If the spacetime is spherically symmetric, then a two-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 called the area-coordinate.) Then,
with the notation of Section 4.1, one obtains
. Based on the
investigations of Misner, Sharp, and Hernandez [365
, 267
], Cahill and McVitte [122] found
contains a contribution from the gravitational ‘field’ too. For example, for fluids it is
not simply the volume integral of the energy density
of the fluid, because that would be
. This deviation can be interpreted as the contribution of the gravitational potential
energy to the total energy. Consequently,
is not a globally monotonic function of
,
even if
. For example, in the closed Friedmann–Robertson–Walker spacetime (where,
to cover the whole three-space,
cannot be chosen to be the area–radius and
)
is increasing for
, taking its maximal value at
, and decreasing for
.
This example suggests a slightly more exotic spherically-symmetric spacetime. Its spacelike slice will
be assumed to be extrinsically flat, and its intrinsic geometry is the matching of two conformally flat
metrics. The first is a ‘large’ spherically-symmetric part of a
hypersurface of the closed
Friedmann–Robertson–Walker spacetime with the line element
, where
is the line
element for the flat three-space and
with positive constants
and
, and the
range of the Euclidean radial coordinate
is
, where
. It contains a maximal
two-surface at
with round-sphere mass parameter
. The scalar
curvature is
, and hence, by the constraint parts of the Einstein equations and by the
vanishing of the extrinsic curvature, the dominant energy condition is satisfied. The other metric is the
metric of a piece of a
hypersurface in the Schwarzschild solution with mass parameter
(see [213
]):
, where
and the Euclidean radial coordinate
runs from
to
, where
. In this geometry there is a minimal surface at
, the scalar
curvature is zero, and the round-sphere energy is
. These two metrics can be matched to
obtain a differentiable metric with a Lipschitz-continuous derivative at the two-surface of the matching
(where the scalar curvature has a jump), with arbitrarily large ‘internal mass’
and arbitrarily
small ADM mass
. (Obviously, the two metrics can be joined smoothly, as well, by an
‘intermediate’ domain between them.) Since this space looks like a big spherical bubble on a
nearly flat three-plane – like the capital Greek letter
– for later reference we will call it
an‘
-spacetime’.
Spherically-symmetric spacetimes admit a special vector field, called the Kodama vector field , such
that
is divergence free [321
]. 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 in the line element given in the
first paragraph above
. If
is a solid ball of radius
, then
the flux of
is precisely the standard round-sphere expression (4.7
) for the two-sphere
[375].
An interesting characterization of the dynamics of the spherically-symmetric gravitational fields can be
given in terms of the energy function given by (4.7
) (or by (4.8
)) (see, e.g., [578
, 352
, 250
]). In
particular, criteria for the existence and formation of trapped surfaces and for the presence and nature of
the central singularity can be given by
. Other interesting quasi-locally–defined quantities
are introduced and used to study nonlinear perturbations and backreaction in a wide class of
spherically-symmetric spacetimes in [483]. For other applications of
in cosmology see, e.g.,
[484, 130].
In the literature there are two kinds of small surfaces. The first is that of the small spheres (both in the
light cone of a point and in a spacelike hypersurface), introduced first by Horowitz and Schmidt [275],
and the other is the concept of small ellipsoids in a spacelike hypersurface, considered first
by Woodhouse in [313
]. A small sphere in the light cone is a cut of the future null cone in
the spacetime by a spacelike hypersurface, and the geometry of the sphere is characterized by
data at the vertex of the cone. The sphere in a hypersurface consists of those points of a given
spacelike hypersurface, whose geodesic distance in the hypersurface from a given point
,
the center, is a small given value, and the geometry of this sphere is characterized by data at
this center. Small ellipsoids are two-surfaces in a spacelike hypersurface with a more general
shape.
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 parametrization completely. Then, in an open neighborhood
of the vertex
,
is a smooth null hypersurface, and hence, for sufficiently small
, the set
is a smooth spacelike two-surface and is homeomorphic to
.
is called a
small sphere of radius
with vertex
. Note that the condition
fixes the boost gauge,
too.
Completing to get a Newman–Penrose complex null tetrad
such that the
complex null vectors
and
are tangent to the two-surfaces
, the components of
the metric and the spin coefficients with respect to this basis can be expanded as a series in
. If, in addition, the spinor constituent
of
is required to be parallelly
propagated along
, then the tetrad becomes completely fixed, yielding the vanishing of
several (combinations of the) spin coefficients. 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 the neighborhood of
or
not.5
These solutions of the GHP equations, with increasing accuracy, are given in [275
, 313
, 118
, 494
].
Obviously, we can calculate the small-sphere limit of various quasi-local quantities built from the matter
fields in the Minkowski spacetime, as well. In particular [494], 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
, are, respectively,
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 spacetime metric, then in its expansion the difference of the power of
and the number of the derivatives in every term must be one, i.e., it must have the form
If the neighborhood 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 linearly on the latter two,
is the zero tensor field. Thus, the
-order term in vacuum is also vanishing. At the fifth
order the only nonzero 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 the energy-momentum,
has the structure
, for angular momentum it is
, while the area of
is
. Therefore, the leading term in the expansion of the angular momentum is
and
order in nonvacuum and vacuum with the energy-momentum and the Bel–Robinson tensors, respectively,
while the first nontrivial correction to the area
is of order
and
in nonvacuum 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 [313]. If
is any smooth function on
with a
nondegenerate minimum at
with minimum value
, then, at least on an open
neighborhood
of
in
, the level surfaces
are smooth
compact two-surfaces homeomorphic to
. Then, in the
limit, the surfaces
look
like small nested ellipsoids centered at
. The function
is usually ‘normalized’ so that
.
A slightly different framework for calculations in small regions was used in [327, 170
, 235
]. Instead of
the Newman–Penrose (or the GHP) formalism and the spin coefficient equations, holonomic (Riemann or
Fermi type normal) coordinates on an open neighborhood
of a point
or a timelike curve
are used, in which the metric, as well as the Christoffel symbols on
, are expressed by the coordinates on
and the components of the Riemann tensor at
or on
. In these coordinates and the
corresponding frames, the various pseudotensorial and tetrad expressions for the energy-momentum
have been investigated. It has been shown that a quadratic expression of these coordinates
with the Bel–Robinson tensor as their coefficient appears naturally in the local conservation
law for the matter energy-momentum tensor [327]; the Bel–Robinson tensor can be recovered
as some ‘double gradient’ of a special combination of the Einstein and the Landau–Lifshitz
pseudotensors [170]; Møller’s tetrad expression, as well as certain combinations of several other classical
pseudotensors, yield the Bel–Robinson tensor [473, 470, 471]. In the presence of some non-dynamical
(background) metric a 11-parameter family of combinations of the classical pseudotensors exists,
which, in vacuum, yields the Bel–Robinson tensor [472, 474]. (For this kind of investigation see
also [465, 468, 466, 467, 469]).
In [235] a new kind of approximate symmetries, namely approximate affine collineations, are introduced
both near a point and a world line, and used to introduce Komar-type ‘conserved’ currents. (For a readable
text on the non-Killing type symmetries see, e.g., [233].) These symmetries turn out to yield a
nontrivial gravitational contribution to the matter energy-momentum, even in the leading
order.
Near spatial infinity we have the a priori and
falloff for the three-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
legitimate.6
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 falloff conditions yield that the spin coefficients tend to their flat spacetime value as and the
curvature components to zero like
. Expanding the spin coefficients and curvature components as a
power series of
, one can solve the field equations asymptotically (see [65
, 61] 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 nontrivial 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 falloff properties
.
Thus, we can define the asymptotically constant spinor fields to be those
that satisfy
, where
is the intrinsic Levi-Civita derivative operator on
. Note
that this implies that, with the notation of Eq. (4.6
), 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 [413, 414, 415, 426]
(see also [208]), 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 origin cut
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 neighborhood of
in the spacetime in the
following way. Let
be the family of smooth outgoing null hypersurfaces in a neighborhood 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
,
two-surfaces
(or simply
if no confusion can
arise) are spacelike topological two-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’.7
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 a detailed discussion
of the coordinate freedom left at the various stages in the introduction of these coordinate systems, see, for
example, [394
, 393
, 107].
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 [394
] chose
to be parallelly propagated along
. This choice yields the vanishing of a
number of spin coefficients (see, for example, the review [393
]). The asymptotic solution of
the Einstein–Maxwell equations as a series of
in this coordinate and tetrad system is
given in [394, 179, 425
], where all the nonvanishing 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, the
rotation with the highest accuracy was done for the solutions of the Einstein–Maxwell system by
Shaw [455
].
In contrast to the spatial-infinity case, the ‘natural’ definition of the asymptotically constant spinor
fields yields identically zero spinors in general [106]. 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 [106], half of these conditions can be
imposed. Namely, at future null infinity
(and at past null infinity
) can always be imposed asymptotically, and 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
) [496
].
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 antiholomorphic 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 two-surface twistor equation
(see also
Section 7.2.1). These operators, together with others reproducing the asymptotic spinors, are discussed
in [496
].
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, [393, 426
]):
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 and in axi-symmetric spacetimes there is. The former is the unit sphere
integral on the cut of the leading term of the Weyl spinor component
, weighted by appropriate
(spin-weighted) spherical harmonics:
Instead of the Bondi type coordinates above, one can introduce other ‘natural’ coordinates in a
neighborhood of . Such is the one based on the outgoing asymptotically–shear-free null geodesics [27
].
While the Bondi-type coordinate system is based on the null geodesic generators of the outgoing null
hypersurfaces
, and hence, in the rescaled metric these generators are orthogonal to the cuts
, the
new coordinate system is based on the use of outgoing null geodesic congruences that extend to
but are not orthogonal to the cuts of
(and hence, in general, they have twist). The
definition of the new coordinates
is analogous to that of the Bondi-type coordinates:
labels the intersection point of the actual geodesic and
, while
is the affine
parameter along the geodesic. The tangent
of this null congruence is asymptotically null rotated
about
: In the NP basis
above
, where
and
is a complex valued function (with spin
weight one) on
. Then Aronson and Newman show in [27
] that if
is chosen to satisfy
, then the asymptotic shear of the congruence is, in fact, of order
, and by an
appropriate choice for the other vectors of the NP basis many spin coefficients can be made
zero. In this framework it is the function
that plays a role analogous to that of
, and,
indeed, the asymptotic solution of the field equations is given in terms of
in [27]. This
can be derived from the solution
of the good-cut equation, which, however, is not
uniquely determined, but depends on four complex parameters:
. It is this
freedom that is used in [325
, 326
] to introduce the angular momentum at future null infinity (see
Section 3.2.4). Further discussion of these structures, in particular their connection with the
solutions of the good-cut equation and the
-space, as well as their applications, is given
in [324, 325, 326, 5].
In the weak field approximation of general relativity [525, 36, 534, 426
, 303] 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 (2.5
) 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 [481
, 277
, 426
].
pp-waves spacetimes are defined to be those that admit a constant null vector field , and they
interpreted as describing pure plane-fronted gravitational waves with parallel rays. If matter is present, then
it is necessarily pure radiation with wave-vector
, i.e.,
holds [478]. 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 [8] 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 [485].
Since the final state of stellar evolution (the neutron star or black hole state) is expected to be described
by an asymptotically flat, stationary, axisymmetric 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 [534]. Thus,
axisymmetric two-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 [418
],
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 two-surfaces in these spacetimes are the spacelike cross sections of the event
horizon [80].
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 as well. 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 [220]. (In the presence of matter fields, e.g., a self-interacting scalar
field, the falloff properties of the metric can be weakened such that the ‘charges’ defined at
infinity and corresponding to the asymptotic symmetry generators remain finite [265].) 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 [42]. A comparison and
analysis of the various definitions of mass for asymptotically anti-de Sitter metrics is given
in [150].
Extending the spinorial proof [349] of the positivity of the total energy in asymptotically anti-de Sitter spacetime, Chruลciel, Maerten and Tod [149] give an upper bound for the angular momentum and center-of-mass in terms of the total mass and the cosmological constant. (Analogous investigations show that there is a similar bound at the future null infinity of asymptotically flat spacetimes with no outgoing energy flux, provided the spacetime contains a constant–mean-curvature, hyperboloidal, initial-data set on which the dominant energy condition is satisfied. In this bound the role of the cosmological constant is played by the (constant) mean curvature of the hyperboloidal spacelike hypersurface [151].) Thus, it is natural to ask whether or not a specific quasi-local energy-momentum or angular momentum expression has the correct limit for large spheres in asymptotically anti-de Sitter spacetimes.
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