There are several mathematically-inequivalent definitions of asymptotic flatness at spatial
infinity [208, 475
, 37
, 65
, 200]. 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 as well. 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
[433
, 64
]. 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 [23, 25, 24, 26] 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 [433
, 64
], with the asymptotic translations
(see [144, 52, 399, 145]). In its standard form, this is the
limit of a two-surface integral of
the first derivatives of the induced three-metric
and of the extrinsic curvature
for spheres
of large coordinate radius
. Explicitly:
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 center-of-mass, discussed below, form an anti-symmetric tensor). One
has 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 [377], and he was able to recover the ADM energy-momentum in a natural way (see
Section 11.3).
Another form of the ADM energy-momentum is based on Møller’s tetrad superpotential [223]: Taking
the flux integral of the current
on the spacelike hypersurface
, by Eq. (3.7
) the flux
can be rewritten as the
limit of the two-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 [379]. 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). (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 two-sphere integral of certain parts of the
conformally rescaled spacetime curvature [28, 29, 43]. This expression is a special case of the more general
‘Riemann tensor conserved quantities’ (see [223
]): If
is any closed spacelike two-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 at the limit of the
two-sphere integral of (twice of) Komar’s superpotential with the Killing vector
of stationarity [223
]
(see also [60
]), as well. (See also the remark following Eq. (3.15
) below.) 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 Eqs. (3.11
) and (3.12
) 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 has been used to give an
alternative, ‘four-dimensional’ proof of the positivity of the ADM energy [276
]. (For a more
detailed recent review of the various forms of the ADM energy and linear momentum, see, e.g.,
[293
].)
In stationary spacetime the notion of the mechanical energy with respect to the world lines of stationary observers (i.e., the integral curves of the timelike Killing field) can be introduced in a natural way, and then (by definition) the total (ADM) energy is written as the sum of the mechanical energy and the gravitational energy. Then the latter is shown to be negative for certain classes of systems [308, 348].
The notion of asymptotic flatness at spatial infinity is generalized in [398]; here the background flat
metric on
is allowed to have a nonzero deficit angle
at infinity, i.e., the corresponding
line element in spherical polar coordinates takes the form
. Then, a
canonical analysis of the minimally-coupled Einstein–Higgs field is carried out on such a background, and,
following a Regge-Teitelboim–type argumentation, an ADM-type total energy is introduced. It is shown
that for appropriately chosen
this energy is finite for the global monopole solution, though the standard
ADM energy is infinite.
The value of the Hamiltonian of Beig and Ó Murchadha [64], together with the appropriately-defined
asymptotic rotation-boost Killing vectors [497
], define the spatial angular momentum and center-of-mass,
provided
and, in addition to the familiar falloff conditions, certain global integral conditions are
also satisfied. These integral conditions can be ensured by the explicit parity conditions of
Regge and Teitelboim [433
] 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 the latter must be odd parity functions of
(see also [64
]).
Thus, in what follows we assume that
. Then the value of the Beig–Ó Murchadha
Hamiltonian parametrized by the asymptotic rotation Killing vectors is the spatial angular
momentum of Regge and Teitelboim [433
], while that parametrized by the asymptotic boost
Killing vectors deviates from the center-of-mass of Beig and Ó Murchadha [64
] by a term,
which is the spatial momentum times the coordinate time. (As Beig and Ó Murchadha pointed
out [64
], the center-of-mass term of the Hamiltonian of Regge and Teitelboim is not finite on
the whole phase space.) The spatial angular momentum and the new center-of-mass form an
anti-symmetric Lorentz four-tensor, which transforms in the correct way under the four-translation of the
origin of the asymptotically Cartesian coordinate system, and is conserved by the evolution
equations [497].
The center-of-mass of Beig and Ó Murchadha was re-expressed recently [57] as the
limit of
two-surface integrals of the curvature in the form (3.14
) with
proportional to the lapse
times
, where
is the induced two-metric on
(see Section 4.1.1). The geometric notion of
center-of-mass introduced by Huisken and Yau [280] is another form of the Beig–Ó Murchadha
center-of-mass [156].
The Ashtekar–Hansen definition for the angular momentum is introduced in their specific conformal
model of spatial infinity as a certain two-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 it
would fall off in metrics with
falloff (but no global integral, e.g., a parity condition had to be
imposed) [37
, 28].
If the spacetime admits a Killing vector of axisymmetry, then the usual interpretation of the
corresponding Komar integral is the appropriate component of the angular momentum (see, e.g., [534]).
However, the value of the Komar integral (with the usual normalization) 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 axisymmetry it is
instead of
the expected
(‘factor-of-two anomaly’) [305
]. We return to the discussion of the Komar integral in
Sections 4.3.1 and 12.1.
The study of the gravitational radiation of isolated sources led Bondi to the observation that
the two-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
. Indeed, this energy is not constant in
time, but decreases with
, showing that gravitational radiation carries away positive energy
(‘Bondi’s mass-loss’) [91, 92]. The set of transformations leaving the asymptotic form of the metric
invariant was identified as a group, currently known as the Bondi–Metzner–Sachs (or BMS) group,
having a structure very similar to that of the Poincaré group [440]. The only difference is that
while the Poincaré group is a semidirect product of the Lorentz group and a four 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
four-parameter subgroup in the latter can be identified in a natural way as the group of the translations.
This makes it possible to compare the Bondi–Sachs four-momenta defined on different cuts of
scri, and to calculate the energy-momentum carried away by the gravitational radiation in an
unambiguous way. (For further discussion of the flux, see the fourth paragraph of Section 3.2.4.)
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 [394
]. However, this energy (currently known
as the Newman–Unti energy) does not seem to have the same significance as the Bondi (or
Bondi–Sachs [426
] or Trautman–Bondi [147
, 148
, 146
]) 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 Bondi–Sachs energy-momentum, has a remarkable uniqueness
property [147, 148].
Without additional conditions on , Komar’s expression does not reproduce the Bondi–Sachs
energy-momentum in nonstationary spacetimes either [557
, 223]: For the ‘obvious’ choice for
(twice
of) Komar’s expression yields the Newman–Unti energy. This anomalous behavior 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 [285, 342
, 343
, 276
]. However, in nonstationary spacetimes the spinor fields that are asymptotically
constant at null infinity are vanishing [106
]. Thus, the spinor fields in the Nester–Witten 2-form must
satisfy a weaker boundary condition at infinity such that the spinor fields themselves are the spinor
constituents of the BMS translations. The first such condition, suggested by Bramson [106
], was to require
the spinor fields to be the solutions of the asymptotic twistor equation (see Section 4.2.4). One can impose
several such inequivalent conditions, and all of 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 [496
].
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 [146] and used to recover the
Bondi–Sachs energy-momentum.
Similar to the ADM case, the simplest proofs of the positivity of the Bondi energy [446] are probably
those that are based on the Nester–Witten 2-form [285] and, in particular, the use of two-component
spinors [342, 343, 276, 274, 436]: 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.
Update
Converting the integral of the Nester–Witten 2-form into a (positive definite) 3-dimensional integral on
, a strictly positive lower bound can be given both for the ADM and Bondi–Sachs masses.
Although total energy-momentum (or mass) in the form of a two-surface integral cannot be a
introduced in closed universes (i.e., when
is compact with no boundary), a non-negative
quantity
, based on this positive definite expression, can be associated with
. If the
matter fields satisfy the dominant energy condition, then
if and only if the spacetime is
flat and topologically
is a 3-torus; moreover its vanishing is equivalent to the existence of
non-trivial solutions of Witten’s gauge condition. This
turned out to be recoverable as the first
eigenvalue of the square of the Sen–Witten operator. It is the usefulness and the applicability of this
in practice which tell us if this is a reasonable notion of total mass of closed universes or
not [503].
At null infinity we have a generally accepted definition for angular momentum only in stationary or axi-symmetric, but not in general, radiative spacetime, where there are various, mathematically inequivalent suggestions for it (see Section 4.2.4). Here we review only some of those total angular momentum definitions that can be ‘quasi-localized’ or connected somehow to quasi-local expressions, i.e., those that can be considered as the null-infinity limit of some quasi-local expression. We will continue their discussion in the main part of the review, namely in Sections 7.2, 11.1 and 9.
In their classic paper Bergmann and Thomson [78] 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 [107
, 108, 109
]. (For another derivation of this
superpotential from Møller’s Lagrangian (3.5
) see [496
].) If
,
, is a normalized
spinor dyad corresponding to the orthonormal frame in Eq. (3.5
), then the integral of the spinor
form of the anti-self-dual part of this superpotential on a closed orientable two-surface
is
The construction based on the Winicour–Tamburino linkage (3.15) can be associated with any BMS
vector field [557, 337, 45]. 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 parametrized 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 two-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.
Motivated by Penrose’s idea that the ‘conserved’ quantities at null infinity should be searched for in the
form of a charge integral of the curvature (which will be discussed in detail in Section 7), a
general expression , associated with any BMS generator
and any cut
of scri,
was introduced [174
]. For real
this is real; it is vanishing in Minkowski spacetime; it
reproduces the Bondi–Sachs energy-momentum for BMS translations; it yields nontrivial results for
proper supertranslations; and for BMS rotations the resulting expressions can be interpreted
as angular momentum. It was shown in [453, 173] that the difference
for any two cuts
and
can be written as the integral of some local function on the
subset of scri bounded by the cuts
and
, and this is precisely the flux integral of [44].
Unfortunately, however, the angular momentum introduced in this way still suffers from the same
supertranslation ambiguity. A possible resolution of this difficulty could be the suggestion by Dain and
Moreschi [169] in the charge integral approach to angular momentum of Moreschi [369, 370]. Their
basic idea is that the requirement of the vanishing of the supermomenta (i.e., the quantities
corresponding to the proper supertranslations) singles out a four–real-parameter family of cuts,
called nice cuts, by means of which the BMS group can be reduced to a Poincaré subgroup
that yields a well-defined notion of angular momentum. For further discussion of certain other
angular momentum expressions, especially from the points of view of numerical calculations, see
also [204].
Another promising approach might be that of Chruลciel, Jezierski, and Kijowski [146], which is based on a Hamiltonian analysis of general relativity on asymptotically hyperboloidal 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 four-space of origins, they appear to be the generators with respect to some fixed ‘center-of-the-cut’, and the corresponding angular momentum appears to be the intrinsic angular momentum.
In addition to the supertranslation ambiguity in the definition of angular momentum, there could be another potential ambiguity, even if the angular momentum is well defined on every cut of future null infinity. In fact, if, for example, the definition of the angular momentum is based on the solutions of some linear partial differential equation on the cut (such as Bramson’s definition, or the ones discussed in Sections 7 and 9), then in general there is no canonical isomorphism between the spaces of the solutions on different cuts, even if the solution spaces, as abstract vector spaces, are isomorphic. Therefore, the angular momenta on two different cuts belong to different vector spaces, and, without any natural correspondence between the solution spaces on the different cuts, it is meaningless to speak about the difference of the angular momenta. Thus, we cannot say anything about, e.g., the angular momentum carried away by gravitational radiation between two retarded time instants represented by two different cuts.
One possible resolution of this difficulty was suggested by Helfer [264]. He followed the twistorial
approach presented in Section 7 and used a special bijective map between the two-surface twistor spaces on
different cuts. His map is based on the special structures available only at null infinity. Though this map is
nonlinear, it is shown that the angular momenta on the different cuts can indeed be compared.
Another suggestion for (only) the spatial angular momentum was given in [501
]. This is based
on the quasi-local Hamiltonian analysis that is discussed in Section 11.1, and the use of the
divergence-free vector fields built from the eigenspinors with the smallest eigenvalue of the
two-surface Dirac operators. The angular momenta, defined in these ways on different cuts, can also
be compared. We give a slightly more detailed discussion of them in Sections 7.2 and 11.1,
respectively.
The main idea behind the recent definition of the total angular momentum at future null infinity of
Kozameh, Newman and Silva-Ortigoza, suggested in [325, 326
], is analogous to finding the center-of-charge
(i.e., the time-dependent position vector with respect to which the electric dipole moment is
vanishing) in flat-space electromagnetism: By requiring that the dipole part of an appropriate
null rotated Weyl tensor component
be vanishing, a preferred set of origins, namely a
(complex) center-of-mass line can be found in the four–complex-dimensional solution space
of the good-cut equation (the
-space). Then the asymptotic Bianchi identities take the
form of conservation equations, and certain terms in these can (in the given approximation)
be identified with angular momentum. The resulting expression is just Eq. (4.15
), to which
all the other reasonable angular momentum expressions are expected to reduce in stationary
spacetimes. A slightly more detailed discussion of the necessary technical background is given in
Section 4.2.4.
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