In general spacetimes the twistor equations have only the trivial solution. Thus to be able to associate a
kinematical twistor to a closed orientable spacelike 2-surface in general, the conditions on the spinor
field
had to be relaxed. Penrose’s suggestion [307
, 308
] is to consider
in Equation (43
) to be
the symmetrized product
of spinor fields that are solutions of the ‘tangential projection to
’ of
the valence 1 twistor equation, the so-called 2-surface twistor equation. (The equation obtained as the
‘tangential projection to
’ of the valence 2 twistor equation (46
) would be under-determined [308].)
Thus the quasi-local quantities are searched for in the form of a charge integral of the curvature:
The 2-surface twistor equation that the spinor fields should satisfy is just the covariant spinor equation
. By Equation (25
) its GHP form is
, which is a first order elliptic
system, and its index is
, where
is the genus of
[43]. Thus there are at least four (and in
the generic case precisely four) linearly independent solutions to
on topological 2-spheres.
However, there are ‘exceptional’ 2-spheres for which there exist at least five linearly independent
solutions [221]. For such ‘exceptional’ 2-spheres (and for higher genus 2-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 [370]). The
space
of the solutions to
is called the 2-surface twistor space. In fact, in
the generic case this space is 4-complex-dimensional, and under conformal rescaling the pair
transforms like a valence 1 contravariant twistor.
is called a 2-surface
twistor determined by
. If
is another generic 2-surface with the corresponding 2-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 2-surface data on
.
For the solutions and
of the 2-surface twistor equation, the spinor identity (24
) reduces to Tod’s
expression [307
, 313
, 377
] for the kinematical twistor, making it possible to re-express
by the
integral of the Nester-Witten 2-form [356
]. 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 2-surface twistor space. As
is shown in [220, 223
, 375
],
is constant on
for any two 2-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 2-surfaces are called non-contorted, while those that cannot be embedded are
called contorted. One natural candidate for the Hermitian metric could be the average of
on
[307
]:
, which reduces to
on non-contorted
2-surfaces. Interestingly enough,
can also be re-expressed by the integral (55
) of the
Nester-Witten 2-form [356
]. Unfortunately, however, neither this metric nor the other suggestions
appearing in the literature are conformally invariant. Thus, for contorted 2-surfaces, the definition of the
quasi-local mass as the norm of the kinematical twistor (cf. Equation (52
)) is ambiguous unless a natural
is found.
If is non-contorted, then the scalar product
defines the totally anti-symmetric
twistor
, and for the four independent 2-surface twistors
, …,
the contraction
, and hence by Equation (49
) the determinant
, is constant on
. Nevertheless,
can be constant even for contorted 2-surfaces for which
is not. Thus, the totally
anti-symmetric twistor
can exist even for certain contorted 2-surfaces. Therefore, an alternative
definition of the quasi-local mass might be based on Equation (53
) [371
]. 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 is also expected, then we should ensure
the Hermiticity of the kinematical twistor. But to formulate the Hermiticity condition (51),
one also needs the infinity twistor. However,
is not constant on
even if it is non-contorted, 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 [356
]), but the resulting twistor would not be simple. In fact, even on 2-surfaces in de Sitter and
anti-de Sitter spacetimes with cosmological constant
the natural definition for
is
[313
, 311, 371
], while on round spheres in spherically symmetric spacetimes it is
[363
]. Thus no natural simple infinity
twistor has been found in curved spacetime. Indeed, Helfer claims that no such infinity twistor can
exist [197]: Even if the spacetime is conformally flat (whenever 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 non-simple
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
2-surface twistor space itself depends on the geometry of
, and hence all the structures thereon
also.
Since in the Hermiticity condition (51) only the special combination
of the infinity
and metric twistors (the so-called ‘bar-hook’ combination) appears, it might still be hoped
that an appropriate
could be found for a class of 2-surfaces in a natural way [377
].
However, as far as the present author is aware of, no real progress has been achieved in this
way.
Obviously, the kinematical twistor vanishes in flat spacetime and, since the basic idea came from the
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 [222]. He
considers a 1-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 2-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 [307, 313
]. Then every element of
the construction is built from conformally rescaled quantities of the non-physical spacetime. Since
is
shear-free, the 2-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 2-surface twistor space [377
]. 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
2-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 2-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 4-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’. As was shown in [129
], Penrose’s angular
momentum can be considered as a supertranslation of previous definitions. The corresponding
angular momentum flux through a portion of the null infinity between two cuts was calculated
in [129, 196] and it was shown that this is precisely that given by Ashtekar and Streubel [29] (see
also [336, 337
, 128]).
The other way of determining the null infinity limit of the energy-momentum and angular momentum is
to calculate them for the large spheres from the physical data, instead of the spheres at null infinity from
the conformally rescaled data. These calculations were done by Shaw [338, 340
]. 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 Equation (15
)). The
specific radiative solution for which the Penrose mass has been calculated is that of Robinson and
Trautman [371
]. The 2-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 the incoming
radiation.
The large sphere limit of the 2-surface twistor space and the Penrose construction were investigated by Shaw in the Sommers [344], the Ashtekar-Hansen [23], and the Beig-Schmidt [48] models of spatial infinity in [334, 335, 337]. Since no gravitational radiation is present near the spatial infinity, the large spheres are (asymptotically) non-contorted, 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 [339].
The Penrose mass in asymptotically anti-de-Sitter spacetimes was studied by Kelly [234]. He calculated
the kinematical twistor for spacelike cuts of the infinity
, which is now a timelike 3-manifold in the
non-physical spacetime. Since
admits global 3-surface twistors (see the next Section 7.2.5),
is
non-contorted. 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 4-vector
coming from the Penrose definition is shown to coincide with that of Ashtekar and Magnon [27]. Therefore,
the energy-momentum 4-vector is future pointing and timelike if there is a spacelike hypersurface extending
to
on which the dominant energy condition is satisfied. Consequently,
. Kelly showed that
is also non-negative and in vacuum it coincides with
. In fact [377
],
holds.
The Penrose mass has been calculated in a large number of specific situations. Round spheres are always
non-contorted [375], thus the norm-mass can be calculated. (In fact, axi-symmetric 2-surfaces in spacetimes
with twist-free rotational Killing vector are non-contorted [223].) The Penrose mass for round spheres
reduces to the standard energy expression discussed in Section 4.2.1 [371
]. 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 2-surfaces [368]. Interestingly enough, although these spheres cannot be shrunk to a point
(thus the mass cannot be interpreted as ‘the 3-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 3-volume of a sphere in flat space with the same area as
[376
]. In conformally flat spacetimes [371
] the 2-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 (say) spacelike hypersurface
containing
. These twistors are the so-called 3-surface twistors [371
, 373
], which are solutions of
certain (overdetermined) elliptic partial differential equations, the so-called 3-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 [373]. Such hypersurfaces are called non-contorted. It might be interesting to note
that the non-contorted 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 3-surface twistor connection on the twistor bundle over
[49, 361]. Returning to the
quasi-local mass calculations, Tod showed that in vacuum the kinematical twistor for a 2-surface
in a non-contorted
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 non-contorted,
is also non-contorted, and hence the norm-mass is well-defined. This
implies that the Penrose mass in the Schwarzschild solution is the Schwarzschild mass for any
non-contorted 2-surface that can be deformed into a round sphere, and it is zero for those that do not
link the black hole [375
]. Thus, in particular, the Penrose mass can be zero even in curved
spacetimes.
A particularly interesting class of non-contorted 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 2-surfaces in such
a hypersurface [371
]. He found that the mass is zero if
does not link any black hole, it is the mass
of the
-th black hole if
links precisely the
-th hole, it is
if
links precisely the
-th and the
-th holes, where
is some appropriate measure of the distance of
the 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’ [45]. He found a symmetric trace-free and divergence-free tensor field
and, for any conformal
Killing vector
of the data set, defined the 2-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 [371
] (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 axi-symmetric 2-surfaces (cylinders) in various
LRS Bianchi and Kantowski-Sachs cosmological models [376] and more general cylindrically symmetric
spacetimes [378]. In all these cases the 2-surfaces are non-contorted, and the construction works. A
technically interesting feature of these calculations is that the 2-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 they are 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
axi-symmetric Brill waves on the Schwarzschild background was presented in [69
]. 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
showed [374, 375
] that for static black holes the Penrose inequality holds if the mass of the 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
non-contorted 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 was tested numerically
in [69].
Apart from the cuts of in radiative spacetimes, all the 2-surfaces discussed so far were
non-contorted. The spacelike cross section of the event horizon of the Kerr black hole provides a
contorted 2-surface [377
]. 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 showed clearly some unwanted feature of the original
construction [372, 235
, 398
], and forced its modification.
Update
First, although the small spheres are contorted in general, the leading term of the pointwise
Hermitian scalar product is constant:
for any
2-surface twistors
and
[372
, 235
]. Since in
non-vacuum spacetimes the kinematical twistor has only the ‘4-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
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