preferable as we can directly read of vTBTATbw = (ABv)Tbw.3

Definition 1.5. Two square matrices A, B are called congruent iff there exists an invertible matrix T such that B = T TAT . Two square matrices À, B are called similar iff there exists an invertible matrix T such that B = T −1AT .

0 0 Lemma 1.2. Consider a bilinear space (V,W ; b). Let E = ei i∈I and E = ei i∈I be J0 K 0 i J K two bases of V , and T a change of basis matrix from E to E , ei0 = eiT i0 . Let further 0 0 F = fj j∈J and F = fj j∈J be two bases of W , and P a change of basis matrix from F 0J K0 j J K to F , fj0 = ejP j0 . Then T [b]E0F 0 = T [b]EF P. (1.18) For W = V , the matrix of a bilinear space (V, b) is well-deﬁned up to a congruence.

i Remark 1.7. For a vector v = eiv ∈ V , the ‘covariant’ (change of basis) transformation 0 0 i 0i0 −1 i i ei0 = eiT i0 corresponds to the ‘contravariant’ coordinate transformation v = (T ) iv , 0 0 0i0 i −1 k i −1 v = ei0 v such that T k0 (T ) j = δ j, that is, TT = I where we assume that T is j 0 j non-singular. Similarly for w = fjw ∈ W , we will have fj0 = fjP j0 , where in general dim V 6= dim W .

0 0 Proof. We evaluate the (i , j )-component of [b]E0F 0

0 0 k l k l T k l b(ei0 , fj0 ) = b ekT i0 , flP j0 = T i0 bklP j0 = (T )i0 bklP j0 (1.19)

0 0 T which is exactly the (i , j )-component of T [b]EF P . Finally, when W = V , it is obvious that T the matrices [b]E0 and [b]E are congruent as it holds [b]E0 = T [b]E T for an invertible T .

Remark 1.8. Compare congruence with similarity. Similar matrices A and B = T −1AT correspond to the same linear transformation under two diﬀerent bases, with T being the change of basis matrix.

3By this we retain the notion of the composition of maps, and also the order of application of the corresponding non-commutative inner matrix product. 6 Chapter 1. Quadratic Spaces, Fundamentals

1.3 Regular Spaces

Deﬁnition 1.6. Let (V,W ; b) a bilinear space. The bilinear form b is called non-degenerate with respect to W , iﬀ

(∀v ∈ V ) b(v, w) = 0 ⇒ w = 0W , (1.20) and similarly, b is called non-degenerate with respect to V , iﬀ

(∀w ∈ W ) b(v, w) = 0 ⇒ v = 0V . (1.21)

Iff both conditions are satisfied, we say that b is non-degenerate (or non-singular or regular). A bilinear space (V,W ; b) is called regular iff b is non-degenerate; otherwise it is called singular.

1.3.1 Canonical Dualities Theorem 1.1. Let b be a bilinear form on V × W . Let further V ∗ and W ∗ be duals of V ∗ ∗ and W , respectively. (i) Then the maps b[· : V → W and b·[ : W → V deﬁned by

◦ 1 b[· : v 7→ b[·(v) := b(v, ·), b(v, ·)(w) = b(v, w), ∀w ∈ W, (1.22) ◦ 2 b·[ : w 7→ b·[(w) := b(·, w), b(·, w)(v) = b(v, w), ∀v ∈ V, (1.23) are the homomorphisms. (ii) If b is non-degenerate with respect to V , then b[· is injective, ∗ i.e., b[· : V,→ W is the monomorphism. If b is non-degenerate with respect to W , then b·[ ∗ is injective, i.e., b·[ : W,→ V is the monomorphism. Theorem 1.2. Let (V,W ; b) be a regular bilinear space. If one of the vector spaces V or W is ﬁnite-dimensional, then it is also the other. Moreover, it also holds dim V = dim W ∗ ∗ as well as that the maps b[· and b·[ are isomorphisms from V to W and from W to V , respectively.

A detailed proof of the last two theorems can be found in [Per91]. Now, let V be a finite-dimensional vector space over K and V ∗ = L(V,K) its dual. Then, as a corollary of the last theorem, the map V × V ∗ → K defined by (v, ω) 7→ ω(v) ∗∗ ∗∗ ∗∗ establishes the natural isomorphism from V to V , written πV : V → V (V is often called the bi-dual of V ). It is noteworthy that without assuming V to be finite-dimensional, the surjectivity of the canonical map could not be proven in Theorem 1.2.

Deﬁnition 1.7. Let (V,W ; b) be a regular bilinear space. Since b[· and b·[ are bijective, −1 ∗ −1 ∗ then there exist the inverses (b[·) : W → V and (b·[) : V → W , which we denote ]· −1 ·] −1 as b := (b[·) or b := (b·[) and call the canonical isomorphisms (or dualities). More ]· ·] speciﬁc, we call b[· and b·[ the left and right lowering while b and b the left and right ] ·] raising isomorphisms. We write for short b[ := b·[ and b := b . Remark 1.9. These dualities are often called musical because they are written with symbols like [ : V → V ∗ and ] : V ∗ → V [Ber03, p. 696]. They already appeared in 1971 in [BGM71, p. 21] wherein referred to as “isomorphismes canoniques réciproques (‘isomorphismes musi- ˆ caux’)”. Ibid. uses [ to denote the right lowering b·[. Conversely, [Sch85] uses b to denote the T T left lowering b[·; see also [Lee03]. Observe that b[· = (b )·[ and b·[ = (b )[· where the equality holds up to the natural isomorphism. 1.3 Regular Spaces 7

Lemma 1.3. Let (V, b) be a quadratic space. If E = ei i∈I is an ordered basis of V and ∗ i ∗ J K ∗ E = e i∈I is the dual basis of V , then the matrix [b[]EE∗ of b[ with respect to bases E, E J K is the matrix [b]E with respect to E. The form b is non-degenerate iﬀ det[b]E 6= 0.

The Inverse Bilinear Form

Let (V, b) be a quadratic space. The kernel of b is the subset of V

ker(b) = { v ∈ V | (∀w ∈ V ) b(v, w) = 0 }, (1.24) that is, it is the same as the kernel of the linear map b[ = b·[ = b[·. Thus, the equivalent statement that b is non-degenerate iﬀ ker(b) = {0V }, or alternatively that for each v 6= 0V there exists w with b(v, w) 6= 0. This means that, in a regular space V , only the zero vector 0V is perpendicular to all other vectors (for orthogonality see Deﬁnition 1.15).

Deﬁnition 1.8. Let (V,W ; b) be a regular bilinear space. We can introduce the inverse map −1 ∗ ∗ −1 b : W × V → K that is bilinear in each slot such that b (b[·(v), b·[(w)) = b(v, w) for all v ∈ V , w ∈ W , or equivalently, b−1(ω, υ) = b(b]·(ω), b·](υ)) for all υ ∈ V ∗, ω ∈ W ∗.

∗ ]· Remark 1.10. The consistency of this deﬁnition can be veriﬁed using b[· : V → W , b : ∗ ∗ ·] ∗ W → V , b·[ : W → V and b : V → W by the commutative diagram on Figure 1.1. The order of the slots in b−1 enables one to compose the canonical dualities in the usual sense such that b−1b = I in matrix notation.

V × W b K

b]· b·] b−1 W ∗ × V ∗

Figure 1.1: The inverse of a bilinear form b−1(ω, υ) = b(b]·(ω), b·](υ))

1.3.2 Inner Product and Norm

Deﬁnition 1.9. Let (V, b) be a regular quadratic space over K = R. The non-degenerate symmetric bilinear form b induces a (real) inner product, deﬁned by

hv, wi := b(v, w). (1.25)

Similarly, the quadratic form qb associated with b induces a norm of a vector, deﬁned by

2 kvk := qb(v). (1.26) where from the polarization identity, we have

1 h 2 2 2i hv, wi = 2 kv + wk − kvk − kwk . (1.27)

A regular quadratic space is thus an inner product space. Since an inner product naturally induces the associated norm, a regular quadratic space is also a normed vector space. 8 Chapter 1. Quadratic Spaces, Fundamentals

Remark 1.11. Since we identiﬁed the inner product with the non-degenerate symmetric bilinear form, bilinear forms will be sometimes written using the inner product notation h·, ·i for convenience. In the case when we have to deal with several diﬀerent vector spaces, we shall mark the inner product or the norm with the symbol that denotes the corresponding vector space, e.g., h·, ·iV or k·kV .

1.4 Transpose and Adjoint Transformations

The transposition, as a permutation of two slots of a multilinear map, is well-defined up to a natural isomorphism between a vector space and its bidual. Earlier we defined the transpose of a bilinear form, here we define the transpose of a linear map between vector spaces. Moreover we introduce the adjoint, which under special circumstances (the orthonormal basis) has the same associated matrix as the transpose. These two concepts are important for the considerations of the square-root isometry studied in Chapter2, and as such, to the rest of this work.

1.4.1 Transpose

We firstly observe that every linear map S : V → W induces a bilinear map s : W ∗ ×V → K defined by s(ω, ·) = ω(S(·)) = (ω ◦ S)(·). By interchanging the slots of s, we can introduce sT : W × V ∗ → K which depends on the bilinear map s, and subsequently on the starting linear map S. This allows us to define the transpose of S:

Deﬁnition 1.10. Let S : V → W be a linear map from V to W . For any linear form ω : W → K, ω ∈ W ∗, the composition ω ◦ S : V → K is a linear form belonging to V ∗ and induces a map ST : W ∗ → V ∗ deﬁned by ST(ω) = ω ◦ S or

ST(ω)(v) = ω(S(v)), ∀v ∈ V, (1.28)

The map ST is called the transpose (or pullback or dual) of S.

The transpose map ST is obviously linear. After identifying vector spaces with their ∗∗ ∗∗ biduals by using the natural isomorphisms πV : V → V and πW : W → W , we have T T (S ) πV = πW S. It can also be veriﬁed that, with respect to any bases E of V and F of W , it holds T T [S]EF = [S ]F ∗E∗ , (1.29)

T T where [S]EF = [S]F ∗E and [S]F ∗E = [S ]EF ∗ . Furthermore [Kal07, pp. 225-227], it also holds that S and ST has the same rank, ρ(S) = ρ(ST).

1.4.2 Adjoint

Theorem 1.3. Let b and b0 be two non-degenerate bilinear forms on V × W and V 0 × W 0, respectively. (i) Then for every linear transformation S ∈ Hom(V,V 0) there exists a unique map S0 : W 0 → W such that

b0(S(v), w0) = b(v, S0(w0)), ∀v ∈ V, ∀w0 ∈ W 0, (1.30) 1.4 Transpose and Adjoint Transformations 9

which is also linear. (ii) Moreover, if [S]E0E is the matrix of S with respect to bases E and 0 0 0 0 0 0 E of V and V , then the matrix [S ]FF 0 of S with respect to bases F and F of W and W , is given by T 0 0 0 −1 T 0 [S]E0E [b ]E0F 0 = [b]EF [S ]FF 0 (S = b S b ). (1.31) Remark 1.12. We can construct the analogue statements starting from S ∈ Hom(W, W 0), S ∈ Hom(V 0,V ) or S ∈ Hom(W 0,W ). Also, if E0 is orthonormal to E and F 0 orthonormal to F, as in the case of dual bases, the matrices of the adjoint and transpose coincide.

0 0 0∗ T 0∗ Proof. Consider the commutative diagram on Figure 1.2. We have b[ : W → V , S : V → 0 ∗ ] ∗ 0 0 0 T 0 0 ] T 0 V , b[ : W → V , b : V → W and S : W → W , that is b[ ◦ S = S ◦ b[ or S = b ◦ S ◦ b[. 0 0 Because b and b are non-degenerate, the maps b[ and b[ are isomorphisms, thus to prove that the adjoint S0 is isomorphic to the transpose ST is straightforward. The linearity can be proven by the explicit evaluation [Per91][Lan02, p. 223]. 0 0 e0 0 For (ii), starting from [b]EF = bef , [b ]E0F 0 = be0f 0 , [S]E0E = S e and [S ]FF 0 = 0f 0 0 J K J K J K S f 0 , where e, f, e , f range over the corresponding index sets of the respective bases, we 0 J KT e 0 0f T 0 get (S )e (b )e0f 0 = bef S f 0 , that is S b = bS. Finally, as the matrix of b is non-singular, we obtain S0 = b−1STb0.

Deﬁnition 1.11. If two linear transformations S and S0 satisfy the relations from Theorem 1.3, we say that S0 is adjoint to S relative to b and b0. 4

b] V V ∗ W b[ S ST S0

0 0∗ 0 V V 0 W b[

0 ] T 0 Figure 1.2: The adjoint transformation S = b ◦ S ◦ b[

In addition, (i) If V 0 = W , W 0 = V then S : V → W is called self-adjoint iff S0 = S and skew-adjoint iff S0 = −S. (ii) If W = V , W 0 = V 0 then S : V → V 0 is called orthogonal iff S0 = S−1 (see Figure 1.3).

b b V ∗ [ W V ∗ [ V

ST S0 = S ST S S0 = S−1

∗ 0∗ 0 W 0 V V 0 V b[ b[

T 0 T 0 b[ ◦ S = S ◦ b[ b[ = S ◦ b[ ◦ S S : V → WS : V → V 0 b : V × W → K b : V × V → K b0 : W × V → K b0 : V 0 × V 0 → K

Figure 1.3: The self-adjoint (left) and the orthogonal transformation (right)

4S0 is also called the transpose of S with respect to b and b0 [Lan02, p. 524]. 10 Chapter 1. Quadratic Spaces, Fundamentals

Example 1.3. Let f and g be two bilinear forms with the same signature, (V, f) =∼ (V, g) =∼ m,n 2 ] p −1 (R , η). Deﬁne S ∈ End(V ) as S = g f[, or up to a sign: S = g f in matrix notation. Then S is self-adjoint relative to g, that is S0 = g−1STg = S. See Section 2.2.

Remark 1.13. The adjoint allows us to deﬁne the orthogonal group over a regular quadratic space without reference to matrices as the set of all endomorphisms of V for which the adjoint equals the inverse: S0 = S−1. For example, in case of the Minkowski space, in matrix notation the adjoint of the Lorentz transformation is Λ0 = η−1ΛTη which is equal to Λ−1. The transpose and adjoint, when treated as a unary operation on transformations, share the properties of the matrix transpose. Let the 0-operation denotes an adjoint. Then, for example, it can be shown [Per91, Lan02] that

(A + B)0 = A0 + B0, (αA)0 = αA0, (A0)0 = A, (CA)0 = A0C0, (1.32) for all A, B ∈ Hom(V1,V2), C ∈ Hom(V2,V3) and α ∈ K.

1.5 Isometries and Conformal Maps

We can construct a new bilinear space by structure transport (or structure transfer) from an existing bilinear space (V, f) to another vector space W using a structure-preserving map called the homomorphism (if the map admits an inverse, it is called the isomorphism). The concept of isometry is motivated by structure transfer which preserves distances.

1.5.1 Isometries and Pullbacks

Deﬁnition 1.12. Let (V, f) and (W, b) be two quadratic spaces. An injective linear transformation S ∈ Hom(V,W ) is called an isometry, written S :(V, f) ,→ (W, b), iﬀ

f(v1, v2) = b(S(v1),S(v2)), ∀v1, v2 ∈ V. (1.33)

The quadratic spaces (V, f) and (W, b) are called isometric (or isomorphic), written

(V, f) =∼ (W, b), (1.34) iﬀ there exists a bijective isometry S : V → W .

W × W b K (W, b)

⇐⇒ S S f S V × V (V, f)

Figure 1.4: The isometry S :(V, f) ,→ (W, b).

Remark 1.14. An equivalent deﬁnition can be given with respect to metric spaces [BBI01].

Consider two metric spaces X and Y with metrics dX and dY . A map S : X → Y is called an isometry or distance preserving iﬀ dY (S(x1),S(x2)) = dX (x1, x2) for all x1, x2 ∈ X. 1.5 Isometries and Conformal Maps 11

An isometry is clearly injective and every isometry between metric spaces is a topological embedding. A bijective isometry is also called global isometry (or isometric isomorphism or congruence mapping).

Remark 1.15. In [HSMvS13], a local transformation S : TP (M) → TP (M) on diﬀerentiable manifolds (M, f) and (M, g) deﬁned by f = STgS is denoted as the ‘generalized vielbein’. The vielbeins are treated in Section 3.4 on page 39. Remark 1.16. Compare Figure 1.4 with Figure 1.1 and note g] :(V ∗, g−1) =∼ (V, g). It is easily proved that the composition of isometries is an isometry and that the set of bijective isometries V → V forms a group with respect to composition. Furthermore, the bijective isometry is an equivalence relation. In terms of a given norm, S : V,→ W is an 2 2 isometry if and only if kS(v)kW = kvkV , ∀v ∈ V . As a precomposition, an isometry is also kind of a pullback that provides somewhat more convenient notation.

Definition 1.13. Let V and W be vector spaces over a field K. Given a multilinear form b : W × W × · · · × W → K on W and an injective linear transformation S ∈ Hom(V,W ), the pullback S∗b of b by S is a multilinear form S∗b on V defined by

∗ S b(v1, v2, . . . , vn) = b(S(v1),S(v2),...,S(vn)), ∀v1, v2, . . . , vn ∈ V. (1.35)

Hence, S∗ is a linear transformation from multilinear forms on W to forms on V . Clearly, ∗ ∗ ∗ the composition of two pullbacks S1 and S2 is given by (S2 ◦ S1) b = S1 (S2 b) and, if S is bijective, then from f = S∗b it follows b = (S−1)∗f.

Subsequently, the isometry S :(V, f) ,→ (W, b) can simply be written f = S∗b. Similarly, ∗ in terms of the inner product, S is an isometry iﬀ the pullback S h·, ·iW of the inner product on W is equal to the inner product h·, ·iV on V . Remark 1.17. Note the special case when b is a linear form on W , that is a linear functional b ∈ W ∗; then S∗b(v) = b(S(v)) = (b ◦ S)(v) is an element of V ∗, and the pullback by S : V → W is a linear map between dual spaces S∗ : W ∗ → V ∗. In such case, the pullback S∗ is also the transpose (or dual) map of S, written ST (see Deﬁnition 1.10).

Example 1.4. Let (M, g) and (M 0, g0) be two manifolds, and let ϕ : M → M 0 be a diffeomorphism. Then ϕ is called an isometry iff g = ϕ∗g0, where ϕ∗g0 denotes the pullback of 0 the rank (0, 2) metric tensor g by ϕ. Equivalently, in terms of the pushforward ϕ∗, we have 0 5 g(v, w) = g (ϕ∗v, ϕ∗w) for any two vector fields v, w on M. If ϕ is a local diffeomorphism such that g = ϕ∗g0, then ϕ is called a local isometry [Lee03].

1.5.2 Conformal Maps There is another, somewhat more general way to transfer structure from an existing bilinear space to a vector space. We can require that the vector space homomorphism only preserves orthogonality of vectors. It is obvious that such transformation need not be isometric.

5 0 Let x ∈ M; the diﬀerential of ϕ at x is a linear map dϕx : Tx(M) → Tϕ(x)(M ) from the tangent space 0 0 of M at x to the tangent space of M at x = ϕ(x). The application of dϕx to a tangent vector v ∈ Tx(M) m n is called the pushforward of v by ϕ, denoted ϕ∗v. In the case ϕ : R → R , the total derivative of ϕ at x is represented by the Jacobian (matrix) of ϕ at x with respect to the canonical basis. 12 Chapter 1. Quadratic Spaces, Fundamentals

Deﬁnition 1.14. Let (V, f) and (W, b) be two bilinear spaces. A linear transformation S : V → W is called conformal mapping (or homothety or similitude or dilation) iﬀ there exists a scalar α2 > 0 such that

2 b(S(αv1),S(αv2)) = α f(v1, v2), ∀v1, v2 ∈ V. (1.36)

2 Hence, w1 ⊥ w2 ⇔ v1 ⊥ v2 (see Deﬁnition 1.15). The scalar α is called the conformal factor (or similitude factor).

With respect to composition, the conformal maps form a group called the conformal group. It can be veriﬁed that a conformal map preserves the angles between vectors. On a real inner product space with a non-singular bilinear form (Section 1.3.2), the proof follows from the deﬁnition of the angle between two vectors

hx, yiV = kxkV kykV cos ∠(x, y). (1.37)

Finally, we observe that a conformal map between (V, f) and (W, b) can be viewed as an isometry between (V, α2f) and (W, b). Clearly, a conformal mapping is an isometry if and only if its conformal factor equals 1. Remark 1.18. In differential geometry, conformal factor needs not to be constant throughout a manifold, that is, it may depend on the position and accordingly form the conformal vector field when combined with a local isometry; this field can be considered as a natural generalization of the Killing vector field.

1.6 Orthogonality and Orthogonal Splitting

Definition 1.15. Let (V, b) be a quadratic space. Two vectors v, w ∈ V are orthogonal iff b(v, w) = b(w, v) = 0. Two subspaces X,Y ⊆ V are called orthogonal iff b(v, w) = 0 for all v ∈ X, w ∈ Y . We denote orthogonality by ⊥, e.g., v ⊥ w or X ⊥ Y . To each subspace X ⊆ V , the orthogonal or perpendicular subspace X⊥ is defined as

X⊥ := { v ∈ V | (∀w ∈ X) b(v, w) = 0 } = { v ∈ V | v ⊥ X }. (1.38)

V ⊥ ≡ ker(b) is called the radical of (V, b), more correctly denoted as Rad V .

We say that V has the orthogonal splitting V = V1 ⊥ V2 ⊥ · · · ⊥ Vk into subspaces Vi, iﬀ V is the direct sum V = V1 ⊕ V2 ⊕ · · · ⊕ Vk with the Vi pair-wise orthogonal, Vi ⊥ Vj, i 6= j.

Lemma 1.4. Given a quadratic space (V, b), let X,Y be subsets of V . Then

1◦ X⊥ is a subspace of V, (1.39) 2◦ X ⊆ Y ⇒ Y ⊥ ⊆ X⊥, (1.40) 3◦ X ⊆ (X⊥)⊥. (1.41)

◦ Proof. All three statements are direct consequences of the deﬁnition. E.g. for 1 , if v1, v2 ∈ ⊥ X , then v1 ⊥ w and v2 ⊥ w for all w ∈ V , and consequently (α1v1 + α2v2) ⊥ w for any ⊥ α1, α2 ∈ K, thus α1v1 + α2v2 ∈ X . 1.6 Orthogonality and Orthogonal Splitting 13

Lemma 1.5. Orthogonal vectors are sent to orthogonal vectors by an isometry. Let S ∈ Hom(V,W ) be an isometry of from (V, f) to (W, b); then for each subset X ⊂ V , it holds S(X⊥) ⊆ S(X)⊥. Moreover, if S is bijective, then S(X⊥) = S(X)⊥.

Proof. Clearly, v1 ⊥ v2 ⇔ f(v1, v2) = 0 implies b(S(v1),S(v2)) = 0 ⇔ S(v1) ⊥ S(v2), and since S is injective, we have S(X⊥) ⊆ S(X)⊥. If S is also surjective, equality holds.

1.6.1 Orthogonal Group

In this section we turn our attention to the geometry of quadratic spaces. In particular we investigate the isometries and the corresponding group that preserve quadratic forms, the orthogonal group.

Deﬁnition 1.16. With respect to composition, the isometries S ∈ End(V ) form a group called the orthogonal group or the automorphism group of (V, b), denoted by O(V, b) (or Aut(V, b)). The elements of this group are called the orthogonal transformations.

Remark 1.19. The last deﬁnition of the orthogonal transformation is compatible with Re- mark 1.13, Theorem 1.3 and Deﬁnition 1.11 since the assertions b(S(v), w0) = b(v, S−1(w0)) and b(S(v),S(w)) = b(v, w) are equivalent for an invertible S ∈ Aut(V ).

Lemma 1.6. If (V, b) =∼ (V 0, b0), then O(V, b) =∼ O(V 0, b0).

Proof. We note that S ∈ Hom(V,V 0) is bijective; hence a map deﬁned by O(V, b) 3 T 7→ STS−1 ∈ O(V 0, b0) is an isomorphism of the orthogonal groups.

Theorem 1.4. Two quadratic spaces are isometric if and only if their associated matrices (with respect to arbitrary bases) are congruent.

0 0 0 0 Proof. Let (V, b) and (V , b ) be two quadratic spaces with bases E = ei i∈I and E = ei i∈I , respectively. Let further the linear transformation S : V → V 0 haveJ theK associatedJ matrixK i 0 i [S]EE0 = S j deﬁned by S(ej) = eiS j (see [Kal07, p. 105]). The map S is bijective if it has rank dimJV andK if it holds

T 0 T T v [b]E w = b(v, w) = b (S(v),S(y)) = v [S]EE0 [b]E0 [S]EE0 w, ∀v, w ∈ V. (1.42)

T Thus [b]E = [S]EE0 [b]E0 [S]EE0 , noting [S]EE0 = [S]E0∗E .

In the particular case when (V, b) = (V 0, b0) and E = E0, we get the condition b = STbS for the matrix of the isometry S , and subsequently an isomorphism

O(V, b) =∼ { S ∈ GL(dim V,K) | b = STbS }. (1.43)

Let K×2 denote the subgroup of squares of the multiplicative group K× of a field K.A square class of a field K is an element of the quotient group K×/K×2, also denoted as the square class group. Each square class is a subset of the non-zero elements (a coset of K×) consisting of the elements of the form xy2 where x is a fixed element and y ∈ K×. For ×2 example, R/R has two elements which are represented by 1 and −1. Now, consider a quadratic space (V, b). Let [b]E be the matrix of b with respect to some T 0 basis E and [b]E0 = T [b]E T be the matrix with respect to another basis E , where T is the 14 Chapter 1. Quadratic Spaces, Fundamentals change of basis matrix. Since T is invertible and det T T = det T , then the determinants ×2 det[b]E and det[b]E0 differ only by an element of K . Accordingly, (V, b) is singular if and × ×2 only if det[b]E = 0. If (V, b) is regular, then det[b]E is the element of K /K and we can define

det(V, b) := det[b]E . (1.44) Because isometric spaces have congruent associated matrices, the equality of the determinant is a necessary condition for isometry between spaces, thus the following lemma holds:

Lemma 1.7. If (V, b) is a regular quadratic space over K and S ∈ O(V, b), then det S ∈ K×/K×2.

This determinant induces a homomorphism det : O(V, b) → K×/K×2. In particular for K = R, det : O(V, b) → {±1}. The kernel of this homomorphism (that is, the isometries with det = 1) is called the special orthogonal group, denoted by SO(V, b).

1.7 Isotropy and Hyperbolic Splitting

Before continuing with the orthogonal decomposition, we shall highlight yet another two important aspects of quadratic spaces: the isotropy of vectors and the accompanying emer- gence of hyperbolic structure. In fact, according to [Sch85], the modern algebraic theory of quadratic forms begin with Witt’s paper [Wit37] wherein these two concepts play the prominent role.6

Definition 1.17. Let (V, b) a regular bilinear space. A non-zero vector v ∈ V is called isotropic iff b(v, v) = 0; otherwise, v is called non-isotropic or anisotropic. The space (V, b) is said to be isotropic iff it contains at least one isotropic vector, and totally isotropic iff every non-zero vector in V is isotropic. The isotropy index (or Witt index) of a quadratic space is the maximum of all the dimensions of the totally isotropic subspaces (or the dimension of a maximal isotropic subspace).

Remark 1.20. In view of Deﬁnition 1.20, a quadratic space (V, b) is anisotropic if and only if the quadratic form qb is either positive or negative deﬁnite. More generally, if (V, b) is regular and has the signature (m, n), then its isotropy index is the minimum of m and n.

Example 1.5. In the Minkowski spacetime, isotropic vectors are null-vectors given by a light-cone.

Deﬁnition 1.18. Two vectors v and w of a bilinear space (V, b) form a hyperbolic pair iﬀ b(v, w) = b(w, v) = 1 and b(v, v) = b(w, w) = 0. Observe that if {u, v} is a hyperbolic pair, then v 6= 0 and w 6= 0, since b(v, w) = 1; hence it follows that v and w are isotropic vectors. Moreover, the vectors v and w are linearly independent. A hyperbolic plane is a two-dimensional quadratic space that is spanned by a hyperbolic pair.

6Witt’s theorem states that any isometry between two subspaces of a regular quadratic space may be expanded to an isometry of the whole space. 1.8 Orthogonal Decomposition 15

Lemma 1.8. Let (V, b) be a 2-dimensional bilinear space. Then the following assertions are equivalent: 1◦ V is a hyperbolic plane, (1.45) 2◦ V is isotropic and regular, (1.46) 3◦ (V, b) =∼ −1, 1 =∼ −α, α , ∀α ∈ K×, (1.47) L M L M 4◦ det(V, b) ≡ −1 (mod K×2). (1.48)

Proposition 1.1. Every regular isotropic quadratic space is split by a hyperbolic plane. Every regular isotropic bilinear space is universal (Def. 1.3).

The proof of both the last theorem and the lemma above are given in [Sch85]. Recall × that the quadratic space (V, b) is universal iﬀ qb(V ) = K, i.e., that for all scalars α ∈ K there is a vector v ∈ V such that qb(v) = α.

Theorem 1.5 (The Witt Decomposition, 1937). Let (V, b) be a (regular or not) quadratic space over a ﬁeld K. Then it admits a decomposition

(V, b) ' Rad V ⊕ (Va, ba) ⊕ (Vh, bh) (1.49)

⊥ where Rad V = V = ker b is the radical of b, (Va, ba) is an anisotropic quadratic space and (Vh, bh) is a split quadratic space (which is a direct sum of hyperbolic planes).

m,n m,m+k Example 1.6. The Witt decomposition splits the Minkowski space R = R into m copies of the hyperbolic plane plus a k-dimensional definite space (either positive or negative definite). The number of different copies of the hyperbolic plane is the Witt index. The 2 Witt decomposition can be used to construct quadratic spaces. Let begin with R having 2 2 2 2 the quadratic form x1 + x2. After adding one hyperbolic plane −t1 + x3 = 1, we get the 1,3 2 2 2 2 Minkowski spacetime R with the quadratic form −t1 + x1 + x2 + x3 of Witt index 1. If 2 2 2,4 we further add a second hyperbolic plane −t2 + x4 = 1, we get R with the quadratic form 2 2 2 2 2 2 −t1 − t2 + x1 + x2 + x3 + x4 of Witt index 2.

1.8 Orthogonal Decomposition

Lemma 1.9. A quadratic space (V, b) is regular if and only if Rad V = {0V }.

Proof. We observe that Rad V = V ⊥ = ker(b), therefore dim(V ) = dim(V ⊥) + dim(Im(b)). ⊥ Because V is regular iﬀ b[ is an isomorphism, then V is regular iﬀ dim(V ) = 0, that is, ⊥ V = {0V }.

Lemma 1.10. Let (V, b) be a regular quadratic space, and X,Y be subspaces of V . Then

1◦ dim X + dim X⊥ = dim V, (1.50) 2◦ (X⊥)⊥ = X, (X + Y )⊥ = X⊥ ∩ Y ⊥, (X ∩ Y )⊥ = X⊥ + Y ⊥. (1.51)

◦ ⊥ ∗ Proof. We can deduce 1 because the image of b[(X ) ⊆ V is the annihilator of the subspace ∗ X. Namely, we know that b[ : V → V is an isomorphism for a regular quadratic space. ∗ ∗ Since X ⊆ V , the canonical projection π : V → X is surjective and the kernel of πb[ : V → 16 Chapter 1. Quadratic Spaces, Fundamentals

∗ ∗ ⊥ 7 ⊥ ∗ V → X is X (as πb[ is also surjective) which implies that dim(V ) = dim(X )+dim(X ), where dim(X∗) = dim(X). To prove 2◦, we use X ⊆ (X⊥)⊥.

Lemma 1.11. If X is a regular subspace of (V, b), then we have orthogonal splitting V = X ⊥ X⊥ and we call X⊥ the orthogonal complement of X.

Proof. Since the orthogonality between X and X⊥ is obvious, we only need to show that V is the direct sum of X and X⊥, i.e. that V = X ⊕ X⊥, which is equivalent to X ∩ X⊥ = 0 ⊥ ⊥ and V = X +X . Firstly, if X is a subspace of the bilinear space (V, b), then X = ker(πb[) where π : V ∗ → X∗ is the canonical projection. Since x⊥ ∈ X⊥ means by deﬁnition ⊥ ⊥ ⊥ ⊥ [b[(x )](y) = b(y, x ) = 0 for all y ∈ X, we have [b[(x )]|X×X = 0 and hence x ∈ ker(πb[). ⊥ So, as X is regular, {0V } = ker(πb[) = X ∩ X . On the other hand, if x ∈ V with ∗ ω = [b[(x)]|X×X ∈ V , then there is a y ∈ X with ω = bX[(y) (because, yet again, X is regular). Therefore

b(v, x) = b[(x)(v) = f(v) = bX[(y)(v) = b(v, y) ⇒ b(v, x − y) = 0 (1.52) for all v ∈ X. The last equation enables us to write x = y + y⊥ ∈ V where y ∈ X and y⊥ = x − y ∈ X⊥.

On the basis of this lemma one can diagonalize any regular symmetric bilinear form and establish an orthogonal basis. But we will introduce a useful notation.

Deﬁnition 1.19. If α ∈ K, we shall use α to denote8 the one-dimensional bilinear space L M with the associated 1 × 1 matrix α . In general, we deﬁne α1, α2, . . . , αn := A where A J K L M is the matrix having the diagonal entries A = δijαj ; moreover, if all diagonal elements are n J K equal α, we write for short α := δijα . L M J K Theorem 1.6. Every quadratic space (V, b) permits an orthogonal basis.

Proof. We shall prove that (V, b) is the orthogonal sum of one-dimensional subspaces. The proof is on n = dim V by induction, and it eﬀectively constructs an orthogonal basis. Since V ⊥ is an orthogonal summand of V = V ⊕ V ⊥, we may assume that V is regular. If b 6= 0, there exist vectors v, w ∈ V with b(v, w) 6= 0. Now, because

1 b(v, w) = 2 [qb(v + w) − qb(v) − qb(w)] 6= 0 (1.53) there must be some e1 ∈ {v + w, v, w} for which qb(e1) 6= 0. Consequently, the one- dimensional subspace Ke1 is non-degenerate and using the last lemma we can decompose ⊥ V = (Ke1) ⊕ (Ke1) . The proof is completed by using the mathematical induction on ⊥ (Ke1) , obtaining e2, and so on up to en. Hence, we have constructed the orthogonal basis E = e1,, e2, ..., en where b(ei, ej) = 0 J K for all i 6= j and which spans V . On the diagonal of b we will have λi = b(ei, ei) 6= 0, and the matrix of b relative to basis E will be λ1, λ2, . . . , λn . We can always rescale ei such L M that b(ei, ei) is 1 for K = C and ±1 for K = R or Q. This will lead us to the classiﬁcation of quadratic spaces in Section 1.9.

7Again, rank-nullity theorem: If T ∈ Hom(V,U), then dim(Im(T )) + dim(ker(T )) = dim(V ). 8In mathematical literature, the angle brackets h· · · i are used instead of ··· . Here we avoid such notation as it might give rise to confusion. L M 1.9 Classiﬁcation of Real Quadratic Spaces 17

Lemma 1.12. [Sch85] There is a canonical bijection between the isometry classes of quadratic spaces and congruence classes of symmetric matrices.

Proof. Consider the vector space Kn of column vectors with the components in K. To every symmetric matrix B ∈ Mat(n, K), B = BT, we can associate a symmetric bilinear form T bB(v, w) = v Bw with the corresponding bilinear space denoted (cf. Deﬁnition 1.19)

n B := (K , bB). (1.54) L M n 0 0 j Let further S : V → K be a linear transformation deﬁned by ei = S(ei) where ei = δi n 0 J jK J K is the canonical basis of K (for a ﬁxed i, ei is a column vector with the elements δi ). Since S is a bijective isometry, we have (V, b) =∼ B . (1.55) L M By this we have established the isomorphism

Sym(V,K) =∼ Sym(n, K) := { B ∈ Mat(n, K) | B = BT } (1.56) where n = dim V . Hence, each problem about quadratic forms can be converted into the terminology of symmetric matrices.

Corollary 1.1. Every invertible symmetric matrix in A ∈ GL(n, K) is congruent to a diagonal matrix.

Proof. Using Theorem 1.6, construct an orthogonal basis E = e1, e2, . . . , en for the bilinear J K space A . Then from to the last lemma, [bA]E is a diagonal matrix congruent to A. L M Since the automorphism of Kn given by permuting vectors in the canonical basis is an isometry, it immediately follows:

Corollary 1.2. Let π be an arbitrary permutation of (1, . . . , n). Then for arbitrary scalars × αi ∈ K ∼ 2 2 λ1, . . . , λn = α1λπ(1), . . . , αnλπ(n) . (1.57) L M L M 1.9 Classiﬁcation of Real Quadratic Spaces

×2 We have seen that the quotient group R/R can be represented by the elements from {−1, 1}. By Theorem 1.6, any regular real quadratic space (V, b) can be decomposed as

V =∼ −1 m ⊥ +1 n (1.58) L M L M for some m, n. The integer m is called the negative index (or just index) of V and is denoted by ρ−(V ).9 Similarly, n = ρ+(V ) is the positive index of V . The diﬀerence ρ+(V ) − ρ−(V ) is called the signature of V .10

9It is usual that the index of a bilinear form is deﬁned as the maximum possible dimension of a negative deﬁnite subspace. 10 Equivalently, if [b]E is a matrix of b with respect to some basis E of a real quadratic space (V, b), then the signature (or inertia) (m, n, p) of b is the number of positive, negative and zero eigenvalues of [b]E counted with their algebraic multiplicity. If the space is regular, then there will be no zero eigenvalues. 18 Chapter 1. Quadratic Spaces, Fundamentals

Both ρ+(V ) and ρ−(V ), and hence the signature, depend only on the isometry class of V . Namely, the Witt decomposition states that m, n are uniquely given by b; this is a consequence of Witt’s cancellation theorem [O’M73].

Theorem 1.7 (Sylvester’s Law of Inertia, 1852). Let V and W be two non-degenerate ∼ + + − − quadratic spaces over R. Then V = W if and only if ρ (V ) = ρ (W ) and ρ (V ) = ρ (W ).

Definition 1.20. Let (V, b) be a real quadratic space. A quadratic form qb is called positive definite iff qb(v) > 0, ∀v ∈ V , and negative definite iff qb(v) < 0, ∀v ∈ V .

Lemma 1.13. If a quadratic form is either positive or negative definite, then it is regular; moreover, ρ+(V ) = dim V for positive definite forms and ρ−(V ) = dim V for negative definite forms.

Remark 1.21. It can also be verified that ρ+(V ) is geometrically characterized as the maximal dimension of a positive definite subspace of V and ρ−(V ) is the maximal dimension of a negative definite subspace of V [O’M73, p. 155]. Remark 1.22. From earlier we know that (V, b) =∼ (V 0, b0) implies O(V, b) =∼ O(V 0, b0), which means that, for K = R, we can use Theorem 1.7 to classify isometries based on the unique − + signature (m, n) where m = ρ (V ) and n = ρ (V ). For K = C we have a different story. The complex versions of the orthogonal groups, O(m, n, C), are all isomorphic to each other ∼ if they have the same dimension, O(m, n, C) = O(m + n, C) [Sze04].

Taxonomy

m n m,n Deﬁnition 1.21. We denote a regular real quadratic space −1 ⊥ +1 by (R , ηm,n) L M L M m n where the bilinear form ηm,n is given by the diagonal matrix [ηm,n]E = (−1) ⊕ (+1) with m+n n respect to the canonical basis E of R . In the special case when m = 0, we denote +1 n ˆ ˆ L M by (R , δn) where δn ≡ η0,n. m,n 11 (i) A regular quadratic space (R , ηm,n), m ≥ 1, n ≥ 1 is called the Minkowski space. n ˆ ˆ A regular quadratic space (R , δn) is called the Euclidean space and the form δn is called the n standard bilinear form of R . ∼ m,n (ii) Isometries under which (V, b) = (R , ηm,n) will be called the vielbeins. m,n ∼ (iii) Since O(R , η) = O(m, n), the corresponding orthogonal group is simply denoted O(m, n), where T O(m, n) ≡ { S ∈ GL(m + n, R) | S ηm,nS = ηm,n }. (1.59) Moreover, the corresponding special orthogonal group will be denoted by SO(m, n), while its identity component by SO0(m, n). If m = 0, we write O(n) ≡ O(0, n) and SO(n) ≡ SO(0, n).

m,n m,n Unless confusion might arise, both R and (R , η) will often be used to denote m,n ˆ ˆ (R , ηm,n), and similarly η to denote ηm,n. Notice that ηm,n = (−δm) ⊕ δn. With respect m,n to a given basis, the m + n decomposition of quadratic spaces isometric to R will be elaborated in Section 3.3. The structure of the orthogonal group O(m, n) will be analyzed in Section 3.1. All further work will be concerned with the regular quadratic spaces isometric m,n to (R , η).

11 1,3 The special case R is called the Minkowski spacetime. It is usual to refer to a quadratic space isometric 1,n to R as the Lorentzian space. Chapter 2

The Square-Root Isometry

In this chapter we introduce the square-root isometry as an isometry which is also a self- adjoint transformation and which, as such, has special properties. We then explore the action of the square-root isometry group on quadratic spaces and its possible orbits. The key proposition of this chapter is the so-called symmetrization condition which asserts the correspondence between the self-adjoint isometries and the underlying orthogonal transformations on coupled quadratic spaces.

2.1 Coupled Quadratic Spaces

Deﬁnition 2.1. Quadratic spaces (V, f), (W, g),... that are mutually isometric through a m,n common Minkowski space (R , η) will be called coupled quadratic spaces.

We begin with the general conditions imposed on the isometries that couples two ∼ ∼ m,n ∗ ∗ quadratic spaces (V, f) = (W, g) = (R , η). Let f = L η and g = E η be two isome- m,n m,n tries such that L :(V, f) ,→ (R , η) and E :(W, f) ,→ (R , η), and let further S be an isometry (V, f) ,→ (W, g). As shown earlier, the matrix of a quadratic space is well-deﬁned m,n up to a congruence. This means that, in general, between the two copies of (R , η) where one is relative to f and the other is relative to g, we have an orthogonal transformation Λ ∈ O(m, n). Subsequently, we arrive at the composition of isometries corresponding to the commutative diagram shown on Figure 2.1.

(V, f) S (W, g)

L E

m,n Λ ∈ O(m, n) m,n (R , η) (R , η)

∼ ∼ m,n Figure 2.1: Coupled quadratic spaces (V, f) = (W, g) = (R , η)

Lemma 2.1. Let (V, f) and (W, g) be two quadratic spaces coupled through the Minkowski m,n m,n space (R , η). The orthogonal transformations Λ ∈ O(m, n) on R are related to the isometries S :(V, f) ,→ (W, g) by ES = ΛL. (2.1)

Mikica Kocic, The Square-Root Isometry, MSc thesis, SU 2014 Printed: 2015-03-11 20 Chapter 2. The Square-Root Isometry

Moreover, if E and L are bijective, then for every Λ there exists S and conversely, for every S there exists Λ S = E−1ΛL, Λ = ESL−1. (2.2)

The existence of Λ and S is assured by Lemma 1.2 and the assertion can be veriﬁed by the commutative diagram, from which we can construct Table 2.1. In the rest of this chapter, we shall narrow our considerations to a single vector space W = V .

Table 2.1: Isometries of coupled quadratic spaces

m,n m,n m,n ∗ T Λ: R → R η(x, y) = η (Λ(x), Λ(y)) x, y ∈ R η = Λ η η = Λ ηΛ

m,n ∗ T L : V → R f(v1, v2) = η (L(v1),L(v2)) v1, v2 ∈ V f = L η f = L ηL

m,n ∗ T E : W → R g(w1, w2) = η (E(w1),E(w2)) w1, w2 ∈ W g = E η g = E ηE

∗ T S : V → W f(v1, v2) = g (S(v1),S(v2)) v1, v2 ∈ V f = S g f = S gS

Corollary 2.1. Let (V, g) be a regular quadratic space isometric to the Minkowski space m,n ∗ (R , η) where g = E η. Between the orthogonal transformations Λ ∈ O(m, n) and X ∈ O(V, b) we have a bijective similarity transformation

X = E−1ΛE, Λ = EXE−1. (2.3)

Degrees of Freedom

∼ m,n 2 An arbitrary isometry E :(V, g) = R , i.e. a vielbein E, has d independent components as the part of the associated matrix, where d = dim V = m + n. On the other hand, a symmetric bilinear g form on V has only d(d + 1)/2 independent components. The diﬀerence d2 − d(d + 1)/2 = d(d − 1)/2 is a part of the orthogonal transformation Λ ∈ O(m, n). The dimensionality dim O(m, n) = d(d − 1)/2 can be seen from the constraints on the components on the associated congruence matrix, or from the dimensionality of the corresponding Lie algebra o(m, n) that is constructed using skew-symmetric d×d square matrices whose entries are reals (see Section 3.1).

2.2 The Square-Root Isometry

Let (V, f) and (V, g) be two quadratic spaces. Consider a map S : V → V deﬁned by ] S ◦ S = g ◦ f[ 2 ] S (v) = g (f[(v)), ∀v ∈ V. (2.4) 2 ] Clearly, starting from S = g f[, the map S has the associated matrix (N.B. matrix square- roots ar not unique!) q S = g−1f. (2.5)

It can be shown that such transformation is in fact an isometry. However, before proving such assertion and examining the properties of S in more details, we need to acquire some tools to handle the matrix functions F (XY ) and F (YX). 2.2 The Square-Root Isometry 21

F (XY ) Matrix Toolbox

We begin by noting that the Jordan canonical structures of the matrices XY and YX are closely related. Namely, according to the theorem of Flanders [Fla51], for any X ∈ Mat(m × n, C) and Y ∈ Mat(n × m, C), the non-zero eigenvalues of XY and YX are the same, and XY and YX share the equal Jordan structure (but their singular Jordan block sizes can diﬀer by 1). It can be shown that for an arbitrary function F , the expressions F (XY ) and F (YX) are also related. This is based on the claim that, for all X and Y for which the products XY and YX are deﬁned, and for an arbitrary polynomial P , it holds

XP (YX) = P (XY ) X. (2.6)

To do the transition from a polynomial P to a function F , we need the following theorem: Theorem 2.1. [Hig08, p. 21] Let F be a matrix function deﬁned on the spectra of both X ∈ Mat(n, C) and Y ∈ Mat(m, C). Then there exists exactly one polynomial P such that F (X) = P (X) and P (Y ) = F (Y ).

Corollary 2.2. Consider X ∈ Mat(m × n, C) and Y ∈ Mat(n × m, C). Let F be a function deﬁned on the spectra of both XY and YX. Then

XF (YX) = F (XY ) X. (2.7)

Proof. Obviously by the last theorem, there exist only one polynomial P for which F (XY ) = P (XY ) and F (YX) = P (YX). Thus, using Eq. (2.6)

XF (YX) = XP (YX) = P (XY ) X = F (XY ) X, (2.8) which concludes the proof.

Remark 2.1. The last corollary is helpful for interchanging F (XY ) and F (YX) within an expression (for example, see Lemma 3.1). − An immediate consequence of the last corollary is, when XY has no eigenvalues on R , the relation X (YX)1/2 = (XY )1/2 X. (2.9)

− Notice that, if XY does not have eigenvalues on R , neither YX does and furthermore, X and Y are necessarilly square [Hig08].

Corollary 2.3. Let X ∈ Mat(n, C), Y ∈ Mat(n, C) be two symmetric matrices and let F be a function deﬁned on the spectra of both XY and YX. Then

XF (YX) = (XF (YX))T . (2.10)

Proof. For any polynomial P , we have P (XY )T = P (Y TX T) thus

XP (YX) = P (XY ) X = P (Y TX T)TX = (X TP (Y TX T))T = (XP (YX))T (2.11) and accordingly by Theorem 2.1

XF (YX) = (XF (YX))T . (2.12) 22 Chapter 2. The Square-Root Isometry

Adjointness of the Square-Root

At this point, we are ready to investigate the properties of the square-root map.

Proposition 2.1. Consider quadratic spaces (V, f) and (V, g) where g is non-degenerate. Let S be an operator on V deﬁned by the matrix square-root S = pg−1f. Then, in matrix notation

1◦ fS = (fS)T,S = f −1STf, S = S0, (2.13) 2◦ gS = (gS)T,S = g−1STg, S = S0, (2.14) 3◦ f = STgS, (2.15) 4◦ f = (gS)Tg−1(gS), f = hg−1h, h ≡ gS, S = g−1h. (2.16)

Proof. The proof of 1◦ directly follows from Corollary 2.3 wherein we set X := f, Y := g−1 √ and F (x) := x q q T fS = f g−1f = f g−1f = (fS)T. (2.17)

Then we prove 3◦ starting from S2 = g−1f and from the established symmetry STf = fS

f = gS2 ⇒ STf = STg ⇒ S2fS = STgS2 ⇒ f = STgS. (2.18)

For 2◦ we again begin with fS = (fS)T and substitute f from 3◦

fS = STf ⇒ (STgS) S = ST (STgS) ⇒ gS = STg ⇒ gS = (gS)T. (2.19)

Finally, 4◦ follows from 2◦

f = gS2 = gS g−1g S = (gS)Tg−1(gS). (2.20)

Corollary 2.4. Let S be an isometry (V, f) ,→ (V, g), f = S∗g. Then S is self-adjoint relative to g or f, if and only if S2 = g−1f.

In view of the assertion 3◦ of Proposition 2.1, S is an isometry, which justiﬁes the following deﬁnition:

Definition 2.2. Let (V, f) be a quadratic space and (V, g) be a regular quadratic space. 2 An injective linear transformation S ∈ End(V ) defined by f[ = g[S , or equivalently by the matrix satisfying S2 = g−1f, (2.21) √ will be called the square-root isometry from (V, f) to (V, g), written (V, f) ,→(V, g). Iff f is non-degenerate, S is bijective.

As earlier pointed out in Remark I.1 on page vii, the solution to matrix equation S2 = g−1f, denoted by S = pg−1f, is not unique. The set of these alternative square-roots will be studied in Section 5.1 on page 54.

Symmetrizing quadratic space. Observe that (V, h), where h ≡ gS, is a quadratic space. It is obvious that the correspondence S ↔ h is bijective for a non-degenerate g. 2.3 The Square-Root Group Actions and Orbits 23

Hence, we can start from any quadratic space (V, h) and construct a well-deﬁned S := g−1h in which case (notice the symmetry under the interchange of f and g)

−1 T −1 ∗ −1 f = hg h = h g h, f = h[ g , (2.22) −1 T −1 ∗ −1 g = hf h = h f h, g = h[ f . (2.23) √ Deﬁnition 2.3. If S is a square-root isometry (V, f) ,→(V, g), then a quadratic space (V, h) deﬁned by h = gS will be called the symmetrizing quadratic space.

2.3 The Square-Root Group Actions and Orbits

Consider the set of all regular quadratic spaces on a real vector space V

Q = { (V, b) | b ∈ Sym(V ; R), ker b = {0V }}. (2.24)

Since V is fixed, throughout this section we shall sloppily use the bilinear form b to represent the whole space (V, b); for example, it will be often written b ∈ Q instead of writing the more formal (V, b) ∈ Q. Now, noting that the bijective isometries between different bilinear spaces (V, b) in Q are automorphisms of Q, it is trivial to verify the following assertion: Lemma 2.2. Let C be the set of all bijective isometries on Q. Then C is a group under the composition of functions. Definition 2.4. Let ϕ be a function Q × C → Q defined by ϕ : ((V, b),S) 7→ (V, b0), b0 = STbS, where S is an isometry S :(V, b0) ,→ (V, b). Further on, we write for short ϕ(S, b) := ϕ ((V, b),S) in which case ϕ(b, S) = S∗b. It can be verified that ϕ satisfies1

1◦ ϕ(b, S) ∈ Q, ∀S ∈ C, ∀b ∈ Q, (2.25) ◦ 2 ϕ(b, idV ) = b, ∀b ∈ Q, (2.26) ◦ 3 ϕ(ϕ(b, S1),S2) = ϕ(b, S1S2), ∀S1,S2 ∈ C, ∀b ∈ Q, (2.27) where idV is the identity transformation on Q deﬁned by idV :(V, b) 7→ (V, b). The function ϕ deﬁnes the (right) group action of C on Q. In addition, for every b ∈ Q, the orbit of b is

Orb(b) = C(b) := { ϕ(b, S) ∈ Q | ∀S ∈ C }, (2.28) and the stabilizer of b is Stab(b) := { S ∈ C | ϕ(b, S) = b }. (2.29)

We call b a ﬁxed point for the action iﬀ ϕ(b, S) = b, ∀S ∈ C, that is, when Orb(b) = {b} or Stab(b) = C. It can be proven2 that the stabilizer of b is a subgroup of C (also termed the isotropy group or isotropy subgroup at b) and that the orbit of b is the equivalence class of b under

1 ∗ ∗ ∗ For the composition ϕ(ϕ(b, S1),S2) = S2 (S1 b) = (S1 ◦ S2) b = ϕ(b, S1S2) (see Deﬁnition 1.13). 2These assertions are valid for any group [GN04]. 24 Chapter 2. The Square-Root Isometry

the equivalence relation on Q in which b1 is equivalent to b2 if and only if b2 = ϕ(b1,S) for some S ∈ C. With respect to the action Q × C → Q, the orthogonal group O(m, n) is the m,n stabilizer of the Minkowski space (R , η) ∈ Q, and, more generally, the orthogonal group O(V, b) is the stabilizer of a quadratic space (V, b). The orbit of η can be identiﬁed with the coset space of C/ O(m, n).3 Obviously, the orbit of η is the complete space Q, hence the action ϕ is transitive.4 This means that we can obtain any element in Q by a suitible vielbein. An immediate question is: Under what conditions can the square-root isometries be composed to form a subgroup of C and moreover, transitively stay closed under composition when acting on a subset of the quadratic spaces in Q constructed on V ? Our focus is therefore on group actions of the square-root isometry with a single orbit.

Starting from bi ∈ Q, i, j = 1, 2, 3 and Sij ∈ C, deﬁned by

T 2 b2 = S12b1S12 = b1S12, (2.30) T 2 b3 = S13b1S13 = b1S13, (2.31) T 2 b3 = S23b2S23 = b2S23, (2.32)

T unless we allow b1 = X b1X and XS13 = S12S23, it is easy to see that the necessary condition for transitivity is S12S23 = S23S12, i.e. that Sij commute. One obvious choice is then to consider matrix functions of S over the similar spectra. We know that any two diagonalizable operators are simultaneously diagonalizable if and only if they commute [Rom07, Hig08], hence the set CS of all functions F (S) forms a commuting family of operators acting on V . What we ﬁrst need is to ensure is that these operators stay the square-root isometries.

Lemma 2.3. Let (V, f) and (V, g) be two regular (real) quadratic spaces and S a bijective √ square-root isometry S :(V, f) ,→(V, g). Let further F be a (real-valued, regular) function deﬁned on the spectrum of S with respect to some basis. Then F induces a (bijective) square- root isometry √ F (S):(V, g0) ,→(V, g). (2.33) where g0 = F T(S) g F (S).

Proof. Since F (ST) = F T(S) and S = pg−1f, by Corollary 2.3 and Proposition 2.1 we can assert that gF (S) is symmetric. Therefore F (S) is self-adjoint relative to g, and it holds

g0 = gF 2(S), (2.34) hence, F (S) is a square-root isometry.

Remark 2.2. In the context of theories of massive spin-2 ﬁelds coupled to gravity, the function √ F (S) = I + S2 (yielding g0 = g(I + S2)) was ﬁrst analyzed in [HSMvS13].

3More generally, consider a group G and its action on a set X. The orbit of x ∈ X can be naturally identified with the coset space of G/H, where H is the stabilizer of x [GN04]. For example, let p be a timelike vector; then Stab(p) = SO(n), so the homogeneous space SO0(1, n)/ SO(n), which is the oriented hyperbolic space, corresponds to the momentum space of a massive particle. See Remark 3.2 on page 32. 4An action of a group G on a set X is called transitive iff the set is nonempty and it has only one orbit, i.e., if there exists x ∈ X such that Orb(x) = X, which is iff Orb(x) = X, ∀x ∈ X. Furthermore, it can be shown that the group action is transitive if and only if X is non-empty and if for any x, y ∈ X there exists a g ∈ G such that ϕ(g, x) = y (or ϕ(x, g) = y) where ϕ is a left (or right) group action [GN04]. 2.3 The Square-Root Group Actions and Orbits 25

Remark 2.3. According to the last lemma, we can construct a square-root isometry group only as a subset of C for a fixed S as the generator. This also fixes g. However, because gS is a symmetric bilinear form, every bijective square-root isometry S has the associated symmetrizing quadratic form h defined through S(g) = g−1h, and we can equally view h as the generator.

Definition 2.5. Let CS be a set of all regular functions F of S defined in the context of Lemma 2.3 for which F (S) is invertible. To every F ∈ CS we can associate a quadratic space (V, gF 2) which implies that we can define the corresponding set of regular quadratic spaces based on S as a subset of Q defined by

0 0 2 QS(g) := { (V, g ) ∈ Q | g = gF (S), ∀F ∈ CS) }, (2.35)

QS(g) will be called the coupled set of quadratic spaces generated by S (or by h where S = g−1h).

We note that CS includes the identity operator I ≡ idV and S itself. Thus:

Lemma 2.4. CS is a group under composition of functions. In view of Deﬁnition 2.4, QS ×CS → QS is the (right) group action of square-root isometries CS acting on the coupled set QS (g).

The orbit of g in QS given by the action of CS on g is

CS(g) := { ϕ(g, F ) ∈ QS(g) | ∀F ∈ CS }, (2.36) therefore C (g) = Q (g) which means the group action of C is transitive. This implies that S S S √ for every g1, g2 ∈ QS(g) there exists T ∈ CS such that T :(V, g1) ,→(V, g2).

Proposition 2.2. Let two regular quadratic spaces (V, g1) and (V, g2) be elements of the 2 −1 same coupled set QS(g) (or the orbit CS(g)). Then the square-root isometry T = g1 g2 generates the coupled set QT (g1) and QS(g) = QT (g1).

Proof. Consider S2 := g−1g where g , g ∈ Q (g). Let further F ,F ∈ C be regular ij i j i j √S i j √S functions on K = R such that Fi(S):(V, gi) ,→(V, g) and Fj(S):(V, gj) ,→(V, g) are bijective, which corresponds to the commutative diagram shown on Figure 2.2.

Sij (V, gi) (V, gj)

Fj(S) Fi(S) (V, g)

2 −1 Figure 2.2: The coupled quadratic spaces Sij := gi gj

Consequently, −1 −1 2 2 2 2 2 Sij = gFi (S) gFj (S) = Fi (S) Fj (S). (2.37)

Because CS is a commuting family and FiFj = FjFj, we obtain up to a sign

−1 −1 Sij = Fij(S) := (Fi(S)) Fj(S) = Fj(S)(Fi(S)) , (2.38) 26 Chapter 2. The Square-Root Isometry

where Fij ∈ CS. The last expression can be solved for S in terms of Sij, which concludes the 0 proof. The proposition holds even for subsets of CS ⊂ CS, if Fij retains the same algebraic 0 0 shape as Fi,Fj ⊂ CS, i.e. if Fij ⊂ CS.

Since I,S ∈ CS, both (V, g) and (V, f) are elements of QS(g), therefore (V, g) and (V, f) are elements of both QS(g) and QT (g1). This means that g and f can be reconstructed from any two g1, g2 ∈ QS(g). exp Remark 2.4. Consider a subset CS ⊂ CS deﬁned by

exp CS := { exp(ξS) ∈ CS | ∀ξ ∈ R }, (2.39)

exp exp Clearly, CS is a subgroup of CS since exp(ξ1S) exp(ξ2S) = exp ((ξ1 + ξ2)S) ∈ CS , 0 ∀ξ1, ξ2 ∈ R, the identity element is idV ∈ CS for ξ = 0 and the inverse of ξ is −ξ. No- exp 2 exp tice that CS cannot act on g and give f = gS (i.e., f∈ / CS (g)). Remark 2.5. We know that two arbitrary (square-root) isometries do not commute; however, the operators built from the same S constitute a commuting family of square-root isometries. This can be compared to the orthogonal transformations which do not commute in general, yet they commute when composed around the same ﬁxed axis (this axis is unique in special cases; cf. Euler’s rotation theorem). Furthermore, it can be compared to loxodromic transformations which are a conjugacy class of the restricted Lorentz group SO0(1, 3); namely, ∼ the loxodromic element of Spin0(1, 3) = SL(2, C) is ! exp (ξ/2) 0 P = P P = P P = , ξ = α + iβ ∈ , (2.40) 3 2 1 1 2 0 exp (−ξ/2) C and has the ﬁxed points ξ = 0 and ξ = ∞ on the Riemann sphere. The spinor map SL(2, C) → SO0(1, 3) converts P3 to the Lorentz transformation Λ3 = Λ2Λ1 = Λ1Λ2.

The Continuation Set. An interesting subset of QS(g) can be obtained from a subset 0 CS ⊂ CS deﬁned by 0 CS := { αI + βS ∈ CS | ∀α, β ∈ R }, (2.41) 0 2 0 where g = gF (S) ∈ QS is given by the symmetric bilinear form

0 2 2 2 g = g(αI + βS) = α g + 2αβgS + β f, ∀α, β ∈ R. (2.42)

This bilinear form corresponds to the eﬀective composite metric [dRHR14] wherein a new type of matter coupling was proposed for massive and bimetric theories (using this eﬀective composite metric, of course). In terms of vielbeins f = LTηL and g = ETηE, by using ES = ΛL from Lemma 2.1, we have

g0 = (αI + βS)TETηE(αI + βS) = (αE + βΛ)Tη(αE + βΛ). (2.43) thus g0 = (αE + βΛL)∗η. 2 0 2 0 2 −1 Now, let g1 = g (α1I +β1S) ∈ QS and g2 = g (α2I +β2S) ∈ QS. Then for S12 := g1 g2 2 we have g3 := g1(ξ1I + ξ2S12) and after some algebraic manipulation (the derivation is on p. 75) 2 g3 = g ((ξ1α1 + ξ2α2) I + (ξ1β1 + ξ2β2) S) , (2.44) 2.4 Coupled Symmetrization 27

∗ or in terms of vielbeins g3 = ((ξ1α1 + ξ2α2) E + (ξ1β1 + ξ2β2)ΛL) η. 00 0 Furthermore, let CS be a subset of CS such that

00 1 1 CS = { Fξ(S) ∈ CS | Fξ(S) = 2 (1 − ξ)I + 2 (1 + ξ)S, ∀ξ ∈ R }. (2.45) which generates 00 2 00 QS = { gξ ∈ QS | gξ = gFξ (S),Fξ(S) ∈ CS }. (2.46) 00 0 It is clear that CS can be deﬁned in terms of CS by setting:

1 1 ξ1 = 2 (1 − ξ), ξ2 = 2 (1 + ξ), (2.47) 1 1 α1 = 2 (1 − α), β1 = 2 (1 + α), (2.48) 1 1 α2 = 2 (1 − β), β2 = 2 (1 + β). (2.49)

After the composition Eq. (2.44), we obtain

2 1 0 1 0 g3 = g 2 (1 − ξ )I + 2 (1 + ξ )S , (2.50)

0 1 1 00 where ξ = 2 (1 − ξ)α + 2 (1 + ξ)β. Therefore g3 ∈ QS. 0 00 Observe that CS and CS are not subgroups of CS.

2 1 2 2 Remark 2.6. If we compactify R, then we have Fξ(S) = 4 ξ (I − S) in the limit ξ → ±∞, 2 i.e., Fξ(S) deﬁnes an isometry (I − S) up to the conformal factor ξ /4 (which corresponds to −S branch).

The function Fξ(S) is continuous with respect to the scalar ξ ∈ R. After redefining −1 S = g h, the isometry Fξ(S) geometrically behaves like an one-parameter ‘rotation’ of the T 2 5 null-cones { x ∈ V | x gξx = 0, gξ = gFξ (S) } with respect to a fixed h. Such defined set 00 QS will be called the continuation set of g with respect to h (or S). The continuation set will later be used in Chapter5.

2.4 Coupled Symmetrization

Earlier in Section 2.1 we established ES = ΛL as the relation between the isometries of m,n coupled quadratic spaces (V, f), (V, g) and (R , η). Here we narrow the condition on S for the case when S is self-adjoint, S = S0, either relative to g or to f (which are the equivalent assertions according to Proposition 2.1).

m,n ∗ Theorem 2.2. Let (V, g) and (V, f) be quadratic spaces isometric to (R , η) where f = L η and g = E∗η, and S an isometry from (V, f) to (V, g). For S to be self-adjoint, it is necessary m,n and suﬃcient that there exists an orthogonal transformation Λ ∈ O(m, n) on R such that the symmetrization condition holds

ETηΛL = (ETηΛL)T . (2.51)

5The ‘rotation’ in the sense of continuously tilting and deforming the null-cones in a complex way. The null-cones are treated in Section 3.6 on page 44. 28 Chapter 2. The Square-Root Isometry

Proof. From ES = ΛL ⇒ S = E−1ΛL and g = ETηE

gS = ETηΛL. (2.52)

On the other hand, the assertion S = S0 is equivalent to g(S(·), ·) = g(·,S(·)), i.e., gS = (gS)T, thus ETηΛL = (ETηΛL)T = LTΛTηE = LTηΛ−1E, (2.53) which completes the proof.

Remark 2.7. By simple algebraic manipulation, we can obtain the equivalent statements 6 T ηΛLE−1 = ηΛLE−1 , (2.54) −1 −1T ηL Λ−1E = ηL Λ−1E . (2.55)

Deﬁnition 2.6. We shall call the relation in Eq. (2.51) the vielbein symmetrization condition.

Remark 2.8. The relation in Eq. (2.51) is sometimes refered to as the “Deser-van Nieuwen- huizen gauge” as ﬁrst appeared in [DvN74]. It is also considered in [DMZ12] for the case m = 1 where m + n = 2, 3, 4. p Corollary 2.5. Over K = R, the square-root S = g−1f exists if and only if there exists Λ ∈ O(m, n) such that ETηΛL = (ETηΛL)T.

Proof. By using the last theorem and Corollary 2.4.

Remark 2.9. Theorem 2.2 and Corollary 2.5 establishes the basis independent correspondence between the existence of an square-root isometry and an orthogonal transformation. Although it seems that we are tied to some basis (as we rely on matrix notation), we are in fact working with transformations and their compositions. For example, ηΛLE−1 means −1 η[ ◦ Λ ◦ L ◦ E , and the symmetrization condition states

−1 −1 0 ] −1 T Λ ◦ L ◦ E = (Λ ◦ L ◦ E ) = η ◦ (Λ ◦ L ◦ E ) ◦ η[. (2.56)

The origin of the vielbein symmetrization condition is the symmetry of gS, i.e., without the self-adjointness S = S0, the map S would be just an unconstrained isometry. On the other hand, the symmetric part of the orthogonal transformation is its boost (see the polar decomposition theorem in 3.1.2 on page 31). Hence, there is a relation between the square- root isometry and the boosts. This will be elaborated in Chapter4, where the square- root symmetriziation condition will establish a one-to-one correspondence between the ratio of the shifts of two vielbeins (or the diﬀerence between their shift parameters) and the corresponding boost parameter of the symmetrizing orthogonal transformation.7 Morever, m,n 2 −1 by setting f = g = η and V = R , the square-root isometry S = g f becomes an involutory orthogonal transformation Λ2 = I, Λ ∈ O(m, n). This will eventually lead us to 2 2 −1 the set of alternative square-roots S2 satisfying S = S2 = g f, which will be explored and used in Chapter5.

6AB . . . XY Z is symmetric is equivalent to Z−1,TAB . . . XY , Y −1,TZ−1,TAB . . . X etc. are symmetric. 7For the correspondence see Eq. (4.10). The shift tranformation is stipulated by Deﬁnitions 3.3 and 3.6. Chapter 3

Geometry of Decompositions

The square-root symmetrization condition (2.51) relates the three basic ingredients: quadratic spaces, isometries and orthogonal transformations (Figure 2.1 on page 19). In Chapter1, we elaborated how a real isotropic quadratic space admits both the orthogonal and hyperbolic splittings. At the same time, it was also demonstrated that any regular quadratic space m,n over reals is isometric to R , for unique m, n. Based on these premises, here we proceed and further develop the orthogonal decompositions of quadratic spaces in addition to the m,n decompositions of their isometries to R (vielbeins). This is done both in matrix notation as well as in the component form. At the end, this will give rise to the so-called m + n decomposition, which will later be used to study the square-root symmetrization condition in more detail. As our interest lies in the geometry of such decompositions, we explore the structure of the orthogonal transformations ﬁrst.

3.1 Structure of O(m, n)

The structure of the orthogonal group can be, with the purpose of determining its isometry classes, examined in the context of Lie groups and Lie algebra [Hal03, Sze04, Lan02]. Namely, since the orthogonal group O(V, b) is a closed subgroup of GL(V ),1 it is a Lie group. On the other hand, if a subgroup G ⊂ GL(V ) is a Lie group, then the associated Lie algebra is the subspace of all endomorphisms S : V → V such that exp(λS) ∈ G for all scalars λ in K. Proposition 3.1. The Lie algebra of the orthogonal group O(V, b) reads 2

o(V, b) = { S ∈ End(V ) | b(Sv, w) + b(v, Sw) = 0, ∀v, w ∈ V }. (3.1)

A detailed proof is presented in [Mei13]. Consequently, S ∈ o(V, b) if and only if S is a skew-adjoint transformation relative to b, where

dim o(V, b) = d(d − 1)/2, d = dim V. (3.2)

1 A matrix Lie group G is a subgroup of the general linear group GL(n; C) such that, if any sequence of matrices in G converges to some matrix A, then either A ∈ G, or A is singular, i.e., such that G is a closed subset of GL(n; C). This is equivalent to stating that a matrix Lie group is a closed subgroup of the general linear group. Most of the interesting subgroups of GL(n; C) are closed, hence the condition that a subgroup is closed should be regarded as a pretty formal. A typical counterexample is the set of all non-singular matrices whose entries are real rational numbers, which is not closed [Hal03]. 2Using the exponential map of matrices, exp : End(V ) ⊇ o(V ) → O(V ) ⊆ GL(V ).

Mikica Kocic, The Square-Root Isometry, MSc thesis, SU 2014 Printed: 2015-03-11 30 Chapter 3. Geometry of Decompositions

Thus, the dimension of O(m, n) agrees with the dimension of the associated Lie algebra

dim o(m, n) = d(d − 1)/2, d = m + n. (3.3)

As it will be shown, the orthogonal transformation O(m, n) admits the polar decomposition. The polar decomposition O(m, n) is, however, based on the polar decomposition of real matrices, which will be discussed ﬁrst.

3.1.1 Polar Decomposition of O(n)

Theorem 3.1 (Polar Decomposition). Let V be a ﬁnite-dimensional vector space over K = C and let S be any linear operator on S ∈ End(V ). Then there exist a unitary operator H U ∈ End(V ),U U = idV and a non-negative operator P ∈ End(V ), kP k ≥ 0 such that S = PU. The non-negative operator P is unique. If S is invertible, the operator U is also unique.

The proof is given in [HK71, pp. 342–343]. In terms of matrices, the corresponding statement is that every non-singular matrix S ∈ Mat(n, C) can be uniquely factored as 1 S = PU, where P is Hermitian positive definite and U is unitary. If V = C , dim V = 1, we recover the polar form of a complex number z = ρeiθ, ρ ≥ 0 (in such case Hermitian positive semi-definite matrices are non-negative real numbers, and unitary matrices are points on the unit circle). Since our primary interest are real quadratic spaces, we need to work out the polar decomposition of the general linear group over reals. We shall follow the prescriptions given in [Mei13]. Let Sym(n) be the vector space of all real symmetric n × n square matrices, and let Sym+(n) be its subspace of positive definite matrices. The map

+ exp : gl(n, R) ⊃ Sym(n) → Sym (n) ⊂ GL(n, R), (3.4) and its inverse log : Sym+(n) → Sym(n) are both diﬀeomorphisms. Consequently, the map Sym(n) × O(n) → GL(n, R), deﬁned by

X (X,R) 7→ S = e R,X ∈ Sym(n),R ∈ O(n),S ∈ GL(n, R), (3.5) is a diﬀeomorphism. The inverse map GL(n, R) → Sym(n) × O(n) deﬁned by

S 7→ (X,R),X = log |S| ,R = |S|−1 S, |S| = (SST)1/2, (3.6) is called the polar decomposition of the general linear group GL(n, R). We shall see that O(n) is a maximal compact subgroup of GL(n, R).

Proposition 3.2. [Mei13] Let G be a closed subgroup of GL(n, R) that is invariant under the transpose A 7→ AT. Consider the restriction K = G∩O(n); then K is a maximal compact subgroup of G.

Proof. Clearly, the polar decomposition for GL(n, R) restricts to G. Now, introduce

+ P = G ∩ Sym (n) ⊂ GL(n, R) and p = g ∩ Sym(n) ⊂ gl(n, R). (3.7) 3.1 Structure of O(m, n) 31

The diffeomorphism exp : Sym(n) → Sym+(n) restricts to exp : p → P and its inverse restricts to log : P → p, both staying diffeomorphisms. Thus, the polar decomposition for GL(n, R) restricts to a diffeomorphism K×p → G of which inverse is the polar decomposition of G. On the other hand, for all X 6= 0 ∈ Sym(n), the closed subgroup of GL(n, R) generated by eX is non-compact. Namely, suppose keX k > 1 (if not, we can replace X with −X). Then

lim keαX k = lim keX kα = ∞, (3.8) α→∞ α→∞ Therefore, K is a maximal compact subgroup of G.

3.1.2 Polar Decomposition of O(m, n)

The same recipe from the proof of Proposition 3.2 can be applied to the orthogonal group m+n ˆ ˆ ˆ ˆ O(m, n). Let R be a quadratic space with the bilinear form δ ≡ δm+n = δm ⊕ δn. m+n Consider the endomorphism J ∈ End(R ) deﬁned by

η(v, w) = δ(v, J(w)) (3.9) where η = (−δˆm) ⊕ δˆn. We obviously have J = (−Im) ⊕ In in block form, because S acts as m the minus identity map when restricted to R ⊕ {0Rn } and the identity map when restricted n m+n to {0Rm } ⊕ R . Thus an endomorphism of R commutes with J if and only if it preserves m n the decomposition R ⊕ R . On the other hand, for any matrix Λ ∈ O(m, n) it is necessary and suﬃcient that Λ0JΛ = J where the adjoint Λ0 = δˆ−1ΛTδˆ is relative to δˆ (not relative to η). This follows from η(·, ·) = η(Λ(·), Λ(·)) = δˆ(Λ(·),J ◦ Λ(·)) and η(·, ·) = δˆ(·,J(·)), i.e. from η = ΛTδJˆ Λ and η = δJˆ in matrix notation, so that

δˆ−1ΛTδˆ JΛ = J ⇒ Λ0JΛ = J. (3.10)

Since δˆ is Euclidean, the adjoint and transpose coincide Λ0 = ΛT in terms of matrices, so we can sloppily write ΛTJΛ = J. In the same manner, X ∈ o(m, n) iﬀ X TJ + JX = 0. Thus, as O(m, n) is invariant under the involution Λ 7→ ΛT, the polar decomposition can be applied. Now, in view of Proposition 3.2, let G = O(m, n). Elements B in K = G ∩ O(m + n) are characterized by BTJB = J and BTB = I, that is BJ = JB when combined. Since B m n and J commute, B preserves the decomposition R ⊕ R , therefore B ∈ O(m) × O(n) and consequently K = O(m) × O(n) is the maximal subgroup of O(m, n). Furthermore, for all X ∈ p = Sym(m + n) ∩ o(m, n) we have X TJ + JX = 0 and X T = X, that is, X is symmetric and has the oﬀ-diagonal block form, which means that p has the correct characterization and, consequently, O(m, n) admits the polar decomposition. 3 Similar assertions can be proved for SO(m, n) and SO0(m, n), and we arrive at the following proposition [Mei13]:

Proposition 3.3. Relative to the polar decomposition of GL(m + n, R), the maximal subgroups of

G = O(m, n), SO(m, n), SO0(m, n), (3.11)

3 SO0(m, n) admits the polar decomposition because SO(m)×SO(n) is the identity component of S(O(m)× O(n)). 32 Chapter 3. Geometry of Decompositions are, respectively,

K = O(m) × O(n),S(O(m) × O(n)), SO(m) × SO(n). (3.12)

As the consequence, assuming both m and n are nonzero, the Lie group O(m, n) comprises four connected components while SO(m, n) comprises two connected components (in- tuitively, as manifolds, these groups consist of a number of topologically separate pieces). Moreover, the group O(m, n) is compact only when m = 0 (or n = 0), and in such case it has two connected components. The set of these four connected components of O(m, n) has a group structure of the quotient group H = O(m, n)/ SO0(m, n). Each element in O(m, n) can be written as the semidirect product SO0(m, n) o H of an element of SO0(m, n) (which is a normal subgroup of O(m, n)) and an element of the discrete group H = {I,P,T,PT }, where T and P are given by

T := (−Im) ⊕ In,P := Im ⊕ (−In). (3.13) The operators T and P reverse the respective orientations on the m and n dimensional subspaces on which the form is deﬁnite. Notice that T and P commute. The action of H can be seen in the change of sign of the determinants det Rm and det Rn in Eq. (3.16) below. Remark 3.1. The special case when m = 1, n ≥ 1 is called the Lorentz group. The Lorentz transformations which have determinant −1 are said to be improper and which have determinant +1 are called proper. The Lorentz transformations that preserve the direction of time are said to be orthochronous. The Lorentz transformations that are both proper and orthochronous are said to be restricted, and they form a subgroup denoted by SO0(1, n). The discrete transformation T is called the time reversal and P is called the space inversion. 1,n 1,n Remark 3.2. Again, consider R . Now, deﬁne Hn(α) = { v ∈ R | −α = η(v, v) } where α > 0; then, Hn is a hyperboloid of two sheets. It can be shown [Gal05] that Hn(α) has + − two connected components Hn (α) and Hn (α), as well that Λ ∈ SO(1, n) preserves Hn(α), + − ∀α > 0. Moreover, any Λ ∈ SO0(1, n) preserves both Hn (α) and Hn (α). The action + + + SO0(1, n) × Hn (α) → Hn (α), α > 0, is transitive, and Hn (1) is a homogeneous space + ∼ Hn (1) = SO0(1, n)/ SO(n) that corresponds to the momentum space of a massive particle.

m,n Proposition 3.4. For the orthogonal transformation Λ ∈ O(m, n) on R , we have the polar decomposition 4

Λ = BR,B = B0 = δˆ−1BTδ,ˆ R−1 = R0 = δˆ−1RTδ,ˆ (3.14)

√ ! ! I + p0p p0 0 x0 m √ X B = 0 ,B = e ,X = , (3.15) p In + pp x 0 ! R 0 m 0 ˆ−1 T ˆ R = , p = δm p δn, (3.16) 0 Rn where p is an m×n matrix with real entries.

4This is known as the left polar decomposition (writing the positive deﬁnite matrix on the left), whereas the decomposition Λ = R0B0 (which we started from) is known as the right polar decomposition. Of course, RB 6= BR, in general, but it can be veriﬁed there will exist two other transformations B0 and R0 such that R0B0 = BR. To use the left form BR is somewhat more practical. 3.1 Structure of O(m, n) 33

Remark 3.3. p0p and pp0 are square matrices which are not necessarily invertible.

T T Proof. Observe that R δRˆ = δˆ is the same as R ηR = η since R = Rm ⊕ Rn. For the elements of the space B ∈ exp(p) where p = Sym(m + n) ∩ o(m, n), the proof is done by induction, realizing ! ! (xTx)k 0 0 (xTx)kxT X2k = ,X2k+1 = . (3.17) 0 (xxT)k x(xTx)k 0

From this we obtain an expression for eX as a function of xTx that will contain cosh(xTx) and sinh(xTx). This function should be interpreted as the entire holomorphic function in terms of the power series expansion [Mei13]; see also [Gal05]. Otherwise, the proof is trivial.

Remark 3.4. The diagonal elements in Eq. (3.15) can be written

q q ˆ−1 T ˆ ˆ−1 ˆ T ˆ Λt = Im + δm p δnp = δm δm + p δnp, (3.18) and similarly q q −1 T −1 −1 T Λs = In + pδˆm p δˆn = δˆn + pδˆm p δˆn, (3.19) Therefore, we have  q  ˆ−1 ˆ T ˆ ˆ−1 T ˆ ! δm δm + p δnp δm p δn Rm 0 Λ = BR =  q  . (3.20) −1 −1 T 0 R p δˆn + pδˆm p δˆn n

Deﬁnition 3.1. In view of Proposition (3.4), we refer to the symmetric part B of Λ ∈ m n O(m, n) as the hyperbolic rotation or the boost between R and R (where the direction depends on the sign of p), and to the orthogonal part R of Λ as the circular rotation or m n simply the rotation on the individual R , R subspaces. q q ˆ−1 ˆ T ˆ ˆ−1 ˆ−1 T ˆ Lemma 3.1. Let Λt = δm δm + p δnp and Λs = δn + pδm p δn. Then

ˆ−1 T ˆ 0 ˆ−1 T ˆ 0 Λt = δm Λt δm = Λt, Λs = δn Λs δn = Λs, (3.21) and 0 ˆ−1 T ˆ ˆ−1 T ˆ 0 Λtp = Λtδm p δn = δm p δnΛs = p Λs, Λsp = pΛt. (3.22)

Proof. Consider more general case where A ∈ Mat(n × m, C), B ∈ Mat(m × n, C). Let F be a function deﬁned on the spectrum of αIn + AB as well on α ∈ C. Then from Corollary 2.2, we obtain F (αIn + AB) A = AF (αIn + BA).

Corollary 3.1. The inverse of the boost B(p) is B−1(p) = B(−p).

Proof. By the last lemma, also noting that Λt(−p) = Λt(p) and Λs(−p) = Λs(p), we evaluate B(p)B(−p) = I.

2 2 2 2 Lemma 3.2. kΛtk ≥ 1 and kΛsk ≥ 1, and also kΛ|Rm⊕{0}k ≥ 1 and kΛ|{0}⊕Rn k ≥ 1.

T T Proof. Clearly, from kp δˆspk ≥ 0 and kpδˆtp k ≥ 0.

Remark 3.5. Notice the rank ρ(p0p) = ρ(pp0) = ρ(p) = ρ(p0) ≤ min (m, n) [HJ94, Lan02]. 34 Chapter 3. Geometry of Decompositions

Reparametrization. Lemma 3.1 allows us to reparametrize the symmetric part of the orthogonal transformation B by introducing a new parameter ±v deﬁned through

p = ±Λsv = ±vΛt. (3.23)

In terms of the new parameter v, the block-diagonal parts of B become

2 ˆ−1 T ˆ ˆ−1 T T ˆ ˆ−1 T ˆ ˆ−1 T ˆ Λt = Im + δm p δnp = Im + δm Λt v δnvΛt = Im + δm Λt δmδm v δnvΛt. (3.24)

0 ˆ−1 T ˆ −1 After using Λt = δm Λt δm = Λt and then multiplying both sides with Λt , we obtain q q −1 ˆ−1 T ˆ −1 ˆ−1 T ˆ Λt = Im − δm v δnv, Λs = In − vδm v δn (3.25) which yields the √ ! ! I + p0p p0 Λ ±Λ v0 m √ t t B = 0 = . (3.26) p In + pp ±vΛt Λs

This reparametrization reduces the parameters space of p (which is R) to the open ‘ball’ T kv δˆnvk < 1.

Example 3.1. For m = 1, n ≥ 1 (the Lorentz group), p can be considered as the n- dimensional column vector representing a boost parameter p = −γv, where the gamma 2 −1/2 T factor reads γ = Λt = (1 − |v| ) (this implies the rank ρ(pp ) = 1). The sign in front of γv is selected to represent the passive Lorentz transformations. The parameter space of the n-velocity v is 0 ≤ |v|2 < 1 and the Lorentz boost reads

T ! c ! γ −γv A γ −γv δcb B = γ2 0 ,B B = a γ2 a c . (3.27) −γv In + γ+1 vv −γv In + γ+1 v v δcb

1,n n The range of indices is A, B, · · · ∈ {0, 1, 2 . . . , n} for R , and a, b, · · · ∈ {1, 2, . . . , n} for R . 5 Observe that the velocity v is the eigenvector of Λs with the eigenvalue γ because Λsv = γv.

3.2 Partitioned Matrices

Consider the Minkowski space

m,n m n m ˆ n ˆ (R , η) = −1 ⊥ +1 = (R , −δm) ⊥ (R , δn). (3.28) L M L M m+n The matrix of the bilinear form η with respect to the canonical basis of R is partitioned ! −δˆm 0 η = = (−δˆm) ⊕ δˆn, (3.29) 0 δˆn

m+n m n so that the form η naturally decomposes the underlying vector space R into R ⊕ R . This was used in the last section to study the structure of O(m, n).

5 a a 0 ˆ−1 T ˆ 0 00 c Notice that the matrix v 0 becomes a column vector v . Hence, v = δm v δn reads (v )a = δ v0 δca = c v δca = va. 3.2 Partitioned Matrices 35

A similar decomposition can be applied to any regular quadratic space (V, b) that is m,n isometric to (R , η). This may be done through bunching of basis vectors of the vector 6 space V into two groups, ‘t’ and ‘s’, such that V = Vt ⊕ Vs, m = dim Vt, n = dim Vs. In such basis, the matrix associated to the bilinear form b will be block partitioned. Now, since m, n ≥ 1, the space (V, b) is necessarily isotropic and because every regular isotropic quadratic space is split by a hyperbolic plane (see Proposition 1.1), we can argue that, after the splitting, the restriction of b on Vs will be positive-definite in some basis and, if the matrix of b has no off-diagonal diagonal elements, the restriction of b on Vt will be negative-definite at the same time. However, in most cases there will be some off-diagonal elements and to be able to deal with these we need to acquire some additional matrix tools. Remark 3.6. Of course, we can always diagonalize a regular quadratic form and then, by using Corollary 1.2, permute the basis to achieve such decomposition. This is not always possible in general when we consider coupled quadratic spaces that are sharing a common basis.

Deﬁnition 3.2. Let M be a (m+n)×(m+n) matrix over K partitioned as

! AB M = , (3.30) CD where A ∈ Mat(m×m, K), B ∈ Mat(m×n, K), C ∈ Mat(n×m, K) and D ∈ Mat(n×n, K). If A is non-singular, the Schur complement of A is a n×n matrix deﬁned by,

S := D − CA−1B (3.31) and similarly, if D is non-singular, the Schur complement of D is a m×m matrix deﬁned by

T := A − BD−1C. (3.32)

Proposition 3.5. Consider a partitioned matrix M as stipulated in the last deﬁnition. If A and S are both non-singular, then

!−1 ! AB A−1 + A−1BS−1CA−1 −A−1BS−1 = . (3.33) CD −S−1CA−1 S−1

If B and T are both non-singular, then

!−1 ! AB T −1 −T −1BD−1 = . (3.34) CD −D−1CT −1 D−1 + D−1CT −1BD−1

Proof. The proof is by direct veriﬁcation; see [Ber09, Section 2.8] and [GVL96, Mey00].

As an immediate consequence we have the following two assertions.

Corollary 3.2 (Sherman-Morrison-Woodbury formula). Let A ∈ Mat(n × n, K), U ∈ Mat(n × m, K), V ∈ Mat(m × n, K) and C ∈ Mat(m × m, K). If A and C−1 + VA−1U are

6The allusion to ‘time’ and ’space’ is deliberate (the alternative would be to use ‘v’ and ‘h’ for verti- cal/horizontal). 36 Chapter 3. Geometry of Decompositions non-singular then A + UCV is non-singular and

−1 (A + UCV )−1 = A−1 − A−1U C−1 + VA−1U VA−1. (3.35)

Corollary 3.3. Let T ∈ Mat(m × m, K), X ∈ Mat(m × n, K), Y ∈ Mat(n × m, K) and S ∈ Mat(n×n, K). If T and S are both non-singular, then

! ! T −1 + XSYXS T −TX M = ,M −1 = . (3.36) SYS −YTS−1 + YTX

Proof. At ﬁrst, substituting T −1 → T , B → BD, C → DC, A = T +BD−1C → T −1 +BDC into Eq. (3.34) yields

!−1 ! T −1 + BDCBD T −TB = . (3.37) DCD −CTD−1 + CTB

Finally, after substitution B → X, C → Y and D → S we obtain Eq. (3.36).

Remark 3.7. Both M and M −1 can further be decomposed ! ! ! ! T −1 + XSYXS I X T −1 0 I 0 M = = m m , (3.38) SYS 0 In 0 S YIn ! ! ! ! −1 T −TX Im 0 T 0 Im −X M = −1 = −1 . (3.39) −YTS + YTX −YIn 0 S 0 In

Here, the corresponding linear transformations T and S can be considered as operators acting ∼ ∼ on the respective subspaces VT = Mat(m×m, K) and VS = Mat(n×n, K) while X and Y as the transformations between such subspaces, i.e., X : VS → VT and Y : VT → VS. Notice that, if −T and S are positive deﬁnite, we can use the Cholesky decomposition −1 T T −1 T = −Et Et, S = Es Es to additionally split the block diagonal matrix T ⊕ S as

! !T ! ! T −1 0 E 0 −I 0 E 0 = t m t . (3.40) 0 S 0 Es 0 In 0 Es

−2 T Example 3.2. Let T = −α ∈ R, β = Y = X ∈ Mat(n × 1, R) and γ = S ∈ R. Then

!−1 ! ! −α2 + βTγβ βTγ −α−2 α−2βT −1 βT = = α−2 . (3.41) γβ γ α−2β γ−1 − α−2ββT β α2γ−1 − ββT

This is an expression of the so-called 3+1 decomposition of a metric tensor in GR.

3.3 m+n Decomposition of Quadratic Forms

m,n Let (V, g) be a regular quadratic space isometric to (R , η) and E = ei i∈I be an ordered basis of V where I is a ﬁnite index set. A decomposition J K

V = Vt ⊕ Vs, m = dim Vt, n = dim Vs, (3.42) 3.3 m+n Decomposition of Quadratic Forms 37

implies that we split the basis E = Et ∪ Es (without permuting the elements) such that Et and Es span Vt and Vs, respectively; i.e.

Et = ei i∈It , Es = ej j∈Is , I = It ∪ Is, (3.43) J K J K

In such decomposition, we denote the identity maps on Vt and Vs as It and Is. t −1 Now, let g = g| be the restriction of g on V and also g = g | ∗ the restriction s Vs s Vt −1 ∗ of g on Vt . Subsequently, in matrix notation we have gs ≡ gσ1σ2 , σ1, σ2 ∈ Es and t τ1τ2 J K −1 g ≡ g , τ1, τ2 ∈ Et. Since gs is non-degenerate, we can introduce ν = gs gσ1τ2 as a J K J K linear transformation ν : Vt → Vs. Using Eq. (3.38), we can write

! !T ! ! gt,−1 + νTg ν νTg I 0 gt,−1 0 I 0 g = s s = t t , (3.44) gsν gs ν Is 0 gs ν Is

  T t t T ! t ! ! −1 g −g ν It 0 g 0 It 0 g =   = . (3.45) t −1 t T −1 −νg gs + νg ν −ν Is 0 gs −ν Is

Hence, the linear transformation containing the oﬀ-diagonal ν in Eq. (3.44) acts as an isom- t,−1 etry from (V, g) to the quadratic space (V, g ⊕ gs).

Deﬁnition 3.3. Let n be a linear transformation Vt → Vs where V = Vt ⊕ Vs. A map Ξ(ν) ∈ End(V ) deﬁned by ! I 0 Ξ(ν) := t (3.46) ν Is is called the shear transformation, which we also refer to as the shift. The linear map

ν ∈ Hom(Vt,Vs) will be called the shift parameter.

Clearly, det Ξ(ν) = 1 for all ν ∈ Hom(Vt,Vs), so any shift is invertible and its inverse reads Ξ−1(ν) = Ξ(−ν). (3.47)

Therefore, the shift induces a bijective isometry on any quadratic space on V . Moreover, the shifts composes additively and also commute

Ξ(ν1) ◦ Ξ(ν2) = Ξ(ν1 + ν2) = Ξ(ν2) ◦ Ξ(ν1), ∀ν1, ν2 ∈ Hom(Vt,Vs), (3.48)

and, since Ξ(0Hom(Vt,Vs)) = idV , they form an abelian group. This group acts on V as the ‘s’-part translation relative to the ‘t’-part, i.e., it acts as a partial boost on the ‘s’-part translating all the vectors parallel to Vs. More precisely, a shift Ξ(ν) acts only on the ‘s’ part x of some vector (t, x) ∈ V as 7

! ! ! It 0 t t = , t ∈ Vt, x ∈ Vs. (3.49) ν Is x x + νt

As a geometric transformation, the shift (shear) preserves the (m+n)-dimensional measure (volume) of any set in V , which follows from det Ξ(ν) = 1.

7 1,1 In the simplest example R , t and x are scalars. Notice also that, if we assign a physical dimension T −1 to the elements of Vt and L to the elements of Vs, the shift parameter ν will have dimension LT (where the dimension of the quadratic form is T−2 ⊕ L−2). 38 Chapter 3. Geometry of Decompositions

Deﬁnition 3.4. Consider a regular quadratic space (V, g) on V = Vt⊕Vs with the restrictions t −1 g = g| and g = g | ∗ . A symmetric bilinear form deﬁned by s Vs Vt

t,−1 gˆ = g ⊕ gs, (3.50)

−1 t −1 will be called the core of the bilinear form g, and its inverse reads gˆ = g ⊕ gs . Theorem 3.2. Any regular quadratic space (V, g) admits the m+n decomposition

∗ t,−1 g = Ξ(ν) g ⊕ gs , (3.51)

∗ −1 T ∗ −1 t −1 that is, g = Ξ(ν) gˆ and g = Ξ(−ν ) gˆ . Moreover, det g = detg ˆ = (det g ) det gs.

t ∗ Deﬁnition 3.5. If both g and gs are positive deﬁnite, the m+n decomposition g = Ξ(ν) gˆ is said to be proper and the core gˆ will be called the (proper) Minkowski core.

∼ m,n Theorem 3.3. A regular quadratic space (V, g) = (R , η) admits a proper m+n decomposition if and only if there exists a vielbein Eˆ in the lower-triangular form such that g = Eˆ∗η.

Ξ(ν) (V, g) (V, gˆ)

et ⊕ es Eˆ m,n (R , η)

∗ ∗ Figure 3.1: A proper m + n decomposition g = Ξ(ν) (et ⊕ es) η

t t Proof. Let gs (or −g ) be positive deﬁnite. Then −g (or gs) is necessarily positive deﬁnite t t,−1 T ˆ T ˆ so both −g and gs admit the Cholesky decompositions g = −et δtet and gs = es δses, t,−1 ∗ˆ ∗ˆ i.e., g = −et δt and gs = es δs, such that et ∈ End(Vt) and es ∈ End(Vs) are bijective operators with the corresponding matrices in the lower-triangular form relative to a given basis of V . Subsequently

!T ˆ ! ! et 0 −δt 0 et 0 ∗ gˆ = , gˆ = (et ⊕ es) η. (3.52) 0 es 0 δˆs 0 es

When combined with the shift Ξ(ν), we can deﬁne the vielbein Eˆ in the lower-triangular form ! ∗ et 0 g = Eˆ η, Eˆ = (et ⊕ es) ◦ Ξ(ν), Eˆ = . (3.53) esν es Conversely, for an arbitrary non-singular vielbein in the lower triangular form ! e 0 Eˆ = t , (3.54) est es

ˆ −1 we have det E = det et det es 6= 0, therefore we can introduce the shift ν = es est. As the result, the core deﬁned by Ξ(−ν)∗g will be a Minkowski core.

Deﬁnition 3.6. A vielbein in the lower-triangular form will be called the proper vielbein. 3.4 Vielbeins in Diﬀerential Geometry 39

2 Remark 3.8. det g = −(det et det es) , and the inverse of a proper vielbein Eˆ is given by

−1 ! ˆ−1 −1 −1 −1 ˆ−1 et 0 E = ((et ⊕ es) ◦ Ξ(ν)) = Ξ(−ν) ◦ (et ⊕ es ), E = −1 −1 . (3.55) −νet es

3+1 Decomposition in GR

A well-known application is the 3+1 decomposition of spacetime in general relativity [Gou12]. The 3+1 formalism is based on the slicing of the four-dimensional spacetime by three- dimensional spacelike hypersurfaces, eﬀectively amounting to a decomposition of spacetime into “space”+“time”.8 This space+time splitting is, however, not an inferred structure of GR but relies on arbitrary choice of a time coordinate. The metric components in terms of 3+1 quantities are given

! 2 k l k ! g00 g0j −N + β γklβ β γlj gµν = = l , (3.56) gi0 gij γilβ γij or in terms of line elements

2 µ ν 2 2 i i j j ds = gµνdx dx = −N dt + (dx + β dt)γij(dx + β dt). (3.57)

In this decomposition, N is the lapse or lapse function, β is the shift vector and γ is the ij ij spatial metric (notice that gij = γij but g 6= γ ). When compared to Eq. (3.44), we have t 00 −2 t gs ≡ γ, g ≡ g = −N and ν ≡ β. The major difference is that, in our treatment, −g may or may not be positive definite. Only if it is positive definite, it will allow Cholesky t,−1 T ˆ decomposition g = −et δtet, in which case a triangular et will serve as the lapse variable.

Historical note. The 3+1 formalism is known from a long time ago. According to [Gou12] and [CB14], it originates from studies by Darmois in the 1920s [Dar27], Lichnerowicz in the 1930–1940s [Lic39, Lic44, Lic52] and Choquet-Bruhat in the 1950s [FB52, FB56]9. In 1952, Choquet-Bruhat was able to show that the Cauchy problem arising from the 3+1 decomposition has locally a unique solution. In the late 1950s and early 1960s, the 3+1 formalism was employed to develop Hamiltonian formulations of general relativity by Dirac [Dir58, Dir59], and Arnowitt, Deser and Misner (the so-called ‘ADM formalism’) [ADM08]. Around the same time, Wheeler coined the names lapse and shift [Whe64]. Finally, the global existence and uniqueness theorem for the Einstein equations was proven by Choquet-Bruhat and Geroch in 1969 [CBG69].

3.4 Vielbeins in Diﬀerential Geometry

∼ m,n By Definition 1.21, for a regular quadratic space (V, g) = (R , η), we have identified the pullbacks (or isometries) E under which g = E∗η as the vielbeins of g. On the other hand in differential geometry, a vielbein is an orthonormal frame locally attached to every point in

8More properly, in our notation m+n, this would be a 1+3 decomposition. We retain more common name, however, to avoid the confusion with 1+3 formalism where the basic structure is obtained by splitting the spacetime by a set of one-dimensional mostly time-like curves. 9Yvonne Choquet-Bruhat was Fourès-Bruhat at that time. 40 Chapter 3. Geometry of Decompositions

the spacetime (M, g) on a manifold M equipped with a metric gµν(x). These orthonormal frames are also called non-coordinate or non-holonomic bases. ] To be more precise, let P be a point in a diﬀerentiable manifold M and let further ∂µ|P µ J K and dx[ |P be the (ordered) coordinate bases for the tangent TP (M) and the cotangent space ∗ J K TP (M), respectively. Remark 3.9. We use the Greek indices µ, ν, . . . to denote the components relative to the coordinate bases. These indices are often called the coordinate (or world or spacetime) indices. In such bases the metric g(x) reads

µ ν µ ν (µ ν) g(x) = gµν(x)dx[ dx[ , dx[ dx[ := dx[ ⊗ dx[ . (3.58)

] µ µ ] µ Although the bases ∂µ|P and dx[ |P satisfy the biorthogonality relation dx[ (∂ν) = δν , they are not orthonormalJ K since inJ generalK

] ] g(∂µ, ∂ν) = gµν 6= ηµν. (3.59)

] However, we can introduce a set of tangent vector fields EA|P that span TP (M) at every point P and satisfy J K ] ] g(EA,EB) = ηAB, (3.60) m,n where ηAB is the Minkowski metric of (R , η). Equally we introduce their dual one-forms A A ] A −1 A B E[ |P building a dual basis defined by E[ (EB) = δB where obviously g (E[ ,E[ ) = J AB K ] A η . Such constructed, the vector and covector fields, EA(x) and E[ (x), form the local orthonormal bases called vielbeins.10 Remark 3.10. We use the roman indices A, B, . . . to denote the components relative to the orthonormal bases. These indices are often called the Lorentz indices. In such bases the metric g can be decomposed

A B A B (A B) g(x) = ηABE[ (x)E[ (x),E[ E[ := E[ ⊗ E[ . (3.61)

The components of the vielbeins are given by

µ −1 µ A A A µ dx[ = (E ) AE[ ,E[ = E µdx[ , (3.62) ] ] A ] ] −1 µ ∂µ = EAE µ,EA = ∂µ(E ) A, (3.63)

A −1 where E is the matrix with the coeﬃcients (E µ) while E is its inverse. After organizing ] ] ] ] µ A ∂ ≡ ∂µ and E = EA as row vectors while dx[ ≡ dx[ and E[ ≡ E[ as column vectors,J theK equivalentJ statementK in matrix notation is J K J K

−1 dx[ = E E[,E[ = E dx[, (3.64) ∂] = E] E,E] = ∂] E−1. (3.65)

10 1,3 ‘Vielbein’ is the generic name. In the case V =∼ R , where d = m + n = 4, these frames are rather called tetrads or vierbeins (and similarly dreibeins when d = 3, and zweibeins when d = 2). Notice that, by A ] A some authors, the bases E[ and EA are called the non-coordinate bases, while the coeﬃcients E µ and µ J K J K EA of such bases are called the vielbeins; e.g., see [Nak03, p. 283]. 3.4 Vielbeins in Diﬀerential Geometry 41

µ ] µ A ] A From this we can reassert dx[ (∂ν) = δν and E[ (EB) = δB

µ ] −1 µ A ] B −1 µ A ] B −1 µ A B dx[ (∂ν) = (E ) AE[ EBE ν = (E ) A E[ (EB) E ν = (E ) AδBE ν. (3.66)

Subsequently, the metric g can be expressed in terms of the vielbeins as g(·, ·) = η(E(·),E(·)) where the components of the metric are given by

] ] ] A ] B A ] ] B A B gµν = g(∂µ, ∂ν) = g(EAE µ,EBE ν) = Eµ g(EA,EB)E ν = Eµ ηABE ν (3.67) or in matrix notation g = ETηE. Similarly

] ] ] ] gµν = η E(∂µ),E(∂ν) = η(Eµ,Eν), (3.68)

] ] where Eµ = E(∂µ) are the components of the vielbein in the coordinate basis. Hence, a vielbein is nothing more than a local bijective isometry

∼ m,n E(x):(TP (M), g(x)) = (R , η), (3.69) or the pullback g(x) = E(x)∗η. A −1 µ If we view the ﬁelds E = E µ as fundamental (having the inverses E = E A ), J AK B J K we may take the equation gµν = Eµ ηABE ν as a deﬁnition for a metric gµν in terms of a A vielbein E µ.

Table 3.1: Overview of the vielbein notation in component form

] ] ] ] A B T gµν = g(∂µ, ∂ν) = η(Eµ,Eν) = Eµ ηABE ν, g = E ηE

Matrix notation Components In the coordinate basis

A A A µ E E µ E = E µdx Vielbeins [ [ −1 µ −1 µ ] ] µ E E A≡ (E ) A EA = ∂µE A

µ ν g gµν g = gµνdx dx Metric [ [ −1 µν −1µν −1 µν ] ] g g ≡ g g = g ∂µ∂ν

µ ν (µ ν) ] ] ] ] Here dx[ dx[ := dx[ ⊗ dx[ and ∂µ∂ν := ∂(µ ⊗ ∂ν) as in [CBDMDB77].

A Remark 3.11. The vielbein components E µ have both the coordinate index µ as well as the Lorentz index A, which may give rise to confusion. The vielbein is not a usual 2-tensor on a tensor space constructed over a single vector space. As it we have seen, it is rather a m,n ∗ multilinear map E ∈ R ⊗ TP (M) which can also be thought of as a linear transformation A m,n ] ] µ A A µ A E µ : TP (M) → R . Alternatively, in view of EA = ∂µE A and E[ = E µdx[ , E µ can be regarded as the coordinate transformation.

Remark 3.12. In the coordinate basis, we raise and lower indices with gµν. Conversely, in the orthonormal basis, we raise and lower indices with ηAB. Accordingly, the coordinate indices of a vielbein are transformed by general coordinate transformations (GCT), while the Lorentz indices by local Lorentz transformations (LLT). 42 Chapter 3. Geometry of Decompositions

3.5 Decomposition of Vielbeins

Notice how distributed are the degrees of freedom of a regular quadratic space isometric to m,n ˆ (R , η): The number of independents components of a proper vielbein E = (et ⊕ es) ◦ Ξ(ν) reads 1 1 1 2 m(m + 1) + 2 n(n + 1) + mn = 2 (m + n)(m + n + 1), (3.70) which is in agreement with the degrees of freedom of the regular quadratic form in question. Taking in account the additional components of the orthogonal transformation Λ ∈ O(m, n) m,n 2 which preserves the quadratic form (R , η), we get (m+n) components in total. The question is: Under which conditions can an arbitrary vielbein E be decomposed into a proper vielbein Eˆ = (et ⊕ es) ◦ Ξ(ν) and an orthogonal transformation Λ such that Eˆ = ΛE?

Ξ(ν) (V, g) (V, gˆ)

Eˆ E et ⊕ es

m,n Λ m,n (R , η) (R , η)

Figure 3.2: The proper m+n decomposition of a vielbein

Let E be an arbitrary vielbein, in partitioned form given by ! e eT E = 0 1 . (3.71) e2 e3

If we apply the orthogonal transformation ΛE = Eˆ (Figure 3.2), we expect to obtain

! ! ! ! Λ p0 R 0 e eT e 0 t t 0 1 = t . (3.72) p Λs 0 Rs e2 e3 esν es where Rt ∈ O(m), Rs ∈ O(m). Thus the triangularization condition reads

T 0 ΛtRte1 + p Rse3 = 0. (3.73)

0 0 Using the reparametrization p = −vΛt, p = −Λtv (see Eq. (3.23)), the triangularization condition reads

0 T −Λtv Rse3 = −ΛtRte1, (3.74) 0 T −1ˆ−1 T ˆ v = Rte1e3 δs Rs δs. (3.75)

0 ˆ−1 T ˆ Since v = δt v δs, we have

ˆ 0ˆ−1 T ˆ T −1ˆ−1 T T ˆ−1 −1,T T ˆ v = (δtv δs ) = (δtRte1e3 δs Rs ) = Rsδs e3 e1Rt δt, (3.76) which gives (the derivation is on page 75)

−1 0 T T ˆ −1 ˆ Rt (Im − v v)Rt = Im − e1(e3δse3) e1δt. (3.77) 3.5 Decomposition of Vielbeins 43

√ −1 0 −1 0 0 Having Λt = Im − v v and Rt (Im − v v)Rt = Im − w w, w = vRt, we get the condition T T ˆ −1 ˆ ke1(e3δse3) e1δtk < 1, which is independent on the choice of rotations Rt and Rs and moreover, independent of the sign of det Rt and det Rt (notice also that kδˆtk = 1 and that T T ˆ −1 e1(e3δse3) e1 is symmetric). Hence, we proved the following statement:

∼ m,n Lemma 3.3. Given some basis of a regular quadratic space (V, g) = (R , η), an arbitrary vielbein ! e eT E = 0 1 . (3.78) e2 e3 can be put into lower-triangular form by an orthogonal transformation, iﬀ

T T ˆ −1 0 ≤ e1(e3δse3) e1 < 1. (3.79)

∼ m,n Proposition 3.6. The condition on a basis of a regular quadratic space E :(V, g) = (R , η) to admit the proper m+n decomposition of g is the same as for an arbitrary vielbein E to be put into lower-triangular form by an orthogonal transformation.

Proof. Assume a basis where the proper m+n decomposition of the quadratic form g reads ! gt,−1 + νTg ν νTg g = s s . (3.80) gsν gs

Given an arbitrary vielbein E, the partitioned form of g = ETηE is, on the other hand, ! −eTδˆ e + eTδˆ e −eTδˆ eT + eTδˆ e g = 0 t 0 2 s 2 0 t 1 2 s 3 (3.81) ˆ T ˆ ˆ T T ˆ −e1δte0 + e3δse2 −e1δte1 + e3δse3

Combining the last two expressions gives the matrix equation ! ! gt,−1 + νTg ν νTg −eTδˆ e + eTδˆ e −eTδˆ eT + eTδˆ e s s = 0 t 0 2 s 2 0 t 1 2 s 3 , (3.82) ˆ T ˆ ˆ T T ˆ gsν gs −e1δte0 + e3δse2 −e1δte1 + e3δse3 from which we obtain the core elements (the full derivation is on page 75)

T −1 t,−1 T −1 ˆ−1 T T ˆ −1 T −1 −g = e0 − e1e3 e2 δt − e1(e3δse3) e1 e0 − e1e3 e2 , (3.83) T ˆ −1,T ˆ T −1 gs = e3 δs − e3 e1δte1e3 e3. (3.84)

We know that the quadratic form g admits a proper m + n decomposition iﬀ −gt,−1 and −1,T ˆ T −1 gs are both positive deﬁnite. From kgsk > 0 we obtain ke3 e1δte1e3 k < 1. Since the expression of −gt,−1 has the form ATBA, the requirement kATBAk > 0 is equivalent to kBk > 0 because kAATk ≥ 0. Therefore,

T T ˆ −1 0 ≤ e1(e3δse3) e1 < 1. (3.85)

This concludes the proof.

Remark 3.13. By using a local orthogonal transformation alone, an arbitrary vielbein cannot always be put into triangular form. However, we may always ﬁnd a general coordinate transformation to meet the conditions (3.79) and (3.85). 44 Chapter 3. Geometry of Decompositions

3.6 Quadrics

The geometrical version of algebra dealing with quadratic forms is the projective geometry of quadratic surfaces [Hit03]. Consider a vector space V over a ﬁeld K. The set of one- dimensional vector subspaces of V is called the projective space of V , denoted by P (V ).

This is equivalent to say that the projective space P (V ) is the quotient of V \{0V } by the equivalence relation v ∼ w iff v = αw for some α ∈ K×. The equivalence class of v ∈ V , denoted by [v], is the set of elements that are related to a by ∼, i.e., [v] = {αv ∈ V \{0V } | × α ∈ K }. Thus, in view of group actions, P (V ) is the orbit space (V \{0V })/K where K acts on V \{0V } by scalar multiplication. For v ∈ V \{0V }, any one-dimensional subspace K×v of V is the set of multiples of v. In such case, v is called the representative vector for × the point [v] ∈ P (V ); thus [v] = [αv], ∀α ∈ K . Let further ei be an ordered basis of V J K i where dim V = d + 1 and i ∈ {0, 1, . . . d}. For any non-zero vector v = eiv , we can write [v] ∈ P (V ) simply as [vi]; these coordinates are called the homogeneous coordinates. Clearly, [v0, v1, . . . , vd] = v0[1, v1/v0, . . . , vd/v1] for v0 ∈ K×, which implies that any subset U of P (V ) containing points [vi] with v0 6= 0 is uniquely represented by [1, v1≤i≤d/v0], therefore U =∼ Kd where d + 1 = dim V . This justifies a definition that, if the vector space V has dimension d + 1, then P (V ) is a projective space of dimension d. Definition 3.7. Let (V, b) be a quadratic space. A quadratic surface or quadric is the set of points [v] ∈ P (V ) in a projective space P (V ) whose representative vectors v are isotropic, b(v, v) = 0. The set of all isotropic vectors N (V, b) := { v ∈ V \{0V } | b(v, v) = 0 } is called the null-cone. The dimension of the quadric is dim P (V ) − 1. The quadric is called non-singular iff b is non-degenerate. Remark 3.14. Since b(αv, αv) = α2v and [αv] = [v], the quadrics defined by b and α2b are the same for all α ∈ K. Note also that, strictly speaking, {0V } ∈/ N (V, b). 1,3 2,2 Example 3.3. The contour plots of the null-cones N (R ) and N (R ) are shown on 1,3 1,3 Figure 3.3. Given (t, x, y, z) ∈ R , the t-contours of N (R ) are the concentric 2-spheres, 2,2 2,2 while for (t1, t2, x, y) ∈ R , the t1-contours of N (R ) are the concentric hyperboloids of one sheet.

z t2

y y x x

1,3 2,2 Figure 3.3: The null-cones of R and R (3D-contour plots). 3.6 Quadrics 45

∼ m,n We shall narrow our considerations to K = R. Let (V, g) = (R , η) be a regular ∗ m,n quaratic space such that g = E η. The null-cone N (R , η), which is deﬁned by

T T ˆ T ˆ m,n w ηw = −t δtt + x δsx = 0, ∀w = (t, x) ∈ R \{0Rm+n }, (3.86) can also be parametrized as

T T t δˆtt = x δˆsx = α, α > 0. (3.87)

The parameter α foliates the null-cone by ‘Cliﬀord’ tori Sm−1 × Sn−1 giving the topology + m−1 n−1 11 R ×S ×S , where the dimension of the null-cone hypersurface is obviously m+n−1. ∗ t,−1 On the other hand, for g = Ξ(ν) g ⊕ gs , the null-cone N (V, g) is given by

T T t,−1 T Q = w gw = t g t + (x + νt) gs(x + νt) = 0, ∀w = (t, x) ∈ V \{0V }. (3.88)

Furthermore, if a given basis admits the proper m+n decomposition of g, we have

T T ˆ T T ˆ t et δtett = (x + νt) es δses(x + νt) = α, α > 0. (3.89)

m,n Now, the null-cone N (R , η) can be equivalently given as the set of ‘rays’

m 0 T ˆ ˆ m n x = v t, t ∈ R \{0Rm }, v v = Im, v δsv = δt, v : R → R , (3.90)

T T T T where x δˆsx = t v δˆsvt = t δˆtt, provided that there are enough degrees of freedom left in 0 0 v v to achieve v v = Im. This is only possible if n ≥ m, which we may assume without loss m n 0 of generality. In such case, the parameter space { v ∈ End(R , R ) | v v = Im } is invariant under the orthogonal transformation O(m) × O(n),12 that is

0 T ˆ ˆ T T ˆ T ˆ v v = Im ⇔ v δsv = δt ⇔ v Rs δsRsv = Rt δtRt. (3.91)

Combining Eqs. (3.87) and (3.89), we obtain

t = ett, x = es(x + νt). (3.92)

From Eq. 3.90 we then get the null-cone N (V, g) in terms of rays given as

−1 T ˆ x = (−ν + es vet) t, t ∈ Vt \{0Vt }, kv δsvk = 1. (3.93)

We have thus proved the following theorem: ∼ m,n ∗ Theorem 3.4. Let (V, b) = (R , η), g = E η be a regular quadratic space admitting the proper decomposition E = (et ⊕es)◦Ξ(ν) in a given basis, where V = Vt ⊕Vs and 1 ≤ m ≤ n. For 0 ≤ α ≤ 1, deﬁne

−1 T ˆ N (V, g; α) := { (t, x) ∈ V | x = (−ν + es vet) t, t ∈ Vt \{0Vt }, kv δsvk = α }. (3.94)

Then the null-cone is given by N (V, g; α = 1).

11Observe that (SO(m) × SO(n)) × Sm−1 × Sn−1 → Sm−1 × Sn−1 is a transitive group action. 12Under any two independent ‘t’- and ‘s-’-subspace rotations. 46 Chapter 3. Geometry of Decompositions

Deﬁnition 3.8. The interior of the null-cone is Nint(V, g) := N (V, g; 0 ≤ α < 1) and its center by N0(V, g) := N (V, g; α = 0).

The center of the null-cone N (V, g) is deﬁned through the center of a quadric Q that is obtained by intersecting the null-cone by Vs-hyperplane at constant t ∈ Vt, where the center of the quadric Q is determined from the equation ∂Q/∂x = 0 [Coo45]. Clearly, this gives x + νt = 0. What about when the given basis does not admit the proper decomposition? In t,−1 such case, either −g or gs are indeﬁnite and the quadrics that are obtained by intersecting n the null-cone by Vs-hyperplane will not be an ellipsoid in Vs but rather R -hyperboloid or n R -paraboloid (the latter without a center). This can be seen by introducing an equivalent to Eq. (3.94) which is somewhat more general

T t,−1 T N (V, g; α) := { (t, x) ∈ V | αt g t + (x + νt) gs(x + νt) = 0 }. (3.95)

As before, setting the parameter α to one gives the null-cone of (V, g) while the range

0 ≤ α < 1 parametrizes the Vt-subspace interior of the null-cone. Finally, the limit α → 0 T colapses the Vt-subspace part into a central ray for which (x + νt) gs(x + νt) = 0. When gs is deﬁnite, then necessarily x + νt = 0. Theorem 3.4 enables us to give a geometrical interpretation of the vielbein decomposition

E = (et ⊕ es) ◦ Ξ(ν) as shown on Figure 3.4.

gs ν −gt,−1

et es Ξ(ν)

reshape Vt reshape Vs shift

t E = (et ⊕ es) ◦ Ξ(ν) t ( m,n, η) (V, g) x R x

Figure 3.4: Decomposition of a proper vielbein acting on a null-cone

Here Ξ(ν) shifts the null-cone in the ‘s’-subspace preserving its hypervolume, es distorts n−1 m−1 the S sphere and et distorts the S sphere (where et and es commute). Notice the direction of arrows showing the vielbein transformation and the decomposition: the pullback of η by E is in the opposite direction of the decomposition. Namely, the decomposition corresponds to the pullback of g by E−1 giving η, i.e., η = (E−1)∗g, which accordingly shifts the center of the null-cone for −ν. Chapter 4

Symmetrization Condition

In this chapter we explore the square-root symmetrization condition in terms of vielbeins. We also give the geometrical and topological interpretation of coupled quadratic spaces in terms of their null-cones.

4.1 Decomposition of the Symmetrization Condition

Consider two quadratic spaces (V, f) and (V, g) coupled through the Minkowski space ( m,n, η) √ R such that f = L∗η and g = E∗η. Let S be the square-root isometry (V, f) ,→(V, g) with the symmetrization condition

h = gS = ETηΛL = (ETηΛL)T . (2.51)

Suppose we are in the basis of the vector space V that admits a proper m+n decomposition for both g and f. In such case the vielbeins E and L can be put into the proper vielbeins Eˆ and Lˆ by some orthogonal transformations Λ1 and Λ2 (see Figure 4.1).

Ξ(µ) Ξ(ν) (V, fˆ) (V, f) S (V, g) (V, gˆ)

Lˆ Eˆ mt ⊕ ms L E et ⊕ es

m,n Λ2 m,n Λ m,n Λ1 m,n (R , η) (R , η) (R , η) (R , η) Λˆ ∈ O(m, n)

Figure 4.1: Quadratic spaces (V, f) =∼ (W, g) coupled by the square-root isometry S

ˆT ˆ ˆ ˆT ˆ ˆ T ˆ −1 Accordingly, Eq. (2.51) becomes E ηΛL = (E ηΛL) , where Λ = Λ1 ΛΛ2; hence, we can study the symmetrization condition only in terms of the proper vielbeins Eˆ and Lˆ symmetrized by the orthogonal transformation Λˆ. Therefore, we shall work under the premise “provided a basis that admits a simultaneous proper m + n decomposition of two coupled quadratic spaces (V, f) and (V, g)” and suppress the hats over the variables (Eˆ → E, Lˆ → L, Λˆ → Λ) in the rest of this chapter. (The existence of such basis will be discussed in the following chapter in more detail.) In terms of the proper vielbeins, Eq. (2.51) is equivalent to ηΛLE−1 = ηΛLE−1T,

Mikica Kocic, The Square-Root Isometry, MSc thesis, SU 2014 Printed: 2015-03-11 48 Chapter 4. Symmetrization Condition where in matrix notation ! ! ! ! ! −δˆ 0 Λ p0 R 0 m 0 e−1 0 ηΛLE−1 = t t t t t (4.1) ˆ −1 −1 0 δs p Λs 0 Rs msµ ms −νet es ! −δˆ Λ R m e−1 − δˆ p0 R m (µ − ν)e−1 −δˆ p0 R m e−1 = t t t t t t s s t t s s s . (4.2) ˆ −1 ˆ −1 ˆ −1 δsp Rtmtet + δsΛs Rsms(µ − ν)et δsΛs Rsmses

According to the commutative diagram on Figure 4.1, the symmetrization condition formally corresponds to

−1 −1 S = Ξ(−ν) ◦ (et ⊕ es ) ◦ Λ ◦ (mt ⊕ ms) ◦ Ξ(µ), (4.3) 0 ] T ] S = S = g ◦ S ◦ g[ ⇔ S ◦ S = g ◦ f[, (4.4)

−1 −1 X = Λ ◦ (mt ⊕ ms) ◦ Ξ(µ) ◦ Ξ(−ν) ◦ (et ⊕ es ), (4.5) 0 ] T X = X = η ◦ X ◦ η[, Λ = B ◦ (Rt ⊕ Rs). (4.6)

Observe that (Rt ⊕ Rs) ◦ (mt ⊕ ms) = (Rtmt) ⊕ (Rsms). Proposition 4.1. The symmetrization condition is invariant under the same shift acting both on (V, f) and (V, g). Proof. (Ξ(µ) ◦ Ξ(λ)) ◦ (Ξ(ν) ◦ Ξ(λ))−1 = Ξ(µ − ν), hence Eq. (4.5) remains unaltered.

Theorem 4.1. Provided a basis that admits the proper m + n decomposition of two regular √ quadratic spaces (V, f) and (V, g) coupled by the square-root isometry S :(V, f) ,→(V, g), the symmetrization condition is equivalent to

◦ ˆ −1 ˆ −1T 1 δsΛs (Rsms) es = δsΛs (Rsms) es , (4.7) ◦ ˆ −1 −1 ˆ −1 −1T 2 δtΛt (Rtmt) et = δtΛt (Rtmt) et , (4.8) ◦ −1 −1 3 ν − es v et = µ + (Rsms) v (Rtmt), (4.9)

−1 −1 where the parameter v = Λs p = pΛt is conditioning the ‘t’- and ‘s’-parts of the boost by Eq. (3.25) q q −1 ˆ−1 T ˆ −1 ˆ−1 T ˆ Λt = Im − δt v δsv, Λs = In − vδt v δs. (3.25)

T The parameter space of v is the open set kv δˆsvk < 1. Proof. After some work (the full derivation is on page 77), the symmetrization condition yields the system of equations (4.7)-(4.9). Solving the coupled system gives Rs, Rt and v, and consequently the orthogonal transformation Λ. The existence of the solution is provided by Theorem 2.2 and Corollary 2.5. The parameter space of v is obviously the same as for the symmetric part (boost) of the orthogonal transformation Λ.

Remark 4.1. Eq. (4.9) can also be written in terms of the shift parameter diﬀerence

−1 −1 ν − µ = es v et + (Rsms) v (Rtmt) . (4.10)

Remark 4.2. The alternative form of Eq. (4.8) is

ˆ −1 ˆ −1 T δt (Rtmt) et Λt = δt (Rtmt) et Λt . (4.11) 4.1 Decomposition of the Symmetrization Condition 49

Remark 4.3. The existence of Rs and Rt is ensured by the polar decomposition. Namely, both Eq. (4.7) and Eq. (4.8) have the form ARB where R−1 = R0 = δˆ−1RTδˆ (labeled with the appropriate indices indicating the corresponding subspace, either Vs or Vt). Since the symmetrization of ARB is equivalent to the left (or right) polar decomposition of XR (or RY ) where X = B−1,TA (or Y = BA−1,T), we know the solution for R in terms of X (or Y ); see Section 3.1.1. The polar decomposition of δARBˆ yields (the full derivation is on page 79) r −1 T R = δˆ−1B−1,TδAˆ δˆ−1B−1,TδAˆ δˆ−1 δˆ−1B−1,TδAˆ δ,ˆ (4.12) where the expression inside the square-root is by construction positive deﬁnite (so R exists). −1,T −1,T T Consequently, for A = Λs, B = ms es we obtain r −1 T ˆ−1 −1,T T ˆ ˆ−1 −1,T T ˆ ˆ−1 ˆ−1 −1,T T ˆ ˆ Rs = δs ms es δsΛs δs ms es δsΛs δs δs ms es δsΛs δs, (4.13)

−1 −1,T −1,T T and similarly, for A = Λt , B = mt et , we get r −1 T ˆ−1 −1,T T ˆ −1 ˆ−1 −1,T T ˆ −1 ˆ−1 ˆ−1 −1,T T ˆ −1 ˆ Rt = δt mt et δtΛt δt mt et δtΛt δs δt mt et δtΛt δt. (4.14)

T Remark 4.4. As we shall see, νc is the shift parameter of the quadratic form gS = E ηΛL (which corresponds to the center of the null-cone quadric of h = gS)

−1 −1 νc = ν − es v et = µ + (Rsms) v (Rtmt). (4.15)

Example 4.1. Consider the case when m = 1 and n ≥ 1. The shift parameters and the boost then become n-dimensional column vectors νi, µi and va, respectively.1 Also, the

‘t’-part of vielbeins become one dimensional scalars, i.e., the lapses et ≡ N and mt ≡ M, and the ‘t’-rotation Rt ∈ O(1) reduces to Rt = ±1 representing the discrete time reversal. Since the ‘t’-parts are automatically symmetric, we get the system of two equations only

◦ ˆ −1 ˆ −1T 1 δsΛs (Rsms) es = δsΛs (Rsms) es , (4.16) ◦ −1 −1 2 ν − µ = N es ± M (Rsms) v, (4.17) or, in the component form

b c d −1 i Xae = Xea,Xae := δabΛ cR dm i[e ] e, (4.18) i i −1 i a c −1i a ν − µ = N [e ] a ± M R cm i v , (4.19) J K a

a p a 2 a c a a γ2 a c −2 p a b where Λ b = δ b + γ v v δcb b = δ b + γ+1 v v δcb, γ = 1 − v δabv and 0 ≤ a b J K v δabv < 1 (see also Example 3.1). Finally, the polar decomposition of the spatial part is provided by Eq. (4.13)

hq ib a −1 a b cd T e a ab −1,T µ T c d R f = [A ] b A cδ [A ]d δef ,A e := δ [m ]b [e ]µ δcdΛ e. (4.20) J K f

1 n The shifts vectors belong to vector space Vs, while the boost to the Minkowski space R . They have a i diﬀerent bases and diﬀerent sets of indices. Notice also that In 6= Is as In = δ b while Is = δ j . J K J K 50 Chapter 4. Symmetrization Condition

4.2 The Symmetrizing Quadratic Space

As earlier noted in Chapter2, the symmetrization condition (2.51) is equivalent to h = hT where h = gS = ETηΛL is the symmetric bilinear form of the symmetrizing quadratic space (V, h) (see Deﬁnition 2.6). We can then rather claim the propositions from Eqs. (4.7)-(4.9) in terms of h.

Theorem 4.2. Provided a basis that admits the proper m + n decomposition of two regular √ quadratic spaces (V, f) and (V, g) coupled by the square-root isometry S :(V, f) ,→(V, g), the t,−1 symmetrizing quadratic space (V, h ≡ gS) can be properly decomposed h = (h ⊕hs)◦Ξ(νc), where T ˆ hs = es δsΛs(Rsms), (4.21)

t,−1 T ˆ −1 −h = et δtΛt (Rtmt), (4.22) −1 νc = ν − es v et (4.23) −1 = µ + (Rsms) v (Rtmt). (4.24)

Consequently, the assertion h = hT is equivalent to Theorem 4.1.

Proof. Let w ∈ V = Vt ⊕ Vs be any non-zero vector such that ! t Q = wThw, w = 6= 0 , t ∈ V , x ∈ V . (4.25) x V t s

Starting from h = ETηΛL, we obtain (the full derivation is on page 79)

!T ! ! ! ! e 0 −δˆ 0 Λ p0 R 0 m 0 Q = wT t t t t t w (4.26) esν es 0 δˆs p Λs 0 Rs msµ ms

!T ! ! I 0 −eTδˆ Λ−1R m 0 I 0 = wT t t t t t t t w. (4.27) T ˆ νc Is 0 es δsΛsRsms νc Is

t,−1 On the other hand, from h = (h ⊕ hs) ◦ Ξ(νc) we have

T T t,−1 Q = w Ξ(νc) (h ⊕ hs)Ξ(νc)w. (4.28)

Finally, since the non-zero w ∈ V is arbitrary (and consequently Ξ(νc)w is arbitrary), the assertion holds.

Corollary 4.1. The Minkowski core of the symmetrizing quadratic space (V, h) reads ˆ T ˆ −1 ˆ h = (et ⊕ es) −δtΛt ⊕ δsΛs (Rtmt ⊕ Rsms) . (4.29)

4.3 Topology of the Symmetrization Condition

When studying the proper m + n decomposition of a vielbein E in Section 3.6 (where E =

(et ⊕ es) ◦ Ξ(ν)), we interpreted the various components et, es and ν as the geometrical elements of the null-cone that is associated to the quadratic space (V, g), g = E∗η. Moreover, 4.3 Topology of the Symmetrization Condition 51 we identiﬁed the interior of the null-cone by Eq. (3.94) for 0 ≤ α < 1 as

−1 T ˆ N (V, g; α) = { (t, x) ∈ V = Vt⊕Vs | x = (−ν+es vet) t, t ∈ Vt\{0Vt }, kv δsvk = α }, (4.30)

∗ In the case of yet another quadratic space (V, f), f = L η, L = (mt ⊕ms)◦Ξ(µ), we similarly have

−1 T ˆ N (V, f; α) = { (t, x) ∈ V = Vt ⊕ Vs | x = (−µ + ms vmt) t, t ∈ Vt \{0Vt }, kv δsvk = α }. (4.31) Now, if (V, g) and (V, f) are coupled by the square-root isometry S, then the shift parameter of the symmetrizing quadratic space h = gS reads (Eqs. (4.23) and (4.24))

−1 −1 T ˆ νc = ν − es vsym et = µ + (Rsms) vsym (Rtmt), kvsymδsvsymk < 1. (4.32)

Here we used vsym to denote a boost parameter that is obtained through the symmetrization of h (instead of v, to avoid confusion with the dummy variables in N (V, g; α) and N (V, f; α)). Then the center of the null-cone of the symmetrizing quadratic space reads

N0(V, h) = { (t, x) ∈ V | x = −νct } (4.33) −1 = { (t, x) ∈ V | x = (−ν + es vsymet) t } (4.34) −1 = { (t, x) ∈ V | x = (−µ − (Rsms) vsym(Rtmt)) t }. (4.35)

Hence, there is a topological relation between the null-cones of the coupled quadratic spaces and the center of the symmetrizing quadratic space.

Theorem 4.3. Provided a basis that admits the proper m + n decomposition of two coupled √ quadratic spaces (V, f) and (V, g), the square-root isometry S :(V, f) ,→(V, g) exists if and only if N (V, f) ∩ N (V, g) contains an open set for a ﬁxed t ∈ Vt.

−1 Proof. Suppose that vsym exists. The expression ms vmt in N (V, f; α) is invariant up to ar- −1 bitrary rotations Rt ∈ O(m) and Rs ∈ O(n). Namely, under the redeﬁnition v2 := −Rs vRt, ˆ−1 T ˆ we have v2 = −δs Rs δsvRt and subsequently

−1 −1 −1 ms vmt = (Rsms) v (Rtmt) = ms v2mt, (4.36) T ˆ T T ˆ ˆ−1 T ˆ T T ˆ v2δsv2 = Rt v δsRsδs Rs δsvRt = Rt v δsvRt. (4.37)

−1 T ˆ Therefore, arbitrary rotations do not inﬂuence ms vmt and kv δsvk, so we can write

−1 T N (V, f; α) = { (t, x) ∈ V | x = (−µ − (Rsms) v(Rtmt) t, kv δˆsvk = α }. (4.38)

T Because kv δˆsvk ≤ 1 deﬁnes the closed interior of the null-cone, we conclude from Eqs. (4.30), (4.34), (4.35) and (4.38) that the ray x = −νct which represents the center of the symmetrizing quadratic space (V, h), belongs to both N (V, g) and N (V, f), that is,

N0(V, h) ⊂ N (V, g) ∩ N (V, f). (4.39)

T ˆ The condition kvsymδsvsymk < 1 is satisﬁed iﬀ there exists an open set O such that N0(V, h) ⊂ T ˆ O ⊂ N (V, g) ∩ N (V, f). The boundary is at kvsymδsvsymk → 1, in which case the null-cones ‘touches’ at N (V, g) ∩ N (V, f) = N0(V, h). 52 Chapter 4. Symmetrization Condition

Conversely, suppose that N (V, g) ∩ N (V, f) 6= ∅ is open; then there is 0 ≤ α0 < 1 such that N (V, g; α0) ∩ N (V, f; α0) 6= ∅, where

−1 T ˆ N (V, g; α0) = { (t, x) ∈ V | x = (−ν + es v1et) t, kv1δsv1k = α0 }, (4.40) −1 T ˆ N (V, f; α0) = { (t, x) ∈ V | x = (−µ + ms v2mt) t, kv2δsv2k = α0 }. (4.41)

T ˆ T ˆ −1 Using α0 = kv1δsv1k = kv2δsv2k, the redeﬁnition v2 := −Rs v1Rt provides the existence of the symmetrization condition

−1 −1 −ν + es v1et = −µ − (Rsms) v1(Rtmt), (4.42) −1 −1 ν − µ = es v1et + (Rsms) v1(Rtmt). (4.43)

T ˆ In the limitting case α0 → 1, we have kv1δsv1k → 1 and the null-cones touches. ∼ ∼ 1,2 Example 4.2. Two intersecting null-cones of (V, g) = (V, f) = (R , η) are shown on Fig- ure 4.2. The red line on the left image depicts the center ray N0(V, h), while the middle ellipse on the right image represents the null-cone N1(V, h) of the symmetrizing space h.

νc h = gS

g f

Figure 4.2: Intersecting null-cones of quadratic spaces.

Remark 4.5. The fact that the Vt-interirors of two null-cones intersect while sharing a common Vs-subspace (‘provided a basis that admits the proper m + n decomposition’) is a topological property which does not change under the general coordinate transformations. As we shall see in the followng chapter, the existence of the square-root ensures that the local null-cones intersect in such way to have both a common ‘t’-like vector and a common ‘s’-like hypersurface. The local orthogonal transformations preserve the toplogical structure of the m,n quadratic spaces isometric to R . These properties are of the key role when considering the causality of spacetimes coupled through a square-root isometry. Chapter 5

Simultaneous Proper Decomposition

In the last chapter, the assumption was that we work in the basis which simultaneously admits the proper m+n decomposition of two coupled quadratic spaces. Of course, this presupposition is questionable in general. Nevertheless, Theorem 2.2 and Corollary 2.5 establish the basis independent correspondence between the existence of a square-root isometry and the associated orthogonal transformation. This means that we might have chosen a basis where we do not have the proper decomposition of both quadratic spaces at the same tame, invalidating any usage of Theorems 4.1, 4.2 and 4.3 despite the fact that the square-root isometry exists. With the square-root in hand, however, there is a constructive way how to ﬁnd such proper basis. To motivate such claim we shall employ continuation sets (the continuation sets are discussed on page 27). Let S be the square-root isometry from (V, f) to (V, g). Consider a continuation set of g relative to S represented by the bilinear form

2 1 1 gξ = g 2 (1 − ξ) I + 2 (1 + ξ) S , ξ ∈ R. (5.1)

Now, continuously vary ξ starting from −1 to 1 at all times staying in the basis where gξ is diagonal, i.e., start from the basis where gξ=−1 = g is diagonal, and go towards the other end where gξ=1 = f is diagonal. Notice that the square-root isometry will always pair-wise exist between gξ, g and f for all ξ ∈ R (even outside the segment [−1, 1]). In the neighborhood of ξ = −1, both gξ and g admit the proper decomposition. On the other hand, in the neighborhood of ξ = 1, both gξ and f admit the proper decomposition. Notice that gξ is a continuous function of ξ. We could argue that there will be a basis wherein gξ would be diagonal for some ξ ∈ [−1, 1], and which would admit the proper decomposition of both g and f simultaneously. This would always be true provided that the square-root isometry is unique, which is not. The matrix square-root is a multivalued function with multiple branches, and the proper basis will lie outside the segment [−1, 1] if we selected an inappropriate branch (where ξ → ∞ corresponds to −S). Nevertheless, as it will be shown in the course of this chapter, the null-cones of the symmetrizing spaces that are associated with these alternative primary and nonprimary square-roots touch each other “tiling” the complete space V . Hence, there will always exist a “nice” square-root isometry that will provide the simultaneous proper decomposition.

Mikica Kocic, The Square-Root Isometry, MSc thesis, SU 2014 Printed: 2015-03-11 54 Chapter 5. Simultaneous Proper Decomposition

Remark 5.1. An example of a continuation set is shown on Figure 5.1. The ellipses represent the Vs-slices of the null-cones of g (on the left), gξ (in the middle) and f (on the right). These slices are ellipsoid quadrics in a more general case. The curve which goes through the centra of the quadrics is parametrized by ξ. The straight lines denote the asymptotes at those values of the continuous parameter ξ where the quadric of gξ becomes centerless (paraboloid-like), that is, when the positive deﬁnite gξ,s turns into indeﬁnite.

gξ

g f

Figure 5.1: The continuation set of two coupled quadratic spaces.

The rest of this chapter is organized as follows. We firstly investigate the multivaluedness of the square-root isometry and construct the related reflection group with a finite set of generators. We then give propositions for vector space coverings by null-cones, and present a constructive proof for finding the proper basis which admits the simultaneous proper decomposition of two coupled quadratic spaces. Finally, we provide the conditions for the simultaneous diagonalization of quadratic forms by congruence and state the canonical pair form theorem for symmetric matrices, and then prove the proper basis theorem.

5.1 Alternative Square-Roots

Let (V, f) and (V, g) be regular quadratic spaces isometric to the Minkowski space coupled √ through the square-root isometry S :(V, f) ,→(V, g) with the associated symmetrizing space (V, h), h = gS. Let further f = L∗η, g = E∗η, h = H∗η and

f = gS2 = hg−1h, h = gS, h = hT. (5.2)

p −1 Consider yet another square-root isometry S2 = g f such that

2 −1 T f = gS2 = h2g h2, h2 = gS2, h2 = h2. (5.3)

−1 −1 Combining Eqs. (5.2) and (5.3) we get f = h2g h2 = hg h; thus

−1 T −1 g = (h h2) g(h h2). (5.4)

−1 −1 After introducing X ∈ Aut(V ) deﬁned by X := h h2 = S S2, we have

T T g = X gX, h2 = hX, h2 = h2. (5.5) 5.1 Alternative Square-Roots 55

Consequently, X is an orthogonal transformation of (V, g) that symmetrizes hX.

T T Continuing from h2 = h2 and h = gS = X gXS, we obtain

hX = X Th ⇐⇒ X TgXSX = X TgS, (5.6) which gives the algebraic Riccati equation

XSX = S. (5.7)

The trivial solution X = ±IV corresponds to h2 = ±h. Notice that any solution X to 2 2 k k XSX = S implies X SX = XSX, that is, X SX = S for all k ∈ N0. In particular for k = 2, we have X2SX2 = S and accordingly (X2S)2 = S2, from which get the property

2 X = IV . (5.8)

2k 2k+1 Furthermore, X = IV and X = X for all k ∈ N0. Hence, the orthogonal transformation X ∈ O(V, g) in Eq. (5.5) is necessarily involutory with the eigenvalues in {±1}.

Premultiplying Eq. (5.7) by S yields (SX)2 = S2, from which we get the solution

X = S(S2)−1/2 = X−1 = S−1(S2)1/2. (5.9)

Here, any square-root of S2 = g−1f can be taken. The principal solution is X = sign(S), and clearly, X is real because S is real. (For a more general treatment of quadratic matrix equations see [SLB74, Pot66]. See also Section 2.9 Nonlinear Matrix Equations in [Hig08].)

2 After multiplying Eq. (5.7) by X (from any side) then using X = IV , we obtain

XS = SX, (5.10)

−1 −1 thus X and S commute as expected. Notice also that X = S S2 = S2S and S = XS2 = S2X, and that any two solutions to Eq. (5.5) commute. This is compliant with SS2 = S2S. Plugging in g = X TgX into f = STgS, then employing the last commutation relation gives f = STX TgXS = X TSTgSX from which we obtain

f = X TfX. (5.11)

T T 2 Finally, h2 = hX = (hX) = X h and X = IV implies

h = X ThX. (5.12)

Therefore, at the same time: X ∈ O(V, g), X ∈ O(V, f) and X ∈ O(V, h).

Observe that any involutory orthogonal transformation is also self-adjoint; in this case: X−1 = X = X0 = g−1X Tg = f −1X Tf = h−1X Th.

Further on, we can express X in terms of vielbeins and orthogonal transformations on m,n R . By Corollary 2.1 for all X ∈ O(V, g) there exist Λ, Λ1, Λ2 ∈ O(m, n) such that

−1 −1 −1 X = E ΛE = L Λ1L = H Λ2H (5.13) 56 Chapter 5. Simultaneous Proper Decomposition

2 Since E, L and H are bijective, the involution X = IV is equivalent to

2 2 2 Λ = Λ1 = Λ2 = I. (5.14)

Therefore, all of Λ, Λ1 and Λ2 are involutory and

T T T ηΛ = (ηΛ) , ηΛ1 = (ηΛ1) , ηΛ2 = (ηΛ2) .

T −1 T Then, from h2 = hX = H ηH E ΛH, the symmetrization condition h2 = h2 can be written in terms of Λ and T := HE−1 as

T TηT Λ = (T TηT Λ)T , (5.15) ηT ΛT −1 = (ηT ΛT −1)T, (5.16) which again yields the algebraic Riccati equation Λ T 0T Λ = T 0T . Here, the adjoint T 0 is given relative to η, that is, T 0 := η−1T Tη. Finally, we arrive at the theorem which summarizes the results obtained in the course of this section.

Proposition 5.1. Let X be an involutory orthogonal transformation that symmetrizes gSX. −1 2 2 Then, in matrix notation, f g = S = S2 and

2 h = gS, h2 = gS2,S2 = SX = XS, X = IV , (5.17) T T −1 −1 −1 g = X gX, gX = (gX) , g = hf h = h2f h2,X = E ΛE, (5.18) T T −1 −1 −1 f = X fX, fX = (fX) , f = hg h = h2g h2,X = L Λ1L, (5.19) T T −1 h = X hX, hX = (hX) ,X = H Λ2H, (5.20) T T −1 h2 = X h2X, h2X = (h2X) , Λ2 = T ΛT , (5.21) f = hS, h = gS, T ≡ HE−1, (5.22)

2 2 2 where Λ, Λ1, Λ2 ∈ O(m, n) and Λ = Λ1 = Λ2 = I. Thus, X ∈ O(V, g), X ∈ O(V, f) and X ∈ O(V, h).

5.2 Reﬂection Group of the Square-Root Isometry

± ± Given any X ∈ End(V ), define Π (X) := (IV ± X)/2. It can be easily proven that Π are ± ± ± 2 projectors if and only if X is involutory (Π Π = Π iff X = IV ). These projectors are + − + − pair-wise orthogonal, ker Π ∩ker Π = {0V }, and they split the space, V = ker Π ⊕ker Π . Consequently Π± are diagonalizable with the eigenvalues {1, 0}, thus X has the eigenvalues 2 {1, −1}. Therefore, X = IV is a reflection and always diagonalizable. What about the square-root S solving S2 = A where A is arbitrary? Not all matrices are diagonalizable. However, any matrix A ∈ Mat(n, C) can be written in the Jordan canonical (or normal) form [HJ94, Hig08]

−1 J = Z AZ = diag(J1,J2,...,Jp), (5.23)

Pp where J is the direct sum of p ≤ n Jordan blocks Jk ∈ Mat(nk, C), n = k=1 nk that depend 5.2 Reﬂection Group of the Square-Root Isometry 57

s on the spectrum {λi}i=1 of A (having s ≤ p distinct eigenvalues)   λk 1  .   λ ..  J := J (λ ) =  k  ∈ Mat(n , ) (5.24) k nk k  .  k C  ..   1  λk

Given any scalar-valued function f, denote

 0 1 (n −1)  f(λk) f (λk) ··· f k (λk) (nk−1)!  . .   f(λ ) .. .  f(J ) := f(J (λ )) =  k  . (5.25) k nk k    .. 0   . f (λk)  f(λk)

We say that f is deﬁned on the spectrum of A iﬀ all derivatives in f(Jk) exist for all Jordan blocks Jk. In such case, we can introduce a matrix-valued function f(A) given by

−1 −1 f(A) := Z f(J) Z = Z diag(f(Jk)) Z . (5.26)

Such deﬁned f(A) is called the primary matrix function.1 √ Now, specialize the stem function f to be the square-root f(x) = x. Since f has two ± branches, we shall write Eq. (5.25) as

(jk) (jk) p Lk := Lk (Jk) = f(Jnk (λk)) = ± Jk, (5.27)

(1) (2) where jk ∈ {1, 2} denotes one of the ± branches of f. Thus Lk = −Lk . Let s ≤ p be the number of diﬀerent eigenvalues of A. According to the classiﬁcation of square-roots by Theorem 1.26 in [Hig08] (see also [Hig87] and [HJ94, p. 468]), A has exactly 2s square-roots given by

(j1) (j2) (jp) −1 s Sj = Z diag(L1 ,L2 ,...,Lp ) Z , j ∈ {1,..., 2 }, (5.28) satisfying the condition ji = jk whenever λi = λk, and corresponding to all the possible combinations of j1, . . . , jp ∈ {1, 2}. These square-roots are called primary, as they are primary functions. If s < p, then A in addition has nonprimary square-roots given by parametrized families

(j1) (j2) (jp) −1 −1 s p Sj(U) = ZU diag(L1 ,L2 ,...,Lp ) U Z , j ∈ {2 + 1,..., 2 }, (5.29)

2 where for each j there are i and k (that depend on j) such that λi = λk for ji 6= jk. Here,

1There are several ways of defining a matrix function f(A) [HJ94, Hig08]. The most common are via the Jordan canonical form, via the Hermite interpolation and via the Cauchy integral using the resolvent. These definitions are equivalent and produce primary matrix functions. On the other hand, nonprimary matrix functions are obtained by solving matrix equation and inexpressible by these (three) definitions. 2For an eigenvalue that appears in more than one Jordan block, the equal choice of branch is not made in each block. A matrix is non-derogatory iff each of its distinct eigenvalues only appears in one Jordan block. Hence, all the square-roots of a non-singular non-derogatory matrix are primary. 58 Chapter 5. Simultaneous Proper Decomposition each family is parametrized by an arbitrary invertible matrix U that commutes with J; thus, every family Sj(U) contains infinitely many elements sharing the same spectrum. Notice also that, because some eigenvalues λi and λk of a nonprimary square-root must be λi = −λk, only the spectrum of a primary square-root can lie in the open right half-plane. − For a matrix A without eigenvalues on R , this provides the existence and uniqueness of the principal square-root A1/2 whose all of the eigenvalues are in the open right half-plane (Theorem 1.29 in [Hig08]). Theorem 5.1. Let (V, f) and (V, g) be regular quadratic spaces coupled through the square- √ root isometry S :(V, f) ,→(V, g). Let further (V, h) be the associated symmetrizing space, h = gS. The set of all involutory automorphisms X on a vector space V which are primary functions that satisfy the equation XSX = X and symmetrize hX, forms a finite abelian group under the composition of functions. This group will be called the (primary) reflection group of the square-root isometry S, denoted by Γ(S). The reflection group Γ(S) is a common subgroup of O(V, g), O(V, f) and O(V, h), h = gS. Remark 5.2. The theorem still holds if we extend the reflection group by including nonprimary functions, provided that we fix U in Eq. (5.29). This group is then called the nonprimary reflection group of S, denoted by Γ(S, U).

2 2 2 2 Proof. Besides S2 = S , consider another square-root S3 = S with h3 = gS3 = hY . The equation S = XSX = YSY yields the involutory solutions X and Y that commute with each other as both are the primary functions of S (or even nonprimary with a fixed U). Thus 2 XY = YX and XYXY = (XY ) = IV , so the composition law is closed and commutative. Finally, the identity element IV ≡ idV symmetrizes gS and for every X there exists an inverse since X−1 = X. The subgroup property relative to the orthogonal groups O(V, g), O(V, f) and O(V, h), follows from Proposition 5.1. The finiteness of the reflection group Γ(S) can be shown by investigating the properties of the Jordan canonical form of the matrix square-root S. Using Eq. (5.27) define

(ji−ki) ji−ki (ji) (ki),−1 Pi := (−1) Ii = Li Li , ji, ki ∈ {1, 2}, i ∈ {1, . . . , p}, (5.30) where Ii is the identity matrix of the equal size as the Jordan block Ji. Then clearly

−1 (j1−k1) (j2−k2) (jp−kp) −1 −1 s Xjk = SjSk = ZU diag(P1 ,P2 ,...,Pp ) U Z , j, k ∈ {1,..., 2 }, (5.31) where we included any fixed invertible matrix U that commutes with J for the sake of 2 −1 generality. Furthermore, Xjk = In, Xjk = Xjk , Xjj = In and XjkSk = SkXjk. Since Pi at a fixed slot i differs only in sign, provided the constraint j = j whenever λ = λ , the i k √ i k s number of distinct elements Xjk is 2 . Hence, the reflection group Γ( A) is finite. (ji−ki) Let X be a subset of {Xjk} which contains a single negative Pi block, i.e., it has only one block ji 6= ki. The number of elements in X is obviously s. Now, each element of {Xjk} can be obtained as the combination of finitely many elements of X (and their inverses), hence the set X is a generating set of the finite reflection group. This generating set is minimal, but not unique as any element Xj ∈ X can be replaced by its combination with any other element Xk ∈ X , Xj → XjXk (this substitution can be repeated many times p with different Xj and Xk). If we include the nonprimary functions with a fixed U, we get 2 as the size of Γ(S, U), with p as the size of its minimal generating set. 5.3 Symmetrizing Space Covered by Null-Cones 59

5.3 Symmetrizing Space Covered by Null-Cones

m,n Lemma 5.1. Consider the Minkowski space (R , η). Let the orthogonal transformation Λ ∈ O(m, n) be an involution Λ2 = I with exactly two negative eigenvalues such that η and m,n m,n ηΛ have the same signature n − m. Then the null-cones N (R , η) and N (R , ηΛ) touch m+n each other twice from outside covering the vector space R . Consequently, any vector that is ‘t’-like with respect to η will be ‘s’-like with respect to ηΛ and vice versa.

Proof. Earlier we have shown that any involution can be diagonalized. Let Λ = TPT −1 be an eigendecomposition of Λ where P is diagonal with the elements ±1 and T a nonsingular similarity transformation. From ηΛ = ΛTη, we conclude

T TηΛT = T TηT T −1ΛT = T TηT P = P TT TηT, (5.32) which also implies T TηT = P TT TηT P. On the other hand, set g = ηΛ and f = η in Theo- rem 5.3, and deﬁne Z := TR−1 where R is an orthogonal transformation R that diagonalizes T T T T ηT = R dηR (note that T is not an orthogonal transformation); then, we have Z ηZ = dη T and Z ηΛZ = dηP . In the coordinate system w = Zw, the plane of reﬂection for dη with respect to P 2 = I is given relative to the intersection

m,n T { w ∈ R | w dη(I − P )w = 0 } = N (dη − dηP ). (5.33)

i j An arbitrary reflection with two negative elements P i = P j = −1 preserves the signature n−m if and only if one index refers to ‘t’ and the other to ‘s’ subspace. Such reflection is then equivalent to the permutation of the coordinates xi and xj, i.e., it “swaps” the respective i j axes. This is because P transforms dii < 0 and djj > 0 into diiP i > 0 and djjP j < 0, respectively (with no summation on repeated indices here). Therefore, the null-cones N (dη) ∼ m−1,n−1 and N (dηP ) touch each other, and the intersection is a degnerated null-cone = N (R ) with the coordinates xi and xj excluded. In particular for m = 1, the intersection is the k set of two lines obtained from N (dη − dηP ) by setting all coordinates x to zero for which k P k = 1 and crossing at the origin (see the remark on page 80). Notice also that, formally, the null-cones exclude the zero-vector by definition, which we must add by hand to convert rays into lines. Now, apply the inverse coordinate transformation w = Zw. The isometry

Z will map the intersection N (dη) ∩ N (dηP ) into N (η) ∩ N (ηΛ) preserving the rank of the intersection (that is, lines will be mapped into lines, etc.).

m,n Theorem 5.2. Let (V, g) be a regular quadratic space isometric to (R , η) and X ∈ O(V, g) 2 an involutory orthogonal transformation, X = IV with exactly two negative eigenvalues, such that g and gX have the same signature n − m. Then the null-cones N (V, g) and N (V, gX) touch each other twice from outside, covering the space so that any vector which is ‘t’-like with respect to g is ‘s’-like with respect to gX and vice versa.

∼ m,n Proof. Let E be the vielbein of the isometry (V, g) = (R , η). According Corollary 2.1, m,n −1 for every X ∈ O(V, g) there exists Λ ∈ O(R , η) such that X = E ΛE. Furthermore, 2 2 the involution X = IV is necessary and suﬃcient for the involutory Λ = I. Let us employ change of basis using the vielbein E on the null-cones. The ﬁrst null-cone transforms as

T m,n E : N (V, g) = N (V,E ηE) → N (R , η), (5.34) 60 Chapter 5. Simultaneous Proper Decomposition while the second transforms as

T −1 T m,n E : N (V, gX) = N V,E η EE ΛE = N (V,E ηΛE) → N (R , ηΛ). (5.35)

m,n m,n Thus, E maps N (V, g) ∩ N (gX) into N (R , η) ∩ N (R , ηΛ) so, by Lemma 5.1, the null-cones N (V, g) and N (V, gX) touch each other since E is bijective.

Corollary 5.1. Let (V, f) and (V, g) be regular quadratic spaces coupled by the square-root √ isometry S :(V, f) ,→(V, g) with the associated symmetrizing space (V, h) where h = gS. Consider a subset of the reﬂection group Γ(S) such that each reﬂection X in the subset has exactly two negative eigenvalues and also preserves the signature of hX. Then, all the null-cones N (V, hX) touch each other covering the complete space.

Proof. By Proposition 5.1 and Theorem 5.2.

Remark 5.3. An interpretation of the null-cone “tilings” can be given in terms of packings on d−1 m,n the sphere. For this purpose, introduce the unit sphere S on (R , η) where d ≡ m + n d−1 d−1 (notice that the projective space P R is S with antipodal points identiﬁed [Hit03]). An intersection of a null-cone N (η) and the unit sphere will be a torus Sm−1 × Sn−1 (e.g., in 1,2 1 the case case R , the intersection will be a set of two disconnected circles, {−1, 1} × S ). The similarity transformation T from Lemma 5.1 will deform the spheres in Sm−1 × Sn−1 of

N (η), into the “ellipses” of N (dη), which will be reﬂected by P into the ﬁnal N (dηP ). The collection of intersections for all possible P will pack Sm−1 × Sn−1 on the unit sphere Sd−1 m−1,n−1 d−1 d−1 touching at N (R ) ∩ S (for m = 1, these become points on S ). For a quadratic space (V, g = E∗η), the vielbein E will deform the shapes, yet preserving the packing.

5.4 Intersecting Null-Cones

Again, consider regular quadratic spaces (V, f) and (V, g) coupled through the square-root √ isometry S :(V, f) ,→(V, g) with the associated symmetrizing space (V, h), h = gS. By Theorem 1.6 there always exists a basis where the symmetric bilinear form h is diagonal, h = ht ⊕ hs. Here ht and hs denote the restrictions of h on Vt and Vs, respectively. Assume ∗ t,−1 −1 ∗ t −1 (not necessarily proper) decompositions g = Ξ(ν) g ⊕ gs , g = Ξ(−ν) g ⊕ gs and ∗ t,−1 ] ] −1 f = Ξ(µ) f ⊕ fs . Then from g[ = h[f h[ and f[ = h[g h[, i.e., from g = hf h and f = hg−1h in matrix notation, we obtain the constraint ! ! gt,−1 + νTg ν νTg h f th h f tµTh s s = t t t s , (5.36) t −1 t T gsν gs hsµf ht hs fs + µf µ hs ! ! f t,−1 + µTf µ µTf h gth h gtνTh s s = t t s s . (5.37) t −1 t T fsµ fs hsνg ht hs gs + νg ν hs

Because the shift parameter of h is zero, the Vt-center ray of h is a hypersurface in Vt at x = 0, and the condition for the null-cones N (g) and N (f) to contain the center ray of h are (from Eq. (3.95))

T t,−1 T 0 = t α1g + ν gsν t, (5.38)

T t,−1 T 0 = t α2f + µ fsµ t, (5.39) 5.5 Simultaneous Diagonalization by Congruence 61

such that α1, α2 ≤ 1 for any non-zero t ∈ Vt. The equivalent statement is that the Vt-center ray of h is ‘t’-like relative to both g and f

T t,−1 T t g + ν gsν t < 0, (5.40)

T t,−1 T t f + µ fsµ t < 0, (5.41)

T T After expressing ν gsν and µ fsµ from the ‘tt’-parts in Eqs. (5.36) and (5.36), we get

T t t,−1 ν gsν = htf ht − g , (5.42) T t t,−1 µ fsµ = htg ht − f . (5.43)

Substituting back into the null-cone equations yields

T t,−1 T t (1 − α1)t g t = t htf htt, (5.44) T t,−1 T t (1 − α2)t f t = t htg htt, (5.45)

t,−1 for all t ∈ Vt. Hence the constraint α1, α2 < 1 is equivalent to the requirement that −g t,−1 and −f are simultaneously definite (ht does not influence the definiteness of the right- hand side). In the case when both −gt,−1 and −f t,−1 are positive definite, we have the simultaneous proper decomposition. Beware that, for m < n, a negative definite −gt,−1 does not imply a positive definite gs but rather an indefinite one. However, by using two-axis reflections X from Corollary 5.1, we can then go to a basis where the alternative symmetrizing space hj = gSj = hX is diagonal (like, e.g., −hS). Since the null-cones N (h) and N (hX) touch each other, the Vt-center ray of hX is orthogonal to the center ray of h, thus, in one of these alternative basis, −gt,−1 and −f t,−1 will be positive definite admitting the simultaneous proper decomposition. However, a more decisive proof is by finding the basis where all the forms g, f and h = gS are simultaneously diagonal (or near diagonal) and assert the relation between such diagonalized g and f.

5.5 Simultaneous Diagonalization by Congruence

Simultaneous diagonalization of matrices usually refers to diagonalization by similarity. Nonetheless, two symmetric matrices can be simultaneously diagonalized not only via a similarity transformation, but also by congruence. This is a well-known result, perhaps as earliest obtained in classical mechanics wherein used to ﬁnd normal mode coordinates in the study of small oscillations of coupled oscillators. Consider a pencil B −λA of two matrices A and B, with A nonsingular. The generalized eigenvalue equation (B − λA)x = 0 is equivalent to (A−1B − λI)x = 0, hence A−1B may share the eigenvectors with A and B. Further elaboration leads to the following claim.3

Theorem 5.3. Let (V, f) and (V, g) be two quadratic spaces where g is regular. Deﬁne A := ] −1 g f[ ∈ End(V ) with the matrix g f relative to a given basis. Then there exist T ∈ Aut(V ) T T such that T gT = dg and T fT = df are both diagonal if and only if A is diagonalizable.

3For a more general statement, see [HHJ86] and Theorem 4.5.15 in [HJ90, p 228]. 62 Chapter 5. Simultaneous Proper Decomposition

T T Proof. If T gT = dg and T fT = df are both diagonal then T diagonalizes A via similarity

−1 −1,T −1 −1 −1,T −1 −1 −1 g f = (T dgT ) (T df T ) = T dg df T . (5.46)

Conversely, assume that there is a nonsingular Z and a diagonal D such that A = ZDZ−1 = −1 g f. Without loss of generality, suppose that D has p distinct values λk, k = 1, . . . , p where the multiple values are grouped together

D = λ1I1 ⊕ λ2I2 ⊕ · · · ⊕ λpIp, (5.47)

Pp eﬀectively decomposing the space V = V1 ⊕ V2 ⊕ · · · ⊕ Vp so that dim V = k=1 nk where dim Vk = nk is the multiplicity of λk. Here Ik denotes the identity map on Vk. Then,

fZ = gZD and Z TfZ = Z TgZD. (5.48)

Let us accordingly partition Z, f and g. We shall use indices to designate blocks 1, . . . , p instead of the individual elements. Choose any oﬀ-diagonal block Xij : Vj → Vi of X = Z TfZ = Z TgZD with i 6= j. From the symmetry of Z TfZ and Z TgZ, we obtain

X T X T X T X T X T ZikgklZljλj = ZikfklZlj = ZjkfklZli = ZjkgklZliλi = ZikgklZljλi. (5.49) k,l k,l k,l k,l k,l

P T Since i 6= j implies λi 6= λj, then necessarily k,l ZikgklZkj = 0, which further yields P T T T k,l ZikfklZkj = 0. Hence, Z fZ and Z gZ are both block diagonal; that is

T T Z fZ = f1 ⊕ f2 ⊕ · · · ⊕ fp = λ1g1 ⊕ λ2g2 ⊕ · · · ⊕ λpgp = Z gZD. (5.50)

T Becaise g is symmetric, we can diagonalize each block gk = RkdkRk using a vielbein Rk. Now combining dg = d1 ⊕ d2 ⊕ · · · ⊕ dp and R = R1 ⊕ R2 ⊕ · · · ⊕ Rp, we can express

T T T T Z gZ = R dgR,Z fZ = R dgDR. (5.51)

−1 Therefore, T = ZR diagonalizes both f and g, and dg = df D.

Remark 5.4. Observe that df and dg are two bilinear forms diagonalized on V while D is a diagonal operator on V .

Theorem 5.4. Let (V, f) and (V, g) be regular quadratic spaces coupled through the square- √ root isometry S :(V, f) ,→(V, g). Let further (V, h) be the associated symmetrizing space, −1 h = gS. Take b to be any of f, g or h. If Z ∈ Aut(V ) diagonalizes S such that S = ZDSZ , ∗ ∗ ∗ −1 and the vielbein R ∈ Aut(V ) diagonalizes Z b such that Z b = R db, then T := ZR is an isometry which simultaneously diagonalizes all of f, g and h. Moreover, T diagonalizes 0 T 2 any pencil α1f + α2g + α3h and any quadratic space g = F (S) gF (S) = gF (S) that is an element of the coupled set QS(g) (see Deﬁnition 2.5).

Proof. If T diagonalizes g and f, it also diagonalizes S2 = g−1f as well any other function F (S) deﬁned on the spectrum of S, provided that S exists. Observe that the generalized eigenvalue equation (h − λg)w = 0, w ∈ V is equivalent to (S − λIV )w = 0. Now, replace f with h = gS in Theorem 5.3. Then A = g−1f becomes A = g−1gS = S. This completes the proof for f, g and h. Diagonalization of the pencil α1f + α2g + α3h is the consequence of 5.5 Simultaneous Diagonalization by Congruence 63

T 0 T −1 2 2 the linearity of T . Finally, we have T g T = T gT T F (S) T = dgF (DS) for the elements of the coupled set.

In general, the product of two diagonalizable matrices is not diagonalizable, therefore we cannot always diagonalize g−1f. Nevertheless, for a non-diagonalizable g−1f we can employ the real Jordan canonical form and assert the canonical pair theorem for symmetric matrices.

Theorem 5.5. Let f and g be real quadratic forms, with g nonsingular, and let A = g−1f have a real Jordan canonical form

−1 J = Z AZ = diag(J1,...,Jq,Cq+1,...,Cp) ∈ Mat(n, R), (5.52) where the Jordan blocks Jk ∈ Mat(nk, R) are of type as in Eq. (5.24) corresponding to real −1 eigenvalues λk ∈ R of A = g f, and the Jordan blocks Ck ∈ Mat(2nk, R) are of type as in Eq. (5.53)   Λk I2  .  ! !  Λ ..  a b 1 0  k  k k Ck :=  .  , Λk = ,I2 = , (5.53)  ..  −bk ak 0 1  I2  Λk

−1 corresponding to the pairs of conjugate eigenvalues ak ± ibk of g f. Then there exists a nonsingular congruence matrix Z ∈ Mat(n, R) such that

T Z gZ = diag(1E1, . . . , qEq,Eq+1,...,Ep), (5.54) T Z fZ = diag(1E1J1, . . . , qEqJq,Eq+1Cq+1,...,EpCp), (5.55) where k ∈ {±1} are auxiliary signature terms, and Ek is the permutation matrix

 1 .

. Ek :=  . , (5.56) 1 where dim Ek = dim Jk for k ≤ q and dim Ek = dim Ck for k > q.

Proof. The proof is given in [Uhl76] (originated from [Mut05]); see also Theorem 4.5.19 in [HJ90], Theorem 0.4 in [Uhl73], and Chapter XII in [Gan59].

Theorem 5.6. Let (V, f) and (V, g) be regular quadratic spaces coupled through the square- √ root isometry S :(V, f) ,→(V, g) where S is a primary function of g−1f. Then there exists a basis which admits the simultaneous proper decomposition of f and g.

Proof. For simplicity, we shall assume we are in the basis where all the Jordan blocks in −1 p −1 Theorem 5.5 are trivial for A = g f and S = g f. This implies that Ek are either 0 1 scalars 1 for k ≤ q or the permutation matrices P2 := ( 1 0 ) for k > q). Then

T Z gZ = diag(1, . . . , q,P2I2,...,P2I2), (5.57) T Z fZ = diag(1λ1, . . . , qλq,P2Λq+1,...,P2Λp). (5.58)

−1 − Clearly, when g f is without eigenvalues on R , S is a primary function and the principal square-root (g−1f)1/2 exist whose all of the eigenvalues are in the open right half-plane 64 Chapter 5. Simultaneous Proper Decomposition

(Theorem 1.29 in [Hig08]). In such case the auxiliary signature terms k of g and f are of the same sign since necessarily λk > 0 for k ≤ q. On the other hand, for k > q, the signature 2 2 2 terms for Λk are ﬁxed to 1, and because det Λk = |Λk| = ak + bk > 0, the determinants of the corresponding Jordan blocks of f and g are of the same sign, i.e., the blocks have the −1 same signature (wherein bk ak acts as a shift parameter).

Remark 5.5. A necessary condition that a matrix A has a real square-root is that, for every negative eigenvalue, A comprises an even number of Jordan blocks of each size. Moreover, if A has negative eigenvalues then none of its primary square-roots is real [HJ94, Theorem 6.4.14],

[Hig08, Theorem 1.23]. This, for example, means that A = −I2n has a real nonprimary −1 0 1 0 square-root for all n ∈ N. More specifically, take a look at g = 0 1 , f = 0 −1 , S = 0 1 −1 0 . In such case, the intersection of the null-cones is not an open set. Notice also that, J 0 0 I 2 if two Jordan blocks J of A are of the equal size, then 0 J = J 0 . Remark 5.6. The presence of non-trivial Jordan blocks is related to the geometry of null- cones. It can be verified that the null-cones of g and f share a common tangent hyperplane for each Jordan block dim Jk > 1. On the other hand, a special condition can be achieved which ensures diagonalization of g−1f. When d = m + n ≥ 3, the null-cones of g and f are disjoint (N (g) ∩ N (f) = ∅) if and only if there exists a positive definite pencil α1f + α2g for −1 some α1, α2 ∈ R, and in such case, g f is always diagonalizable [Uhl79]. As we have shown, the square-root isometry does not exist if the null-cones are disjoint. Thus, N (g)∩N (f) = ∅ ensures that the null-cones of g and f can not have a common tangent hyperplane. This effectively eliminates the Jordan blocks of size > 1 making diagonalization possible, which is compliant with [Uhl79]. Further elaboration of this claim is beyond the scope of this document (it is a part of our other work), and we only provide an example here.

∼ ∼ 1,n 1,1 Example 5.1. Let (V, g) = (V, f) = (R , η) where (we shall write only the upper-left R block): ! ! ! 0 1 0 ξ ξ 1 − ξ g = , f = , g−1f = , ξ ∈ , ξ 6= 0. (5.59) 1 1 ξ 1 0 ξ R The Jordan canonical form J and the square-root S of g−1f are given by

! √ 1  ξ 1 ξ √ (1 − ξ) J = ,S =  2 ξ√  . (5.60) 0 ξ 0 ξ

The intersection of the null-cones of g and f is the ray which coincides with the axis x0 and the null-cones touch each other sharing the common tangent plane x1 = 0. The (real) square-root does not exist for ξ < 0. On the other hand, the associated vielbeins and the symmetrizing orthogonal transformation read

! !  √1 (1 + ξ) √1 (1 − ξ) 1 0 ξ 0 −1 2 ξ 2 ξ E = ,L = , Λ = ESL =  , (5.61) 1 1 ξ 1 √1 (1 − ξ) √1 (1 + ξ) 2 ξ 2 ξ which, for ξ > 0, can be reparametrized ! γ γv 1 − ξ 1 1 + ξ Λ = , v = ∈ (−1, 1), γ = √ = √ ≥ 1. (5.62) γv γ 1 + ξ 1 − v2 2 ξ Chapter 6

Application

The major application of the square-root isometry is in bimetric theory where it is used to construct a ghost-free action. In this chapter, we specialize to regular quadratic spaces 1,n isometric to R that correspond to tangent spaces at points of a diﬀerentiable manifold, where we mostly take n = 3. We ﬁrstly review bimetric theory and then provide a summary of the ghost analysis in the Hamiltonian framework. Finally, we show the equivalence of bimetric theory and its reformulation in terms of vielbeins.

6.1 Review of Bimetric Theory

Bimetric Action

Consider a diﬀerentiable manifold M equipped with two interacting spin-2 ﬁelds gµν(x) and fµν(x). The bimetric action is given by [HR12b, HR12c]

√ S = m2 d4x −g R(g) + m2 d4x p−f R(f) − HR g ˆ f ˆ √ 4 q − 2m4 d4x −g X β e g−1f. (6.1) sc ˆ n n n=0

Here, for the free-field terms, mg and mf are the corresponding Planck masses for the spaces (M, g) and (M, f). The interaction term is parametrized by a mass scale msc and interaction parameters βn, and is given in terms of en(X) which denote the elementary symmetric polynomials of a matrix X.1 The dependence on the square-root isometry S = pg−1f is essential for the absence of the Boulware-Deser ghost. The action (6.1) has a mass spectrum that is well-defined around proportional back- 2 grounds f¯µν = c g¯µν. The perturbations of the two metrics subsequently diagonalize into 2 2 2 a massless δG ∝ δg/mf + δf/mg and a massive fluctuation δM ∝ δf − c δg [HSMvS13]. Therefore, we can consider the metrics gµν and fµν are mixture of massless and massive

1The elementary symmetric polynomials appear in Vieta’s formulas, and also, for a matrix X with the eigenvalues λi, in the expansion of the determinant

4 X X det(I + X) = ek(X), ek(X) = λi1 ··· λik .

k=0 1≤i1<···

Mikica Kocic, The Square-Root Isometry, MSc thesis, SU 2014 Printed: 2015-03-11 66 Chapter 6. Application modes where the metric with a larger Planck mass dominates as a massless component. The matter couplings which can be added to Eq. (6.1) without reintroducing the ghosts issue, are of the form,

√ S = d4x g L (g, φ ) + d4xpf L (f, φ ). (6.2) matter ˆ g g ˆ f f

Here, φg and φf are different types of matter fields which minimally couple to metrics g and f in the standard way. As pointed out before, the metric gµν is mostly massless when mg mf , in which case gµν can be regarded as the usual gravitational metric with MPl = mg [HR12b, HSMvS13], while fµν is an extra field that modifies gravity.

Hamiltonian Analysis

The Hamiltonian analysis is done using the ADM formalism [ADM08] and the full aspects of the ghost analysis of bimetric theory is given in [HR12a, HRSM12, HR12b, HR12c]. Suppose we are provided with a coordinate basis that simultaneously admits proper m+n decomposition of the two metrics. Then, in matrix notation we have the decomposition (see Section 3.3, and “3+1 Decomposition in GR” on page 39)

! ! −N 2 + νTgν˜ νTg˜ −M 2 + µTfµ˜ µTf˜ g = , f = . (6.3) gν˜ g˜ fµ˜ f˜

Here, N and M denote the corresponding lapses, νi and µi are the shift three-vectors, and g˜ij and f˜ij denote the metrics reduced to the spatial subspace with i, j ∈ Is ranging over the index set Is = {1, . . . , n}. The transition to the Hamiltonian formalism is done by introducing the momenta πij ij and p canonically conjugate to g˜ij and f˜ij, after which, in the phase space, the Lagrangian density reads

ij ij ˜ 0(g) i (g) 0(f) i (f) 4 ˜ ˜ L = π ∂tg˜ij +p ∂tfij +NR +ν Ri +MR +µ Ri −2mscV (N, M, ν, µ, g,˜ f) . (6.4)

Note that this density is given up to surface terms. For the purpose of counting degrees of freedom, let m = 1, n = 3 and d = m + n = 4 . The 2 × (m + n) = 2d = 8 components of lapses and shifts N, M, νi, µi occur without time derivatives and, as such, they are non-dynamical. On the other hand, g˜ij and f˜ij contain 2×n(n+1)/2 = 12 dynamical ﬁelds (yielding a total of 2n(n+1) = 24 degrees of freedom in the phase space). These degrees of freedom also include some ghost modes that have to be eradicated by gauge ﬁxing, in addition to those constraints coming out from the equations of motion for N, M, νi, µi. The basis of the Boulware-Deser argument for the presence of a ghost [BD72b] is that, because V˜ is nonlinear in the N, M, νi, µi, then the corresponding 8 equations of motion could potentially depend on all the non-dynamical variables and govern them in terms of g˜ij and f˜ij instead of becoming constraints on g˜ij and f˜ij. Nevertheless, the other possibility is that these equations do not determine all of N, M, νi, µi thus providing some of them to instead impose constraints on g˜ij and f˜ij. The analysis is outlined below. It can be shown that the 8 equations of motion emerging out from N, M, νi, µi can tie only 3 combinations of these variables in terms of g˜ij and f˜ij, then the rest of 5 equations will be independent on the non-dynamical variables, and as such, they become constraints on g˜ij 6.1 Review of Bimetric Theory 67

and f˜ij, thus eliminating some of the undesired extra modes [HR12b]. Let us denote these 3 combinations by ni(N, M, ν, µ, g,˜ f˜). Now, we can substitute 3 of the non-dynamical variables in action, for example νi, with these new ni(N, M, ν, µ, g,˜ f˜) (retaining the 3-dimensional general covariance). For the consistency, the remaining lapses and shifts µi, N and M must appear linearly as Lagrange multipliers that impose the constraint equations (5 in total). i (g) i Because the action in Eq. (6.4) already has a term ν Ri , then the expression for ν in terms of ni must be linear in the remaining µi, N and M. For the action in Eq. (6.1), a redeﬁnition of shifts that works is [HRSM12, HR12b]

i i i i j ν − µ = (Mδ j + ND j) n , (6.5)

ν − µ = (MIs + ND) n, (6.6)

i i where the matrix D = D j is speciﬁed in Eq. (6.12) below. After introducing n and eliminating νi, the actionJ becomesK [HR12b]

S = d4x πij∂ g˜ + pij∂ f˜ + µiC(µ) + MC + NC , (6.7) ˆ t ij t ij i M N

(µ) (g) (f) i where the variables Ci = Ri + Ri do not dependent of n . The explicit forms of CN and CM are given in Eqs. (6.15) and (6.16) below. The important is that the theory is linear in the µi, M and N relative to ni.

Because the equations of motion for ni are linear in the lapses M and N, it appears that ni depend on M and N, which conﬂicts the assumption. However, this is only a deception because ni enter only through νi, that is

δS δS δνj i = j i = 0 , (6.8) δn δν ν=ν(n) δn or explicitly, ∂ ∂ ∂ N C + M C = C(ν) Mδj + N (Dn)i = 0 . (6.9) ∂ni N ∂ni M j i ∂nj (ν) In bimetric theory, Cj is independent of M and N, which may not be true in general. Because the Jacobian matrix element δνj/δni is invertible, it follows that ni are determined by (ν) ˜ Cj (n, g,˜ f, π, p) = 0 , (6.10) and as such, ni do not dependent of the lapses and shifts as wanted. This property was explicitly verified for the action (6.1)[HR12a, HRSM12, HR12b, HR12c]. Nonetheless, the discussion that is just presented shows that this is always the case for an action which can be made linear in the lapses and shifts by using redefinitions like the one in Eq. (6.5). The Lagrange multipliers in Eq. (6.7) subsequently yield the 5 constraints on the dynamical variables (µ) Ci = 0 , CM = 0 , CN = 0 , (6.11) where the last constraint CN = 0 is in addition followed by a secondary constraint CN(2) ≡ ∂tCN = 0 . This pair of constraints removes the BD ghost and its conjugate momentum [HR12c]. The 4 constraints that are associated with general covariance by gauge fixing eliminate another 8 degrees of freedom in the phase space. Consequently, we arrive at 68 Chapter 6. Application

24 − 2 − 8 = 14 phase space degrees of freedom, which yields 7 dynamical ﬁelds in total. It is noteworthy that some of the initial dynamical equations reduce to further constraints that determine N, M and µi. Moreover, Eq. (6.9) implies that, if all the ni equations i i are fulﬁlled, then necessarily ∂CN /∂n = 0 and ∂CM /∂n = 0, thus all the constraints become ni-independent.

The Square-Root in Bimetric Theory

The prospect of proving the absence of ghost depends on the ability of re-expressing an action which is nonlinear in the lapses and shifts (6.1) to a partly linear form (6.7) by using a redefinition as in Eq. (6.5). In the case of two dynamical metrics g and f, a yet another difficulty is in the appearance of the matrix square-root pg−1f in the action. The i redefinition in Eq. (6.5) settles both of these problems, provided that the matrix D j in Eq. (6.5) satisfies DQD =g ˜−1f˜ and also [HRSM12] J K

q −1 −1 T T D = g˜ fQQ˜ ,Q ≡ xIs + nn f,˜ x ≡ 1 − n fn˜ , (6.12) q i il ˜ m i −1 k i i j k ˜ i ˜ j D j = g˜ flmQ k k[Q ] j,Q j ≡ xδ j + n n fkj . x ≡ 1 − n fijn , (6.13) J K with the additional very important property [HRSM12, Eq. (3.12)]

fD˜ = (fD˜ )T. (6.14)

Remark 6.1. A notation reminder: Here Is stands for the identity transformation on the i spatial subspace (Is = δ j in matrix notation). Notice also that gij =g ˜ij but, in general, ij ij ij J −1K ij ij −1 ij i g 6=g ˜ because g ≡ [g ] √and g˜ ≡ [˜g ] . As usual, A j denotes√ a matrix build from i −1 i J K −1 the elements A j while, e.g., [ B CD] j are the components of B CD. i Consequently, in terms of n , the action takes the form (6.7) where CM and CN are given by [HR12b]

(g) (g) i j 4p CN = R0 + Ri D jn + 2m detg ˜ V . (6.15) (f) (g) i 4 p CM = R0 + Ri n + 2msc detg ˜ U , (6.16) where U and V are emerging from the interaction potential as

2 ! √ X √ T T U := x βn+1en( xD) + β3 e1(D) n fDn˜ − (Dn) fDn˜ + n=0 √ −g˜ T ˜ + β2n fDn + β4 q , (6.17) −f˜ 3 X √ V := βnen( xD) . (6.18) n=0

Vielbein Formulation

A reformulation of bimetric theory in terms of vielbeins which avoids the matrix square-root has been proposed in [HR12d]. As usual, in order to express the interaction potential of the metric formulation in terms of vielbeins, we decompose the two metrics using the pullbacks 6.2 Algebraic Relation to Vielbein Formulation 69 g = E∗η and f = L∗η. The square-root pg−1f can be then evaluated in terms of the vielbeins provided that the following symmetry condition (2.51) is satisﬁed. If this is the case one obtains pg−1f = E−1ΛL, Λ ∈ O(1, n).

6.2 Algebraic Relation to Vielbein Formulation

Consider a diﬀerentiable manifold M equipped with two symmetric rank-2 covariant tensor

(real) fields gµν(x) and fµν(x) having the same signature with index 1. Let P be a point in a differentiable manifold M, and V ≡ TP (M) be a tangent space at P. Let further (V, f) 1,n and (V, g) be coupled quadratic spaces isometric to R , moreover regular at P, such that g = E∗η and f = L∗η. √ Suppose that the square-root isometry S :(V, f) ,→(V, g) given in the bimetric action (6.1) exists. Then, as it was shown in Chapter5, it is always possible to find a coordinate basis that admits proper m + n decomposition of (V, f) and (V, g). In such basis we have (see Example 4.1) E = (N ⊕ e) ◦ Ξ(ν),L = (M ⊕ m) ◦ Ξ(µ). (6.19)

Here {N, ν, e} and {M, µ, m} denote the lapse, the shift vector and the spatial part of the vielbeins E and L, respectively, where

T T g˜ = e δˆse, f˜ = m δˆsm, (6.20) a b ˜ a b g˜ij = ei δabe j, fij = mi δabm j . (6.21)

Now, observe how the redeﬁnition in Eq. (6.5) in bimetric theory resembles the symmetrization condition of the vielbeins given in Eqs. (4.10) and (4.17) on page 49. Introduce one further redeﬁnition of the shift vector ni in terms of new variables va

a a b i v = Rm n, v = R bm in , (6.22) −1 i −1 i a n = (Rm) v, n = [(Rm) ] av , (6.23) where R is the spatial rotation obtained in Eq. (4.20). Note that the correspondence v ↔ n is bijective since we have assumed f to be non-degenerate. This redeﬁnition gives an expression that is equivalent to Eq. (6.5) h i ν − µ = N e−1 + M (Rm)−1 v, (6.24) provided that it holds −1 −1 i a i j e v = Dn, [e ] av = D jn . (6.25) To examine whether this is true, we shall evaluate the variables D, Q and x in Eq. (6.12) in terms of the new variables. We begin with x

T T −1,T T −1 x = 1 − n fn˜ = 1 − v (Rm) m δˆsm(Rm) v, (6.26)

T which, by using R δˆsR = δˆs, becomes in terms of the γ-factor of the Lorentz transformation

T −2 x = 1 − v δˆsv = γ . (6.27) 70 Chapter 6. Application

Now, we know for the spatial part of the Lorentz transformation (note that δˆt ≡ 1 for m = 1)

γ2 1 Λ˜ := δˆ−1 + vvT = δˆ−1 + √ vvT, Λ˜ = Λ˜ T, (6.28) s γ + 1 s x + x

˜ such that (see Eqs. (3.27) and (3.20), take Λs ≡ Λδˆs)

˜ ˆ ˜ T ˆ−1 2 T ˆ−1 −1 T ΛδsΛ = δs + γ vv = δs + x vv . (6.29)

˜ −1 −1 Next, we can express Q in terms of Λ as (notice that (Rm) (Rm) = In, (Rm)(Rm) = Is)

T Q = xIs + nn f˜ (6.30) −1 T −1,T T = xIs + (Rm) v v (Rm) (Rm) δˆsRm (6.31) −1 T = xIs + (Rm) vv δˆsRm (6.32) −1 T = (Rm) (xIn + vv ) δˆs(Rm) (6.33) √ √ −1 ˆ−1 −1 T ˆ = (Rm) x δs + x vv δs x(Rm) (6.34) √ √ −1 ˜ ˜ T = (Rm) xΛδˆsΛ δˆs x(Rm). (6.35)

Finally, for D we have

DQD =g ˜−1f,˜ (6.36) g˜−1 = DQDf˜−1 = DQ f˜−1DT, (6.37) where we used (fD˜ )−1 = (fD˜ )−1,T which follows from Eq. (6.14). Substituting the earlier T T obtained Q and f˜ = m δˆsm = (Rm) δˆs(Rm), yields

h √ √ i h i−1 −1 −1 ˜ ˜ T T T g˜ = D (Rm) xΛδˆsΛ δˆs x(Rm) (Rm) δˆs(Rm) D (6.38) √ h i h i √ −1 ˜ ˆ ˜ T ˆ −1ˆ−1 −1,T T = xD (Rm) ΛδsΛ δs(Rm) (Rm) δs (Rm) D x (6.39) √ h i √ −1 ˜ ˜ T −1,T T = xD (Rm) ΛδˆsΛ (Rm) D x (6.40) √ √ T −1 ˜ ˆ ˆ−1 −1 ˜ ˆ = xD (Rm) Λδs δs xD (Rm) Λδs , (6.41)

T ˆ −1 −1ˆ−1 T,−1 On the other hand, from g˜ = e δse we have g˜ = e δs e , thus √ −1 −1 ˜ e = xD(Rm) Λδˆs. (6.42) √ From this we get the expression for D (notice γ = ( x)−1)

−1ˆ−1 ˜ −1 D = e δs γΛ (Rm) (6.43) √1 −1 1√ T ˆ = x e In − 1+ x vv δs (Rm). (6.44)

−1 −1ˆ−1 ˜ −1 −1 After extracting D (Rm) = e δs γΛ , we can evaluate D (Rm) v as

−1 −1ˆ−1 ˜ −1 −1 D (Rm) v = e δs γΛ v = e v, (6.45) where, in the last step, we used the property that the γ−factor is an eigenvalue of the spatial 6.2 Algebraic Relation to Vielbein Formulation 71

˜ ˆ ˆ−1 ˜ −1 part of the Lorentz boost, Λδsv = γv, that is, δs γΛ v = v (see Example 3.1 on page 34). Because (Rm)−1v ≡ n, from Eq. (6.22) we ﬁnally conclude

Dn = e−1v , (6.46) which completes the ﬁrst part of the proof. The second part is to verify that the spatial symmetrization condition holds. We begin from Eq. (6.14) fD˜ = DTf˜

T T T (Rm) δˆs(Rm) D = D (Rm) δˆs(Rm), (6.47) −1 −1 −1 −1T δˆsγ(Rm)D (Rm) = δˆsγ(Rm)D (Rm) . (6.48)

On the other hand, multiplying Eq. (6.42) by e(Rm)−1 from the right then solving for Λ(˜ Rm)e−1, gives √ −1 −1 ˜ −1 (Rm) = xD(Rm) Λδˆse(Rm) , (6.49) −1 −1 ˜ −1 γ(Rm)D (Rm) = Λδˆse(Rm) . (6.50)

From Eqs. (6.48) and (6.50), we ﬁnally obtain the relation

˜ −1 ˜ −1T δˆsΛδˆs (Rm) e = δˆsΛδˆs (Rm) e , (6.51)

˜ which is exactly the spatial symmetrization condition Eq. (4.16) in terms of Λs = Λδˆs. There- fore, the vielbein symmetrization condition for the spatial part is equivalent to fD˜ = (fD˜ )T. Note that Rm also depends on e and v through R in Eq. (4.20). This however does not influence the ghost-absence argument. Remark 6.2. In terms of the new variables, the action can be reformulated from Eq. (6.7) by using the simple substitution where the spatial metrics are expressed in terms of vielbeins, i i a as well as n and D j in terms of v by using Eq. (6.22) and Eq. (6.43). Obviously, the ghost proof of [HRSM12, HR12b, HR12c] which was reviewed in Section 6.1 will go through as before, just with ni replaced by va. Remark 6.3. The matrix D in Eq. (6.12), which appears in the ghost analysis of massive gravity and bimetric theory, involves a matrix square-root which existence is not obvious. Eq. (4.20) and the presented analysis shows that, provided a coordinate basis that simultaneously admits the proper m + n decomposition of g and f, the matrix D always exists. Remark 6.4. When applying the ADM formalism in bimetric theory [HRSM12, HR12b, HR12c], it was always silently assumed that the 3+1 decomposition can be simultaneously employed for both tensor fields. The existence of the matrix square-root pg−1f ensures the simultaneous 3+1 decomposition by Theorem 5.6, and makes the ADM formalism applicable. a b Remark 6.5. From Theorem 4.1 on page 48 we must have v δabv < 1 so that that the spatial part of the boost Λ˜ and the Lorentz γ-factor are both finite and real. Since γ−2 = x, this √ translates into 0 < x ≤ 1 which is the requirement for the existence of x and of the matrix square-root pg−1f in bimetric theory. Therefore, based on the earlier propositions from Chapter4, bimetric theory and its vielbein formulation are locally equivalent. Moreover, 0 < x ≤ 1 follows from the equations of motion [HRSM12].

Chapter 7

Summary and Discussion

In this work, we explored the geometric and algebraic properties of the square-root isometry that arises when considering coupled isotropic quadratic spaces with the same index. This linear transformation has special features since it is self-adjoint relative to both of the coupled quadratic spaces, that is, it is symmetric, and as such, it has a subtle connection to the symmetric parts of orthogonal transformations (the so-called boosts) and to the involutory orthogonal transformations (which are necessarily self-adjoint). In Chapter2, we have shown that, for an arbitrary isometry to be self-adjoint, it is necessary and sufficient that there exists an orthogonal transformation such that the vielbein symmetrization condition holds (Theorem 2.2 and Eq. (2.51)). As a direct consequence, it turned out that the square-root S = pg−1f exists if and only if there exists an orthogonal transformation that symmetrizes such condition (Corollary 2.5). The important is that all these assertions were independent of basis and, moreover, applicable to quadratic spaces of an arbitrary signature. As a bonus, we also examined the properties of the square-root isometry group and its actions on the set of all quadratic spaces Q, with the conclusion that the orthogonal group O(V, b) is the stabilizer of a quadratic space (V, b) in Q. We also transitively enclosed Q and defined the coupled set QS(g) of quadratic spaces generated by a square-root isometry S of quadratic space g. This enabled us to define a notion of the continuation set, which was later assisted in constructing a basis that admits the simultaneous proper decomposition of quadratic spaces. Chapter3 presented a transition to the basis dependent considerations. Namely, in the course of this chapter, we have built the necessary tools to geometrically and algebraically decompose orthogonal transformations, quadratic spaces and isometries (vielbeins). Further- more, we correlated various parts of such decompositions with the equivalent geometrical transformations of the quadratic surfaces and null-cones of quadratic spaces. The obvious gain from Chapter3 was, on one hand, the ability to calculate, and on the other, to examine the topology of quadratic spaces coupled through the square-root isometry. In Chapter4 we explored the square-root symmetrization condition in terms of decomposed vielbeins (Theorem 4.1). In addition, we used Theorem 4.2 to manifest the properties of the symmetrization condition with respect to the symmetrizing quadratic space (Definition 2.3). Based on this, we have proved that, given a basis that admits the proper decomposition of two coupled quadratic spaces, the square-root isometry exists if and only if their null-cones intersect (Theorem 4.3). This has open the door for a topological interpretation of null-cone couplings induced by the square-root isometry, with a conclusion that the existence of the

Mikica Kocic, The Square-Root Isometry, MSc thesis, SU 2014 Printed: 2015-03-11 74 Chapter 7. Summary and Discussion square-root ensures that the null-cones intersect locally in a such way to have both a common ‘t’-like vector and a common ‘s’-like hypersurface (see Figure 7.1). Finally in Chapter 5, we proved the existence of the basis that admits a simultaneous proper decomposition of quadratic spaces coupled by the square-root isometry (Theorem 5.6).

Figure 7.1: Three possible topological cases: (1) no common Vt, (2) common Vt and Vs and (3) no common Vs.

In Chapter6 we applied the acquired tools to show the equivalence of bimetric theory and its reformulation in terms of vielbeins. This was done by introducing an algebraic redefinition of the shift variables without influencing the absence of ghost arguments. Moreover, the analysis has shown that the spatial part of the symmetrization condition is equivalent to fD˜ = (fD˜ )T, and that the parameter space of the symmetrization condition is not limited by the redefinition. To emphasize again, all the work was done in the realm of the geometric and algebraic properties of the isometries, i.e., with the static aspects only and without any dynamics involved. The natural step forward is to proceed in view of the initial value problem and to examine whether a unique and maximal solution can be assured, provided a well-posed Cauchy problem for the obtained equations of motion. The major concern is the causality, which, as it seems and at least local, is ensured if the square-root exists. Namely, the existence of the square-root ensures that the local null-cones of two fields intersect in a special way so to share both a common timelike vector and a common spacelike hypersurface simultaneously. As we know, the latter is a prerequisite for a well-posed Cauchy problem [CB14, Gou12]. Since global hyperbolicity is a very reasonable condition in the context of GR, the logical extension would be then to try to define a notion of the simultaneous global hyperbolicity for a manifold equipped with two symmetric non-degenerate rank-2 covariant tensor fields coupled by the square-root isometry, and then to investigate the globally hyperbolic development of a specified set of initial data (in addition to the uniqueness of solutions and the causality issues). Appendix A

Tedious Derivations

Equation (2.44)

2 g1 = g (α1I + β1S) , (A.1) 2 g2 = g (α2I + β2S) , (A.2) q q −1 −2 2 S12 = g1 g2 = (α1I + β1S) (α2I + β2S) (A.3) −1 = (α1I + β1S) (α2I + β2S), (A.4) 2 g3 = g1(ξ1I + ξ2S12) (A.5) 2 2 = g (α1I + β1S) (ξ1I + ξ2S12) (A.6) 2 2 −1 = g (α1I + β1S) ξ1I + ξ2(α1I + β1S) (α2I + β2S) (A.7) 2 −1 = g (α1I + β1S) ξ1I + ξ2(α1I + β1S) (α2I + β2S) (A.8) 2 = g (ξ1(α1I + β1S) + ξ2(α2I + β2S)) (A.9) 2 = g ((ξ1α1 + ξ2α2)I + (ξ1β1 + ξ2β2)S) . (A.10)

Equation (3.77)

0 T −1ˆ−1 −1,T T ˆ v v = Rte1e3 δs e3 e1Rt δt, (A.11) 0 T T ˆ −1 T ˆ v v = Rte1(e3δse3) e1Rt δt, (A.12) 0 −1 T −1ˆ−1 −1,T T ˆ −1 Im − v v = RtRt − Rte1e3 δs e3 e1Rt δtRtRt , (A.13) 0 −1 T −1ˆ−1 −1,T ˆ −1 Im − v v = RtRt − Rte1e3 δs e3 e1δtRt , (A.14) −1 0 T −1ˆ−1 −1,T ˆ Rt (Im − v v)Rt = Im − e1e3 δs e3 e1δt, (A.15) −1 0 T T ˆ −1 ˆ Rt (Im − v v)Rt = Im − e1(e3δse3) e1δt. (A.16)

Equation (3.83)

! ! ! ! ! eT eT −δˆ 0 e eT eT eT −δˆ e −δˆ eT g = 0 2 t 0 1 = 0 2 t 0 t 1 (A.17) T ˆ T ˆ ˆ e1 e3 0 δs e2 e3 e1 e3 δse2 δse3 ! −eTδˆ e + eTδˆ e −eTδˆ eT + eTδˆ e = 0 t 0 2 s 2 0 t 1 2 s 3 , (A.18) ˆ T ˆ ˆ T T ˆ −e1δte0 + e3δse2 −e1δte1 + e3δse3

Mikica Kocic, The Square-Root Isometry, MSc thesis, SU 2014 Printed: 2015-03-11 76 Appendix A. Tedious Derivations

!T ! ! ! ! e 0 −δˆ 0 e 0 eT νTeT −δˆ e 0 g = t t t = t s t t (A.19) ˆ T ˆ ˆ esν es 0 δs esν es 0 es δsesν δses ! −eTδˆ e + νTeTδˆ e ν νTeTδˆ e = t t t s s s s s s . (A.20) T ˆ T ˆ es δsesν es δses

The above yields the matrix equation from which

t,−1 T T ˆ T ˆ g + ν gsν = −e0δte0 + e2δse2, (A.21) ˆ T ˆ gsν = −e1δte0 + e3δse2, (A.22) T T ˆ T T ˆ ν gs = −e0δte1 + e2δse3, (A.23) ˆ T T ˆ gs = −e1δte1 + e3δse3. (A.24)

Solving for −gt,−1 yields

t,−1 T ˆ T ˆ −g = e0δte0 − e2δse2 + −1 T ˆ T ˆ T T ˆ ˆ T T ˆ ˆ + e2δse3 − e0δte1 e3δse3 − e1δte1 e3δse2 − e1δte0 (A.25) T ˆ T ˆ = e0δte0 − e2δse2 + −1 T ˆ T ˆ T −1 ˆ −1,T ˆ T −1 −1,T T ˆ ˆ + e2δse3 − e0δte1 e3 δs − e3 e1δte1e3 e3 e3δse2 − e1δte0 (A.26) T ˆ T ˆ = e0δte0 − e2δse2 + −1 T ˆ T ˆ T −1 ˆ −1,T ˆ T −1 ˆ −1,T ˆ + e2δs − e0δte1e3 δs − e3 e1δte1e3 δse2 − e3 e1δte0 . (A.27)

−1,T Introduce w := e3 e1

−1 t,−1 T ˆ T ˆ T ˆ T ˆ T ˆ ˆ T ˆ ˆ −g = e0δte0 − e2δse2 + e2δs − e0δtw δs − wδtw δse2 − wδte0 , (A.28) −gt,−1 = A + B + C + D, where

−1 T ˆ T ˆ T ˆ ˆ T ˆ A = e0δte0 + e0δtw δs − wδtw wδte0 (A.29) −1 T ˆ T ˆ T ˆ−1 ˆ T ˆ−1 ˆ = e0δte0 + e0δtw δs Is − wδtw δs wδte0 (A.30) −1 T ˆ T ˆ T ˆ−1 ˆ−1 T ˆ−1 = e0δte0 + e0δtw δs w δt − w δs w e0 (A.31) −1 T ˆ ˆ−1 T ˆ−1 ˆ−1 T ˆ−1 = e0δt δt − w δs w δt − w δs w e0 + −1 T ˆ T ˆ−1 ˆ−1 T ˆ−1 + e0δtw δs w δt − w δs w e0 (A.32) −1 T ˆ ˆ−1 T ˆ−1 T ˆ−1 ˆ−1 T ˆ−1 = e0δt δt − w δs w + w δs w δt − w δs w e0 (A.33) −1 T ˆ−1 T ˆ−1 = e0 δt − w δs w e0, (A.34) −1 T ˆ T ˆ ˆ ˆ T ˆ B = −e2δse2 + e2δs δs − wδtw δse2 (A.35) / use Corollary 3.2 −1 T ˆ T ˆ ˆ−1 ˆ−1 ˆ−1 T ˆ−1 T ˆ−1 ˆ = −e2δse2 + e2δs δs + δs w δt − w δs w w δs δse2 (A.36) Appendix A. Tedious Derivations 77

−1 T ˆ T ˆ ˆ−1 T ˆ−1 T = −e2δse2 + e2 δs + w δt − w δs w w e2 (A.37) −1 T ˆ−1 T ˆ−1 T = e2w δt − w δs w w e2, (A.38) −1 T ˆ ˆ ˆ T ˆ C = −e2δs δs − wδtw wδte0 (A.39) −1 T ˆ T ˆ−1 ˆ = −e2 Is − wδtw δs wδte0 (A.40) −1 T ˆ−1 T ˆ−1 = −e2w δt − w δs w e0, (A.41) −1 T ˆ T ˆ ˆ T ˆ D = −e0δtw δs − wδtw δse2 (A.42) −1 T ˆ T ˆ−1 ˆ T = −e0δtw Is − δs wδtw e2 (A.43) −1 T ˆ−1 T ˆ−1 T = −e0 δt − w δs w w e2. (A.44)

Thus ﬁnally,

−1 −1 t,−1 T ˆ−1 T ˆ−1 T ˆ−1 T ˆ−1 T −g = e0 δt − w δs w e0 + e2w δt − w δs w w e2 − −1 −1 T ˆ−1 T ˆ−1 T ˆ−1 T ˆ−1 − e2w δt − w δs w e0 − e2w δt − w δs w e0 (A.45) −1 T T ˆ−1 T ˆ−1 T = (e0 − w e2) δt − w δs w (e0 − w e2) (A.46) T −1 T −1 ˆ−1 T −1ˆ−1 −1,T T −1 = e0 − e1e3 e2 δt − e1e3 δs e3 e1 e0 − e1e3 e2 (A.47) T −1 T −1 ˆ−1 T T ˆ −1 T −1 = e0 − e1e3 e2 δt − e1(e3δse3) e1 e0 − e1e3 e2 . (A.48)

Equations (4.7)-(4.9)

! ! ! ! ! −δˆ 0 Λ p0 R 0 m 0 e−1 0 X = ηΛLE−1 = t t t t t (A.49) ˆ −1 −1 0 δs p Λs 0 Rs msµ ms −νet es ! ! −δˆ Λ −δˆ p0 R m e−1 0 = t t t t t t (A.50) ˆ ˆ −1 −1 δsp δsΛs Rsms(µ − ν)et Rsmses ! ! c cT −δˆ Λ m e−1 − δˆ p0R m (µ − ν)e−1 −δˆ p0R m e−1 0 1 = t t t t t s s t t s s s (A.51) ˆ −1 ˆ −1 ˆ −1 c2 c3 δspRtmtet + δsΛsRsms(µ − ν)et δsΛsRsmses

T T From X = X , we get the system of three coupled symmetrizations c0 = c0, c2 = c1 and T c3 = c3, where

ˆ −1 ˆ 0 −1 c0 = −δtΛtRtmtet + δtp Rsms(ν − µ)et , (A.52) T ˆ 0 −1 c1 = −δtp Rsmses , (A.53) ˆ −1 ˆ −1 c2 = δspRtmtet − δsΛsRsms(ν − µ)et , (A.54) ˆ −1 c3 = δsΛsRsmses . (A.55)

T (1) We start from the simplest expression. The symmetrization c3 = c3 of the ‘s’-part connects the boost Λs and rotation Rs (here Λs depends on v through p = Λsv)

ˆ −1 ˆ −1T δsΛsRsmses = δsΛsRsmses . (A.56) 78 Appendix A. Tedious Derivations

This equation enters the system of coupled equations unchanged.

T T (2) The symmetrization c2 = (c1) = c1 implies

ˆ −1 ˆ −1 ˆ 0 −1T δsΛsRsms(ν − µ)et = δspRtmtet + δtp Rsmses (A.57) ˆ −1 ˆ −1 −1,T T ˆ δsΛsRsms(ν − µ)et = δspRtmtet + es (Rsms) δsp. (A.58)

−1 −1ˆ−1 Extract (ν − µ) by multiplying from the left by (Rsms) Λs δs and from the right by et

−1 −1 −1 −1ˆ−1 −1,T T ˆ ν − µ = (Rsms) Λs pRtmt + (Rsms) Λs δs es (Rsms) δs pet. (A.59)

0 ˆ−1 T ˆ ˆ T ˆ Using the symmetrization of c3 and the symmetry of boost Λs = Λs = δs Λs δs, δsΛs = Λs δs, we obtain

ˆ −1 −1,T T T ˆ δsΛsRsmses = es (Rsms) Λs δs (A.60) ˆ −1 −1,T T ˆ δsΛsRsmses = es (Rsms) δsΛs (A.61) −1 −1 −1 −1ˆ−1 −1,T T ˆ es Λs = (Rsms) Λs δs es (Rsms) δs. (A.62)

The right hand side of the last equation is the same as the boxed part in the earlier expression for (ν − µ), thus

−1 −1 −1 −1ˆ−1 −1,T T ˆ ν − µ = (Rsms) Λs pRtmt + (Rsms) Λs δs es (Rsms) δspet. (A.63)

−1 Using the reparametrization v := Λs p, we get the diﬀerence in shifts

−1 −1 ν − µ = (Rsms) v (Rtmt) + es v et. (A.64)

T (3) Finally, the symmetrization c0 = c0 of the ‘t’-part connects the boost Λt and rotation Rt (here Λt depends on v through p = vΛt)

ˆ −1 ˆ 0 −1 −c0 = δtΛtRtmtet − δtp Rsms(ν − µ)et (A.65) ˆ −1 ˆ 0 −1 −1 −1 c0 = δtΛt(Rtmt)et − δtp Rsms (Rsms) v(Rtmt) + es vet et (A.66) ˆ −1 ˆ 0 −1 −1 = δtΛt(Rtmt)et − δtp v(Rtmt)et + Rsmses v (A.67) ˆ −1 ˆ 0 −1 −1 −1 −1 = δtΛt(Rtmt)et − δtp pΛt (Rtmt)et + Rsmses pΛt (A.68) ˆ −1 ˆ 0 −1 −1 ˆ 0 −1 −1 = δtΛt(Rtmt)et − δtp pΛt (Rtmt)et − δtp Rsmses Λs p (A.69) ˆ −1 ˆ 0 −1 −1 T ˆ −1 −1 = δtΛt(Rtmt)et − δtp pΛt (Rtmt)et − p δsRsmses Λs p. (A.70)

ˆ T ˆ T Since δsΛs = Λs δs, we have from c3 = c3

T T ˆ −1 T ˆ −1 Λs δsRsmses = Λs δsRsmses (A.71) ˆ −1 −1 ˆ −1 −1T δsRsmses Λs = δsRsmses Λs , (A.72)

T ˆ −1 −1 T thus p δsRsmses Λs p is symmetric and we only need to ensure that c4 = c4 where

ˆ −1 ˆ 0 −1 −1 c4 = δtΛt(Rtmt)et − δtp pΛt (Rtmt)et (A.73) Appendix A. Tedious Derivations 79

ˆ 2 −1 −1 ˆ 0 −1 −1 = δtΛt Λt (Rtmt)et − δtp pΛt (Rtmt)et (A.74) ˆ 2 0 −1 −1 2 0 = δt Λt − p p Λt (Rtmt)et (use: Λt = It + p p) (A.75) ˆ −1 −1 = δtΛt (Rtmt)et . (A.76)

Consequently from cases (1), (2) and (3), we get the coupled system

ˆ −1 ˆ −1T δsΛs (Rsms) es = δsΛs (Rsms) es , (A.77) ˆ −1 −1 ˆ −1 −1T δtΛt (Rtmt) et = δtΛt (Rtmt) et , (A.78) −1 −1 ν − es v et = µ + (Rsms) v (Rtmt). (A.79)

Notice that −1 −1 ˆ −1 ˆ−1 ˆ−1 T ˆ ˆ−1 −1,T ˆ δtΛt = Λtδt = δt Λt δtδt = Λt δt, (A.80) ˆ −1 −1 hence the statement that δtΛt (Rtmt) et is symmetric is equivalent to

ˆ −1 ˆ −1 T δt (Rtmt) et Λt = δt (Rtmt) et Λt .

Equation (4.12)

Starting from δARBˆ = (δARBˆ )T and using RT = δRˆ −1δˆ−1, we have

δARBˆ = BTRTATδˆ (A.81) δARBˆ = BTδRˆ −1δˆ−1ATδˆ (A.82) δˆ−1B−1,TδARˆ = R−1δˆ−1ATδBˆ −1 (A.83) 2 δˆ−1B−1,TδARˆ = δˆ−1B−1,TδARˆ R−1δˆ−1ATδBˆ −1 (A.84) 2 δˆ−1B−1,TδARˆ = δˆ−1B−1,TδAˆ δˆ−1ATδBˆ −1 (A.85) q δˆ−1B−1,TδARˆ = δˆ−1B−1,TδAˆ δˆ−1ATδBˆ −1, (A.86) which yields the result r −1 T R = δˆ−1B−1,TδAˆ δˆ−1B−1,TδAˆ δˆ−1 δˆ−1B−1,TδAˆ δ.ˆ (A.87)

The square-root exists since CCT ≥ 0.

Equation (4.27)

We begin with Q = wTETηΛEw

!T !T ! ! ! ! t e 0 Λ p0 R 0 m 0 t Q = t η t t t (A.88) x esν es p Λs 0 Rs msµ ms x !T !T ! ! ! t e 0 Λ Λ v0 R m 0 t = t η t t t t (A.89) x esν es vΛt Λs Rsmsµ Rsms x !T ! ! e t Λ v0Λ R m t = t η t s t t (A.90) es(x + νt) vΛt Λs Rsms(x + µt) 80 Appendix A. Tedious Derivations

!T ! e t Λ R m t + v0Λ R m (x + µt) = t η t t t s s s . (A.91) es(x + νt) vΛtRtmtt + ΛsRsms(x + µt)

−1 −1 From the ansatz νc = ν − es v et and νc = µ + (Rsms) v (Rtmt) we get

−1 −1 ν = νc + es v et, µ = νc − (Rsms) v (Rtmt). (A.92)

Substituting back into Q yields

!T ! e t Λ R m t + v0Λ R m (x + ν t) − v0Λ vR m t Q = t η t t t s s s c s t t (A.93) es(x + νct) + vett vΛtRtmtt + ΛsRsms(x + νct) − ΛsvRtmtt !T ! e t Λ R m t + v0Λ R m (x + ν t) − v0vΛ R m t = t η t t t s s s c t t t (A.94) es(x + νct) + vett vΛtRtmtt + ΛsRsms(x + νct) − vΛtRtmtt !T ! e t (I − v0v)Λ R m t + v0Λ R m (x + ν t) = t η m t t t s s s c . (A.95) es(x + νct) + vett ΛsRsms(x + νct)

−2 0 Since Λt = Im − v v, we obtain

!T ! e t Λ−1R m t + v0Λ R m (x + ν t) Q = t η t t t s s s c (A.96) es(x + νct) + vett ΛsRsms(x + νct) T ˆ −1 0 = −t etδt Λt Rtmtt + v ΛsRsms(x + νct) +

T + (es(x + νct) + vett) δˆs (ΛsRsms(x + νct)) (A.97) T T ˆ −1 T T ˆ 0 = −t et δtΛt Rtmtt − t et δtv ΛsRsms(x + νct) + T T T ˆ + t et v δsΛsRsms(x + νct) + T T ˆ + (x + νct) es δsΛsRsms(x + νct). (A.98)

0 ˆ−1 T ˆ ˆ 0 T ˆ However, v = δt v δs we have δtv = v δs which makes the inner two terms to cancel out: T T ˆ −1 T T ˆ Q = −t et δtΛt Rtmt t + (x + νct) es δsΛsRsms (x + νct) (A.99) !T !T ! ! ! t I 0 −eTδˆ Λ−1R m 0 I 0 t = t t t t t t t (A.100) T ˆ x νc Is 0 es δsΛsRsms νc Is x !T ! t t = ΞT(ν )(e ⊕ e )T η Λ−1R m ⊕ Λ R m Ξ(ν ) . (A.101) x c t s t t t s s s c x

Remark on Lemma 5.1

i j Split the components of dη into ‘t’ and ’s’ groups. Let P swap t ↔ x where i and j are m,n m,n some ﬁxed indices. The intersection of N (R , dη) ∩ N (R , dηP ) can be solved from

i 2 P k 2 j 2 P k 2 −(t ) − k6=i(t ) + (x ) + k6=j(x ) = 0, (A.102) i 2 P k 2 j 2 P k 2 (t ) − k6=i(t ) − (x ) + k6=j(x ) = 0. (A.103)

i j P k 2 P k 2 The solution is t = ±x and − k6=i(t ) + k6=j(x ) = 0, which for m = 1 degenerates into xk = 0, k∈ / {i, j}. Notice that the projection on the Sn+m−1-sphere P (tk)2 + P (xk)2 = 1 √ k√ k always yields (ti)2 + (xj)2 = 1 and consequently ti = ±1/ 2, xj = ±1/ 2. References

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adjoint transformation,9 canonical dualities,6 ADM formalism,v, 66, 71 canonical pair theorem, 63 algebraic Riccati equation, 55 Cauchy problem,v, 74 anisotropic vector, 14 change of basis,5 automorphism, viii characteristic of a ﬁeld (math),1 automorphism group, 13 Cholesky decomposition, 36, 38, 39 circular rotation, 33 basis commuting family canonical,5, 11, 17 of operators, 24 coordinate, vii, 40 conformal factor, 12 non-coordinate (or non-holonomic), 40 conformal group, 12 orthogonal, 16 conformal mapping, 12 proper, 53 congruence mapping, 11 bilinear form continuation set, 27, 53 core of, 38 example of, 54 deﬁnition,1 core of a quadratic form, 38 inverse map,7 core, Minkowski, 38 non-degenerate (or regular),6 coupled quadratic spaces, 19 skew-symmetric,2 coupled set, 62 standard, 18 coupled set of quadratic spaces, 25 subform,2 coupled symmetrization, 27 symmetric,2 transpose of,2 decomposition bilinear map,2 m + n, of a quadratic form, 37 bilinear space,2 3+1, in bimetric theory, 66 regular,6 3+1, of a metric tensor, 36, 39 singular,6 Cholesky, 36, 38, 39 subspace of,2 orthogonal, 15 symmetric,2 polar, 49 bimetric theory proper, 38, 43, 45 action, 65 simultaneous proper, 53 constraint equations, 67 The Witt decomposition, 15 Hamiltonian analysis, 66 Deser-van Nieuwenhuizen gauge, 28 history of,v diagonalization new variables, 69 simultaneous, by congruence, 61 vielbein formulation, 68 dilation, 12 boost, 33, 71 discriminant, vii

Mikica Kocic, The Square-Root Isometry, MSc thesis, SU 2014 Printed: 2015-03-11 88 Index dual of a linear map,8 hyperbolic plane, 14 hyperbolic rotation, 33 effective composite metric, 26 hyperbolic splitting, 14 elementary symmetric polynomial, 65 embeding, viii identity component, 18 endomorphism, viii index epimorphism, viii isotropy (or the Witt) index, 14 equations of motion, 66 of a quadratic space, 17 Euclidean space, 18 indices coordinate (or world or spacetime), 40 field (math) Lorentz, 40 characteristic,1 inner product,7 of characteristic two,1,3 involutory transformation, 28, 55 field (physics), iii isometry, 10 ghost, tachyon, iv global, 11 interacting spin-2, 65 local, 11 spin-2, iv of coupled quadratic spaces, overview, 20 fixed point, 23 self-adjoint, 22 square-root (properties), 22 generalized eigenvalue equation, 61 isomorphism, viii geometry canonical (or musical),6 orthogonal,2 isometric, 11 symplectic,2 lowering and raising,6 ghost natural,6 Boulware-Deser, iv, 66 isotropic space, 14 field (defined), iv isotropic vector, 14 global hyperbolicity,v, 74 isotropy index, 14 group action, 23 Jordan block, 56 general linear, vii Jordan canonical form, 56, 58, 63 isotropy (sub)group, 23 Lie, 29 kernel,7 Lorentz, 32, 34 Killing vector field, 12 multiplicative, vii of squares, vii Lagrange multipliers, 67 orthogonal, vii, 10 lapse (function), 39 orthogonal (or automorphism) group, 13 Lie algebra, 29 reflection group, 56, 58 Lie group, 29 special orthogonal, 14 light-cone, 14 square class, 13 linear form square-root isometry, 25 transpose (or dual) map of, 11 linear operator, viii homogeneous coordinates, 44 linear transformation, viii homogeneous space, 24, 32 adjoint,9 homomorphism, viii orthogonal,9 homothety, 12 self-adjoint,9 hyperbolic pair, 14 skew-adjoint,9 Index 89

transpose of,8 orthonormal frame,v, 39 Lorentz transformation, 10, 32 lower-triangular form, 38, 43 packings on the sphere, 60 loxodromic transformations, 26 pencil of two matrices, 61 perpendicular subspace, 12 matrix polar decomposition congruence,5, 20 explicit form for O(m, n), 32 non-derogatory, 57 for symmetrization condition, 49 notation,4 of O(m, n), 31 of bilinear form,4 of GL(n, R) and O(n), 30 partitioned, 34 polarization identity,3 sign function, vii, 55 positive definite, 18 similarity,5, 20 positive index, 17 square-root, vii primary matrix function, 57 matrix function, 57 projective space, 44 metric proper decomposition, 38 spatial, 39 proper Lorentz transformation, 32 tensor, 40 proper vielbein, 38 Minkowski spacetime, 18 proportional backgrounds, 65 monomorphism, viii pullback, 39 multilinear form,2 definition, 11 multilinear map,2 of a linear map,8 pushforward, 11 negative definite, 18 negative index, 17 quadratic form,3 nonprimary matrix function, 57 positive or negative definite, 18 norm of a vector,7 symmetrizing, 25 normed vector space,7 quadratic space null-cone, 44 coupled, 19, 69 center, 46 coupled set of, 25 interior, 46 definition,3 intersecting, 52 kernel of,7 tilings, 60 representing a scalar,3 touching, 59 symmetrizing, 23, 50 null-vector, 14 universal,3 quadratic surface, 44 orbit, 23 quadrics, 44 orthochronous Lorentz transformation, 32 orthogonal radical of a quadratic space, 12 basis (theorem), 16 reflection group, 56, 58 complement of, 16 generating set, 58 geometry,2 nonprimary, 58 group, 10, 13 representative vector, 44 splitting, 12, 16 restricted Lorentz transformation, 32 transformation, 13, 20, 32 transformation, self-adjoint, 55 Schur complement, 35 vectors, subspaces etc., 12 shear transformation, 37 90 Index

Sherman-Morrison-Woodbury formula, 35 symmetrization condition, vi, 28 shift, 37, 49 origin of, 28 shift vector, 39 topology, 51 signature of a quadratic space, 17 symmetrizing quadratic space, 50 similitude, 12 symmetrizing space, 23 similitude factor, 12 symplectic geometry,2 simultaneous proper decomposition, 53 space tachyon (field), iv bi-dual,6 time reversal, 32 bilinear,2 torus Euclidean, 18 Clifford, 45 homogeneous, 24, 32 totally isotropic space, 14 inner product,7 transformation inversion, 32 coordinate,5, 41 isotropic, 14 involutory, 28 Lorentzian, 18 linear, viii loxodromic, 26 Minkowski, 18 orthogonal, 13, 20, 32 projective, 44 shear, 37 quadratic,3 similarity, 20 regular bilinear,6 transitive group action, 24 symmetric bilinear,2 transpose spatial metric, 39 of a bilinear form,2 special orthogonal group, 14, 18 of a linear map,8 spin, of a field, iv spinor map, 26 unitary geometry,2 splitting universal quadratic space,3 hyperbolic, 14 orthogonal, 12, 16 vector square class of a field, 13 isotropic, 44 square-root isotropic and anisotropic, 14 alternative, 22, 54 representative, 44 in bimetric theory, 68 shift vector, 39 nonprimary, viii, 57 vielbein, 18 primary, viii, 57 ‘generalized’, 11 principal, vii, viii, 58, 63 in differential geometry, 39 square-root isometry notation (overview), 41 definition, 22 proper, 38 properties, 22 symmetrization condition, 28, 47 reflection group of, 56 Witt index, 14 stabilizer, 23 Witt, decomposition theorem, 15 standard bilinear form, 18 structure transfer, distance preserving, 10 transport (or transfer), 10 summation convention,4 Sylevester’s Law of Inertia, 18