### 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 [375380]. Essentially, his approach is based on the GHP (Geroch-Held-Penrose) formalism [152]. Although this is a very effective and flexible formalism [152312313206347], 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 [355357358359147]. 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 to the closed, orientable spacelike 2-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 projection to and is and , respectively. The induced 2-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 fibre metric and the projection will be called the Lorentzian vector bundle over . 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 defines two covariant derivatives on . The first, denoted by , is analogous to the induced (intrinsic) covariant derivative on (one-codimensional) hypersurfaces: for any section of . Obviously, annihilates both the fibre metric and the projection . However, since for 2-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 1-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 [331], and is defined simply by . This annihilates only the fibre 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 by , where , , and are the relevant traces. is called the main extrinsic curvature vector of . If , 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 2-surface in which the expansion of the surface is vanishing. If is not null, then defines an orthonormal frame in the normal bundle (see for example [8]). If is non-zero 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 2-surface admits a natural gauge choice in the normal bundle unless is vanishing. Geometrically, is a connection coming from a connection on the -principal fibre 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 pull-back 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 Equation (21).

#### 4.1.3 Convexity conditions

To prove certain statements on quasi-local quantities various forms of the convexity of must be assumed. The convexity of 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 . 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  [380]. is called mean convex [182] 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 gauge dependent, their sign is gauge invariant. Then will be called future convex if and , and past convex if and  [380358]. 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.

#### 4.1.4 The spinor bundle

The connections and determine connections on the pull-back 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 so-called complex Gauss curvature [312] of , by means of which the whole curvature can be characterized: . If the spacetime is space and time orientable, at least on an open neighbourhood 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 1-form , i.e. the vertical part of the connection , was given by Liu and Yau [253]: 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 , 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 2-surface data for charge-type quasi-local quantities (i.e. for 2-surface observables) are the universal structure (i.e. the intrinsic metric , the projection and the connection ) and the extrinsic curvature tensor .

#### 4.1.5 Curvature identities

The complete decomposition of into its irreducible parts gives , the Dirac-Witten operator, and , the 2-surface twistor operator. 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 .

#### 4.1.6 The GHP formalism

A GHP spin frame on the 2-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 non-triviality of the tangent bundle 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 and cannot form a globally defined basis on . 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 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  [152312]: 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 non-empty 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  [152312]. 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 1-form, and they are built both from the connection 1-form for the intrinsic Riemannian geometry of and the connection 1-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 [147] for general, and in [13243356] for spherical topology.

#### 4.1.7 Irreducible parts of the derivative operators

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 2-surface twistor operator, the holomorphy/anti-holomorphy operators or the operators whose kernel defines the asymptotic spinors of Bramson [83], are appropriate direct sums of these elementary operators. Their global properties under various circumstances are studied in [43356363].

#### 4.1.8 -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 [181], let the 2-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 neighbourhood 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 1-form above.) Then the so-called 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 [181] 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 2-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 still a gauge dependent quantity.