4.1 The geometry of spacelike two-surfaces

The first systematic study of the geometry of spacelike two-surfaces from the point of view of quasi-local quantities is probably due to Tod [514Jump To The Next Citation Point, 519Jump To The Next Citation Point]. Essentially, his approach is based on the Geroch–Held–Penrose (GHP) formalism [209Jump To The Next Citation Point]. Although this is a very effective and flexible formalism [209Jump To The Next Citation Point, 425Jump To The Next Citation Point, 426Jump To The Next Citation Point, 277Jump To The Next Citation Point, 479], its form is not spacetime covariant. Since in many cases the covariance of a formalism itself already gives some hint as to how to treat and solve the problem at hand, we concentrate here mainly on a spacetime-covariant description of the geometry of the spacelike two-surfaces, developed gradually in [489, 491, 492Jump To The Next Citation Point, 493, 198Jump To The Next Citation Point, 500Jump To The Next Citation Point]. The emphasis will be on the geometric structures rather than the technicalities. In the last paragraph, we comment on certain objects appearing in connection with families of spacelike two-surfaces. Our standard differential geometric reference is [318, 319].

4.1.1 The Lorentzian vector bundle

The restriction Va (𝒮 ) to the closed, orientable spacelike two-surface 𝒮 of the tangent bundle T M of the spacetime has a unique decomposition to the gab-orthogonal sum of the tangent bundle T𝒮 of 𝒮 and the bundle of the normals, denoted by N 𝒮. Then, all the geometric structures of the spacetime (metric, connection, curvature) can be decomposed in this way. If a t and va are timelike and spacelike unit normals, respectively, being orthogonal to each other, then the projections to T𝒮 and N 𝒮 are Πab := δab − tatb + vavb and Oab := δab − Πab, respectively. The induced two-metric and the corresponding area 2-form on 𝒮 will be denoted by qab = gab − tatb + vavb and c d 𝜀ab = tv 𝜀cdab, respectively, while the area 2-form on the normal bundle will be ⊥ 𝜀ab = tavb − tbva. The bundle a V (𝒮 ) together with the fiber metric gab and the projection Πab will be called the Lorentzian vector bundle over 𝒮. For the discussion of the global topological properties of the closed orientable two-manifolds, see, e.g., [10, 500Jump To The Next Citation Point].

4.1.2 Connections

The spacetime covariant derivative operator ∇e defines two connections on Va (𝒮). The first covariant derivative, denoted by δe, is analogous to the induced (intrinsic) covariant derivative on (one-codimensional) hypersurfaces: a : a f b c a f b c δeX = ΠbΠ e∇f (ΠcX ) + ObΠ e∇f (O cX ) for any section a X of a V (𝒮 ). Obviously, δe annihilates both the fiber metric gab and the projection a Π b. However, since for two-surfaces in four dimensions the normal is not uniquely determined, we have the ‘boost gauge freedom’ ta ↦→ tacosh u + vasinh u, va ↦→ tasinh u + vacosh u. The induced connection will have a nontrivial part on the normal bundle, too. The corresponding (normal part of the) connection one-form on 𝒮 can be characterized, for example, by f a Ae := Π e(∇f ta)v. Therefore, the connection δe can be considered as a connection on Va (𝒮 ) coming from a connection on the O (2) ⊗ O (1,1)-principal bundle of the g ab-orthonormal frames adapted to 𝒮.

The other connection, Δe, is analogous to the Sen connection [447], and is defined simply by ΔeXa := Πfe∇f Xa. This annihilates only the fiber metric, but not the projection. The difference of the connections Δe and δe turns out to be just the extrinsic curvature tensor: a a a b b a ΔeX = δeX + Q ebX − X Qbe. Here a a c a a Q eb := − Π cΔeΠ b = τ etb − ν evb, and c d τab := Π aΠ b∇ctd and c d νab := ΠaΠ b∇cvd are the standard (symmetric) extrinsic curvatures corresponding to the individual normals ta and va, respectively. The familiar expansion tensors of the future-pointing outgoing and ingoing null normals, la := ta + va and na := 1(ta − va) 2, respectively, are 𝜃ab = Qabclc and 𝜃′ = Q nc ab abc, and the corresponding shear tensors σ ab and σ′ ab are defined by their trace-free part. Obviously, τab and νab (and hence the expansion and shear tensors 𝜃ab, ′ 𝜃ab, σab, and ′ σ ab) are boost-gauge–dependent quantities (and it is straightforward to derive their transformation from the definitions), but their combination Qaeb is boost-gauge invariant. In particular, it defines a natural normal vector field to 𝒮 as Qb := Qaab = τtb − νvb = 𝜃′lb + 𝜃nb, where τ, ν, 𝜃 and 𝜃′ are the relevant traces. Q a is called the mean extrinsic curvature vector of 𝒮. If &tidle; ⊥ a b ′ Qb := 𝜀 bQ = νtb − τvb = − 𝜃la + 𝜃na, called the dual mean curvature vector, then the norm of Qa and &tidle;Qa is ab ab 2 2 ′ QaQbg = − &tidle;QaQ&tidle;bg = τ − ν = 2𝜃𝜃, and they are orthogonal to each other: Qa &tidle;Qbgab = 0. It is easy to show that Δa &tidle;Qa = 0, i.e., Q&tidle;a is the uniquely pointwise-determined direction orthogonal to the two-surface in which the expansion of the surface is vanishing. If Qa is not null, then {Q ,Q&tidle; } a a defines an orthonormal frame in the normal bundle (see, e.g., [14Jump To The Next Citation Point]). If Q a is nonzero, but (e.g., future-pointing) null, then there is a uniquely determined null normal Sa to 𝒮, such that QaSa = 1, and hence, {Qa, Sa} is a uniquely determined null frame. Therefore, the two-surface admits a natural gauge choice in the normal bundle, unless Qa is vanishing. Geometrically, Δe is a connection coming from a connection on the O(1,3 )-principal fiber bundle of the gab-orthonormal frames. The curvature of the connections δe and Δe, respectively, are

f abcd = − ⊥𝜀ab(δcAd − δdAc ) + 12𝒮R (Πacqbd − Πadqbc), (4.1 ) a a a a a a F bcd = f bcd − δc (Q db − Qbd ) + δd(Q cb − Qbc ) + +QaceQbde + QecaQedb − QadeQbce − QedaQecb, (4.2 )
where 𝒮 R is the curvature scalar of the familiar intrinsic Levi-Civita connection of (𝒮,qab). The curvature of Δe is just the pullback to 𝒮 of the spacetime curvature 2-form: Fabcd = RabefΠeΠf c d. Therefore, the well-known Gauss, Codazzi–Mainardi, and Ricci equations for the embedding of 𝒮 in M are just the various projections of Eq. (4.2View Equation).

4.1.3 Embeddings and convexity conditions

To prove certain statements about quasi-local quantities, various forms of the convexity of 𝒮 must be assumed. The convexity of 𝒮 in a three-geometry is defined by the positive definiteness of its extrinsic curvature tensor. If, in addition, the three-geometry is flat, then by the Gauss equation this is equivalent to the positivity of the scalar curvature of the intrinsic metric of 𝒮. It is this convexity condition that appears in the solution of the Weyl problem of differential geometry [397]: if 2 (S ,qab) is a 4 C Riemannian two-manifold with positive scalar curvature, then this can be isometrically embedded (i.e., realized as a closed convex two-surface) in the Euclidean three-space ℝ3, and this embedding is unique up to rigid motions [477]. However, there are counterexamples even to local isometric embedability, when the convexity condition, i.e., the positivity of the scalar curvature, is violated [373]. We continue the discussion of this embedding problem in Section 10.1.6.

In the context of general relativity the isometric embedding of a closed orientable two-surface into the Minkowski spacetime ℝ1,3 is perhaps more interesting. However, even a naïve function counting shows that if such an embedding exists then it is not unique. An existence theorem for such an embedding, 1,3 i : 𝒮 → ℝ, (with 2 S topology) was given by Wang and Yau [543Jump To The Next Citation Point], and they controlled these isometric embeddings in terms of a single function τ on the two-surface. This function is just xaTa, the ‘time function’ of the surface in the Cartesian coordinates of the Minkowski space in the direction of a constant unit timelike vector field T a. Interestingly enough, (𝒮, q ) ab is not needed to have positive scalar curvature, only the sum of the scalar curvature and a positive definite expression of the derivative δeτ is required to be positive. This condition is just the requirement that the surface must have a convex ‘shadow’ in the direction Ta, i.e., the scalar curvature of the projection of the two-surface i(𝒮) ⊂ ℝ1,3 to the spacelike hyperplane orthogonal to T a is positive. The Laplacian δ δeτ e of the ‘time function’ gives the mean curvature vector of i(𝒮 ) in ℝ1,3 in the direction a T.

If 𝒮 is in a Lorentzian spacetime, then the weakest convexity conditions are conditions only on the mean null curvatures: 𝒮 will be called weakly future convex if the outgoing null normals la are expanding on 𝒮, i.e., 𝜃 := qab𝜃ab > 0, and weakly past convex if 𝜃′ := qab𝜃′ < 0 ab [519Jump To The Next Citation Point]. 𝒮 is called mean convex [247Jump To The Next Citation Point] if ′ 𝜃𝜃 < 0 on 𝒮, or, equivalently, if &tidle; Qa is timelike. To formulate stronger convexity conditions we must consider the determinant of the null expansions D := det∥𝜃ab∥ = 12(𝜃ab𝜃cd − 𝜃ac𝜃bd)qabqcd and D ′ := det∥𝜃 ′ab∥ = 12(𝜃′ab𝜃′cd − 𝜃′ac𝜃′bd)qabqcd. Note that, although the expansion tensors, and in particular the functions 𝜃, 𝜃′, D, and D ′ are boost-gauge–dependent, their sign is gauge invariant. Then 𝒮 will be called future convex if 𝜃 > 0 and D > 0, and past convex if ′ 𝜃 < 0 and ′ D > 0 [519Jump To The Next Citation Point, 492Jump To The Next Citation Point]. These are equivalent to the requirement that the two eigenvalues of 𝜃ab be positive and those of 𝜃′ab be negative everywhere on 𝒮, respectively. A different kind of convexity condition, based on global concepts, will be used in Section 6.1.3.

4.1.4 The spinor bundle

The connections δe and Δe determine connections on the pullback SA (𝒮) to 𝒮 of the bundle of unprimed spinors. The natural decomposition Va (𝒮) = T 𝒮 ⊕ N 𝒮 defines a chirality on the spinor bundle A S (𝒮 ) in the form of the spinor A AA ′ γ B := 2t vBA ′, which is analogous to the γ5 matrix in the theory of Dirac spinors. Then, the extrinsic curvature tensor above is a simple expression of QAeB := 12(ΔeγAC )γC B and γAB (and their complex conjugate), and the two covariant derivatives on SA(𝒮 ) are related to each other by ΔeλA = δeλA + QAeB λB. The curvature F A Bcd of Δ e can be expressed by the curvature fA Bcd of δ e, the spinor A Q eB, and its δe-derivative. We can form the scalar invariants of the curvatures according to

1( ) ( ) f := fabcd-- 𝜀ab − i⊥𝜀ab 𝜀cd = iγABf BAcd𝜀cd = 𝒮R − 2iδc 𝜀cdAd , (4.3 ) 2 F := F 1-(𝜀ab − i⊥𝜀ab)𝜀cd = iγA F B 𝜀cd = f + 𝜃𝜃′ − 2σ ′σe (qab + i𝜀ab) . (4.4 ) abcd2 B Acd ea b
f is four times the complex Gauss curvature [425Jump To The Next Citation Point] of 𝒮, by means of which the whole curvature A f Bcd can be characterized: A i A f Bcd = − 4fγ B𝜀cd. If the spacetime is space and time orientable, at least on an open neighborhood of 𝒮, then the normals ta and va can be chosen to be globally well defined, and hence, N 𝒮 is globally trivializable and the imaginary part of f is a total divergence of a globally well-defined vector field.

An interesting decomposition of the SO (1, 1) connection one-form Ae, i.e., the vertical part of the connection δe, was given by Liu and Yau [338Jump To The Next Citation Point]: There are real functions α and γ, unique up to additive constants, such that Ae = 𝜀efδfα + δeγ. α is globally defined on 𝒮, but in general γ is defined only on the local trivialization domains of N 𝒮 that are homeomorphic to ℝ2. It is globally defined if H1 (𝒮) = 0. In this decomposition α is the boost-gauge–invariant part of Ae, while γ represents its gauge content. Since e e δeA = δeδ γ, the ‘Coulomb-gauge condition’ e δeA = 0 uniquely fixes Ae (see also Section 10.4.1).

By the Gauss–Bonnet theorem one has ∮ ∮ 𝒮 𝒮 fd𝒮 = 𝒮 Rd 𝒮 = 8π (1 − g), where g is the genus of 𝒮. Thus, geometrically the connection δe is rather poor, and can be considered as a part of the ‘universal structure of 𝒮’. On the other hand, the connection Δe is much richer, and, in particular, the invariant F carries information on the mass aspect of the gravitational ‘field’. The two-surface data for charge-type quasi-local quantities (i.e., for two-surface observables) are the universal structure (i.e., the intrinsic metric qab, the projection a Π b and the connection δe) and the extrinsic curvature tensor a Q eb.

4.1.5 Curvature identities

The complete decomposition of Δ ′λ AA B into its irreducible parts gives Δ ′ λA A A, the Dirac–Witten operator, and B 1 CD 𝒯E ′EA λB := ΔE ′(EλA ) + 2γEA γ ΔE ′CλD, the two-surface twistor operator. The former is essentially the anti-symmetric part ΔA ′[AλB ], the latter is the symmetric and (with respect to the complex metric γAB) trace-free part of the derivative. (The trace γAB ΔA ′AλB can be shown to be the Dirac–Witten operator, too.) A Sen-Witten–type identity for these irreducible parts can be derived. Taking its integral one has

∮ ′ ′ [( ) ( ) ( ) ( ) ] ∮ γ¯A B ΔA ′A λA ΔB ′B μB + 𝒯A ′CDE λE 𝒯B′CDF μF d𝒮 = − i2 λAμBFABcd, (4.5 ) 𝒮 𝒮
where λA and μA are two arbitrary spinor fields on 𝒮, and the right-hand side is just the charge integral of the curvature A F Bcd on 𝒮.

4.1.6 The GHP formalism

A GHP spin frame on the two-surface 𝒮 is a normalized spinor basis 𝜀AA := {oA, ιA}, A = 0,1, such that the complex null vectors ma := oA ¯ιA ′ and m¯a := ιA¯oA′ are tangent to 𝒮 (or, equivalently, the future-pointing null vectors a A A ′ l := o ¯o and a A A ′ n := ι ¯ι are orthogonal to 𝒮). Note, however, that in general a GHP spin frame can be specified only locally, but not globally on the whole 𝒮. This fact is connected with the nontriviality of the tangent bundle T 𝒮 of the two-surface. For example, on the two-sphere every continuous tangent vector field must have a zero, and hence, in particular, the vectors ma and m¯a cannot form a globally-defined basis on 𝒮. Consequently, the GHP spin frame cannot be globally defined either. The only closed orientable two-surface with a globally-trivial tangent bundle is the torus.

Fixing a GHP spin frame A {𝜀A } on some open U ⊂ 𝒮, the components of the spinor and tensor fields on U will be local representatives of cross sections of appropriate complex line bundles E (p,q) of scalars of type (p,q) [209Jump To The Next Citation Point, 425Jump To The Next Citation Point]: A scalar ϕ is said to be of type (p,q) if, under the rescaling oA ↦→ λoA, ιA ↦→ λ −1ιA of the GHP spin frame with some nowhere-vanishing complex function : λ U → ℂ, the scalar transforms as p¯q ϕ ↦→ λ λ ϕ. For example, : a b 1 ρ = 𝜃abm ¯m = − 2𝜃, ′ ′ a b 1 ′ ρ := 𝜃abm m¯ = − 2𝜃, a b a b σ := 𝜃abm m = σabm m, and ′ ′ a b ′ a b σ := 𝜃abm ¯ m¯ = σab ¯m ¯m are of type (1,1), (− 1,− 1 ), (3,− 1), and (− 3,1), respectively. The components of the Weyl and Ricci spinors, ψ0 := ψABCDoAoBoC oD, ψ1 := ψABCDoAoBoC ιD, ψ2 := ψABCDoAoB ιCιD, …, ϕ := ϕ ′oA ¯oB′ 00 AB, ϕ := ϕ ′oA ¯ιB′ 01 AB, …, etc., also have definite (p,q )-type. In particular, Λ := R ∕24 has type (0, 0). A global section of E (p,q) is a collection of local cross sections ′ ′ {(U, ϕ),(U ,ϕ ),...} such that ′ {U, U ,...} forms a covering of 𝒮 and on the nonempty overlappings, e.g., on U ∩ U ′, the local sections are related to each other by ϕ = ψp ¯ψqϕ ′, where ψ : U ∩ U ′ → ℂ is the transition function between the GHP spin frames: oA = ψo ′A and ιA = ψ −1ι′A.

The connection δe defines a connection ???e on the line bundles E(p,q) [209Jump To The Next Citation Point, 425Jump To The Next Citation Point]. The usual edth operators, ??? and ??? ′, are just the directional derivatives ??? := ma???a and ???′ := ¯ma???a on the domain U ⊂ 𝒮 of the GHP spin frame {𝜀AA }. These locally-defined operators yield globally-defined differential operators, denoted also by ??? and ???′, on the global sections of E (p,q). It might be worth emphasizing that the GHP spin coefficients β and ′ β, which do not have definite (p,q)-type, play the role of the two components of the connection one-form, and are built both from the connection one-form for the intrinsic Riemannian geometry of (𝒮,qab) and the connection one-form Ae in the normal bundle. ??? and ???′ are elliptic differential operators, thus, their global properties, e.g., the dimension of their kernel, are connected with the global topology of the line bundle they act on, and, in particular, with the global topology of 𝒮. These properties are discussed in [198] in general, and in [177, 58Jump To The Next Citation Point, 490Jump To The Next Citation Point] for spherical topology.

4.1.7 Irreducible parts of the derivative operators

Using the projection operators π±AB := 12(δAB ± γAB ), the irreducible parts ΔA ′AλA and 𝒯E′EAB λB can be decomposed further into their right-handed and left-handed parts. In the GHP formalism these chiral irreducible parts are

− Δ − λ := ???λ1 + ρ ′λ0, Δ+ λ := ???′λ0 + ρλ1, (4.6 ) 𝒯 − λ := ???λ0 + σ λ1, − 𝒯 +λ := ???′λ1 + σ′λ0,
where λ := (λ0,λ1) and the spinor components are defined by λA =: λ1oA − λ0 ιA. The various first-order linear differential operators acting on spinor fields, e.g., the two-surface twistor operator, the holomorphy/antiholomorphy operators or the operators whose kernel defines the asymptotic spinors of Bramson [106Jump To The Next Citation Point], are appropriate direct sums of these elementary operators. Their global properties under various circumstances are studied in [58Jump To The Next Citation Point, 490Jump To The Next Citation Point, 496Jump To The Next Citation Point].

4.1.8 SO (1,1 )-connection one-form versus anholonomicity

Obviously, all the structures we have considered can be introduced on the individual surfaces of one or two-parameter families of surfaces, as well. In particular [246Jump To The Next Citation Point], let the two-surface 𝒮 be considered as the intersection 𝒩 + ∩ 𝒩 − of the null hypersurfaces formed, respectively, by the outgoing and the ingoing light rays orthogonal to 𝒮, and let the spacetime (or at least a neighborhood of 𝒮) be foliated by two one-parameter families of smooth hypersurfaces {ν+ = const.} and {ν− = const.}, where ν± : M → ℝ, such that 𝒩 + = {ν+ = 0} and 𝒩 − = {ν− = 0}. One can form the two normals, n ±a := ∇a ν±, which are null on 𝒩 + and 𝒩 −, respectively. Then we can define β±e := (Δen ±a)na∓, for which β + β = Δ n2 +e −e e, where n2 := g na nb ab + −. (If n2 is chosen to be 1 on 𝒮, then β −e = − β+e is precisely the SO (1,1)-connection one-form Ae above.) Then the anholonomicity is defined by 1 f 1 ωe := 2n2[n− ,n+ ] qfe = 2n2(β+e − β −e). Since ωe is invariant with respect to the rescalings ν+ ↦→ exp (A)ν+ and ν− ↦→ exp(B )ν− of the functions, defining the foliations by those functions A, B : M → ℝ, which preserve ∇ [an ±b] = 0, it was claimed in [246Jump To The Next Citation Point] that ωe depends only on 𝒮. However, this implies only that ω e is invariant with respect to a restricted class of the change of the foliations, and that ωe is invariantly defined only by this class of the foliations rather than the two-surface. In fact, ωe does depend on the foliation: Starting with a different foliation defined by the functions ¯ν+ := exp(α )ν+ and ¯ν− := exp (β)ν− for some α, β : M → ℝ, the corresponding anholonomicity ¯ωe would also be invariant with respect to the restricted changes of the foliations above, but the two anholonomicities, ω e and ¯ω e, would be different: 1 ω¯e − ωe = 2Δe (α − β). Therefore, the anholonomicity is a gauge-dependent quantity.


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