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].
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
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
[380
, 358
]. 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 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
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
.
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
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
[152
, 312
]: 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
[152
, 312
]. 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 [132, 43
, 356
] 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
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.
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