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4.1 The geometry of spacelike 2-surfaces

The first systematic study of the geometry of spacelike 2-surfaces from the point of view of quasi-local quantities is probably due to Tod [375Jump To The Next Citation Point380Jump To The Next Citation Point]. Essentially, his approach is based on the GHP (Geroch-Held-Penrose) formalism [152Jump To The Next Citation Point]. Although this is a very effective and flexible formalism [152Jump To The Next Citation Point312Jump To The Next Citation Point313Jump To The Next Citation Point206Jump To The Next Citation Point347], its form is not spacetime covariant. Since in many cases the covariance of a formalism itself already gives some hint how to treat and solve the problem at hand, here we concentrate mainly on a spacetime-covariant description of the geometry of the spacelike 2-surfaces, developed gradually in [355357358Jump To The Next Citation Point359147Jump 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 2-surfaces.

4.1.1 The Lorentzian vector bundle

The restriction a V (S) to the closed, orientable spacelike 2-surface S of the tangent bundle TM of the spacetime has a unique decomposition to the gab-orthogonal sum of the tangent bundle T S 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 ta and va are timelike and spacelike unit normals, respectively, being orthogonal to each other, then the projection to T S and N S is a a a a TT b := db- t tb + v vb and a a a Ob := db - TT b, respectively. The induced 2-metric and the corresponding area 2-form on S will be denoted by qab = gab- tatb + vavb and eab = tcvdecdab, respectively, while the area 2-form on the normal bundle will be _L eab = tavb- tbva. The bundle Va(S) together with the fibre metric gab and the projection TTa b will be called the Lorentzian vector bundle over S. For the discussion of the global topological properties of the closed orientable 2-manifolds, see for example [5].

4.1.2 Connections

The spacetime covariant derivative operator \~/ e defines two covariant derivatives on Va(S). The first, denoted by de, is analogous to the induced (intrinsic) covariant derivative on (one-codimensional) hypersurfaces: a a f b c a f b c deX := TT bTT e \~/ f (TT cX ) + O bTT e \~/ f (OcX ) for any section a X of a V (S). Obviously, de annihilates both the fibre metric gab and the projection TTab. However, since for 2-surfaces in four dimensions the normal is not uniquely determined, we have the ‘boost gauge freedom’ ta '--> ta coshu + va sinh u, va '--> tasinh u + va coshu. The induced connection will have a nontrivial part on the normal bundle, too. The corresponding (normal part of the) connection 1-form on S can be characterized, for example, by Ae := TTfe( \~/ f ta)va. Therefore, the connection de can be considered as a connection on Va(S) coming from a connection on the O(2) ox O(1, 1)-principal bundle of the gab-orthonormal frames adapted to S.

The other connection, De, is analogous to the Sen connection [331], and is defined simply by DeXa := TTf \~/ f Xa e. This annihilates only the fibre metric, but not the projection. The difference of the connections D e and d e turns out to be just the extrinsic curvature tensor: a a a b b a DeX = deX + Q ebX - X Qbe. Here a a c a a Q eb := - TT cDeTT b = t etb- n evb, and c d tab := TT aTT b \~/ ctd and nab := TTcaTTdb \~/ 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 c hab = Qabcl and ' c hab = Qabcn, and the corresponding shear tensors sab and ' sab are defined by their trace-free part. Obviously, tab and nab (and hence the expansion and shear tensors hab, h'ab, sab, and s'ab) are boost-gauge dependent quantities (and it is straightforward to derive their transformation from the definitions), but their combination Qa eb is boost-gauge invariant. In particular, it defines a natural normal vector field to S by a ' Qb := Q ab = ttb- nvb = hlb + hnb, where t, n, h and ' h are the relevant traces. Qa is called the main extrinsic curvature vector of S. If ' Q~b := ntb - tvb = - hla + hna, then the norm of Qa and Q~a is QaQbgab = - Q~a ~Qbgab = t2 - n2 = 2hh', and they are orthogonal to each other: Qa ~Qbgab = 0. It is easy to show that Da Q~a = 0, i.e. ~Qa is the uniquely pointwise determined direction orthogonal to the 2-surface in which the expansion of the surface is vanishing. If Q a is not null, then {Q ,Q~ } a a defines an orthonormal frame in the normal bundle (see for example [8Jump To The Next Citation Point]). If Qa is non-zero 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 2-surface admits a natural gauge choice in the normal bundle unless Qa is vanishing. Geometrically, D e is a connection coming from a connection on the O(1,3)-principal fibre bundle of the gab-orthonormal frames. The curvature of the connections de and De, respectively, are

f abcd = - _L eab(dcAd - ddAc) + 1SR (TTacqbd- TTadqbc) , (20) a a a a 2 a a F bcd = f bcd- dc (Q db- Qbd ) + dd(Q cb - Qbc ) + +QaceQbde + QecaQedb - QadeQbce - QedaQecb, (21)
where S R is the curvature scalar of the familiar intrinsic Levi-Civita connection of (S,qab). The curvature of De is just the pull-back to S of the spacetime curvature 2-form: f F abcd = RabefTTecTT d. Therefore, the well known Gauss, Codazzi-Mainardi, and Ricci equations for the embedding of S in M are just the various projections of Equation (21View Equation).

4.1.3 Convexity conditions

To prove certain statements on quasi-local quantities various forms of the convexity of S must be assumed. The convexity of S in a 3-geometry is defined by the positive definiteness of its extrinsic curvature tensor. If the embedding space is flat, then by the Gauss equation this is equivalent to the positivity of the scalar curvature of the intrinsic metric of S. 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. h := qabhab > 0, and weakly past convex if h':= qabh' < 0 ab [380Jump To The Next Citation Point]. S is called mean convex [182Jump To The Next Citation Point] if hh'< 0 on S, or, equivalently, if ~Q a is timelike. To formulate stronger convexity conditions we must consider the determinant of the null expansions 1 D := det||hab|| = 2(habhcd- hachbd)qabqcd and 1 D':= det||h'ab||= 2(h'abh'cd - h'ach'bd)qabqcd. Note that although the expansion tensors, and in particular the functions h, h', D, and D' are gauge dependent, their sign is gauge invariant. Then S will be called future convex if h > 0 and D > 0, and past convex if ' h < 0 and ' D > 0 [380Jump To The Next Citation Point358Jump To The Next Citation Point]. These are equivalent to the requirement that the two eigenvalues of a h b be positive and those of 'a h b 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 de and De determine connections on the pull-back SA(S) to S of the bundle of unprimed spinors. The natural decomposition Va(S) = T S o+ N S defines a chirality on the spinor bundleA S (S) in the form of the spinor A : AA' ' g B = 2t vBA, which is analogous to the g5 matrix in the theory of Dirac spinors. Then the extrinsic curvature tensor above is a simple expression of QAeB := 12(DegAC )gC B and gAB (and their complex conjugate), and the two covariant derivatives on SA(S) are related to each other by DecA = decA + QAeBcB. The curvature F ABcd of De can be expressed by the curvature fA Bcd of de, the spinor A Q eB, and its de-derivative. We can form the scalar invariants of the curvatures according to

1( ab _L ab) cd A B cd S ( cd ) f := fabcd 2 e - i e e = ig Bf Acde = R - 2idc e Ad , (22) 1 ( ) ( ) F := Fabcd-- eab- i _L eab ecd = igABF BAcdecd = f + hh'- 2s'easeb qab + ieab . (23) 2
f is four times the so-called complex Gauss curvature [312Jump To The Next Citation Point] of S, by means of which the whole curvature f ABcd can be characterized: f ABcd = - ifgABecd 4. If the spacetime is space and time orientable, at least on an open neighbourhood of S, then the normals t a and v a 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 1-form Ae, i.e. the vertical part of the connection de, was given by Liu and Yau [253Jump To The Next Citation Point]: There are real functions a and g, unique up to additive constants, such that A = e fd a + d g e e f e. a is globally defined on S, but in general g is defined only on the local trivialization domains of N S that are homeomorphic to 2 R. It is globally defined if H1(S) = 0. In this decomposition a is the boost-gauge invariant part of Ae, while g represents its gauge content. Since deAe = dedeg, the ‘Coulomb-gauge condition’ deAe = 0 uniquely fixes Ae (see also Section 10.4.1).

By the Gauss-Bonnet theorem gf gf S S f dS = S R dS = 8p(1 - g), where g is the genus of S. Thus geometrically the connection de is rather poor, and can be considered as a part of the ‘universal structure of S’. On the other hand, the connection De is much richer, and, in particular, the invariant F carries information on the mass aspect of the gravitational ‘field’. The 2-surface data for charge-type quasi-local quantities (i.e. for 2-surface observables) are the universal structure (i.e. the intrinsic metric qab, the projection a TT b and the connection de) and the extrinsic curvature tensor a Q eb.

4.1.5 Curvature identities

The complete decomposition of DAA'cB into its irreducible parts gives DA'AcA, the Dirac-Witten operator, and T ' Bc := D ' c + 1g gCDD ' c EEA B E(E A) 2 EA E C D, the 2-surface twistor operator. A Sen-Witten-type identity for these irreducible parts can be derived. Taking its integral one has

gf gf A'B'[( A)( B) ( E )( CDF )] i A B g DA'Ac DB'Bm + TA'CD cE TB' mF dS = - 2 c m FABcd, (24) S S
where cA and mA are two arbitrary spinor fields on S, and the right hand side is just the charge integral of the curvature F ABcd on S.

4.1.6 The GHP formalism

A GHP spin frame on the 2-surface S is a normalized spinor basis A : A A eA = {o ,i }, A = 0,1, such that the complex null vectors a A A' m := o i and a A A' m := i o are tangent to S (or, equivalently, the future pointing null vectors ' la := oAoA and ' na := iAiA 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 non-triviality of the tangent bundle TS of the 2-surface. For example, on the 2-sphere every continuous tangent vector field must have a zero, and hence, in particular, the vectors a m and a m cannot form a globally defined basis on S. Consequently, the GHP spin frame cannot be globally defined either. The only closed orientable 2-surface with globally trivial tangent bundle is the torus.

Fixing a GHP spin frame {eA } A 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) [152Jump To The Next Citation Point312Jump To The Next Citation Point]: A scalar f is said to be of type (p,q) if under the rescaling oA '--> coA, iA '--> c -1iA of the GHP spin frame with some nowhere vanishing complex function c : U --> C the scalar transforms as f '--> cpcqf. For example r := habmamb = - 1 h 2, r':= h'mamb = - 1h' ab 2, s := h mamb = s mamb ab ab, and s':= h'mamb = s' mamb ab ab are of type (1,1), (- 1,-1), (3,- 1), and (- 3,1), respectively. The components of the Weyl and Ricci spinors, A B C D y0 := yABCDo o o o, A B C D y1 := yABCDo o o i, A B C D y2 := yABCDo o i i, …, ' f00 := fAB'oAoB, ' f01 := fAB'oAiB, …, etc., also have definite (p,q)-type. In particular, /\ := R/24 has type (0, 0). A global section of E(p, q) is a collection of local cross sections {(U, f),(U ',f'),...} such that {U, U',...} forms a covering of S and on the non-empty overlappings, e.g. on ' U /~\ U the local sections are related to each other by p q ' f = y y f, where y : U /~\ U '--> C is the transition function between the GHP spin frames: oA = yo'A and iA = y -1i'A.

The connection de defines a connection ke on the line bundles E(p,q) [152Jump To The Next Citation Point312Jump To The Next Citation Point]. The usual edth operators, k and ' k, are just the directional derivatives : a k = m ka and ': a k = m ka on the domain U < S of the GHP spin frame A {eA}. These locally defined operators yield globally defined differential operators, denoted also by k and ' k, on the global sections of E(p, q). It might be worth emphasizing that the GHP spin coefficients b and b', which do not have definite (p,q)-type, play the role of the two components of the connection 1-form, and they are built both from the connection 1-form for the intrinsic Riemannian geometry of (S, qab) and the connection 1-form Ae in the normal bundle. k and k' 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 [147] for general, and in [13243Jump To The Next Citation Point356Jump To The Next Citation Point] for spherical topology.

4.1.7 Irreducible parts of the derivative operators

Using the projection operators ±A 1 A A p B := 2(dB ± g B), the irreducible parts A DA'Ac and B TE'EA cB can be decomposed further into their right handed and left handed parts. In the GHP formalism these chiral irreducible parts are

- ' + ' -D c := kc1 + rc0, D c := k c0 + rc1, - + ' ' (25) T c := kc0 + sc1, - T c := k c1 + s c0,
where c := (c0,c1) and the spinor components are defined by cA =: c1oA - c0iA. The various first order linear differential operators acting on spinor fields, e.g. the 2-surface twistor operator, the holomorphy/anti-holomorphy operators or the operators whose kernel defines the asymptotic spinors of Bramson [83Jump To The Next Citation Point], are appropriate direct sums of these elementary operators. Their global properties under various circumstances are studied in [43Jump To The Next Citation Point356Jump To The Next Citation Point363Jump To The Next Citation Point].

4.1.8 SO(1, 1)-connection 1-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, too. In particular [181Jump To The Next Citation Point], let the 2-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 neighbourhood of S) be foliated by two one-parameter families of smooth hypersurfaces {n+ = const.} and {n- = const.}, where n± : M --> R, such that + N = {n+ = 0} and - N = {n- = 0}. One can form the two normals, n ±a := \~/ an ±, which are null on N + and N -, respectively. Then we can define b± e := (Den ±a)na ±, for which b + b = D n2 +e -e e, where n2 := g na nb ab + -. (If n2 is chosen to be 1 on S, then b -e = -b+e is precisely the SO(1, 1) connection 1-form Ae above.) Then the so-called anholonomicity is defined by we := 21n2[n- ,n+]fqfe = 21n2(b+e - b-e). Since we is invariant with respect to the rescalings n+ '--> exp(A)n+ and n- '--> exp(B)n - of the functions defining the foliations by those functions A, B : M --> R which preserve \~/ [an ±b] = 0, it was claimed in [181Jump To The Next Citation Point] that we depends only on S. However, this implies only that we is invariant with respect to a restricted class of the change of the foliations, and that we is invariantly defined only by this class of the foliations rather than the 2-surface. In fact, we does depend on the foliation: Starting with a different foliation defined by the functions n+ := exp(a)n+ and n- := exp(b)n - for some a, b : M --> R, the corresponding anholonomicity w e would also be invariant with respect to the restricted changes of the foliations above, but the two anholonomicities, we and we, would be different: 1 we - we = 2De(a - b). Therefore, the anholonomicity is still a gauge dependent quantity.

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