The restriction to the closed, orientable spacelike two-surface of the tangent bundle of the spacetime has a unique decomposition to the -orthogonal sum of the tangent bundle of and the bundle of the normals, denoted by . Then, all the geometric structures of the spacetime (metric, connection, curvature) can be decomposed in this way. If and are timelike and spacelike unit normals, respectively, being orthogonal to each other, then the projections to and are and , respectively. The induced two-metric and the corresponding area 2-form on will be denoted by and , respectively, while the area 2-form on the normal bundle will be . The bundle together with the fiber metric and the projection will be called the Lorentzian vector bundle over . For the discussion of the global topological properties of the closed orientable two-manifolds, see, e.g., [10, 500].

The spacetime covariant derivative operator defines two connections on . The first covariant derivative, denoted by , is analogous to the induced (intrinsic) covariant derivative on (one-codimensional) hypersurfaces: for any section of . Obviously, annihilates both the fiber metric and the projection . However, since for two-surfaces in four dimensions the normal is not uniquely determined, we have the ‘boost gauge freedom’ , . The induced connection will have a nontrivial part on the normal bundle, too. The corresponding (normal part of the) connection one-form on can be characterized, for example, by . Therefore, the connection can be considered as a connection on coming from a connection on the -principal bundle of the -orthonormal frames adapted to .

The other connection, , is analogous to the Sen connection [447], and is defined simply by . This annihilates only the fiber metric, but not the projection. The difference of the connections and turns out to be just the extrinsic curvature tensor: . Here , and and are the standard (symmetric) extrinsic curvatures corresponding to the individual normals and , respectively. The familiar expansion tensors of the future-pointing outgoing and ingoing null normals, and , respectively, are and , and the corresponding shear tensors and are defined by their trace-free part. Obviously, and (and hence the expansion and shear tensors , , , and ) are boost-gauge–dependent quantities (and it is straightforward to derive their transformation from the definitions), but their combination is boost-gauge invariant. In particular, it defines a natural normal vector field to as , where , , and are the relevant traces. is called the mean extrinsic curvature vector of . If , called the dual mean curvature vector, then the norm of and is , and they are orthogonal to each other: . It is easy to show that , i.e., is the uniquely pointwise-determined direction orthogonal to the two-surface in which the expansion of the surface is vanishing. If is not null, then defines an orthonormal frame in the normal bundle (see, e.g., [14]). If is nonzero, but (e.g., future-pointing) null, then there is a uniquely determined null normal to , such that , and hence, is a uniquely determined null frame. Therefore, the two-surface admits a natural gauge choice in the normal bundle, unless is vanishing. Geometrically, is a connection coming from a connection on the -principal fiber bundle of the -orthonormal frames. The curvature of the connections and , respectively, are

where is the curvature scalar of the familiar intrinsic Levi-Civita connection of . The curvature of is just the pullback to of the spacetime curvature 2-form: . Therefore, the well-known Gauss, Codazzi–Mainardi, and Ricci equations for the embedding of in are just the various projections of Eq. (4.2).

To prove certain statements about quasi-local quantities, various forms of the convexity of must be assumed. The convexity of 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 . It is this convexity condition that appears in the solution of the Weyl problem of differential geometry [397]: if is a 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 , 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 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, , (with topology) was given by Wang and Yau [543], and they controlled these isometric embeddings in terms of a single function on the two-surface. This function is just , the ‘time function’ of the surface in the Cartesian coordinates of the Minkowski space in the direction of a constant unit timelike vector field . Interestingly enough, is not needed to have positive scalar curvature, only the sum of the scalar curvature and a positive definite expression of the derivative is required to be positive. This condition is just the requirement that the surface must have a convex ‘shadow’ in the direction , i.e., the scalar curvature of the projection of the two-surface to the spacelike hyperplane orthogonal to is positive. The Laplacian of the ‘time function’ gives the mean curvature vector of in in the direction .

If is in a Lorentzian spacetime, then the weakest convexity conditions are conditions only on the mean null curvatures: will be called weakly future convex if the outgoing null normals are expanding on , i.e., , and weakly past convex if [519]. is called mean convex [247] if on , or, equivalently, if is timelike. To formulate stronger convexity conditions we must consider the determinant of the null expansions and . Note that, although the expansion tensors, and in particular the functions , , , and are boost-gauge–dependent, their sign is gauge invariant. Then will be called future convex if and , and past convex if and [519, 492]. These are equivalent to the requirement that the two eigenvalues of be positive and those of be negative everywhere on , respectively. A different kind of convexity condition, based on global concepts, will be used in Section 6.1.3.

The connections and determine connections on the pullback to of the bundle of unprimed spinors. The natural decomposition defines a chirality on the spinor bundle in the form of the spinor , which is analogous to the matrix in the theory of Dirac spinors. Then, the extrinsic curvature tensor above is a simple expression of and (and their complex conjugate), and the two covariant derivatives on are related to each other by . The curvature of can be expressed by the curvature of , the spinor , and its -derivative. We can form the scalar invariants of the curvatures according to

is four times the complex Gauss curvature [425] of , by means of which the whole curvature can be characterized: . If the spacetime is space and time orientable, at least on an open neighborhood of , then the normals and can be chosen to be globally well defined, and hence, is globally trivializable and the imaginary part of is a total divergence of a globally well-defined vector field.An interesting decomposition of the connection one-form , i.e., the vertical part of the connection , was given by Liu and Yau [338]: There are real functions and , unique up to additive constants, such that . is globally defined on , but in general is defined only on the local trivialization domains of that are homeomorphic to . It is globally defined if . In this decomposition is the boost-gauge–invariant part of , while represents its gauge content. Since , the ‘Coulomb-gauge condition’ uniquely fixes (see also Section 10.4.1).

By the Gauss–Bonnet theorem one has , where is the genus of . Thus, geometrically the connection is rather poor, and can be considered as a part of the ‘universal structure of ’. On the other hand, the connection is much richer, and, in particular, the invariant 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 , the projection and the connection ) and the extrinsic curvature tensor .

The complete decomposition of into its irreducible parts gives , the Dirac–Witten operator, and , the two-surface twistor operator. The former is essentially the anti-symmetric part , the latter is the symmetric and (with respect to the complex metric ) trace-free part of the derivative. (The trace 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

where and are two arbitrary spinor fields on , and the right-hand side is just the charge integral of the curvature on .

A GHP spin frame on the two-surface is a normalized spinor basis , , such that the complex null vectors and are tangent to (or, equivalently, the future-pointing null vectors and are orthogonal to ). Note, however, that in general a GHP spin frame can be specified only locally, but not globally on the whole . This fact is connected with the nontriviality of the tangent bundle 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 and cannot form a globally-defined basis on . 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 on some open , the components of the spinor and tensor fields on will be local representatives of cross sections of appropriate complex line bundles of scalars of type [209, 425]: A scalar is said to be of type if, under the rescaling , of the GHP spin frame with some nowhere-vanishing complex function , the scalar transforms as . For example, , , , and are of type , , , and , respectively. The components of the Weyl and Ricci spinors, , , , …, , , …, etc., also have definite -type. In particular, has type . A global section of is a collection of local cross sections such that forms a covering of and on the nonempty overlappings, e.g., on , the local sections are related to each other by , where is the transition function between the GHP spin frames: and .

The connection defines a connection on the line bundles [209, 425]. The usual edth operators, and , are just the directional derivatives and on the domain of the GHP spin frame . These locally-defined operators yield globally-defined differential operators, denoted also by and , on the global sections of . It might be worth emphasizing that the GHP spin coefficients and , which do not have definite -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 and the connection one-form 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 . These properties are discussed in [198] in general, and in [177, 58, 490] for spherical topology.

Using the projection operators , the irreducible parts and can be decomposed further into their right-handed and left-handed parts. In the GHP formalism these chiral irreducible parts are

where and the spinor components are defined by . 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 [106], are appropriate direct sums of these elementary operators. Their global properties under various circumstances are studied in [58, 490, 496].

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 [246], let the two-surface be considered as the intersection of the null hypersurfaces formed, respectively, by the outgoing and the ingoing light rays orthogonal to , and let the spacetime (or at least a neighborhood of ) be foliated by two one-parameter families of smooth hypersurfaces and , where , such that and . One can form the two normals, , which are null on and , respectively. Then we can define , for which , where . (If is chosen to be 1 on , then is precisely the -connection one-form above.) Then the anholonomicity is defined by . Since is invariant with respect to the rescalings and of the functions, defining the foliations by those functions , which preserve , it was claimed in [246] that depends only on . However, this implies only that is invariant with respect to a restricted class of the change of the foliations, and that is invariantly defined only by this class of the foliations rather than the two-surface. In fact, does depend on the foliation: Starting with a different foliation defined by the functions and for some , the corresponding anholonomicity would also be invariant with respect to the restricted changes of the foliations above, but the two anholonomicities, and , would be different: . Therefore, the anholonomicity is a gauge-dependent quantity.

Living Rev. Relativity 12, (2009), 4
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