Dynamical and Hamiltonian formulation of General Relativity Domenico Giulini Institute for Theoretical Physics Riemann Center for Geometry and Physics Leibniz University Hannover, Appelstrasse 2, D-30167 Hannover, Germany and ZARM Bremen, Am Fallturm, D-28359 Bremen, Germany Abstract This is a substantially expanded version of a chapter-contribution to The Springer Handbook of Spacetime, edited by Abhay Ashtekar and Vesselin Petkov, published by Springer Verlag in 2014. It introduces the reader to the reformulation of Einstein's field equations of General Relativity as a constrained evolutionary system of Hamiltonian type and discusses some of its uses, together with some technical and conceptual aspects. Attempts were made to keep the presentation self contained and accessible to first-year graduate students. This implies a certain degree of explicitness and occasional reviews of background material. Contents 1 Introduction 1 Introduction 1 The purpose of this contribution is to explain how 2 Notation and conventions 2 the field equations of General Relativity|often simply referred to as Einstein's equations|can be 3 Einstein's equations 4 understood as dynamical system; more precisely, 4 Spacetime decomposition 9 as a constrained Hamiltonian system. 5 Curvature tensors 15 In General Relativity, it is often said, spacetime becomes dynamical. This is meant to say that 6 Decomposing Einstein's equations 22 the geometric structure of spacetime is encoded in 7 Constrained Hamiltonian systems 27 a field that, in turn, is subject to local laws of prop- agation and coupling, just as, e.g., the electromag- 8 Hamiltonian GR 36 netic field. It is not meant to say that spacetime as a whole evolves. Spacetime does not evolve, space- 9 Asymptotic flatness and global charges 48 time just is. But a given spacetime (four dimen- 10 Black-Hole data 54 sional) can be viewed as the evolution, or history, of space (three dimensional). There is a huge re- 11 Further developments, problems, and outlook 60 dundancy in this representation, in the sense that 12 Appendix: Group actions on manifolds 61 apparently very different evolutions of space rep- resent the same spacetime. However, if the result- References 66 ing spacetime is to satisfy Einstein's equations, the Index 72 evolution of space must also obey certain well de- 1 fined restrictions. Hence the task is to give pre- Dirac's [54] from 1958. He also noticed its con- cise mathematical expression to the redundancies strained nature and started to develop the cor- in representation as well as the restrictions of evolu- responding generalization of constrained Hamilto- tion for this picture of spacetime as space's history. nian systems in [53] and their quantization [55]. This will be our main task. On the classical side, this developed into the more This dynamical picture will be important for geometric Dirac-Bergmann theory of constraints posing and solving time-dependent problems in [76] and on the quantum side into an elaborate General Relativity, like the scattering of black holes theory of quantization of systems with gauge re- with its subsequent generation and radiation of dundancies; see [80] for a comprehensive account. gravitational waves. Quite generally, it is a key Dirac's attempts were soon complemented by an technology to extensive joint work of Richard Arnowitt, Stanley Deser, and Charles Misner - usually and henceforth • formulate and solve initial value problems; abbreviated by ADM. Their work started in 1959 • integrate Einstein's equations by numerical by a paper [3] of the first two of these authors and codes; continued in the series [4] [6] [5] [9] [8] [7] [10] [11] [12] [14] [15] [13] of 12 more papers by all three. A • characterize dynamical degrees of freedom; comprehensive summary of their work was given in • characterize isolated systems and the associ- 1962 in [16], which was republished in 2008 in [17]; ation of asymptotic symmetry groups, which see also the editorial note [104] with short biogra- will give rise to globally conserved `charges', phies of ADM. like energy and linear as well as angular mo- A geometric discussion of Einstein's evolution mentum (Poincar´echarges). equations in terms of infinite-dimensional symplec- tic geometry has been worked out by Fischer and Moreover, it is also the starting point for the canon- Marsden in [57]; see also their beautiful summaries ical quantization program, which constitutes one and extended discussions in [59] and [58]. More main approach to the yet unsolved problem of on the mathematical aspects of the initial-value Quantum Gravity. In this approach one tries to problem, including the global behavior of gravita- make essential use of the Hamiltonian structure of tional fields in General Relativity, can be found the classical theory in formulating the correspond- in [39], [45], and [106]. Modern text-books on the ing quantum theory. This strategy has been ap- 3+1 formalism and its application to physical prob- plied successfully in the transition from classical lems and their numerical solution-techniques are to quantum mechanics and also in the transition [27, 77]. The Hamiltonian structure and its use in from classical to quantum electrodynamics. Hence the canonical quantization program for gravity is the canonical approach to Quantum Gravity may discussed in [34, 92, 107, 113]. be regarded as conservative, insofar as it tries to apply otherwise established rules to a classical the- ory that is experimentally and observationally ex- 2 Notation and conventions tremely well tested. The underlying hypothesis here is that we may quantize interaction-wise. This From now on \General Relativity" will be abbrevi- distinguishes this approach from string theory, the ated by \GR". Spacetime is a differentiable man- underlying credo of which is that Quantum Gravity ifold M of dimension n, endowed with a metric only makes sense on the basis of a unified descrip- g of signature ("; +; ··· ; +). In GR n = 4 and tions of all interactions. " = −1 and it is implicitly understood that these Historically the first paper to address the prob- are the \right" values. However, either for the lem of how to put Einstein's equations into the sake of generality and/or particular interest, we form of a Hamiltonian dynamical system was will sometimes state formulae for general n and 2 ", where usually n ≥ 2 (sometimes n ≥ 3) and of f. If f is a diffeomorphism we can not only push- either " = −1 (Lorentzian metric) or " = +1 (Rie- forward vectors and pull back co-vectors, but also mannian metric; also called Euclidean metric). vice versa. Indeed, if Y is a vector field on N one ∗ −1 The case " = 1 has been extensively considered can write f Y := (f )∗Y and call it the pull back in path-integral approaches to Quantum Gravity, of Y by the diffeomorphism f. Likewise, if β is a −1 ∗ then referred to as Euclidean Quantum Gravity. co-vector field on M, one can write f∗β := (f ) β The tangent space of M at point p 2 M will be and call it the push forward of β. In this fashion ∗ denoted by TpM, the cotangent space by Tp M, and we can define both, push-forwards and pull-backs, u the tensor product of u factors of TpM with d fac- of general tensor fields T 2 ΓTd M by linearity and ∗ u ∗ tors of Tp M by Tpd M. (Mnemonic in components: applying f∗ or f tensor-factor wise. u = number of indices \upstairs", d = number of Note that the general definition of metric is as u 0 indices \downstairs".) An element T in Tpd M is follows: g 2 ΓT2 M, such that gp is a symmet- called a tensor of contravariant rank u and covari- ric non-degenerate bilinear form on TpM. Such ant rank d at point p, or simply a tensors of rank a metric provides isomorphisms (sometimes called (u;d) at p. T is called contravariant if d = 0 and the musical isomorphisms) u > 0, and covariant if u = 0 and d > 0. A tensor ∗ with u > 0 and d > 0 is then referred to as of mixed [ : TpM ! Tp M 1 ∗ 0 [ type. Note that TpM = Tp0M and Tp M = Tp1M. X 7! X := g(X; · ) ; (1a) The set of tensor fields, i.e. smooth assignments of ∗ u ] : Tp M ! TpM an element in Tpd M for each p 2 M, are denoted u ! 7! !] := [−1(!) : (1b) by ΓTd M. Unless stated otherwise, smooth means C1, i.e. continuously differentiable to any order. Using ] we obtain a metric g−1 on T ∗M from the For t 2 ΓT uM we denote by t 2 T uM the eval- p p d p pd metric g on T M as follows: uation of t at p 2 M. C1(M) denotes the set of p p all C1 real-valued functions on M, which we often −1 ] ] ] gp (!1;!2) := gp(!1;!2) = !1(!2) : (2) simply call smooth functions. 1 If f : M ! N is a diffeomorphism between We also recall that the tensor space Tp1M is nat- manifolds M and N, then f∗p : TpM ! Tf(p)N urally isomorphic to the linear space End(TpM) denotes the differential at p. The transposed (or of all endomorphisms (linear self maps) of TpM. dual) of the latter map is, as usual, denoted by Hence it carries a natural structure as associative ∗ ∗ ∗ fp : Tf(p)N ! Tp M. If X is a vector field algebra, the product being composition of maps on M then f∗X is a vector field on N, called denoted by ◦. As usual, the trace, denoted Tr, the push forward of X by f. It is defined by and the determinant, denoted det, are the natu- (f∗X)q := f∗f −1(q)Xf −1(q), for all q 2 N.