Chapter summary. The aim of this chapter is to introduce the basic mathematical language and technical machinery of the theory of general relativity following variational methods. We focus especially on developing the canonical formulation of general relativ-ity, also known as the Hamiltonian or (3 + 1) formulation. In essence, this provides a way of turning the second-order field equations of the theory for the spacetime metric into a first-order time-evolution problem for the induced spatial three-metric and its conjugate momentum. There are, however, also constraint equations in addition to these (constrain-ing permissible initial conditions as well as their development subject to the dynamical equations), and their proper handling requires a great deal of subtlety. The existence of constraints in general relativity is in fact directly related to—and offers fruitful insight on—the gauge freedom available in the theory.
We begin in Section 2.1with a brief introduction. We generally discuss four broad areas of application for canonical general relativity: mathematical relativity, numerical relativity, quantum gravity, and the issue of gravitational energy-momentum. We return to each of these in the final section of this chapter with specific examples, once we have developed the mathematical tools in detail.
In Section 2.2, we present the Lagrangian formulation of classical field theories in general, and then general relativity in particular, forming the typical starting point of any canonical analysis. We comment on the appearance of constraint equations already at the Lagrangian level, a proper explanation of which requires the canonical picture.
In Section2.3, we develop the canonical formulation of field theories in general, with a careful accounting of the issue of constraints. To this end, we prescribe here the general recipe for foliating spacetime into constant-time spatial (Cauchy) three-surfaces, such that a notion of time evolution in spacetime can be defined. We also define the canonical phase space and the Hamiltonian equations of motion for general field theories, introducing the basic mathematical methods of symplectic geometry.
Then, in section2.4, we proceed to apply this formalism to general relativity in order to formulate it as a canonical theory. In particular, the canonical variables are the induced three-metric on each spatial slice (associated with the choice of foliation), as well as the lapse function and the shift vector (associated with a choice of a time flow vector field), plus their respective conjugate momenta. The lapse and shift are not dynamical variables:
29
30 Chapter 2. Canonical General Relativity their associated equations are first-order in time, and their conjugate momenta vanish.
These constitute the constraints of general relativity.
Finally, in Section2.5, we return in greater detail to the four broad areas of application of canonical general relativity enumerated in the introductory section, and we provide illustrations with explicit examples.
Relativitat general canònica(chapter summary translation in Catalan). L’objectiu d’aquest capítol és introduir el llenguatge matemàtic bàsic i la maquinària tècnica de la teoria de la relativitat general seguint mètodes variacionals. Ens centrem especialment en desenvolupar la formulació canònica de la relativitat general, també coneguda com la formulació hamiltoniana o (3 + 1). En essència, això proporciona una manera de convertir les equacions de camp de segon ordre de la teoria per al tensor mètric de l’espai-temps en un problema d’evolució temporal de primer ordre per al tensor mètrica espacial tridimen-sional induït i el seu moment conjugat. Tanmateix, també hi ha equacions de restricció a més d’aquestes (restringint les condicions inicials admissibles, així com el seu desenvolu-pament subjecte a les equacions dinàmiques), i el seu correcte maneig requereix molta subtilesa. L’existència de restriccions en la relativitat general està directament relacionada amb - i ofereix una visió útil sobre - la llibertat de mesura disponible en la teoria.
Comencem a la Secció2.1amb una breu introducció. Generalment es discuteixen qua-tre àmplies àrees d’aplicació de la relativitat general canònica: la relativitat matemàtica, la relativitat numèrica, la gravetat quàntica i el tema de l’energia i la quantitat de moviment gravitatòria. Tornem a cadascun d’aquests a la secció final d’aquest capítol amb exemples específics, un cop desenvolupades les eines matemàtiques en detall.
A la secció2.2, presentem la formulació lagrangiana de les teories de camps clàssics en general, i la relativitat general en particular, formant el punt de partida típic de qualsevol anàlisi canònica. Comentem l’aparició d’equacions de restricció ja a nivell lagrangià, una explicació adequada de la qual es requereix la formulació canònica.
A la secció2.3, desenvolupem la formulació canònica de les teories de camps en gen-eral, amb una acurada explicació del problema de les restriccions. Amb aquesta finalitat, prescrivim aquí la recepta general per foliar l’espai-temps en superfícies espacials tridi-mensionals de temps constant (superfícies Cauchy), de manera que es pot definir una no-ció d’evoluno-ció en el temps en l’espai-temps. També definim l’espai de fase canònica i les equacions de moviment hamiltonianes per a les teories generals de camp, introduint els mètodes matemàtics bàsics de la geometria simplectica.
A continuació, a la secció2.4, procedim a aplicar aquest formalisme a la relativitat gen-eral per tal de formular-lo com a teoria canònica. En particular, les variables canòniques són el tensor mètric tridimensional induït en cada llesca espacial (associat a l’elecció de la foliació), així com la funció de lapse i el vector de shift (associats amb una elecció d’un camp vectorial de flux de temps), més els seus moments conjugats respectius. El lapse i el
31 shift no són variables dinàmiques: les seves equacions associades són de primer ordre en el temps i els seus moments conjugats desapareixen. Aquests constitueixen les restriccions de la relativitat general.
Finalment a la secció2.5, tornem amb més detall a les quatre àmplies àrees d’aplicació de la relativitat general canònica enumerades a la secció introductòria, i proporcionem il·lustracions amb exemples explícits.
Relativité générale canonique(chapter summary translation in French). Le but de ce chapitre est de présenter le langage mathématique de base et la machinerie technique de la théorie de la relativité générale suivant les méthodes variationnelles. Nous nous concen-trons particulièrement sur le développement de la formulation canonique de la relativité générale, également connue sous le nom de la formulation hamiltonienne ou (3 + 1). En substance, cela fournit un moyen de transformer les équations de champ de deuxième or-dre de la théorie pour le tenseur métrique de l’espace-temps en un problème d’évolution temporelle de premier ordre pour le tenseur métrique spatial trois-dimensionnel induit et son moment conjugué. Cependant, il existe également des équations de contrainte (restreignant les conditions initiales admissibles ainsi que leur développement en fonc-tion des équafonc-tions dynamiques), et leur traitement correct nécessite beaucoup de subtilité.
L’existence de contraintes dans la relativité générale est en fait directement liée à - et offre un éclairage utile sur - la liberté de jauge disponible dans la théorie.
Nous commençons à la section 2.1 avec une brève introduction. Nous traitons généralement quatre grands domaines d’application de la relativité générale canonique : la relativité mathématique, la relativité numérique, la gravitation quantique et la ques-tion de l’énergie et la quantité de mouvement gravitaques-tionnelles. Nous reviendrons sur chacun d’eux dans la dernière section de ce chapitre avec des exemples spécifiques, une fois que nous aurons développé les outils mathématiques en détail.
Dans la section 2.2, nous présentons la formulation lagrangienne des théories de champ classiques en général, puis de la relativité générale en particulier, constituant le point de départ typique de toute analyse canonique. Nous commentons sur l’apparition d’équations de contraintes déjà au niveau lagrangien, dont l’explication correcte nécessite la formulation canonique.
Dans la section2.3, nous développons la formulation canonique des théories de champ en général, tenant en compte la question des contraintes. À cette fin, nous prescrivons ici la recette générale de feuilleter l’espace-temps en surfaces spatiales trois-dimensionnelles à temps constant (surfaces de Cauchy), de manière à pouvoir définir une notion d’évolution temporelle dans l’espace-temps. Nous définissons également l’espace des phases canon-ique et les équations hamiltoniennes du mouvement pour les théories générales du champ, en introduisant les méthodes mathématiques de base de la géométrie symplectique.
32 Chapter 2. Canonical General Relativity Ensuite, dans la section2.4, nous appliquons ce formalisme à la relativité générale afin de la formuler en tant que théorie canonique. En particulier, les variables canoniques sont le tenseur métrique trois-dimensionnel induite sur chaque tranche spatiale (associée au choix de la foliation), et la fonction de déchéance (lapse) et le vecteur de décalage (shift) (associés au choix d’un champ de vecteurs de flux temporel), ainsi que leurs moments conjugués respectifs. La déchéance et le décalage ne sont pas de variables dynamiques : leurs équations associées sont du premier ordre dans le temps et leurs moments conjugués disparaissent. Celles-ci constituent les contraintes de la relativité générale.
Enfin, dans la section2.5, nous reviendrons plus en détail sur les quatre grands do-maines d’application de la relativité générale canonique énumérés dans la section intro-ductive et nous fournissons des illustrations avec des exemples explicites.
2.1. Introduction
There are a number of diverse motivations for casting GR into a canonical form, and for our choice to introduce the topic in this way. We begin by enumerating four broad areas of interest, and comment more on each, focusing on specific examples of applications, in the final section of this chapter.
(1) Mathematically, canonical methods provide a very useful way to develop the sort of geometrical tools generally used for studying subsets of spacetimes, in particular by splitting them up into (usually, families of lower-dimensional) hy-persurfaces via some established procedure. The classical Hamiltonian approach splits spacetime into spatial slices defined, for example, by the constancy of a time function, and is thus adapted to studying “the entire space” at different instants of time. Similar technical constructions can be employed for studying the dynamics of finite (bounded) systems within a spacetime throughout some span of time; in such a case, one could foliate spacetime, for instance, by the constancy of a radial function in order to study the dynamics of the resulting worldtubes. (We shall see more along these lines in Chapter 5.) Spacetime splittings of this sort, and especially (3 + 1)splittings, have supplied the basic framework for many important results in the mathematics of GR.
(2) Practically, canonical methods form the basis of numerical relativity—that is, formulating the Einstein equation as a suitable set of time-dependent partial differential equations (PDEs) which, given some appropriate initial data, can be evolved on computers to obtain numerical solutions. Simulations of strongly dynamical gravitational systems rely critically on methods of this sort.
2.1. Introduction 33 (3) In going beyond GR, in particular in seeking theories of quantum gravity, canonical methods are often regarded as a key connection between the lan-guages of GR and quantum mechanics. Indeed, typical canonical quantization procedures follow some variant of transforming classical canonical variables into operators on a Hilbert space of quantum states. Loop quantum grav-ity, for example, is a candidate theory of quantum gravity which essentially attempts to do this for gravitational canonical variables defined in a suitable way.
(4) Last but not at all least, from a physical point of view, canonical methods form the traditional starting point for understanding the notion of gravitational energy-momentum. In particular, they can supply definitions of gravitational energy-momentum of an entire spacetime under some specific conditions, e.g.
the Arnowitt-Deser-Misner (ADM) energy-momentum for asymptotically-flat spacetimes. On the other hand, canonical methods are not designed to be able to say much more than this, and in particular, anything about the gravitational energy-momentum of a finite spatial region within some spacetime. For the latter, methods such as the worldtube boundary splittings mentioned in (1) are better designed, yet to this day no general consensus exists among relativists on the “best” way to do this. We will comment more on this in later chapters of this thesis, but we end here by remarking that, nevertheless, any proposed definition for gravitational energy-momentum of a finite system is generally expected to agree with,e.g., the ADM energy-momentum in the flat asymptotic limit.
Canonical GR encompasses a variety of possible formulations of GR in terms of some cho-sen canonical variables (configurations and their conjugate momenta). The first canonical formulation of GR was achieved in 1950, following a quantum gravity motivation, by [Pi-rani and Schild1950, Pirani 1951], and independently the following year by [Anderson and Bergmann1951].
Then, [Dirac1958a,b] formulated the general framework for working with constrained Hamiltonian systems, a topic we systematically develop in this chapter before applying it to GR. Beginning in the following year, Arnowitt, Deser and Misner [Arnowitt et al.1959]
devised the first coordinate-independent canonical formulation of GR, since then known eponymously as theADM formulation. Following a series of further papers in the ensuing years, the authors summarized their results in a 1962 review article [Arnowitt et al.1962], republished more recently [Arnowitt et al.2008]. Today, it remains undoubtedly the most famous basic canonical formulation of GR.
Over the decades, other formulations have been developed in response to the ap-plication needs—e.g., the Ashtekar variable formulation used in quantum gravity, the
34 Chapter 2. Canonical General Relativity Baumgarte-Shapiro-Shibata-Nakamura (BSSN) and generalized harmonic formulations used widely in numerical relativity (upon which we will elaborate further in the final section), and numerous others.
Our presentation of canonical GR in this chapter is based, in its broadest outlines, on Chapter 10 and Appendix E of [Wald1984], in combination with Chapter 3 of [Bojowald 2011] (especially for the general formulation canonical theories and constraints). See also Chapter 4 of [Poisson2007] for many of the the step-by-step computations, largely omitted herein lieuof directly stating the main results.
For mathematical clarifications, we generally refer to [Lee2002] for geometry (see also [Nakahara2003], written with more of a view towards physics), and [Evans1998] for PDE theory.
2.2. Lagrangian formulation
2.2.1. Lagrangian formulation of general field theories. Let(M,g,∇)be any (3 + 1)-dimensional spacetime. Suppose that we are interested in a theory describing a collection of fields ψ = {ψA(xa)}A∈I in M, whereA ∈ I is a general (possibly multi-) index for the fieldsψA(and will be accordingly omitted if understood),i.e. it can include tensor indices, field indices etc. For example, if we are considering only gravity, thenψ= g,i.e. our collection of fields includes only the spacetime metricgab. If we are considering gravity coupled to a matter field, for example Maxwellian electromagnetism, thenψ={g,F}, whereFabis the Faraday tensor.
LetS[ψ]be a functional ofψ. Let{ψ(λ)}λ∈R be a smooth one-parameter family of field values and letδψA = (∂λψ(λ)A )|λ=0. For all such families, suppose moreover that (∂λS[ψ(λ)])|λ=0 exists and also that there exists a smooth fieldχAdual toψA(meaning that ifψA∈Tkl(M), thenχA∈Tlk(M)), such that
∂S
∂λ
λ=0
= ˆ
M eχAδψA. (2.2.1)
Here, for reasons that will become transparent shortly, we choose to write the integral with respect to the Minkowski volume form1,
e= dx0∧dx1∧dx2∧dx3 = d4x . (2.2.2) The factor of√
−g, withg = det(g), multiplying the above to yield the volume form of M,
M =√
−ge, (2.2.3)
1In fact, ecan be chosen to beanyLorentzian volume form the non-vanishing components of which take the values±1. See [Wald1984] for a more detailed discussion as to why the present construction is actually independent of the choice of a volume form satisfying this property.
2.2. Lagrangian formulation 35
Figure 2.1. A compact regionV in a spacetimeM where the variation of physical fields are non-zero.
is absorbed into the definition ofχA. Then,S[ψ]is said to befunctionally differentiableat ψ=ψ(0)and itsfunctional derivativeis defined as
δS δψA
ψ(0)
=χA. (2.2.4)
We will now focus our attention upon a certain class of such functionalsS. LetV be a compact region inM such thatsupp(δψA) ⊂V (i.e. δψAtakes non-zero values only in the interior ofV). See Fig.2.1. We assume thatS has the form
S[ψ] = ˆ
V eL[ψ], (2.2.5)
such that
L[ψ] =√
−gf ψA,∇ψA, . . . ,∇· · ·∇ψA
(2.2.6) wherefis a local function ofψAand a finite number of its derivatives. IfSis functionally differentiable and extremized at the field valuesψAwhich are solutions to the field equa-tions of the theory, thenSis referred to as anaction. We then refer toLas the Lagrangian density, and the specification of such anLis what is meant by a Lagrangian formulation of the theory. Note that we may haveg∈ψ(i.e.the gravitational field may be included in the theory, as in GR), and this is the reason for which we have preferred to simply absorb the√
−g(in this case,ψdependent) factor intoL, and thus to writeSas an integral with respect to the flat volume formeinstead of the natural spacetime volume formM.
All major theories of classical physics, including GR, admit a Lagrangian formulation.
In other words, their field equations are equivalent to the extremization of an actionS[ψ]
with respect to their physical fieldsψ, which in turn can be shown to be equivalent to
36 Chapter 2. Canonical General Relativity a system of PDEs known as the Euler-Lagrange equations. For field theories of typical interest, including GR and Maxwellian electromagnetism (EM),Ldepends onψAand its first derivatives only,i.e.L=√
−gf(ψA,∇ψA). In this case, these equations are These are second-order PDEs forψA.
Now fix a coordinate system{xα}={t, xi}. Clearly, all terms which are second order in the derivatives ofψwill emerge from the final term of the above equation: by implicit differentiation, this is The coefficient of the second term on the RHS above is known as theprincipal symbolof the PDE; theα =t=β component of this term contains all the second time derivatives of the fields. Thus, the Euler-Lagrange equations have the form
0 =WAB∂t2ψB+lA, (2.2.9)
where we have defined
WAB = ∂2L
∂(∇tψA)∂(∇tψB), (2.2.10) andlA indicates lower (i.e. first and zeroth) order terms in time derivatives. It is thus apparent that if and only if WAB is non-degenerate in the indicesA, B ∈ I will we be able to obtain a complete set of solutions to the coupled set of equations,i.e. a set of n = card(I)(the cardinality ofI) solutions. If that is the case, we would be able to invertWAB and write the complete system explicitly as
0 =∂t2ψB+ W−1AB
lA. (2.2.11)
IfWAB is degenerate, however, thenψAand its derivatives up to first order in time and second order in space (and in particular, their initial conditions for the time evolution problem) cannot take arbitrary values. Specifically, they must yield anlAwhich is in the image ofWABseen as a linear mapping between vector spaces, the dimension of which is thus less thann. Equivalently,WAB can be seen as a matrix (with indices inI) that has (n−rank(WAB)) null eigenvectorsvjA, withj(in serif font) used to label the set of these eigenvectors (each having components labelled byA ∈I). In other words,vAj are such thatvAj WAB = 0. Multiplying the Euler-Lagrange equation byvjAon the left thus yields the independent equations,
0 =vjAlA. (2.2.12)
These equations are known as constraints, since they do not involve second time deriva-tives of the fields and are thus not regarded as “dynamical”i.e. they do not prescribe the
2.2. Lagrangian formulation 37 time evolution. We will gain a deeper appreciation for what this means once we pass to the Hamiltonian picture, but before we do, let us apply our ideas so far to GR.
2.2.2. Lagrangian formulation of GR. In vacuum, our only field is the gravita-tional field,ψ=g. If, in addition to the requirement thatsupp(δg)⊂V we also assume thatsupp(∇δg) ⊂V, then the Einstein equation can be recovered fully from an action of the form (2.2.5). In particular,
SEH[g] = ˆ
V eLEH[g], LEH = 1 2κ
√−gR , (2.2.13) are the Einstein-Hilbert action and Lagrangian respectively, withκdenoting the Einstein constant,κ = 8πG/c4 = 8π in units ofG = 1 = c. This formulation of GR was first proposed by [Hilbert1915].
In modern Lagrangian formulations of GR, however, it is typical not to assume any-thing about the support of∇δg, and in particular its values on the bounday∂V. Equiva-lently, only the metric components (and not the derivatives thereof) are to be regarded as
In modern Lagrangian formulations of GR, however, it is typical not to assume any-thing about the support of∇δg, and in particular its values on the bounday∂V. Equiva-lently, only the metric components (and not the derivatives thereof) are to be regarded as