There are several mathematically inequivalent definitions of asymptotic flatness at spatial
infinity [151, 344
, 23
, 48
, 148]. The traditional definition is based on the existence of a certain
asymptotically flat spacelike hypersurface. Here we adopt this definition, which is probably the weakest one
in the sense that the spacetimes that are asymptotically flat in the sense of any reasonable
definition are asymptotically flat in the traditional sense too. A spacelike hypersurface
will be
called
-asymptotically flat if for some compact set
the complement
is
diffeomorphic to
minus a solid ball, and there exists a (negative definite) metric
on
,
which is flat on
, such that the components of the difference of the physical and the
background metrics,
, and of the extrinsic curvature
in the
-Cartesian
coordinate system
fall off as
and
, respectively, for some
and
[319
, 47
]. These conditions make it possible to introduce the notion of asymptotic spacetime
Killing vectors, and to speak about asymptotic translations and asymptotic boost rotations.
together with the metric and extrinsic curvature is called the asymptotic end of
.
In a more general definition of asymptotic flatness
is allowed to have finitely many such
ends.
As is well known, finite and well-defined ADM energy-momentum [11, 13, 12, 14] can be associated
with any -asymptotically flat spacelike hypersurface if
by taking the value on the
constraint surface of the Hamiltonian
, given for example in [319
, 47
], with the asymptotic
translations
(see [112, 37, 291, 113]). In its standard form this is the
limit of
a 2-surface integral of the first derivatives of the induced 3-metric
and of the extrinsic
curvature
for spheres of large coordinate radius
. The ADM energy-momentum is an
element of the space dual to the space of the asymptotic translations, and transforms as a
Lorentzian 4-vector with respect to asymptotic Lorentz transformations of the asymptotic Cartesian
coordinates.
The traditional ADM approach to the introduction of the conserved quantities and the Hamiltonian
analysis of general relativity is based on the 3 + 1 decomposition of the fields and the spacetime. Thus it is
not a priori clear that the energy and spatial momentum form a Lorentz vector (and the spatial angular
momentum and centre-of-mass, discussed below, form an anti-symmetric tensor). One had to check a
posteriori that the conserved quantities obtained in the 3 + 1 form are, in fact, Lorentz-covariant. To obtain
manifestly Lorentz-covariant quantities one should not do the 3 + 1 decomposition. Such a
manifestly Lorentz-covariant Hamiltonian analysis was suggested first by Nester [280], and
he was able to recover the ADM energy-momentum in a natural way (see also Section 11.3
below).
Another form of the ADM energy-momentum is based on Møller’s tetrad superpotential [163]: Taking
the flux integral of the current
on the spacelike hypersurface
, by Equation (11
) the
flux can be rewritten as the
limit of the 2-surface integral of Møller’s superpotential on
spheres of large
with the asymptotic translations
. Choosing the tetrad field
to be
adapted to the spacelike hypersurface and assuming that the frame
tends to a constant
Cartesian one as
, the integral reproduces the ADM energy-momentum. The same expression
can be obtained by following the familiar Hamiltonian analysis using the tetrad variables too:
By the standard scenario one can construct the basic Hamiltonian [282]. This Hamiltonian,
evaluated on the constraints, turns out to be precisely the flux integral of
on
.
A particularly interesting and useful expression for the ADM energy-momentum is possible if
the tetrad field is considered to be a frame field built from a normalized spinor dyad ,
, on
which is asymptotically constant (see Section 4.2.3 below). (Thus underlined
capital Roman indices are concrete name spinor indices.) Then, for the components of the ADM
energy-momentum in the constant spinor basis at infinity, Møller’s expression yields the limit of
The ADM energy-momentum can also be written as the 2-sphere integral of certain parts of the
conformally rescaled spacetime curvature [15, 16, 28]. This expression is a special case of the more general
‘Riemann tensor conserved quantities’ (see [163
]): If
is any closed spacelike 2-surface with area
element
, then for any tensor fields
and
one can form the integral
If the spacetime is stationary, then the ADM energy can be recovered as the limit of the
2-sphere integral of Komar’s superpotential with the Killing vector
of stationarity [163
], too. On the
other hand, if the spacetime is not stationary then, without additional restriction on the asymptotic time
translation, the Komar expression does not reproduce the ADM energy. However, by Equations (13
,
14
) such an additional restriction might be that
should be a constant combination of
four future pointing null vector fields of the form
, where the spinor fields
are
required to satisfy the Weyl neutrino equation
. This expression for the ADM
energy-momentum was used to give an alternative, ‘4-dimensional’ proof of the positivity of the ADM
energy [205
].
The value of the Hamiltonian of Beig and Ó Murchadha [47] together with the appropriately defined
asymptotic rotation-boost Killing vectors [364
] define the spatial angular momentum and centre-of-mass,
provided
and, in addition to the familiar fall-off conditions, certain global integral conditions are
also satisfied. These integral conditions can be ensured by the explicit parity conditions of Regge and
Teitelboim [319
] on the leading nontrivial parts of the metric
and extrinsic curvature
: The
components in the Cartesian coordinates
of the former must be even and the components of latter
must be odd parity functions of
(see also [47
]). Thus in what follows we assume that
. Then
the value of the Beig-Ó Murchadha Hamiltonian parameterized by the asymptotic rotation
Killing vectors is the spatial angular momentum of Regge and Teitelboim [319
], while that
parameterized by the asymptotic boost Killing vectors deviate from the centre-of-mass of Beig
and Ó Murchadha [47
] by a term which is the spatial momentum times the coordinate time.
(As Beig and Ó Murchadha pointed out [47
], the centre-of-mass of Regge and Teitelboim is
not necessarily finite.) The spatial angular momentum and the new centre-of-mass form an
anti-symmetric Lorentz 4-tensor, which transforms in the correct way under the 4-translation of the
origin of the asymptotically Cartesian coordinate system, and it is conserved by the evolution
equations [364
].
The centre-of-mass of Beig and Ó Murchadha was reexpressed recently [42] as the
limit of
2-surface integrals of the curvature in the form (15
) with
proportional to the lapse
times
, where
is the induced 2-metric on
(see Section 4.1.1
below).
A geometric notion of centre-of-mass was introduced by Huisken and Yau [209]. They foliate the
asymptotically flat hypersurface by certain spheres with constant mean curvature. By showing the
global uniqueness of this foliation asymptotically, the origin of the leaves of this foliation in some flat
ambient Euclidean space
defines the centre-of-mass (or rather ‘centre-of-gravity’) of Huisken and Yau.
However, no statement on its properties is proven. In particular, it would be interesting to see
whether or not this notion of centre-of-mass coincides, for example, with that of Beig and Ó
Murchadha.
The Ashtekar-Hansen definition for the angular momentum is introduced in their specific conformal
model of the spatial infinity as a certain 2-surface integral near infinity. However, their angular momentum
expression is finite and unambiguously defined only if the magnetic part of the spacetime curvature tensor
(with respect to the timelike level hypersurfaces of the conformal factor) falls off faster than
would follow from the
fall-off of the metric (but they do not have to impose any global integral, e.g. a
parity condition) [23
, 15].
If the spacetime admits a Killing vector of axi-symmetry, then the usual interpretation of the
corresponding Komar integral is the appropriate component of the angular momentum (see for
example [387]). However, the value of the Komar integral is twice the expected angular momentum. In
particular, if the Komar integral is normalized such that for the Killing field of stationarity in the Kerr
solution the integral is
, for the Killing vector of axi-symmetry it is
instead of the expected
(‘factor-of-two anomaly’) [229
]. We return to the discussion of the Komar integral in
Section 12.1.
The study of the gravitational radiation of isolated sources led Bondi to the observation that
the 2-sphere integral of a certain expansion coefficient of the line element of a
radiative spacetime in an asymptotically retarded spherical coordinate system
behaves
as the energy of the system at the retarded time
: This notion of energy is not constant
in time, but decreases with
, showing that gravitational radiation carries away positive
energy (‘Bondi’s mass-loss’) [71, 72]. The set of transformations leaving the asymptotic form of
the metric invariant was identified as a group, nowadays known as the BMS group, having a
structure very similar to that of the Poincaré group [325]. The only difference is that while the
Poincaré group is a semidirect product of the Lorentz group and a 4-dimensional commutative
group (of translations), the BMS group is the semidirect product of the Lorentz group and an
infinite-dimensional commutative group, called the group of the supertranslations. A 4-parameter
subgroup in the latter can be identified in a natural way as the group of the translations. Just at
the same time the study of asymptotic solutions of the field equations led Newman and Unti
to another concept of energy at null infinity [290
]. However, this energy (nowadays known
as the Newman-Unti energy) does not seem to have the same significance as the Bondi (or
Bondi-Sachs [313
] or Trautman-Bondi [115
, 116
, 114
]) energy, because its monotonicity can be proven
only between special, e.g. stationary, states. The Bondi energy, which is the time component of a
Lorentz vector, the so-called Bondi-Sachs energy-momentum, has a remarkable uniqueness
property [115, 116].
Without additional conditions on , Komar’s expression does not reproduce the Bondi-Sachs
energy-momentum in non-stationary spacetimes either [395
, 163]: For the ‘obvious’ choice for
Komar’s expression yields the Newman-Unti energy. This anomalous behaviour in the radiative regime
could be corrected in, at least, two ways. The first is by modifying the Komar integral according to
The Bondi-Sachs energy-momentum can also be expressed by the integral of the Nester-Witten
2-form [214, 255
, 256
, 205
]. However, in non-stationary spacetimes the spinor fields that are
asymptotically constant at null infinity are vanishing [83
]. Thus the spinor fields in the Nester-Witten
2-form must satisfy a weaker boundary condition at infinity such that the spinor fields themselves be the
spinor constituents of the BMS translations. The first such condition, suggested by Bramson [83
], was to
require the spinor fields to be the solutions of the so-called asymptotic twistor equation (see Section 4.2.4).
One can impose several such inequivalent conditions, and all these, based only on the linear first order
differential operators coming from the two natural connections on the cuts (see Section 4.1.2), are
determined in [363
].
The Bondi-Sachs energy-momentum has a Hamiltonian interpretation as well. Although the fields on a
spacelike hypersurface extending to null rather than spatial infinity do not form a closed system, a suitable
generalization of the standard Hamiltonian analysis could be developed [114] and used to recover the
Bondi-Sachs energy-momentum.
Similarly to the ADM case, the simplest proofs of the positivity of the Bondi energy [330] are probably
those that are based on the Nester-Witten 2-form [214] and, in particular, the use of two-components
spinors [255, 256, 205, 203, 321]: The Bondi-Sachs mass (i.e. the Lorentzian length of the
Bondi-Sachs energy-momentum) of a cut of future null infinity is non-negative if there is a
spacelike hypersurface intersecting null infinity in the given cut such that the dominant energy
condition is satisfied on
, and the mass is zero iff the domain of dependence
of
is
flat.
At null infinity there is no generally accepted definition for angular momentum, and there are various, mathematically inequivalent suggestions for it. Here we review only some of those total angular momentum definitions that can be considered as the null infinity limit of some quasi-local expression, and will be discussed in the main part of the review, namely in Section 9.
In their classic paper Bergmann and Thomson [60] raise the idea that while the gravitational
energy-momentum is connected with the spacetime diffeomorphisms, the angular momentum should be
connected with its intrinsic symmetry. Thus, the angular momentum should be analogous with
the spin. Based on the tetrad formalism of general relativity and following the prescription of constructing
the Noether currents in Yang-Mills theories, Bramson suggested a superpotential for the six conserved
currents corresponding to the internal Lorentz-symmetry [84
, 85, 86
]. (For another derivation of this
superpotential from Møller’s Lagrangian (9
) see [363
].) If
,
, is a normalized spinor
dyad corresponding to the orthonormal frame in Equation (9
), then the integral of the spinor
form of the anti-self-dual part of this superpotential on a closed orientable 2-surface
is
The construction based on the Winicour-Tamburino linkage (16) can be associated with any BMS
vector field [395, 252, 30]. In the special case of translations it reproduces the Bondi-Sachs
energy-momentum. The quantities that it defines for the proper supertranslations are called the
super-momenta. For the boost-rotation vector fields they can be interpreted as angular momentum.
However, in addition to the factor-of-two anomaly, this notion of angular momentum contains a huge
ambiguity (‘supertranslation ambiguity’): The actual form of both the boost-rotation Killing vector fields of
Minkowski spacetime and the boost-rotation BMS vector fields at future null infinity depend on
the choice of origin, a point in Minkowski spacetime and a cut of null infinity, respectively.
However, while the set of the origins of Minkowski spacetime is parameterized by four numbers,
the set of the origins at null infinity requires a smooth function of the form
.
Consequently, while the corresponding angular momentum in the Minkowski spacetime has the familiar
origin-dependence (containing four parameters), the analogous transformation of the angular momentum
defined by using the boost-rotation BMS vector fields depends on an arbitrary smooth real
valued function on the 2-sphere. This makes the angular momentum defined at null infinity by
the boost-rotation BMS vector fields ambiguous unless a natural selection rule for the origins,
making them form a four parameter family of cuts, is found. Such a selection rule could be the
suggestion by Dain and Moreschi [125] in the charge integral approach to angular momentum of
Moreschi [272, 273].
Another promising approach might be that of ChruĊciel, Jezierski, and Kijowski [114], which is based on a Hamiltonian analysis of general relativity on asymptotically hyperbolic spacelike hypersurfaces. They chose the six BMS vector fields tangent to the intersection of the spacelike hypersurface and null infinity as the generators of their angular momentum. Since the motions that their angular momentum generators define leave the domain of integration fixed, and apparently there is no Lorentzian 4-space of origins, they appear to be the generators with respect to some fixed ‘centre-of-the-cut’, and the corresponding angular momentum appears to be the intrinsic angular momentum.
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