If the spacetime is spherically symmetric, then a 2-sphere which is a transitivity surface of the
rotation group is called a round sphere. Then in a spherical coordinate system
the spacetime metric takes the form
, where
and
are functions of
and
. (Hence
is the so-called area-coordinate). Then
with the notations of Section 4.1, one obtains
. Based on the
investigations of Misner, Sharp, and Hernandez [268
, 199
], Cahill and McVitte [98] found
Spherically symmetric spacetimes admit a special vector field, the so-called Kodama vector field ,
such that
is divergence free [241]. In asymptotically flat spacetimes
is timelike in the
asymptotic region, in stationary spacetimes it reduces to the Killing symmetry of stationarity (in fact, this
is hypersurface-orthogonal), but in general it is not a Killing vector. However, by
the
vector field
has a conserved flux on a spacelike hypersurface
. In particular, in the
coordinate system
and line element above
. If
is the solid ball
of radius
, then the flux of
is precisely the standard round sphere expression (26
) for the 2-sphere
[278].
An interesting characterization of the dynamics of the spherically symmetric gravitational fields can be
given in terms of the energy function above (see for example [408
, 262
, 185
]). In particular,
criteria for the existence and the formation of trapped surfaces and the presence and the nature of the
central singularity can be given by
.
To define the first, let be a point, and
a future directed unit timelike vector at
. Let
, the ‘future null cone of
in
’ (i.e. the boundary of the chronological future of
).
Let
be the future pointing null tangent to the null geodesic generators of
such that, at the vertex
,
. With this condition we fix the scale of the affine parameter
on the different generators,
and hence by requiring
we fix the parameterization completely. Then, in an open neighbourhood
of the vertex
,
is a smooth null hypersurface, and hence for sufficiently small
the set
is a smooth spacelike 2-surface and homeomorphic to
.
is called a
small sphere of radius
with vertex
. Note that the condition
fixes the boost
gauge.
Completing to a Newman-Penrose complex null tetrad
such that the
complex null vectors
and
are tangent to the 2-surfaces
, the components of
the metric and the spin coefficients with respect to this basis can be expanded as series in
5.
Then the GHP equations can be solved with any prescribed accuracy for the expansion coefficients of the
metric
on
, the GHP spin coefficients
,
,
,
,
, and
, and the higher order
expansion coefficients of the curvature in terms of the lower order curvature components at
. Hence the
expression of any quasi-local quantity
for the small sphere
can be expressed as a series of
,
where the expansion coefficients are still functions of the coordinates,
or
,
on the unit sphere
. If the quasi-local quantity
is spacetime-covariant, then the unit
sphere integrals of the expansion coefficients
must be spacetime covariant expressions
of the metric and its derivatives up to some finite order at
and the ‘time axis’
. The
necessary degree of the accuracy of the solution of the GHP equations depends on the
nature of
and on whether the spacetime is Ricci-flat in a neighbourhood of
or
not6.
These solutions of the GHP equations, with increasing accuracy, are given in [204
, 235
, 94
, 360
].
Obviously, we can calculate the small sphere limit of various quasi-local quantities built from the matter
fields in the Minkowski spacetime, too. In particular [360
], the small sphere expressions for the quasi-local
energy-momentum and the (anti-self-dual part of the) quasi-local angular momentum of the matter fields
based on
, respectively, are
Interestingly enough, a simple dimensional analysis already shows the structure of the leading terms in a
large class of quasi-local spacetime covariant energy-momentum and angular momentum expressions. In
fact, if is any coordinate-independent quasi-local quantity, built from the first derivatives of the
metric, i.e.
, then its expansion is
If a neighbourhood of is vacuum, then the
order term is vanishing, and the fourth order term
must be built from
. However, the only scalar polynomial expression of
,
,
,
and the generator vector
, depending on the latter two linearly, is the zero. Thus the
order term in vacuum is also vanishing. In the fifth order the only non-zero terms are quadratic in the
various parts of the Weyl tensor, yielding
Obviously, the same analysis can be repeated for any other quasi-local quantity. For quasi-local angular
momentum has the structure
, while the area of
is
. Then the leading
term in the expansion of the angular momentum is
and
order in non-vacuum and vacuum,
respectively, while the first non-trivial correction to the area
is of order
and
in
non-vacuum and vacuum, respectively.
On the small geodesic sphere of radius
in the given spacelike hypersurface
one can
introduce the complex null tangents
and
above, and if
is the future pointing unit normal of
and
the outward directed unit normal of
in
, then we can define
and
. Then
is a Newman-Penrose complex null tetrad, and the relevant
GHP equations can be solved for the spin coefficients in terms of the curvature components at
.
The small ellipsoids are defined as follows [235]. If
is any smooth function on
with a
non-degenerate minimum at
with minimum value
, then, at least on an open
neighbourhood
of
in
the level surfaces
are smooth
compact 2-surfaces homeomorphic to
. Then, in the
limit, the surfaces
look
like small nested ellipsoids centred in
. The function
is usually ‘normalized’ so that
.
Near spatial infinity we have the a priori and
fall-off for the 3-metric
and
extrinsic curvature
, respectively, and both the evolution equations of general relativity
and the conservation equation
for the matter fields preserve these conditions. The
spheres
of coordinate radius
in
are called large spheres if the values of
are
large enough such that the asymptotic expansions of the metric and extrinsic curvature are
legitimate7.
Introducing some coordinate system, e.g. the complex stereographic coordinates, on one sphere and then
extending that to the whole
along the normals
of the spheres, we obtain a coordinate system
on
. Let
,
, be a GHP spinor dyad on
adapted to the large
spheres in such a way that
and
are tangent to the spheres and
, the future directed unit normal of
. These conditions fix the spinor dyad
completely, and, in particular,
, the outward directed unit normal to the spheres
tangent to
.
The fall-off conditions yield that the spin coefficients tend to their flat spacetime value like
and
the curvature components to zero like
. Expanding the spin coefficients and curvature components as
power series of
, one can solve the field equations asymptotically (see [48
, 44] for a different
formalism). However, in most calculations of the large sphere limit of the quasi-local quantities only
the leading terms of the spin coefficients and curvature components appear. Thus it is not
necessary to solve the field equations for their second or higher order non-trivial expansion
coefficients.
Using the flat background metric and the corresponding derivative operator
we can define
a spinor field
to be constant if
. Obviously, the constant spinors form a two complex
dimensional vector space. Then by the fall-off properties
. Hence we can define the
asymptotically constant spinor fields to be those
that satisfy
, where
is the
intrinsic Levi-Civita derivative operator. Note that this implies that, with the notations of Equation (25
),
all the chiral irreducible parts,
,
,
, and
, of the derivative of the asymptotically
constant spinor field
are
.
Let the spacetime be asymptotically flat at future null infinity in the sense of Penrose [300, 301, 302, 313]
(see also [151]), i.e. the physical spacetime can be conformally compactified by an appropriate boundary
. Then future null infinity
will be a null hypersurface in the conformally rescaled
spacetime. Topologically it is
, and the conformal factor can always be chosen such that the
induced metric on the compact spacelike slices of
is the metric of the unit sphere. Fixing
such a slice
(called ‘the origin cut of
’) the points of
can be labeled by a
null coordinate, namely the affine parameter
along the null geodesic generators of
measured from
and, for example, the familiar complex stereographic coordinates
, defined first on the unit sphere
and then extended in a natural way along the null
generators to the whole
. Then any other cut
of
can be specified by a function
. In particular, the cuts
are obtained from
by a pure time
translation.
The coordinates can be extended to an open neighbourhood of
in the spacetime in the
following way. Let
be the family of smooth outgoing null hypersurfaces in a neighbourhood of
such that they intersect the null infinity just in the cuts
, i.e.
. Then let
be the
affine parameter in the physical metric along the null geodesic generators of
. Then
forms
a coordinate system. The
,
2-surfaces
(or simply
if no confusion can
arise) are spacelike topological 2-spheres, which are called large spheres of radius
near future null
infinity. Obviously, the affine parameter
is not unique, its origin can be changed freely:
is an equally good affine parameter for any smooth
. Imposing certain
additional conditions to rule out such coordinate ambiguities we arrive at a ‘Bondi-type coordinate
system’8.
In many of the large sphere calculations of the quasi-local quantities the large spheres should be assumed to
be large spheres not only in a general null, but in a Bondi-type coordinate system. For the detailed
discussion of the coordinate freedom left at the various stages in the introduction of these coordinate
systems, see for example [290
, 289
, 84].
In addition to the coordinate system we need a Newman-Penrose null tetrad, or rather a GHP spinor
dyad,
,
, on the hypersurfaces
. (Thus boldface indices are referring to the
GHP spin frame.) It is natural to choose
such that
be the tangent
of the null geodesic generators of
, and
itself be constant along
. Newman and
Unti [290
] chose
to be parallel propagated along
. This choice yields the vanishing of
a number of spin coefficients (see for example the review [289
]). The asymptotic solution of
the Einstein-Maxwell equations as a series of
in this coordinate and tetrad system is
given in [290, 134, 312
], where all the non-vanishing spin coefficients and metric and curvature
components are listed. In this formalism the gravitational waves are represented by the
-derivative
of the asymptotic shear of the null geodesic generators of the outgoing null hypersurfaces
.
From the point of view of the large sphere calculations of the quasi-local quantities the choice of
Newman and Unti for the spinor basis is not very convenient. It is more natural to adapt the GHP spin
frame to the family of the large spheres of constant ‘radius’ , i.e. to require
and
to be tangents of the spheres. This can be achieved by an appropriate null rotation
of the Newman-Unti basis about the spinor
. This rotation yields a change of the spin
coefficients and the metric and curvature components. As far as the present author is aware of, this
rotation with the highest accuracy was done for the solutions of the Einstein-Maxwell system by
Shaw [338
].
In contrast to the spatial infinity case, the ‘natural’ definition of the asymptotically constant spinor
fields yields identically zero spinors in general [83]. Nontrivial constant spinors in this sense could
exist only in the absence of the outgoing gravitational radiation, i.e. when
. In the
language of Section 4.1.7, this definition would be
,
,
and
. However, as Bramson showed [83], half of these
conditions can be imposed. Namely, at future null infinity
(and at past
null infinity
) can always be imposed asymptotically, and it has two
linearly independent solutions
,
, on
(or on
, respectively). The space
of its solutions turns out to have a natural symplectic metric
, and we refer to
as future asymptotic spin space. Its elements are called asymptotic spinors, and the
equations
the future/past asymptotic twistor equations. At
asymptotic
spinors are the spinor constituents of the BMS translations: Any such translation is of the form
for some constant Hermitian matrix
. Similarly, (apart from the
proper supertranslation content) the components of the anti-self-dual part of the boost-rotation BMS vector
fields are
, where
are the standard
Pauli matrices (divided by
) [363
]. Asymptotic spinors can be recovered as the elements of the kernel of several other
operators built from
,
,
, and
, too. In the present review we use only the fact
that asymptotic spinors can be introduced as anti-holomorphic spinors (see also Section 8.2.1),
i.e. the solutions of
(and at past null infinity as holomorphic spinors),
and as special solutions of the 2-surface twistor equation
(see also
Section 7.2.1). These operators, together with others reproducing the asymptotic spinors, are discussed
in [363
].
The Bondi-Sachs energy-momentum given in the Newman-Penrose formalism has already
become its ‘standard’ form. It is the unit sphere integral on the cut of a combination of
the leading term
of the Weyl spinor component
, the asymptotic shear
and its
-derivative, weighted by the first four spherical harmonics (see for example [289, 313
]):
Similarly, the various definitions for angular momentum at null infinity could be rewritten in this
formalism. Although there is no generally accepted definition for angular momentum at null infinity in
general spacetimes, in stationary spacetimes there is. It is the unit sphere integral on the cut of the
leading term of the Weyl spinor component
, weighted by appropriate (spin weighted) spherical
harmonics:
In the weak field approximation of general relativity [382, 22, 387, 313
, 227] the gravitational field is
described by a symmetric tensor field
on Minkowski spacetime
, and the dynamics of the
field
is governed by the linearized Einstein equations, i.e. essentially the wave equation.
Therefore, the tools and techniques of the Poincaré-invariant field theories, in particular the
Noether-Belinfante-Rosenfeld procedure outlined in Section 2.1 and the ten Killing vectors of the
background Minkowski spacetime, can be used to construct the conserved quantities. It turns out that
the symmetric energy-momentum tensor of the field
is essentially the second order term
in the Einstein tensor of the metric
. Thus in the linear approximation the
field
does not contribute to the global energy-momentum and angular momentum of
the matter + gravity system, and hence these quantities have the form (5
) with the linearized
energy-momentum tensor of the matter fields. However, as we will see in Section 7.1.1, this
energy-momentum and angular momentum can be re-expressed as a charge integral of the (linearized)
curvature [349
, 206
, 313
].
pp-waves spacetimes are defined to be those that admit a constant null vector field , and they are
interpreted as describing pure plane-fronted gravitational waves with parallel rays. If matter is present then
it is necessarily pure radiation with wavevector
, i.e.
holds [243]. A remarkable feature of
the pp-wave metrics is that, in the usual coordinate system, the Einstein equations become a two
dimensional linear equation for a single function. In contrast to the approach adopted almost exclusively,
Aichelburg [3] considered this field equation as an equation for a boundary value problem. As we will see,
from the point of view of the quasi-local observables this is a particularly useful and natural standpoint. If a
pp-wave spacetime admits an additional spacelike Killing vector
with closed
orbits,
i.e. it is cyclically symmetric too, then
and
are necessarily commuting and are
orthogonal to each other, because otherwise an additional timelike Killing vector would also be
admitted [351].
Since the final state of stellar evolution (the neutron star or the black hole state) is expected to be
described by an asymptotically flat stationary, axi-symmetric spacetime, the significance of these spacetimes
is obvious. It is conjectured that this final state is described by the Kerr-Newman (either outer or black
hole) solution with some well-defined mass, angular momentum and electric charge parameters [387]. Thus
axi-symmetric 2-surfaces in these solutions may provide domains which are general enough but for which
the quasi-local quantities are still computable. According to a conjecture by Penrose [305
],
the (square root of the) area of the event horizon provides a lower bound for the total ADM
energy. For the Kerr-Newman black hole this area is
. Thus,
particularly interesting 2-surfaces in these spacetimes are the spacelike cross sections of the event
horizon [62].
Update
There is a well-defined notion of total energy-momentum not only in the asymptotically flat, but even in
the asymptotically anti-de-Sitter spacetimes too. This is the Abbott-Deser energy [1], whose positivity has
also been proven under similar conditions that we had to impose in the positivity proof of the ADM
energy [161]. (In the presence of matter fields, e.g. a self-interacting scalar field, the fall-off properties of
the metric can be weakened such that the ‘charges’ defined at infinity and corresponding to the asymptotic
symmetry generators remain finite [198].) The conformal technique, initiated by Penrose, is used to give a
precise definition of the asymptotically anti-de-Sitter spacetimes and to study their general, basic
properties in [27
]. A comparison and analysis of the various definitions of mass for asymptotically
anti-de-Sitter metrics is given in [117]. Thus it is natural to ask whether a specific quasi-local
energy-momentum expression is able to reproduce the Abbott-Deser energy-momentum in this limit or
not.
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