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Home , Arthur Komar, Gerhard Huisken, Hamiltonian constraint, Richard Arnowitt, Robert Geroch, Stanley Deser

Dynamical and Hamiltonian formulation of

Domenico Giulini Institute for Theoretical Riemann Center for Geometry and Physics Leibniz University Hannover, Appelstrasse 2, D-30167 Hannover, 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 , edited by 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 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, - 9 Asymptotic flatness and global charges 48 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 , , 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 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 . 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 , 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 (ε, +, ··· , +). In GR n = 4 and tions of all interactions. ε = −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 ε, where usually n ≥ 2 (sometimes n ≥ 3) and of f. If f is a diffeomorphism we can not only push- either ε = −1 (Lorentzian metric) or ε = +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 ε = 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 ∈ 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 ∈ 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 ∈ 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 ∈ M, are denoted u ω 7→ ω] := [−1(ω) . (1b) by ΓTd M. Unless stated otherwise, smooth means C∞, i.e. continuously differentiable to any order. Using ] we obtain a metric g−1 on T ∗M from the For t ∈ ΓT uM we denote by t ∈ T uM the eval- p p d p pd metric g on T M as follows: uation of t at p ∈ M. C∞(M) denotes the set of p p all C∞ real-valued functions on M, which we often −1 ] ] ] gp (ω1, ω2) := gp(ω1, ω2) = ω1(ω2) . (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 ∈ N. If α rally defined real-valued functions on the space of is a co-vector field on N then f ∗α is a co-vector endomorphisms. For purely co- or contravariant field on M, called the pull back of α by f. It is tensors the trace can be defined by first applying ∗ defined by (f α)p := αf(p) ◦ f∗p, for all p ∈ M. For one of the isomorphisms (1). In this case we write these definitions to make sense we see that f need Trg to indicate the dependence on the metric g. generally not be a diffeomorphism; M and N need Geometric representatives of curvature are of- not even be of the same dimension. More precisely, ten denoted by bold-faced abbreviations of their if α is a smooth field of co-vectors that is defined at names, like Riem and Weyl for the (covariant, i.e. least on the image of f in N, then f ∗α, as defined all indices down) Riemann and Weyl tensors, Sec above, is always a smooth field of co-vectors on M. for the , Ric and Ein for the However, for the push forward f∗ of a general vec- Ricci and Einstein tensors, Scal for the scalar cur- tor field on M to result in a well defined vector field vature, and Wein for the Weingarten map (which on the image of f in N we certainly need injectivity is essentially equivalent to the extrinsic curvature).

3 This is done in order to highlight the geometric in mind when explicit models for T are used and meaning behind some basic formulae, at least the when we speak of “vacuum”, which now means: simpler ones. Later, as algebraic expressions be- −1 come more involved, we will also employ the stan- Tvacuum = TΛ := −κ gΛ . (4) dard component notation for computational ease. The signs are chosen such that a positive Λ ac- counts for a positive energy density and a negative 3 Einstein’s equations pressure if the spacetime is Lorentzian (ε = −1). There is another form of Einstein’s equations In n-dimensional spacetime Einstein’s equations which is sometimes advantageous to use and in 1 which n explicitly enters: form a set of 2 n(n + 1) quasi-linear partial dif- ferential equations of second order for 1 n(n + 1) 2  1  functions (the components of the metric tensor) Ric = κ T − n−2 g Trg(T) . (5) depending on n independent variables (the coor- dinates in spacetime). At each point of spacetime These two forms are easily seen to be mathemati- (event) they equate a purely geometric quantity to cally equivalent by the identities the distribution of energy and momentum carried 1 by the matter. More precisely, this distribution Ein = Ric − 2 g Trg(Ric) , (6a) comprises the local densities (quantity per unit vol- 1 Ric = Ein − g Trg(Ein) . (6b) ume) and current densities (quantity per unit area n−2 and unit time) of energy and momentum. The ge- With respect to a local field of basis vectors ometric quantity in Einstein’s equations is the Ein- {e0, e1, ··· , en−1} we write Ein(eµ, eν ) =: Gµν , stein tensor Ein, the matter quantity is the energy- T(eµ, eν ) =: Tµν , and Ric(eµ, eν ) =: Rµν . Then momentum tensor T. Both tensors are of second (3) and (5) take on the component forms rank, symmetric, and here taken to be covariant (in components: “all indices down”). Their num- Gµν = κ Tµν (7) ber of independent components in n spacetime di- 1 and mensions is 2 n(n + 1)  1 λ Einstein’s equations (actually a single tensor Rµν = κ Tµν − n−2 gµν Tλ (8) equation, but throughout we use the plural to emphasize that it comprises several component respectively. Next we explain the meanings of the equations) state the simple proportionality of Ein symbols in Einstein’s equations from left to right. with T Ein = κ T , (3) 3.1 What aspects of geometry? where κ denotes the dimensionful constant of pro- The left-hand side of Einstein’s equations com- portionality. Note that no explicit reference to the prises certain measures of curvature. As will be dimension n of spacetime enters (3), so that even explained in detail in Section 5, all curvature in- if n 6= 4 it is usually referred to as Einstein’s equa- formation in dimensions higher than two can be tions. We could have explicitly added a cosmolog- reduced to that of sectional curvature. The sec- ical constant term gΛ on the left-hand side, where tional curvature at a point p ∈ M tangent to Λ is a constant the physical dimension of which is span{X,Y } ⊂ TpM is the Gaussian curvature at the square of an inverse length. However, as long p of the submanifold spanned by the geodesics in as we write down our formulae for general T we M emanating from p tangent to span{X,Y }. The may absorb this term into T where it accounts for Gaussian curvature is defined as the product of two a contribution TΛ = −gΛ/κ. This has to be kept principal , each being measured in units

4 of an inverse length (the inverse of a principal ra- matrix, which we represent as follows by splitting dius). Hence the Gaussian curvature is measured off terms involving a time component: in units of an inverse length-squared.  E −cM~  At each point p in spacetime the Einstein and T = . (11) µν 1 ~ Ricci tensors are symmetric bilinear forms on TpM. − c S Tmn Hence Ein and Ric are determined by the values p p Here all matrix elements refer to the matter’s en- Ein (W, W ) and Ric (W, W ) for all W ∈ T M. p p p ergy momentum distribution relative to the rest By continuity in W this remains true if we restrict frame of the observer who momentarily moves W to the open and dense set of vectors which are along e (i.e. with four-velocity u = ce ) and uses not null, i.e. for which g(W, W ) 6= 0. As we will 0 0 the basis {e , e , e } in his/her rest frame. Then see later on, we then have 1 2 3 E = T00 is the energy density, S~ = (s1, s2, s3) the N X1 (components of the) energy current-density, i.e. en- Ein(W, W ) = −g(W, W ) Sec , (9) ergy per unit surface area and unit time interval, ⊥W M~ the momentum density, and finally Tmn the N X2 (component of the) momentum current-density, i.e. Ric(W, W ) = +g(W, W ) Sec . (10) momentum per unit of area and unit time interval. kW Note that symmetry Tµν = Tνµ implies a simple re- lation between the energy current-density and the For the Einstein tensor the sum on the right-hand 1 momentum density side is over any complete set of N1 = 2 (n−1)(n−2) sectional curvatures of pairwise orthogonal planes S~ = c2 M~ . (12) in the orthogonal complement of W in TpM. For the Ricci tensor it is over any complete set of The remaining relations Tmn = Tnm express equal- N2 = n−1 sectional curvatures of pairwise orthog- ity of the m-th component of the current density onal planes containing W . If W is a timelike unit for n-momentum with the n-th component of the vector representing an observer, Ein(W, W ) is sim- current density for m-momentum. Note that the ply (−ε) an equally weighted sum of space- two minus signs in front of the mixed components like sectional curvatures, whereas Ric(W, W ) is ε of (11) would have disappeared had we written times an equally weighted sum of timelike sectional down the contravariant components T µν . In flat µ curvatures. In that sense we may say that, e.g., spacetime, the four equations ∂ Tµν express the Ein(W, W ) at p ∈ M measures the mean Gaussian local conservation of energy and momentum. In curvature of the (local) hypersurface in M that is curved spacetime (with vanishing torsion) we have spanned by geodesics emanating from W orthogo- the identity (to be proven later; compare (90b)) nal to W . It, too, is measured in units of the square µ of an inverse length. ∇ Gµν ≡ 0 (13) implies via (7) µ 3.2 What aspects of matter? ∇ Tµν = 0 , (14) Now we turn to the right-hand side of Einstein’s which may be interpreted as expressing a local con- equations. We restrict to four spacetime dimen- servation of energy and momentum for the matter sions, though much of what we say will apply ver- plus the gravitational field, though there is no such batim to other dimensions. The tensor T on the thing as a separate energy-momentum tensor on right-hand side of (3) is the energy-momentum ten- spacetime for the gravitational field. sor of matter. With respect to an orthonormal Several positivity conditions can be imposed basis {e0, e1, ··· , en−1} with timelike e0 the com- on the energy momentum tensor T. The sim- ponents Tµν := T(eµ, eν ) form a symmetric 4 × 4 plest is known as weak energy-condition and reads

5 T(W, W ) ≥ 0 for all timelike W . It is equivalent to 2013) known with a relative standard uncertainty the requirement that the energy density measured of 1.2 × 10−4 and is thus by far the least well by any local observer is non negative. For a per- known of the fundamental physical constants. c = fect fluid of rest- density ρ and pressure p the 299 792 458 m · s−1 is the vacuum speed of light weak energy-condition is equivalent to both con- whose value is exact, due to the SI-definition of ditions ρ ≥ 0 and p ≥ −c2ρ. The strong energy- meter (“the meter is the length of the path trav- 1  condition says that T − 2 gTrg(T) (W, W ) ≥ 0 eled by light in vacuum during a time interval of again for all timelike W . This neither follows nor 1/299 792 458 of a second”). implies the weak energy-condition. For a perfect 2 The physical dimension of κ is fluid it is equivalent to both conditions p ≥ −c ρ 2 2 time /(mass · length), that is in SI-units and p ≥ −c ρ/3, i.e. to the latter alone if ρ is s2 · kg−1 · m−1 or m−2/(J · m−3), where J = positive and to the former alone if ρ is negative Joule = kg · m2 · s−2. It converts the common (which is not excluded here). Its significance lies physical dimension of all components Tµν , which in the fact that it ensures attractivity of gravity is that of an energy density (Joule per cubic as described by Einstein’s equations. It must, for meter in SI-units) into that of the components example, be violated if matter is to drive infla- of Ein, which is that of curvature (in dimension tion. Note that upon imposing Einstein’s equa- ≥ 2), i.e., the square of an inverse length (inverse tions the weak and the strong energy-conditions square-meter in SI-units). read Ein(W, W ) ≥ 0 and Ric(W, W ) ≥ 0 respec- tively. From (9) and (10) we can see that for fixed If we express energy density as mass density 2 2 2 W these imply conditions on complementary sets of times c , the conversion factor is κc = 8πG/c . sectional curvatures. For completeness we mention It can be expressed in various units that give a the condition of energy dominance, which states feel for the local “curving power” of mass-densities. 3 −3 that T(W, W ) ≥ |T(X,X)| for any pair of or- For that of water, ρW ≈ 10 kg · m , and nu- thonormal vectors W, X where W is timelike (and clear matter in the core of a neutron star (which hence X is spacelike). It is equivalent to the weak is more than twice that of atomic nuclei), ρN ≈ 17 −3 energy-condition supplemented by the requirement 5 × 10 kg · m , we get, respectively: ] that (iW T) be non spacelike for all timelike W . The second requirement ensures locally measured  1 2  1 2 densities of energy currents and momenta of matter κc2 ≈ · ρ−1 ≈ · ρ−1 , (16) 1.5 AU W 10 km N to be non spacelike.

3.3 How do geometry and matter where AU = 1.5 × 1011 m is the astronomical unit relate quantitatively? (mean Earth-Sun distance). Hence, roughly speak- ing, matter densities of water cause curvature radii We return to Einstein’s equations and finally dis- of the order of the astronomical unit, whereas the cuss the constant of proportionality κ on the right- highest known densities of nuclear matter cause hand side of (3). Its physical dimension is that of curvature radii of tens of kilometers. The curva- curvature (m−2 in SI units) divided by that of en- ture caused by mere mass density is that expressed ergy density (J ·m−3 in SI units, where J = Joule). in Ein(W, W ) when W is taken to be the unit time- It is given by like vector characterizing the local rest frame of the matter: It is a mean of spatial sectional curvatures 8πG m−2 −43 in the matter’s local rest frame. Analogous inter- κ := 4 ≈ 2.1 × 10 −3 , (15) c J · m pretations can be given for the curvatures caused where G ≈ 6.67384(80) × 10−11m3 · kg−1 · s−2 is by momentum densities (energy current-densities) Newton’s constant. It is currently (March and momentum current-densities (stresses).

6 3.4 Conserved energy-momentum isometries. We will discuss general Lie-group ac- tensors and globally conserved tions on manifolds in the Appendix at the end quantities of this contribution, containing detailed proofs of some relevent formulae. But in order not to inter- In this subsection we briefly wish to point out rupt the argument too much, let us recall at this that energy-momentum tensors T whose diver- point that an action of G on M is a map gence vanishes (14) give rise to conserved quanti- ties in case the spacetime (M, g) admits non-trivial Φ: G × M → M, (22a) isometries. We will stress the global nature of these (g, m) 7→ Φ(g, m) = Φg(m) , quantities and clarify their mathematical habitat. Conservation laws for the matter alone result which satisfies in the presence of symmetries, more precisely, if Killing fields for (M, g) exist. Recall that a vector Φe = IdM , (22b) field V is called a Killing field iff LV g = 0, where Φg ◦ Φh = Φgh . (22c) LV is the Lie derivative with respect to V . Recall that the Lie derivative can be expressed in terms of Here e ∈ G denotes the neutral element, IdM the the Levi-Civita covariant derivative with respect to identity map on M, and equation (22c) is valid for g, in which case we get the component expression: any two elements g, h of G. In fact, equation (22c) characterizes a left action. In contrast, for a right action we would have Φ instead of Φ on the (LV g)µν = ∇µVν + ∇ν Vµ = 0 . (17) hg gh right-hand side of (22c). Moreover, as the group We consider the one-form JV that results from acts by isometries for the metric g, we also have ∗ contracting T with V : Φhg = g for all h ∈ G. Now, this action defines a map, V , from Lie(G), µ ν JV := iV T = V Tµν dx . (18) the Lie algebra of G, into the vector fields on M. The vector field corresponding to X ∈ Lie(G) is As a result of Killing’s equation (17) it is divergence denoted V X . Its value at a point m ∈ M is defined free, by ∇ J µ = 0 . (19) d µ V X  V (m) := Φ exp(tX), m . (23) This may be equivalently expressed by saying that dt t=0 1 the 3-form ?JV , which is the Hodge dual of the From this it is obvious that V : Lie(G) → ΓT0 M 1-form JV , is closed: is linear. Moreover, one may also show (compare (382b) in Appendix) that this map is a Lie anti- d ? JV = 0 . (20) homomorphism, i.e. that

[X,Y ] X Y Integrating ?JV over some 3-dimensional subman- V = −[V ,V ] . (24) ifold Σ results in a quantity (As shown in the Appendix, a right action would Z have resulted in a proper Lie homomorphism – see Q[V, Σ] := ?JV (21) Σ (382a) –, i.e. without the minus sign on the right- hand side, which however is not harmful.) The left which, because of (20), is largely independent of action of G on M extends to a left action on all Σ. More precisely, if Ω ⊂ M is an oriented domain tensor fields by push forward. In particular, the with boundary ∂Ω = Σ −Σ , then Stokes’ theorem X 1 2 push forward of V by Φg has a simple expression gives Q[V, Σ1] = Q[V, Σ2]. (see (383a) in Appendix) : Suppose now that V arises from a finite- X Adg (X) dimensional Lie group G that acts on (M, g) by Φg∗V = V , (25)

7 where Ad denotes the adjoint representation of nature. In any case, we assume the isometric ac- G on Lie(G). In fact, relation (25) can be di- tion (22) to extend to an action of G on the set of rectly deduced from definition (23). Indeed, writ- matter variables µ, which we denote by µ 7→ Φg∗µ, ing Φ(g, p) = g ·p for notational simplicity, we have like the push-forward on tensor fields. This is also (see Appendix for more explanation) meant to indicate that we assume this to be a left action, i.e. Φg∗ ◦ Φh∗ = Φgh∗. X d  We regard the energy-momentum tensor T as a (Φg∗V )(g · p) = g exp(tX) · p dt t=0 map from the space of matter variables to the space d = g exp(tX) g−1g · p of symmetric second-rank covariant tensor fields on dt t=0 M. We require this map to satisfy the following d    covariance property: = exp tAdg(X) · (g · p) dt t=0 ∗ = V Adg (X)(g · p) . T[Φg∗µ] = Φg∗T[µ] := Φg−1 T , (28) (26) where Φg∗ is the ordinary push-forward of the ten- sor T. Since we take T to be covariant, its push This leads to (25) which we shall use shortly. forward is the pull back by the inverse diffeomor- Returning to the expression (21) we see that, for phism, as indicated by the second equality in (28). fixed Σ, it becomes a linear map from Lie(G) to : R For each specification µ of matter variables we X X can compute the quantitty Q[V , Σ, µ] as in (21). M : Lie(G) → R , M(X) := Q[V , Σ] . (27) Note that we now indicate the dependence on µ Hence each hypersurface Σ defines an element M ∈ explicitly. We are interested in computing how Q Lie∗(G) in the vector space that is dual to the Lie changes as µ is acted on by g ∈ G. This is done as algebra, given that the integral over Σ converges. follws: This is the case for spacelike Σ and energy momen- Z  X  tum tensors with spatially compact support (or at Q V , Σ, Φg∗µ = ? iV X T[Φg∗µ] Σ least sufficiently rapid fall off). The same argument Z ∗ as above using Stokes’ theorem and (20) then shows = ? iV X Φg−1 T[µ] that M is independent of the choice of spacelike Σ Z Σ. In other words, we obtain a conserved quantity ∗  = ? Φ −1 i X T[µ] ∗ g Φg−1∗V M ∈ Lie (G) for G-symmetric (M, g) Σ Z and covariant divergence free tensors T. ∗  = ? Φ −1 i Ad (X) T[µ] g V g−1 So far we considered a fixed spacetime (M, g) and Σ a fixed energy-momentum tensor T, both linked by Z ∗  = Φ −1 ?i Ad (X) T[µ] Einstein’s equations. In this case the vanishing di- g V g−1 Σ vergence (14) is an integrability condition for Ein- Z = ? i Ad (X) T[µ] stein’s equation and hence automatic. However, V g−1 Φ (Σ) it is also of interest to consider the more general g−1  Ad −1 (X)  case where (M, g) is merely a background for some = Q V g , Φg−1 (Σ), µ . matter represented by energy-momentum tensors (29) T[µ], all of which are divergence free (14) with re- spect to the background metric g. Note that we Here we used (28) in the second equality, the gen- ∗ ∗ do not assume (M, g) to satisfy Einstein’s equa- eral formula iV f T = f (if∗V T ) (valid for any dif- tions with any of the T[µ] on the right-hand side. feomorphism f, vector field V , and covariant ten- The µ stands for some matter variables which may sor field T ) in the third equality, (25) in the fourth be fundamental fields and/or of phenomenological equality, the formula ? f ∗F = f ∗ ?F in the fifth

8 equality (valid for any orientation preserving isom- nature does not matter in what follows]. The lin- etry f and any form-field F ; here we assume M ear isometries of (V, η) form the Lorentz group to be oriented), and finally the general formula for Lor ⊂ GL(V ) and the isometries G of (M, g) the integral of the pull back of a form in the sixth can be (non-naturally) identified with the semi- equality. direct product V o Lor, called the Poincar´egroup, Our final assumption is that Q does not de- Poin. Using g we can identify Lie∗(Poin) with pend on which hypersurface Φg(Σ) it is evaluated V ⊕(V ∧V ). The co-adjoint action of (a, A) ∈ Poin on. Since we assume (14) this is guaranteed if all on (f, F ) ∈ Lie∗(Poin) is then given by Φg(Σ) are in the same homology class or, more ∗   generally, if any two hypersurfaces Σ and Φg(Σ) Ad(a,A)(f, F ) = Af , (A ⊗ A)F − a ∧ Af . (32) are homologous to hypersurfaces in the comple- ment of the support of T. A typical situation Note that, e.g., the last term on the right hand side arising in physical applications is that of a source includes the law of change of angular momentum T[µ] with spatially compact support; then any two under spatial translations. In contrast, the adjoint sufficiently extended spacelike slices through the representation on Lie(Poin), the latter also identi- timelike support-tube of T[µ] is homologous to fied with V ⊕ (V ∧ V ), is given by the timelike cylindrical hypersurface outside this support-tube. In this case we infer from (29) that    Ad(a,A)(f, F ) = Af − (A ⊗ A)F a , (A ⊗ A)F ,

 X   Ad −1 (X)  (33) Q V , Σ, Φg∗µ e = Q V g , Σ, µ . (30) where the application of an element in V ∧ V Recall from (27) that for fixed Σ and T we have to an element in V is given by (u ∧ v)(w) := M ∈ Lie∗(G). Given the independence on Σ and u g(v, w) − v g(u, w), and linear extension. Note the depencence of T on µ, we now regard M as a the characteristic difference between (32) and (33), map from the matter variables µ to Lie∗(G). This which lies in the different actions of the subgroup map may be called the momentum map. (Compare of translations, whereas the subgroup of Lorentz the notion of a momentum map in Hamiltonian transformations acts in the same fashion. Physical mechanics; cf. Section 7.) Equation (30) then just momenta transform as in (32), as already exempli- states the Ad∗-equivariance of the momentum map: fied by the non-trivial transformation behavior of angular momentum under spatial translations. For ∗ M ◦ Φg∗ = Adg ◦ M . (31) a detailed discussion of the proper group-theoretic setting and the adjoint and co-adjoint actions, see Here Ad∗ denote the co-adjoint representation of ∗ the recent account [75]. G on Lie(G), which is defined by Adg(α) = α ◦ Adg−1 . From all this we see that the conserved “momentum” that we obtain by evaluating M on 4 Spacetime decomposition the matter configuration µ is a conserved quantity that is globally associated to all of spacetime, not In this section we explain how to decompose a given a particular region or point of it. It is an element spacetime (M, g) into “space” and ‘time”. For this ∗ of the vector space Lie (G) which carries the co- to be possible we need to make the assumption that adjoint representation of the symmetry group G. M is diffeomorphic to the product of the real line R and some 3-manifold Σ: In particular this applies to Special Relativ- ity, where M is the four-dimensional real affine M =∼ R × Σ . (34) space with associated (four-dimensional real) vec- tor space V and g a bilinear, symmetric, non- This will necessarily be the case for globally hy- degenerate form of signature (−, +, +, +) [the sig- perbolic spacetimes, i.e. spacetimes admitting a

9 Cauchy surface [65]. We assume Σ to be orientable, M Σs0 for, if it were not, we could take the orientable Es0 double cover of it instead. Orientable 3-manifolds Σ Es Σs are always parallelizable [111] , i.e. admit three E 00 globally defined and pointwise linearly independent s Σs00 vector fields. This is equivalent to the triviality of the tangent bundle. In the closed case this is known as Stiefel’s theorem (compare [100], problem 12- Figure 1: Spacetime M is foliated by a one-parameter B) and in the open case it follows, e.g., from the family of spacelike embeddings of the 3-manifold Σ. Here well known fact that every open 3-manifold can the image Σs0 of Σ under Es0 lies to the future (above) and 00 0 3 Σ 00 to the past (below) of Σ if s < s < s . ‘Future’ and be immersed in R [117]. Note that orientabil- s s 2 1 ‘past’ refer to the time function t which has so far not been ity is truly necessary; e.g., RP × S is not par- given any metric significance. allelizable. Since Cartesian products of paralleliz- able manifolds are again parallelizable, it follows We distinguish between the abstract 3-manifold that a 4-dimensional product spacetime (34) is also Σ and its image Σs in M. The latter is called the parallelizable. This does, of course, not generalize leaf corresponding to the value s ∈ R. Each point to higher dimensions. Now, for non-compact four- in M is contained in precisely one leaf. Hence there dimensional spacetimes it is known from [64] that is a real valued function t : M → R that assigns to parallelizability is equivalent to the existence of a each point in M the parameter value of the leaf it spin structure, without which spinor fields could lies on: not be defined on spacetime. So we see that the t(p) = s ⇔ p ∈ Σ . (36) existence of spin structure is already implied by s (34) and hence does not pose any further topolog- So far this is only a foliation of spacetime by 3- ical restriction. Note that the only other potential dimensional leaves. For them to be addressed as topological restriction at this stage is that imposed “space” the metric induced on them must be posi- from the requirement that a smooth Lorentz metric tive definite, that is, the leaves should be spacelike is to exist everywhere on spacetime. This is equiv- submanifolds. This means that the one-form dt is alent to a vanishing Euler characteristic (see, e.g., timelike: § 40 in [111]) which in turn is equivalently to the g−1(dt, dt) < 0 . (37) global existence of a continuous, nowhere vanish- ing vector field (possibly up to sign) on spacetime. The normalized field of one-forms is then But such a vector field clearly exists on any Carte- sian product with one factor being . We conclude [ dt R n := p . (38) that existence of a Lorentz metric and a spin struc- −g−1(dt, dt) ture on an orientable spacetime M = R × Σ pose no restrictions on the topology of an orientable Σ. As explained in section 2, we write n[ since we think As we will see later on, even Einstein’s equation of this one form as the image under g of the nor- poses no topological restriction on Σ, in the sense malized vector field perpendicular to the leaves: that some (physically reasonable) solutions to Ein- stein’s equations exist for any given Σ. Topological n[ = g(n, · ) . (39) restrictions may occur, however, if we ask for solu- tion with special properties (see below). The linear subspace of vectors in TpM which are k Now, given Σ, we consider a one-parameter fam- tangent to the leaf through p is denoted by Tp M; ily of embeddings hence

k Es :Σ → M, Σs := Es(Σ) ⊂ M. (35) Tp M := {X ∈ TpM : dt(X) = 0} . (40)

10 The orthogonal complement is just the span of n at For example, letting the horizontal projection of ⊥ p, which we denote by Tp M. This gives, at each the form ω act on the vector X, we get point p of M, the g-orthogonal direct sum k k ] [ P∗ ω(X) = (P ω ) (X) T M = T ⊥M ⊕ T kM. (41) p p p = gP kω],X (47) and associated projections (we drop reference to = gω],P kX the point p) = ωP kX , P ⊥ :TM → T ⊥M, where we merely used the definitions (1) of [ and ] X 7→ ε g(X, n) n , (42a) in the second and fourth equality, respectively, and k P k : TM → T kM, the self-adjointness (44b) of P in the third equal- ity. The analogous relation holds for P ⊥ω(X). It X 7→ X − εg(X, n) n . (42b) ∗ k ⊥ is also straightforward to check that P∗ and P∗ As already announced in Section 2, we introduced are self-adjoint with respect to g−1 (cf. (2)). the symbol Having the projections defined for vectors and ε = g(n, n) (43) co-vectors, we can also define it for the whole ten- sor algebra of the underlying vector space, just in order to keep track of where the signature mat- by taking the appropriate tensor products of these ters. Note that the projection operators (42) are k k maps. All tensor products between P and P∗ will self-adjoint with respect to g, so that for all X,Y ∈ k TM we have then, for simplicity, just be denoted by P , the action on the tensor being obvious. Similarly for gP ⊥X,Y  = gX,P ⊥Y  , (44a) P ⊥. (For what follows we need not consider mixed projections.) The projections being pointwise op- gP kX,Y  = gX,P kY  . (44b) erations, we can now define vertical and horizontal A vector is called horizontal iff it is in the kernel projections of arbitrary tensor fields. Hence a ten- u of P ⊥, which is equivalent to being invariant under sor field T ∈ ΓTd M is called horizontal if and only k P k. It is called vertical iff it is in the kernel of P k, if P T = T . The space of horizontal tensor fields ku ⊥ which is equivalent to being invariant under P . of rank (u,d) is denoted by ΓT d M. All this can be extended to forms. We define As an example, the horizontal projection of the vertical and horizontal forms as those annihilating metric g is horizontal and vertical vectors, respectively: h := P kg := gP k · ,P k ·  = g − εn[ ⊗ n[ . (48) ∗⊥ ∗ k Tp M := {ω ∈ Tp M : ω(X) = 0 , ∀X ∈ Tp M} , k0 (45a) Hence h ∈ ΓT 2 M. Another example of a horizon- ∗k ∗ ⊥ tal vector field is the “acceleration” of the normal Tp M := {ω ∈ Tp M : ω(X) = 0 , ∀X ∈ Tp M} . field n: (45b) a := ∇nn . (49) Using the ‘musical’ isomorphisms (1), the self- Here ∇ denotes the Levi-Civita covariant derivative adjoint projection maps (42) on vectors define self- with respect to g. An observer who moves perpen- adjoint projection maps on co-vectors (again drop- dicular to the horizontal leaves has four-velocity ping the reference to the base-point p) u = cn and four-acceleration c2a. If L denotes the Lie derivative, it is easy to show that the accelera- ⊥ ⊥ ∗ ∗⊥ P∗ :=[ ◦ P ◦ ] : T M → T M, (46a) tion 1-form satisfies k k ∗ ∗k P∗ := [ ◦ P ◦ ] : T M → T M. (46b) [ [ a = Lnn . (50)

11 Moreover, as n is hypersurface orthogonal it is ir- representative of space. Instead of using the folia- rotational, hence its 1-form equivalent satisfies tion by 3-dimensional spatial leaves (35) we could have started with a foliation by timelike lines, plus [ [ dn ∧ n = 0 , (51a) the condition that these lines are vorticity free. which is equivalent to the condition of vanishing These two concepts are equivalent. Depending on horizontal curl: the context, one might prefer to emphasize one or the other. k [ P dn = 0 . (51b) The vector parallel to the worldline at p = Es(q) is, as usual in differential geometry, defined by its Equation (51a) can also be immediately inferred action on f ∈ C∞(M) (smooth, real valued func- directly from (38). Taking the operation in ◦ d (ex- tions): terior derivative followed by contraction with n) as ∂ df(E 0 (q)) f = s . (54) well as the Lie derivative with respect to n of (50) 0 0 ∂t Es(q) ds s =s shows da[ ∧ n[ = 0 , (52a) At each point this vector field can be decomposed into its horizontal component that is tangential to an equivalent expression being again the vanishing the leaves of the given foliation and its normal com- of the horizontal curl of a: ponent. We write k [ P da = 0 . (52b) 1 ∂ = α n + β , (55) This will be useful later on. c ∂t [ Note that a is a horizontal co-vector field, i.e. an where β is the tangential part; see Figure 2. The ku=0 element of ΓT d=1 M. More generally, for a purely covariant horizontal tensor field we have the follow- p0 ing results, which will also be useful later on: Let Σ k0 s+ds T ∈ ΓT d M, then 1 ∂ k c ∂t αn P LnT = LnT, (53a) L T = fL T, (53b) fn n p Σs for all f ∈ C∞(M). Note that (53a) states that the ‘ β Lie derivative in normal direction of a horizontal covariant tensor field is again horizontal. That this Figure 2: For fixed q ∈ Σ its image points p = Es(q) 0 is not entirely evident follows, e.g., from the fact and p = Es+ds(q) for infinitesimal ds are connected by the that a corresponding result does not hold for T ∈ vector ∂/∂t|p, whose components normal to Σs are α (one ku function, called lapse) and β (three functions, called shift) ΓT d M where u > 0. The proofs of (53) just use respectively. standard manipulations. A fixed space-point q ∈ Σ defines the worldline real-valued function α is called the lapse (function) (history of that point) R 3 s 7→ Es(q). The col- and the horizontal vector field β is called the shift lection of all worldlines of all space-points define a (vector-field) . foliation of M into one-dimensional timelike leafs. Each leaf is now labeled uniquely by a space point. 4.1 Decomposition of the metric We can think of “space”, i.e., the abstract mani- 0 fold Σ, as the quotient M/∼, where p ∼ p iff both Let {e0, e1, e2, e3} be a locally defined orthonor- 0 1 2 3 points lie on the same worldline. As any Σs in- mal frame with dual frame {θ , θ , θ , θ }. We call 0 [ tersects each worldline exactly once, each Σs is a them adapted to the foliation if e0 = n and θ = n .

12 0 1 2 3 A local coordinate system {x , x , x , x } is called Orthogonality of the ea implies for the chart adapted if ∂/∂xa are horizontal for a = 1, 2, 3. Note components of the spatial metric (48) that in the latter case ∂/∂x0 is not required to be 3 orthogonal to the leaves (i.e. it need not be par- X h := h∂/∂xm, ∂/∂xn = Aa Aa , (62) allel to n). For example, we may take x0 to be mn m n a=1 proportional to t; say x0 = ct. In the orthonormal co-frame the spacetime met- and its inverse ric, i.e. the field of signature (ε, +, +, +) metrics 3 mn −1 m n X −1 m −1 n in the tangent , has the simple form h := h dx , dx = [A ]a [A ]a . (63) a=1 3 X g = εθ0 ⊗ θ0 + θa ⊗ θa . (56) Inserting (60) into (56) and using (62) leads to a=1 the (3+1)-form of the metric in adapted coordi- The inverse spacetime metric, i.e. the field of signa- nates ture (ε, +, +, +) metrics in the co-tangent spaces, g = εα2 + h(β, β)c2 dt ⊗ dt has the form m m  + cβm dt ⊗ dx + dx ⊗ dt (64) 3 m n −1 X + hmn dx ⊗ dx , g = εe0 ⊗ e0 + ea ⊗ ea . (57) a=1 n [ where βm := hmnβ are the components of β := The relation that expresses the coordinate basis g(β, · ) = h(β, · ) with respect to the coordinate ba- in terms of the orthonormal basis is of the form (in sis {∂/∂xm}. Likewise, inserting (61) into (57) and a self-explanatory matrix notation) using (63) leads to the (3+1)-form of the inverse metric in adapted coordinates (we write ∂t := ∂/∂t  0   a    m ∂/∂x α β e0 and ∂m := ∂/∂x for convenience) m = a , (58) ∂/∂x 0 A ea m −1 −2 −2 g = εc α ∂t ⊗ ∂t a where β are the components of β with respect to −1 −2 m  − εc α β ∂t ⊗ ∂m + ∂m ⊗ ∂t (65) the horizontal frame basis {ea}. The inverse of (58) mn m n is + h + εβ β ∂m ⊗ ∂n .    −1 −1 m  0  e0 α −α β ∂/∂x Finally we note that the volume form on space- = −1 m m , (59) ea 0 [A ]a ∂/∂x time also easily follows from (60) m 0 1 2 3 where β are the components of β with respect dµg = θ ∧ θ ∧ θ ∧ θ to the horizontal coordinate-induced frame basis (66) = αpdet{h } cdt ∧ d3x , {∂/∂xm}. mn The relation for the co-bases dual to those in (58) where we use the standard shorthand d3x = dx1 ∧ is given by the transposed of (58), which we write dx2 ∧ dx3. as:  a  0 a 0 m α β 4.2 Decomposition of the θ θ = dx dx a . (60) 0 Am covariant derivative The inverse of that is the transposed of (59): Given horizontal vector fields X and Y , the covari- ant derivative of Y with respect to X need not be  −1 −1 m 0 m 0 a α −α β horizontal. Its decomposition is written as dx dx = θ θ −1 m . 0 [A ]a (61) ∇X Y = DX Y + nK(X,Y ) , (67)

13 where field. Symmetry follows from the vanishing torsion of ∇, since then k DX Y := P ∇X Y, (68) K(X,Y ) = ε g(n, ∇X Y ) K(X,Y ) := ε g(n, ∇X Y ) . (69) = ε g(n, ∇Y X + [X,Y ]) The map D defines a covariant derivative (in the (74) = ε g(n, ∇ X) sense of Kozul; compare [110], Vol 2) for horizon- Y tal vector fields, as a trivial check of the axioms = K(Y,X) reveals. Moreover, since the commutator [X,Y ] for horizontal X,Y . From (69) one sees that of two horizontal vector fields is always horizontal K(fX, Y ) = fK(X,Y ) for any smooth function (since the horizontal distribution is integrable by f. Hence K defines a unique symmetric tensor construction), we have field on M by stipulating that it be horizontal, i.e. D K(n, ·) = 0. It is called the extrinsic curvature of T (X,Y ) = DX Y − DY X − [X,Y ] the foliation or , the first k  = P ∇X Y − ∇Y X − [X,Y ] (70) fundamental form being the metric. From (69) = 0 and the symmetry just shown one immediately in- fers the alternative expressions due to ∇ being torsion free. We recall that tor- 1 sion is a tensor field T ∈ ΓT2 M associated to each K(X,Y ) = −ε g(∇X n, Y ) = −ε g(∇Y n, X) . covariant derivative ∇ via (75) This shows the relation between the extrinsic cur- ∇ T (X,Y ) = ∇X Y − ∇Y X − [X,Y ] . (71) vature and the Weingarten map, Wein, also called the shape operator, which sends horizontal vectors We have T (X,Y ) = −T (Y,X). As usual, even to horizontal vectors according to though the operations on the right hand side of (71) involve tensor fields (we need to differentiate), X 7→ Wein(X) := ∇X n . (76) the result of the operation just depends on X and Y pointwise. This one proves by simply checking the Horizontality of ∇X n immediately follows from n 1  validity of T (fX, Y ) = fT (X,Y ) for all smooth being normalized: g(n, ∇X n) = 2 X g(n, n) = 0. functions f. Hence (70) shows that D is torsion Hence (75) simply becomes free because ∇ is torsion free. K(X,Y ) = −ε hWein(X),Y  Finally, we can uniquely extend D to all hori- (77) zontal tensor fields by requiring the Leibniz rule. = −ε hX, Wein(Y ) , Then, for X,Y,Z horizontal where we replaced g with h—defined in (48)— (DX h)(Y,Z) since both entries are horizontal. It says that K  is (−ε) times the covariant tensor corresponding = X h(Y,Z) − h(DX Y,Z) − h(Y,DX Z)  (72) to the Weingarten map, and that the symmetry = X g(Y,Z) − g(∇X Y,Z) − g(Y, ∇X Z) of K is equivalent to the self-adjointness of the = (∇X g)(Y,Z) = 0 Weingarten map with respect to h. The Wein- garten map characterizes the bending of the em- due to the metricity, ∇g = 0, of ∇. Hence D is bedded hypersurface in the ambient space by an- metric in the sense swering the following question: In what direction Dh = 0 . (73) and by what amount does the normal to the hy- persurface tilt if, starting at point p, you progress The map K from pairs of horizontal vector fields within the hypersurface by the vector X. The an- (X,Y ) into functions define a symmetric tensor swer is just Weinp(X). Self adjointness of Wein

14 then means that there always exist three (n − 1 assume a minimal and a maximal value, denoted in general) perpendicular directions in the hyper- by kmin(p) = k(p, vmin) and kmax(p) = k(p, vmax) surface along which the normal tilts in the same respectively. These are called the principal curva- direction. These are the principal curvature direc- tures of S at p and their reciprocals are called the tions mentioned above. The principal curvatures principal radii. It is clear that the principal direc- are the corresponding eigenvalues of Wein. tions vmin and vmax just span the eigenspaces of Finally we note that the covariant derivative of the Weingarten map discussed above. In particu- the normal field n can be written in terms of the lar, vmin and vmax are orthogonal. The Gaussian acceleration and the Weingarten map as follows curvature K(p) of S at p is then defined to be the product of the principal curvatures: ∇n = εn[ ⊗ a + Wein . (78) K(p) = kmin(p) · kmax(p) . (81) Recalling (77), the purely covariant version of this is This definition is extrinsic in the sense that essen- [ [ [ ∇n = −ε K − n ⊗ a . (79) tial use is made of the ambient R3 in which S is em- From (48) and (79) we derive by standard manip- bedded. However, Gauss’ theorema egregium states ulation, using vanishing torsion, that this notion of curvature can also be defined in- trinsically, in the sense that the value K(p) can be

Lnh = −2εK . (80) obtained from geometric operations entirely car- ried out within the surface S. More precisely, it is In presence of torsion there would be an addi- a function of the first fundamental form (the met- [ tional term +2(inT )s, where the subscript s de- ric) only, which encodes the intrinsic geometry of [ notes symmetrization; in coordinates [(inT )s]µν = S, and does not involve the second fundamental λ α n Tλ(µgν)α. form (the extrinsic curvature), which encodes how S is embedded into R3. Let us briefly state Gauss’ theorem in mathemat- 5 Curvature tensors ical terms. Let

We wish to calculate the (intrinsic) curvature ten- a b g = gab dx ⊗ dx (82) sor of ∇ and express it in terms of the curvature tensor of D, the extrinsic curvature K, and the be the metric of the surface in some coordinates, spatial and normal derivatives of n and K. Be- and fore we do this, we wish to say a few words on the c 1 cd  definition of the curvature measures in general. Γab = 2 g −∂dgab + ∂agbd + ∂bgda , (83) All notions of curvature eventually reduce to that 3 certain combinations of first derivatives of the met- of curves. For a surface S embedded in R we have the notion of Gaussian curvature which comes ric coefficients, known under the name of Christof- c about as follows: Consider a point p ∈ S and a unit fel symbols . Note that Γab has as many indepen- vector v at p tangent to S. Consider all smooth dent components as ∂agbc and that we can calculate curves passing through p with unit tangent v. It the latter from the former via is easy to see that the curvatures at p of all such ∂ g = g Γn + g Γn . (84) curves is not bounded from above (due to the pos- c ab an bc bn ac sibility to bend within the surface), but there will Next we form even more complicated combinations be a lower bound, k(p, v), which just depends on of first and second derivatives of the metric coeffi- the chosen point p and the tangent direction repre- cients, namely sented by v. Now consider k(p, v) as function of v. a a a a n a n As v varies over all tangent directions k(p, v) will R b cd = ∂cΓdb − ∂dΓcb + ΓcnΓdb − ΓdnΓcb , (85)

15 which are now known as components of the Rie- From (88) and using (71) one may show that the mann curvature tensor. From them we form the Riemann tensor always obeys the first and second totally covariant (all indices down) components: Bianchi identities:

n X Rab cd = ganR b cd . (86) R(X,Y )Z (XYZ) They are antisymmetric in the first and second X n o = (∇ T )(Y,Z) − T X,T (Y,Z) , index pair: Rab cd = −Rba cd = −Rab dc, so that X (XYZ) R12 12 is the only independent component. Gauss’ theorem now states that at each point on S we have (89a) X R12 12 (∇X R)(Y,Z) K = 2 . (87) (XYZ) g11g22 − g 12 X = RX,T (Y,Z) , (89b) An important part of the theorem is to show that the right-hand side of (87) actually makes good (XYZ) geometric sense, i.e. that it is independent of the where the sums are over the three cyclic permu- coordinate system that we use to express the coef- tations of X, Y , and Z. For zero torsion these ficients. This is easy to check once one knows that identities read in component form: Rabcd are the coefficients of a tensor with the sym- X α metries just stated. In this way the curvature of R λ µν = 0 , (90a) a surface, which was primarily defined in terms of (λµν) curvatures of certain curves on the surface, can be X α ∇λR β µν = 0 . (90b) understood intrinsically. In what follows we will see (λµν) that the various measures of intrinsic curvatures of n-dimensional manifolds can be reduced to that of The second traced on (α, µ) and contracted with 2-dimensional submanifolds, which will be called gβν yields (−2) times (13). sectional curvatures. The covariant Riemann tensor is defined by Back to the general setting, we start from the Riem(W, Z, X, Y ) := gW, R(X,Y )Z . (91) notion of a covariant derivative ∇. Its associated curvature tensor is defined by For general covariant derivatives its only symme-  try is the antisymmetry in the last pair. But for R(X,Y )Z = ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] Z. (88) special choices it acquires more. In standard GR For each point p ∈ M it should be thought of we assume the covariant derivative to be metric compatible and torsion free: as a map that assigns to each pair X,Y ∈ TpM of tangent vectors at p a linear map R(X,Y ): ∇g = 0 , (92) T M → T M. This assignment is antisymmetric, p p T = 0 . (93) i.e. R(X,Y ) = −R(Y,X). If R(X,Y ) is applied to Z the result is given by the right-hand side of In that case the Riemann tensor has the symme- (88). Despite first appearance, the right-hand side tries of (88) at a point p ∈ M only depends on the values of X,Y , and Z at that point and hence de- Riem(W, Z, X, Y ) = −Riem(W, Z, Y, X) , (94a) fines a tensor field. This one again proves by show- Riem(W, Z, X, Y ) = −Riem(Z, W, X, Y ) , (94b) ing the validity of R(fX, Y )Z = R(X, fY )Z = Riem(W, X, Y, Z) + Riem(W, Y, Z, X) + R(X,Y )fZ = fR(X,Y )Z for all smooth real- Riem(W, Z, Y, X) = 0 , (94c) valued functions f on M. In other words: All terms involving derivatives of f cancel. Riem(W, Z, X, Y ) = Riem(X, Y, W, Z) . (94d)

16 Equation (94a) is true by definition (88), (94b) is Here X,Y is a pair of linearly independent tangent equivalent to metricity of ∇, and (94c) is the first vectors that span a 2-dimensional tangent subspace Bianchi identity in case of zero torsion. The last restricted to which g is non-degenerate. We will symmetry (94d) is a consequence of the preceding say that span{X,Y } is non-degenerate. This is the three. Together (94a), (94b), and (94d) say that, necessary and sufficient condition for the denomi- at each point p ∈ M, Riem can be thought of nator on the right-hand side to be non zero. The as symmetric bilinear form on the antisymmetric quantity Sec(X,Y ) is called the sectional curva- tensor product TpM ∧ TpM. The latter has di- ture of the manifold (M, g) at point p tangent to 1 mension N = 2 n(n − 1) if M has dimension n, span{X,Y }. From the symmetries of Riem it is and the space of symmetric bilinear forms has di- easy to see that the right-hand side of (99) does 1 mension 2 N(N + 1). From that number we have indeed only depend on the span of X,Y . That is, to subtract the number of independent conditions for any other pair X0,Y 0 such that span{X0,Y 0} = n 0 0 (94c), which is 4 in dimensions n ≥ 4 and zero span{X,Y }, we have Sec(X ,Y ) = Sec(X,Y ). otherwise. Indeed, it is easy to see that (94c) is The geometric interpretation of Sec(X,Y ) is as identically satisfied as a consequence of (94a) and follows: Consider all geodesics of (M, g) that pass (94b) if any two vectors W, Z, X, Y coincide (pro- through the considered point p ∈ M in a direc- portionality is sufficient). Hence the number # of tion tangential to span{X,Y }. In a neighborhood independent components of the curvature tensor is of p they form an embedded 2-surface in M whose Gaussian curvature is just Sec(X,Y ). #Riem = Now, Riem is determined by components of the  1 N(N + 1) − n = 1 n2(n2 − 1) for n ≥ 4 form Riem(X,Y,X,Y ), as follows from the fact  2 4 12 that Riem is a symmetric bilinear form on TM ∧ 6 for n = 3  TM. This remains true if we restrict to those X,Y 1 for n = 2 whose span is non-degenerate, since they lie dense 1 2 2 in TM ∧ TM and (X,Y ) 7→ Riem(X,Y,X,Y ) is = 12 n (n − 1) for all n ≥ 2 . (95) continuous. This shows that the full information of the Riemann tensor can be reduced to certain The Ricci and scalar curvatures are obtained Gaussian curvatures. by taking traces with respect to g: Let {e1, ··· , en} This also provides a simple geometric interpre- be an orthonormal basis, g(ea, eb) = δabεa (no sum- tation of the scalar and Einstein curvatures in mation) with εa = ±1, then terms of sectional curvatures. Let {X1, ··· ,Xn} be any set of pairwise orthogonal non-null vec- n 1 X tors. The 2 n(n − 1) 2-planes span{Xa,Xb} are Ric(X,Y ) = εa Riem(ea, X, ea,Y ) (96) non-degenerate and also pairwise orthogonal. It a=1 then follows from (97) and (99) that the scalar cur- n X vature is twice the sum of all sectional curvatures: Scal = εa Ric(ea, ea) . (97) n a=1 X Scal = 2 Sec(Xa,Xb) . (100) The Einstein tensor is a,b=1 a

1 Ein = Ric − 2 Scal g . (98) The sum on the right-hand side of (100) is the same 1 for any set of 2 n(n − 1) non-degenerate and pair- The sectional curvature is defined by wise orthogonal 2-planes. Hence the scalar curva- ture can be said to be twice the sum of mutually or- Riem(X,Y,X,Y ) thogonal sectional curvatures, or n(n−1) times the Sec(X,Y ) = 2 , (99) g(X,X)g(Y,Y ) − g(X,Y ) mean sectional curvature. Similarly for the Ricci

17 and Einstein curvatures. The symmetry of the their Kulkarni-Nomizu product is defined by Ricci and Einstein tensors imply that they are fully determined by their components Ric(W, W ) and k `(X1,X2,X3,X4) := k(X1,X3) `(X2,X4) ? Ein(W, W ). Again this remains true if we restrict + k(X2,X4) `(X1,X3) to the dense set of non-null W , i.e. g(W, W ) 6= 0. − k(X1,X4) `(X2,X3) Let now {X1, ··· ,Xn−1} be any set of mutually orthogonal vectors (again they need not be nor- − k(X2,X3) `(X1,X4) , malized) in the orthogonal complement of W . As (103) before the 1 (n − 1)(n − 2) planes span{X ,X } 2 a b or in components are non degenerate and pairwise orthogonal. From (96), (98), and (99) it follows that (k `)abcd = kac`bd+kbd`ac−kad`bc−kbc`ad . (104) ? n−1 X The Weyl tensor, Weyl, is of the same type as Ric(W, W ) = g(W, W ) Sec(W, Xa) (101) Riem but in addition totally trace-free. It is ob- a=1 tained from Riem by a projection map, PW , given by and Weyl := PW (Riem) n−1 1  1  X := Riem − n−2 Ric − 2(n−1) Scal g g . Ein(W, W ) = −g(W, W ) Sec(Xa,Xb) . ? a,b=1 (105) a