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
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
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
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
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.
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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