In general spacetimes, the twistor equations have only the trivial solution. Thus, to be able to associate a
kinematical twistor with a closed orientable spacelike two-surface in general, the conditions on the
spinor field
have to be relaxed. Penrose’s suggestion [420
, 421
] is to consider
in Eq. (7.1
) to
be the symmetrized product
of spinor fields that are solutions of the ‘tangential projection to
’
of the valence-one twistor equation, the two-surface twistor equation. (The equation obtained as the
‘tangential projection to
’ of the valence-two twistor equation (7.4
) would be under-determined [421].)
Thus, the quasi-local quantities are searched for in the form of a charge integral of the curvature:
The two-surface twistor equation that the spinor fields should satisfy is just the covariant
spinor equation . By Eq. (4.6
) its GHP form is
,
which is a first-order elliptic system, and its index is
, where
is the genus of
[58]. Thus, there are at least four (and in the generic case precisely four) linearly-independent
solutions to
on topological two-spheres. However, there are ‘exceptional’ two-spheres
for which there exist at least five linearly independent solutions [297]. For such ‘exceptional’
two-spheres (and for higher-genus two-surfaces for which the twistor equation has only the trivial
solution in general) the subsequent construction does not work. (The concept of quasi-local
charges in Yang–Mills theory can also be introduced in an analogous way [509, 183]). The space
of the solutions to
is called the two-surface twistor space. In fact, in the
generic case this space is four-complex-dimensional, and under conformal rescaling the pair
transforms like a valence one contravariant twistor.
is called a two-surface
twistor determined by
. If
is another generic two-surface with the corresponding two-surface
twistor space
, then although
and
are isomorphic as vector spaces, there is
no canonical isomorphism between them. The kinematical twistor
is defined to be the
symmetric twistor determined by
for any
and
from
. Note that
is constructed only from the two-surface data on
.
For the solutions and
of the two-surface twistor equation, the spinor identity (4.5
) reduces to
Tod’s expression [420
, 426
, 516
] for the kinematical twistor, making it possible to re-express
by
the integral of the Nester–Witten 2-form [490
]. Indeed, if
In general, the natural pointwise Hermitian scalar product, defined by ,
is not constant on
, thus, it does not define a Hermitian scalar product on the two-surface twistor space.
As is shown in [296, 299
, 514
],
is constant on
for any two two-surface twistors if and only if
can be embedded, at least locally, into some conformal Minkowski spacetime with its intrinsic metric
and extrinsic curvatures. Such two-surfaces are called noncontorted, while those that cannot be embedded
are called contorted. One natural candidate for the Hermitian metric could be the average of
on
[420
]:
, which reduces to
on noncontorted
two-surfaces. Interestingly enough,
can also be re-expressed by the integral (7.14
) of the
Nester–Witten 2-form [490
]. Unfortunately, however, neither this metric nor the other suggestions
appearing in the literature are conformally invariant. Thus, for contorted two-surfaces, the definition of the
quasi-local mass as the norm of the kinematical twistor (cf. Eq. (7.10
)) is ambiguous unless a natural
is found.
If is noncontorted, then the scalar product
defines the totally anti-symmetric
twistor
, and for the four independent two-surface twistors
, …,
the contraction
, and hence, by Eq. (7.7
), the determinant
, is constant on
. Nevertheless,
can be constant even for contorted two-surfaces for which
is not. Thus, the totally
anti-symmetric twistor
can exist even for certain contorted two-surfaces. Therefore, an
alternative definition of the quasi-local mass might be based on Eq. (7.11
) [510
]. However, although
the two mass definitions are equivalent in the linearized theory, they are different invariants
of the kinematical twistor even in de Sitter or anti-de Sitter spacetimes. Thus, if needed, the
former notion of mass will be called the norm-mass, the latter the determinant-mass (denoted by
).
If we want to have not only the notion of the mass but its reality as well, then we should ensure the
Hermiticity of the kinematical twistor. But to formulate the Hermiticity condition (7.9), one also
needs the infinity twistor. However,
is not constant on
even if it is
noncontorted. Thus, in general, it does not define any twistor on
. One might take its average
on
(which can also be re-expressed by the integral of the Nester–Witten 2-form [490
]),
but the resulting twistor would not be simple. In fact, even on two-surfaces in de Sitter and
anti-de Sitter spacetimes with cosmological constant
the natural definition for
is
[426
, 424, 510
], while on round spheres in spherically-symmetric spacetimes it is
[496
]. Thus, no natural simple infinity
twistor has been found in curved spacetime. Indeed, Helfer claims that no such infinity twistor can
exist [263]: even if the spacetime is conformally flat (in which case the Hermitian metric exists) the
Hermiticity condition would be fifteen algebraic equations for the (at most) twelve real components of the
‘would be’ infinity twistor. Then, since the possible kinematical twistors form an open set in the space of
symmetric twistors, the Hermiticity condition cannot be satisfied even for nonsimple
s. However, in
contrast to the linearized gravity case, the infinity twistor should not be given once and for all on
some ‘universal’ twistor space that may depend on the actual gravitational field. In fact, the
two-surface twistor space itself depends on the geometry of
, and hence all its structures
also.
Since in the Hermiticity condition (7.9) only the special combination
of the infinity
and metric twistors (the ‘bar-hook’ combination) appears, it might still be hoped that an appropriate
could be found for a class of two-surfaces in a natural way [516
]. However, as far as the present
author is aware, no real progress has been achieved in this way.
Obviously, the kinematical twistor vanishes in flat spacetime and, since the basic idea comes from linearized
gravity, the construction gives the correct results in the weak field approximation. The nonrelativistic weak
field approximation, i.e., the Newtonian limit, was clarified by Jeffryes [298]. He considers a one-parameter
family of spacetimes with perfect fluid source, such that in the limit of the parameter
, one gets
a Newtonian spacetime, and, in the same limit, the two-surface
lies in a
hypersurface
of the Newtonian time
. In this limit the pointwise Hermitian scalar product is constant,
and the norm-mass can be calculated. As could be expected, for the leading
-order term
in the expansion of
as a series of
he obtained the conserved Newtonian mass. The
Newtonian energy, including the kinetic and the Newtonian potential energy, appears as a
-order
correction.
The Penrose definition for the energy-momentum and angular momentum can be applied to the cuts
of the future null infinity
of an asymptotically flat spacetime [420, 426
]. Then every element of
the construction is built from conformally-rescaled quantities of the nonphysical spacetime. Since
is
shear-free, the two-surface twistor equations on
decouple, and hence, the solution space admits a
natural infinity twistor
. It singles out precisely those solutions whose primary spinor parts span the
asymptotic spin space of Bramson (see Section 4.2.4), and they will be the generators of the
energy-momentum. Although
is contorted, and hence, there is no natural Hermitian scalar
product, there is a twistor
with respect to which
is Hermitian. Furthermore, the
determinant
is constant on
, and hence it defines a volume 4-form on the two-surface twistor
space [516
]. The energy-momentum coming from
is just that of Bondi and Sachs. The angular
momentum defined by
is, however, new. It has a number of attractive properties. First,
in contrast to definitions based on the Komar expression, it does not have the ‘factor-of-two
anomaly’ between the angular momentum and the energy-momentum. Since its definition is based
on the solutions of the two-surface twistor equations (which can be interpreted as the spinor
constituents of certain BMS vector fields generating boost-rotations) instead of the BMS vector fields
themselves, it is free of supertranslation ambiguities. In fact, the two-surface twistor space on
reduces the BMS Lie algebra to one of its Poincaré subalgebras. Thus, the concept of the
‘translation of the origin’ is moved from null infinity to the twistor space (appearing in the form of
a four-parameter family of ambiguities in the potential for the shear
), and the angular
momentum transforms just in the expected way under such a ‘translation of the origin’. It is shown
in [174] that Penrose’s angular momentum can be considered as a supertranslation of previous
definitions.
The other way of determining the null infinity limit of the energy-momentum and angular momentum is
to calculate them for large spheres from the physical data, instead of for the spheres at null infinity from
the conformally-rescaled data. These calculations were done by Shaw [455, 457
]. At this point it should be
noted that the
limit of
vanishes, and it is
that yields the
energy-momentum and angular momentum at infinity (see the remarks following Eq. (3.14
)). The
specific radiative solution for which the Penrose mass has been calculated is that of Robinson and
Trautman [510
]. The two-surfaces for which the mass was calculated are the
cuts of the
geometrically-distinguished outgoing null hypersurfaces
. Tod found that, for given
,
the mass
is independent of
, as could be expected because of the lack of incoming
radiation.
In [264] Helfer suggested a bijective nonlinear map between the two-surface twistor spaces on the
different cuts of , by means of which he got something like a ‘universal twistor space’. Then he
extends the kinematical twistor to this space, and in this extension the shear potential (i.e., the
complex function
for which the asymptotic shear can be written as
) appears
explicitly. Using Eq. (7.12
) as the definition of the intrinsic-spin angular momentum at scri, Helfer
derives an explicit formula for the spin. In addition to the expected Pauli–Lubanski type term,
there is an extra term, which is proportional to the imaginary part of the shear potential. Since
the twistor spaces on the different cuts of scri have been identified, the angular momentum
flux can be, and has in fact been, calculated. (For an earlier attempt to calculate this flux,
see [262].)
The large sphere limit of the two-surface twistor space and the Penrose construction were investigated by Shaw in the Sommers [475], Ashtekar–Hansen [37], and Beig–Schmidt [65] models of spatial infinity in [451, 452, 454]. Since no gravitational radiation is present near the spatial infinity, the large spheres are (asymptotically) noncontorted, and both the Hermitian scalar product and the infinity twistor are well defined. Thus, the energy-momentum and angular momentum (and, in particular, the mass) can be calculated. In vacuum he recovered the Ashtekar–Hansen expression for the energy-momentum and angular momentum, and proved their conservation if the Weyl curvature is asymptotically purely electric. In the presence of matter the conservation of the angular momentum was investigated in [456].
The Penrose mass in asymptotically anti-de Sitter spacetimes was studied by Kelly [312]. He
calculated the kinematical twistor for spacelike cuts of the infinity
, which is now a
timelike three-manifold in the nonphysical spacetime. Since
admits global three-surface
twistors (see the next Section 7.2.5),
is noncontorted. In addition to the Hermitian scalar
product, there is a natural infinity twistor, and the kinematical twistor satisfies the corresponding
Hermiticity condition. The energy-momentum four-vector coming from the Penrose definition is
shown to coincide with that of Ashtekar and Magnon [42]. Therefore, the energy-momentum
four-vector is future pointing and timelike if there is a spacelike hypersurface extending to
on
which the dominant energy condition is satisfied. Consequently,
. Kelly shows that
is also non-negative and in vacuum it coincides with
. In fact [516
],
holds.
The Penrose mass has been calculated in a large number of specific situations. Round spheres
are always noncontorted [514], thus, the norm-mass can be calculated. (In fact, axisymmetric
two-surfaces in spacetimes with twist-free rotational Killing vectors are noncontorted [299].)
The Penrose mass for round spheres reduces to the standard energy expression discussed in
Section 4.2.1 [510
]. Thus, every statement given in Section 4.2.1 for round spheres is valid for the
Penrose mass, and we do not repeat them. In particular, for round spheres in a
slice of the Kantowski–Sachs spacetime, this mass is independent of the two-surfaces [507].
Interestingly enough, although these spheres cannot be shrunk to a point (thus, the mass cannot be
interpreted as ‘the three-volume integral of some mass density’), the time derivative of the
Penrose mass looks like the mass conservation equation. It is, minus the pressure times the rate
of change of the three-volume of a sphere in flat space with the same area as
[515
]. In
conformally-flat spacetimes [510
] the two-surface twistors are just the global twistors restricted
to
, and the Hermitian scalar product is constant on
. Thus, the norm-mass is well
defined.
The construction works nicely, even if global twistors exist only on a, e.g., spacelike hypersurface
containing
. These are the three-surface twistors [510
, 512
], which are solutions of
certain (overdetermined) elliptic partial-differential equations, called the three-surface twistor
equations, on
. These equations are completely integrable (i.e., they admit the maximal
number of linearly-independent solutions, namely four) if and only if
, with its intrinsic
metric and extrinsic curvature, can be embedded, at least locally, into some conformally-flat
spacetime [512]. Such hypersurfaces are called noncontorted. It might be interesting to note that the
noncontorted hypersurfaces can also be characterized as the critical points of the Chern–Simons
functional, built from the real Sen connection on the Lorentzian vector bundle or from the
three-surface twistor connection on the twistor bundle over
[66, 495]. Returning to the quasi-local
mass calculations, Tod showed that in vacuum the kinematical twistor for a two-surface
in a noncontorted
depends only on the homology class of
. In particular, if
can
be shrunk to a point, then the corresponding kinematical twistor is vanishing. Since
is
noncontorted,
is also noncontorted, and hence the norm-mass is well defined. This implies that the
Penrose mass in the Schwarzschild solution is the Schwarzschild mass for any noncontorted
two-surface that can be deformed into a round sphere, and it is zero for those that do not go
round the black hole [514
]. Thus, in particular, the Penrose mass can be zero even in curved
spacetimes.
A particularly interesting class of noncontorted hypersurfaces is that of the conformally-flat
time-symmetric initial data sets. Tod considered Wheeler’s solution of the time-symmetric vacuum
constraints describing ‘points at infinity’ (or, in other words,
black holes) and
two-surfaces in such a hypersurface [510
]. He found that the mass is zero if
does not go
around any black hole, it is the mass
of the
-th black hole if
links precisely the
-th black hole, it is
if
links precisely the
-th and
the
-th black holes, where
is some appropriate measure of the distance between the
black holes, …, etc. Thus, the mass of the
-th and
-th holes as a single object is less than
the sum of the individual masses, in complete agreement with our physical intuition that the
potential energy of the composite system should contribute to the total energy with negative
sign.
Beig studied the general conformally-flat time-symmetric initial data sets describing ‘points at
infinity’ [62]. He found a symmetric trace-free and divergence-free tensor field
and, for any conformal
Killing vector
of the data set, defined the two-surface flux integral
of
on
. He
showed that
is conformally invariant, depends only on the homology class of
, and, apart
from numerical coefficients, for the ten (locally-existing) conformal Killing vectors, these are
just the components of the kinematical twistor derived by Tod in [510
] (and discussed in the
previous paragraph). In particular, Penrose’s mass in Beig’s approach is proportional to the
length of the
’s with respect to the Cartan–Killing metric of the conformal group of the
hypersurface.
Tod calculated the quasi-local mass for a large class of axisymmetric two-surfaces (cylinders) in various
LRS Bianchi and Kantowski–Sachs cosmological models [515] and more general cylindrically-symmetric
spacetimes [517]. In all these cases the two-surfaces are noncontorted, and the construction works. A
technically interesting feature of these calculations is that the two-surfaces have edges, i.e., they are not
smooth submanifolds. The twistor equation is solved on the three smooth pieces of the cylinder separately,
and the resulting spinor fields are required to be continuous at the edges. This matching reduces the number
of linearly-independent solutions to four. The projection parts of the resulting twistors, the
s, are not continuous at the edges. It turns out that the cylinders can be classified
invariantly to be hyperbolic, parabolic, or elliptic. Then the structure of the quasi-local mass
expressions is not simply ‘density’
‘volume’, but is proportional to a ‘type factor’
as well, where
is the coordinate length of the cylinder. In the hyperbolic, parabolic, and
elliptic cases this factor is
,
, and
, respectively, where
is an
invariant of the cylinder. The various types are interpreted as the presence of a positive, zero, or
negative potential energy. In the elliptic case the mass may be zero for finite cylinders. On the
other hand, for static perfect fluid spacetimes (hyperbolic case) the quasi-local mass is positive.
A particularly interesting spacetime is that describing cylindrical gravitational waves, whose
presence is detected by the Penrose mass. In all these cases the determinant-mass has also
been calculated and found to coincide with the norm-mass. A numerical investigation of the
axisymmetric Brill waves on the Schwarzschild background is presented in [87
]. It was found that
the quasi-local mass is positive, and it is very sensitive to the presence of the gravitational
waves.
Another interesting issue is the Penrose inequality for black holes (see Section 13.2.1). Tod
shows [513, 514
] that for static black holes the Penrose inequality holds if the mass of the black hole is
defined to be the Penrose quasi-local mass of the spacelike cross section
of the event horizon. The
trick here is that
is totally geodesic and conformal to the unit sphere, and hence, it is
noncontorted and the Penrose mass is well defined. Then, the Penrose inequality will be a Sobolev-type
inequality for a non-negative function on the unit sphere. This inequality is tested numerically
in [87].
Apart from the cuts of in radiative spacetimes, all the two-surfaces discussed so far were
noncontorted. The spacelike cross section of the event horizon of the Kerr black hole provides a
contorted two-surface [516
]. Thus, although the kinematical twistor can be calculated for this, the
construction in its original form cannot yield any mass expression. The original construction has to be
modified.
The properties of the Penrose construction that we have discussed are very remarkable and promising.
However, the small surface calculations clearly show some unwanted features of the original
construction [511, 313
, 560
], and force its modification.
First, although the small spheres are contorted in general, the leading term of the pointwise
Hermitian scalar product is constant:
for any
two-surface twistors
and
[511
, 313
]. Since in
nonvacuum spacetimes the kinematical twistor has only the ‘four-momentum part’ in the leading
-order with
, the Penrose mass, calculated with the norm above, is just
the expected mass in the leading
order. Thus, it is positive if the dominant energy
condition is satisfied. On the other hand, in vacuum the structure of the kinematical twistor is
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
![]() This work is licensed under a Creative Commons License. E-mail us: |