To motivate the main idea behind the Brown–York definition [120, 121
], let us first consider a classical
mechanical system of
degrees of freedom with configuration manifold
and Lagrangian
(i.e., the Lagrangian is assumed to be first order and may depend on time explicitly).
For given initial and final configurations,
and
, respectively, the corresponding action
functional is
, where
is a smooth curve in
from
to
with tangent
at
. (The pair
may be called a history or world line in
the ‘spacetime’
.) Let
be a smooth one-parameter deformation of this history,
for which
, and
for some
. Then, denoting the derivative
with respect to the deformation parameter
at
by
, one has the well known expression
The main idea of Brown and York [120, 121
] is to calculate the analogous variation of an appropriate
first-order action of general relativity (or of the coupled matter + gravity system) and isolate
the boundary term that could be analogous to the energy
above. To formulate this idea
mathematically, Brown and York considered a compact spacetime domain
with topology
such that
correspond to compact spacelike hypersurfaces
; these form a smooth
foliation of
and the two-surfaces
(corresponding to
) form a foliation
of the timelike three-boundary
of
. Note that this
is not a globally hyperbolic
domain.14
To ensure the compatibility of the dynamics with this boundary, the shift vector is usually chosen to be
tangent to
on
. The orientation of
is chosen to be outward pointing, while the normals, both
of
and of
, are chosen to be future pointing. The metric and extrinsic curvature
on
will be denoted, respectively, by
and
, and those on
by
and
.
The primary requirement of Brown and York on the action is to provide a well-defined variational
principle for the Einstein theory. This claim leads them to choose for the ‘trace
action’ (or, in the
present notation, the ‘trace
action’) for general relativity [572, 573, 534
], and the action for
the matter fields may be included. (For minimal, nonderivative couplings, the presence of the
matter fields does not alter the subsequent expressions.) However, as Geoff Hayward pointed
out [243], to have a well-defined variational principle, the ‘trace
action’ should in fact be
completed by two two-surface integrals, one on
and the other on
. Otherwise, as a
consequence of the edges
and
, called the ‘joints’ (i.e., the nonsmooth parts of the
boundary
), the variation of the metric at the points of the edges
and
could not be
arbitrary. (See also [242
, 315
, 100
, 119
], where the ‘orthogonal boundaries assumption’ is also
relaxed.) Let
and
be the scalar product of the outward-pointing normal of
and the
future-pointing normal of
and of
, respectively. Then, varying the spacetime metric
(for the variation of the corresponding principal function
) they obtained the following:
Clearly, the trace- action cannot be recovered as the volume integral of some scalar
Lagrangian, because it is the Hilbert action plus a boundary integral of the trace
, and the latter
depends on the location of the boundary itself. Such a Lagrangian was found by Pons [431]. This
depends on the coordinate system adapted to the boundary of the domain
of integration. An
interesting feature of this Lagrangian is that it is second order in the derivatives of the metric, but it
depends only on the first time derivative. A detailed analysis of the variational principle, the
boundary conditions and the conserved charges is given. In particular, the asymptotic properties of
this Lagrangian is similar to that of the
Lagrangian of Einstein, rather than to that of
Hilbert.
The 3 + 1 decomposition of the spacetime metric yields a 2 + 1 decomposition of the metric , as well.
Let
and
be the lapse and the shift of this decomposition on
. Then the corresponding
decomposition of
defines the energy, momentum, and spatial-stress surface densities according to
The quasi-local energy is still not completely determined, because the ‘subtraction term’ in the
principal function has not been specified. This term is usually interpreted as our freedom to shift the zero
point of the energy. Thus, the basic idea of fixing the subtraction term is to choose a ‘reference
configuration’, i.e., a spacetime in which we want to obtain zero quasi-local quantities
(in
particular zero quasi-local energy), and identify
with the
of the reference spacetime. Thus, by
Eq. (10.5
) and (10.6
) we obtain that
For a definition of the Brown–York energy as a quasi-local energy oparator in loop quantum gravity,
see [565].
As we noted, ,
, and
depend on the boost-gauge that the timelike boundary defines on
.
Lau clarified how these quantities change under a boost gauge transformation, where the new boost-gauge
is defined by the timelike boundary
of another domain
such that the particular
two-surface
is a leaf of the foliation of
as well [333
]. If
is another foliation
of
such that
and
is orthogonal to
, then the new
,
, and
are built from the old
,
, and
and the 2 + 1 pieces on
of the canonical
momentum
, defined on
. Apart from the contribution of
, these latter quantities are
Lau repeated the general analysis above using the tetrad (in fact, triad) variables and the Ashtekar
connection on the timelike boundary, instead of the traditional ADM-type variables [331]. Here the energy
and momentum surface densities are re-expressed by the superpotential , given by Eq. (3.6
), in a
frame adapted to the two-surface. (Lau called the corresponding superpotential 2-form the ‘Sparling
2-form’.) However, in contrast to the usual Ashtekar variables on a spacelike hypersurface [30
], the time
gauge cannot be imposed globally on the boundary Ashtekar variables. In fact, while every
orientable three-manifold
is parallelizable [410], and hence, a globally-defined orthonormal
triad can be given on
, the only parallelizable, closed, orientable two-surface is the torus.
Thus, on
, we cannot impose the global time gauge condition with respect to any spacelike
two-surface
in
unless
is a torus. Similarly, the global radial gauge condition in the
spacelike hypersurfaces
(even in a small open neighborhood of the whole two-surfaces
in
) can be imposed on a triad field only if the two-boundaries
are all
tori. Obviously, these gauge conditions can be imposed on every local trivialization domain of
the tangent bundle
of
. However, since in Lau’s local expressions only geometrical
objects (like the extrinsic curvature of the two-surface) appear, they are valid even globally (see
also [332]). On the other hand, further investigations are needed to clarify whether or not the
quasi-local Hamiltonian, using the Ashtekar variables in the radial–time gauge [333], is globally well
defined.
In general, the Brown–York quasi-local energy does not have any positivity property even if
the matter fields satisfy the dominant energy conditions. However, as G. Hayward pointed
out [244], for the variations of the metric around the vacuum solutions that extremalize the
Hamiltonian, called the ‘ground states’, the quasi-local energy cannot decrease. On the other
hand, the interpretation of this result as a ‘quasi-local dominant energy condition’ depends on
the choice of the time gauge above, which does not exist globally on the whole two-surface
.
Booth and Mann [100] shifted the emphasis from the foliation of the domain
to the foliation of the
boundary
. (These investigations were extended to include charged black holes in [101], where the
gauge dependence of the quasi-local quantities is also examined.) In fact, from the point of view of the
quasi-local quantities defined with respect to the observers with world lines in
and orthogonal to
,
it is irrelevant how the spacetime domain
is foliated. In particular, the quasi-local quantities
cannot depend on whether or not the leaves
of the foliation of
are orthogonal to
. As a result, Booth and Mann recovered the quasi-local charge and energy expressions of
Brown and York derived in the ‘orthogonal boundary’ case. However, they suggested a new
prescription for the definition of the reference configuration (see Section 10.1.8). Also, they calculated
the quasi-local energy for round spheres in the spherically-symmetric spacetimes with respect
to several moving observers, i.e., in contrast to Eq. (10.9
), they did not link the generator
vector field
to the normal
of
. In particular, the world lines of the observers are
not integral curves of
in the coordinate basis given in Section 4.2.1 on the round
spheres.
Using an explicit, nondynamic background metric , one can construct a covariant first-order
Lagrangian
for general relativity [306
], and one can use the action
based on this
Lagrangian instead of the trace
action. Fatibene, Ferraris, Francaviglia, and Raiteri [184
] clarified the
relationship between the two actions,
and
, and the corresponding quasi-local
quantities. Considering the reference term
in the Brown–York expression as the action of the
background metric
(which is assumed to be a solution of the field equations), they found that the two
first-order actions coincide if the spacetime metrics
and
coincide on the boundary
. Using
, they construct the conserved Noether current for any vector field
and, by taking its flux integral, define charge integrals
on two-surfaces
.15
Again, the Brown–York quasi-local quantity
and
coincide if the spacetime
metrics coincide on the boundary
and if
has some special form. Therefore, although the two
approaches are basically equivalent under the boundary condition above, this boundary condition is
too strong from both the point of view of the variational principle and that of the quasi-local
quantities. We will see in Section 10.1.8 that even the weaker boundary condition, that requires
only the induced three-metrics on
from
and from
to be the same, is still too
strong.
If we can write the action of our mechanical system into the canonical form
,
then it is straightforward to read off the Hamiltonian of the system. Thus, having accepted the trace
action as the action for general relativity, it is natural to derive the corresponding Hamiltonian in the
analogous way. Following this route Brown and York derived the Hamiltonian, corresponding to the ‘basic’
(or nonreferenced) action
as well [121
]. They obtained the familiar integral of the sum of the
Hamiltonian and the momentum constraints, weighted by the lapse
and the shift
, respectively,
plus
, given by Eq. (10.8
), as a boundary term. This result is in complete agreement with
the expectations, as their general quasi-local quantities can also be recovered as the value of the
Hamiltonian on the constraint surface (see also [100
]). This Hamiltonian was investigated further in [119
].
Here all the boundary terms that appear in the variation of their Hamiltonian are determined and
decomposed with respect to the two-surface
. It is shown that the change of the Hamiltonian under a
boost of
yields precisely the boosts of the energy and momentum surface density discussed
above.
Hawking, Horowitz, and Hunter also derived the Hamiltonian from the trace action
both
with the orthogonal [241
] and nonorthogonal boundary assumptions [242
]. They allowed matter fields
, whose dynamics is governed by a first-order action
, to be present. However, they
treated the reference configuration in a different way. In the traditional canonical analysis of the fields and
the geometry based on a noncompact
(for example in the asymptotically flat case) one has to impose
certain falloff conditions that ensure the finiteness of the action, the Hamiltonian, etc. This finiteness
requirement excludes several potentially interesting field + gravity configurations from our investigations. In
fact, in the asymptotically flat case we compare the actual matter + gravity configurations with the flat
spacetime + vanishing matter fields configuration. Hawking and Horowitz generalized this picture by
choosing a static, but otherwise arbitrary, solution
,
of the field equations, considered the
timelike boundary
of
to be a timelike cylinder ‘near the infinity’, and considered the
action
and those matter + gravity configurations that induce the same value on as
and
. Its limit
as
is ‘pushed out to infinity’ can be finite, even if the limit of the original (i.e., nonreferenced) action is
infinite. Although in the nonorthogonal boundaries case the Hamiltonian derived from the nonreferenced
action contains terms coming from the ‘joints’, by the boundary conditions at
they are canceled
from the referenced Hamiltonian. This latter Hamiltonian coincides with that obtained in the
orthogonal boundaries case. Both the ADM and the Abbott–Deser energy can be recovered from this
Hamiltonian [241
], and the quasi-local energy for spheres in domains with nonorthogonal boundaries in the
Schwarzschild solution is also calculated [242
]. A similar Hamiltonian, including the ‘joints’ or ‘corner’
terms, was obtained by Francaviglia and Raiteri [191
] for the vacuum Einstein theory (and for
Einstein–Maxwell systems in [9
]), using a Noether charge approach. Their formalism, using the
language of jet bundles, is, however, slightly more sophisticated than that common in general
relativity.
Booth and Fairhurst [95] reexamined the general form of the Brown–York energy and angular momentum from a Hamiltonian
point of view.16
Their starting point is the observation that the domain
is not isolated from its environment, thus,
the quasi-local Hamiltonian cannot be time independent. Therefore, instead of the standard
Hamiltonian formalism for the autonomous systems, a more general formalism, based on the
extended phase space, must be used. This phase space consists of the usual bulk configuration and
momentum variables
on the typical three-manifold
and the time coordinate
, the
space coordinates
on the two-boundary
, and their conjugate momenta
and
.
The second important observation of Booth and Fairhurst is that the Brown–York boundary conditions
are too restrictive. The two-metric, lapse, and shift need not be fixed, but their variations corresponding to
diffeomorphisms on the boundary must be allowed. Otherwise diffeomorphisms that are not
isometries of the three-metric on
cannot be generated by any Hamiltonian. Relaxing the
boundary conditions appropriately, they show that there is a Hamiltonian on the extended phase
space, which generates the correct equations of motions, and the quasi-local energy and angular
momentum expression of Brown and York are just (minus) the momentum
conjugate to the
time coordinate
. The only difference between the present and the original Brown–York
expressions is the freedom in the functional form of the unspecified reference term. Because of the
more restrictive boundary conditions of Brown and York, their reference term is less restricted.
Choosing the same boundary conditions in both approaches, the resulting expressions coincide
completely.
The quasi-local quantities introduced above become well defined only if the subtraction term in the
principal function is specified. The usual interpretation of a choice for
is the calibration of the
quasi-local quantities, i.e., fixing where to take their zero value.
The only restriction on that we had is that it must be a functional of the metric
on the
timelike boundary
. To specify
, it seems natural to expect that the principal function
be zero
in Minkowski spacetime [216
, 120
]. Then
would be the integral of the trace
of the extrinsic
curvature of
, if it were embedded in Minkowski spacetime with the given intrinsic metric
.
However, a general Lorentzian three-manifold
cannot be isometrically embedded, even locally,
into the Minkowski spacetime. (For a detailed discussion of this embedability, see [120
] and
Section 10.1.8.)
Another assumption on might be the requirement of the vanishing of the quasi-local quantities, or
of the energy and momentum surface densities, or only of the energy surface density
, in some reference
spacetime, e.g., in Minkowski or anti-de Sitter spacetime. Assuming that
depends on the lapse
and shift
linearly, the functional derivatives
and
depend only on the
two-metric
and on the boost-gauge that
defined on
. Therefore,
and
take the
form (10.10
), and, by the requirement of the vanishing of
in the reference spacetime it follows that
should be the trace of the extrinsic curvature of
in the reference spacetime. Thus, it would be
natural to fix
as the trace of the extrinsic curvature of
, when
is embedded
isometrically into the reference spacetime. However, this embedding is far from unique (since, in
particular, there are two independent normals of
in the spacetime and it would not be
fixed which normal should be used to calculate
), and hence the construction would be
ambiguous. On the other hand, one could require
to be embedded into flat Euclidean
three-space, i.e., into a spacelike hyperplane of Minkowski spacetime. This is the choice of Brown and
York [120
, 121
]. In fact, as we already noted in Section 4.1.3, for two-surfaces with everywhere positive
scalar curvature, such an embedding exists and is unique. (The order of the differentiability of
the metric is reduced in [261] to
.) A particularly interesting two-surface that cannot be
isometrically embedded into the flat three-space is the event horizon of the Kerr black hole, if the
angular momentum parameter
exceeds the irreducible mass (but is still not greater than the
mass parameter
), i.e., if
[463
]. (On the other hand, for its global
isometric embedding into
, see [203].) Thus, the construction works for a large class of
two-surfaces, but certainly not for every potentially interesting two-surface. The convexity condition is
essential.
It is known that the (local) isometric embedability of into flat three-space with extrinsic
curvature
is equivalent to the Gauss–Codazzi–Mainardi equations
and
. Here
is the intrinsic Levi-Civita covariant derivative and
is the
corresponding curvature scalar on
determined by
. Thus, for given
and (actually the flat)
embedding geometry, these are three equations for the three components of
, and hence, if the
embedding exists,
determines
. Therefore, the subtraction term
can also be interpreted as a
solution of an under-determined elliptic system, which is constrained by a nonlinear algebraic equation. In
this form the definition of the reference term is technically analogous to the definition of those in
Sections 7, 8, and 9, but, by the nonlinearity of the equations, in practice it is much more difficult to
find the reference term
than the spinor fields in the constructions of Sections 7, 8, and
9.
Accepting this choice for the reference configuration, the reference gauge potential
will
be zero in the boost-gauge in which the timelike normal of
in the reference Minkowski spacetime is
orthogonal to the spacelike three-plane, because this normal is constant. Thus, to summarize, for convex
two-surfaces, the flat space reference of Brown and York is uniquely determined,
is determined by this
embedding, and
. Then
, from which
can be calculated (if needed).
The procedure is similar if, instead of a spacelike hyperplane of Minkowski spacetime, a spacelike
hypersurface of constant curvature (for example in the de Sitter or anti-de Sitter spacetime) is
used. The only difference is that extra (known) terms appear in the Gauss–Codazzi–Mainardi
equations.
Brown, Lau, and York considered another prescription for the reference configuration as
well [118, 334
, 335
]. In this approach the two-surface
is embedded into the light cone of a point
of the Minkowski or anti-de Sitter spacetime instead of into a spacelike hypersurface of constant
curvature. The essential difference between the new (‘light cone reference’) and the previous
(‘flat space reference’) prescriptions is that the embedding into the light cone is not unique,
but the reference term
may be given explicitly, in a closed form. The positivity of the
Gauss curvature of the intrinsic geometry of
is not needed. In fact, by a result of
Brinkmann [115], every locally–conformally-flat Riemannian
-geometry is locally isometric to an
appropriate cut of a light cone of the
dimensional Minkowski spacetime (see, also, [178
]). To
achieve uniqueness some extra condition must be imposed. This may be the requirement of the
vanishing of the ‘normal momentum density’
in the reference spacetime [334
, 335
], yielding
, where
is the Ricci scalar of
and
is the cosmological constant of
the reference spacetime. The condition
defines something like a ‘rest frame’ in the reference
spacetime. Another, considerably more complicated, choice for the light cone reference term is used
in [118
].
Although the general, nonreferenced expressions are additive, the prescription for the reference term
destroys the additivity in general. In fact, if
and
are two-surfaces such that
is
connected and two-dimensional (more precisely, it has a nonempty open interior, for example, in
),
then in general
(overline means topological closure) is not guaranteed to
be embeddable, the flat three-space, and even if it is embeddable then the resulting reference
term
differs from the reference terms
and
determined from the individual
embeddings.
As noted in [100], the Brown–York energy with the flat space reference configuration is not zero in
Minkowski spacetime in general. In fact, in the standard spherical polar coordinates let
be the
spacelike hyperboloid
,
the hyperplane
and
, the sphere of radius
in the
hyperplane. Then the trace of the
extrinsic curvature of
in
and in
is
and
, respectively.
Therefore, the Brown–York quasi-local energy (with the flat three-space reference) associated with
and
the normals of
on
is
. Similarly, the Brown–York quasi-local energy
with the light cone references in [334
] and in [118
] is also negative for such surfaces with the boosted
observers.
Recently, Shi and Tam [458] have proven interesting theorems in Riemannian three-geometries,
which can be used to prove positivity of the Brown–York energy if the two-surface
is a
boundary of some time-symmetric spacelike hypersurface on which the dominant energy condition
holds. In the time-symmetric case, this energy condition is just the condition that the scalar
curvature be non-negative. The key theorem of Shi and Tam is the following: let
be a compact,
smooth Riemannian three-manifold with non-negative scalar curvature and smooth two-boundary
such that each connected component
of
is homeomorphic to
and the scalar
curvature of the induced two-metric on
is strictly positive. Then, for each component
holds, where
is the trace of the extrinsic curvature of
in
with
respect to the outward-directed normal, and
is the trace of the extrinsic curvature of
in
the flat Euclidean three-space when
is isometrically embedded. Furthermore, if in these
inequalities the equality holds for at least one
, then
itself is connected and
is flat. This
result is generalized in [459] by weakening the energy condition, in which case lower estimates
of the Brown–York energy can still be given. For some rigidity theorems connected with this
positivity result, see [461]; and for their generalization for higher dimensional spin manifolds,
see [329].
The energy expression for round spheres was calculated in [121, 100
]. In the spherically-symmetric
metric discussed in Section 4.2.1, on round spheres the Brown–York energy with the flat space reference
and fleet of observers
on
is
. In particular, it is
for the Schwarzschild solution. This deviates from the standard round
sphere expression, and, for the horizon of the Schwarzschild black hole, it is
(instead of the
expected
). (The energy has also been calculated explicitly for boosted foliations of the
Schwarzschild solution and for round spheres in isotropic cosmological models [119
].) Still in the
spherically-symmetric context the definition of the Brown–York energy is extended to spherical
two-surfaces beyond the event horizon in [347] (see also [443]). A remarkable result is that
while the total energy of the electrostatic field of a point charge in any finite three-volume
surrounding the point charge in Minkowski spacetime is always infinite, the negative gravitational
binding energy compensates the electrostatic energy so that the quasi-local energy is negative
within a certain radius under the event horizon in the Reissner–Nordström spacetime and
tends to
as
. The Brown–York energy is discussed from the point of view of
observers in spherically-symmetric spacetimes (e.g., the connection between this energy and
the effective energy in the geodesic equation for radial geodesics) in [90, 576]. The explicit
calculation of the Brown–York energy with the (implicitly assumed) flat-space reference in
Friedmann–Robertson–Walker spacetimes (as particular examples for the general round sphere case) is
given in [6
].
The Newtonian limit can be derived from the round sphere expression by assuming that is the mass
of a fluid ball of radius
and
is small: It is
. The first term is
simply the mass defined at infinity, and the second term is minus the Newtonian potential energy associated
with building a spherical shell of mass
and radius
from individual particles, bringing them together
from infinity. (For the calculation of the Newtonian limit in the covariant Newtonian spacetime, see [564].)
However, taking into account that on the Schwarzschild horizon
, while at spatial infinity it is
just
, the Brown–York energy is monotonically decreasing with
. Also, the first law of black hole
mechanics for spherically-symmetric black holes can be recovered by identifying
with
the internal energy [120
, 121
]. The thermodynamics of the Schwarzschild–anti-de Sitter black
holes was investigated in terms of the quasi-local quantities in [116
]. Still considering
to
be the internal energy, the temperature, surface pressure, heat capacity, etc. are calculated
(see Section 13.3.1). The energy has also been calculated for the Einstein–Rosen cylindrical
waves [119
].
The energy is explicitly calculated for three different kinds of two-spheres in the slices (in
the Boyer–Lindquist coordinates) of the slow rotation limit of the Kerr black hole spacetime with the flat
space reference [356]. These surfaces are the
surfaces (such as the outer horizon), spheres
whose intrinsic metric (in the given slow rotation approximation) is of a metric sphere of radius
with
surface area
, and the ergosurface (i.e., the outer boundary of the ergosphere). The
slow rotation approximation is defined such that
, where
is the typical spatial
measure of the two-surface. In the first two cases the angular momentum parameter
enters
the energy expression only in the
order. In particular, the energy for the outer
horizon
is
, which is twice the irreducible
mass of the black hole. An interesting feature of this calculation is that the energy cannot be
calculated for the horizon directly, because, as previously noted, the horizon itself cannot be
isometrically embedded into a flat three-space if the angular momentum parameter exceeds the
irreducible mass [463]. The energy for the ergosurface is positive, as for the other two kinds of
surfaces.
The spacelike infinity limit of the charges interpreted as the energy, spatial momentum, and spatial
angular momentum are calculated in [119] (see also [241]). Here the flat-space reference configuration and
the asymptotic Killing vectors of the spacetime are used, and the limits coincide with the standard ADM
energy, momentum, and spatial angular momentum. The analogous calculation for the center-of-mass is
given in [57]. It is shown that the corresponding large sphere limit is just the center-of-mass expression of
Beig and Ó Murchadha [64]. Here the center-of-mass integral is also given in terms of a charge integral of
the curvature. The large sphere limit of the energy for metrics with the weakest possible falloff
conditions is calculated in [181
, 462]. A further demonstration that the spatial infinity limit of the
Brown–York energy in an asymptotically Schwarzschild spacetime is the ADM energy is given
in [180].
Although the prescription for the reference configuration by Hawking and Horowitz cannot be imposed
for a general timelike three-boundary (see Section 10.1.8), asymptotically, when
is pushed out to
infinity, this prescription can be used, and coincides with the prescription of Brown and York. Choosing the
background metric
to be the anti-de Sitter one, Hawking and Horowitz [241
] calculated the
limit of the quasi-local energy, and they found it to tend to the Abbott–Deser energy. (For the
spherically-symmetric Schwarzschild–anti-de Sitter case see also [116
].) In [117
] the null infinity limit of the
integral of
was calculated both for the lapses
, generating asymptotic time
translations and supertranslations at the null infinity, and the fleet of observers was chosen to tend to
the BMS translation. In the former case the Bondi–Sachs energy, in the latter case Geroch’s
supermomenta are recovered. These calculations are based directly on the Bondi form of the
spacetime metric, and do not use the asymptotic solution of the field equations. (The limit of
the Brown–York energy on general asymptotically hyperboloidal hypersurfaces is calculated
in [330].) In a slightly different formulation Booth and Creighton calculated the energy flux of
outgoing gravitational radiation [94
] (see also Section 13.1) and they recovered the Bondi–Sachs
mass-loss.
However, the calculation of the small sphere limit based on the flat-space reference configuration gave
strange results [335]. While in nonvacuum the quasi-local energy is the expected
, in
vacuum it is proportional to
, instead of the Bel–Robinson ‘energy’
.
(Here
and
are, respectively, the conformal electric and conformal magnetic curvatures, and
plays a double role. It defines the two-sphere of radius
[as is usual in the small sphere calculations], and
defines the fleet of observers on the two-sphere.) On the other hand, the special light cone reference used
in [118, 335] reproduces the expected result in nonvacuum, and yields
in vacuum.
The small sphere limit was also calculated in [181] for small geodesic spheres in a time symmetric spacelike
hypersurface.
The light cone reference was shown to work in the large sphere limit near the null
and spatial infinities of asymptotically flat spacetimes and near the infinity of asymptotically
anti-de Sitter spacetimes [334]. Namely, the Brown–York quasi-local energy expression with this
null-cone reference term tends to the Bondi–Sachs, the ADM, and Abbott–Deser energies. The
supermomenta of Geroch at null infinity can also be recovered in this way. The proof is simply a
demonstration of the fact that this light cone and the flat space prescriptions for the subtraction
term have the same asymptotic structure up to order
. This choice seems to work
properly only in the asymptotics, because for small ellipsoids in the Minkowski spacetime this
definition yields nonzero energy and for small spheres in vacuum it does not yield the Bel–Robinson
‘energy’.17
A formulation and a proof of a version of Thorne’s hoop conjecture for spherically symmetric
configuarations in terms of are given in [402
], and will be discussed in Section 13.2.2.
As previously noted, Hawking, Horowitz, and Hunter [241, 242] defined their reference configuration by
embedding the Lorentzian three-manifold isometrically into some given Lorentzian spacetime,
e.g., into the Minkowski spacetime (see also [216]). However, for the given intrinsic three-metric
and
the embedding four-geometry the corresponding Gauss and Codazzi–Mainardi equations form a system of
equations for the six components of the extrinsic curvature
[120
]. Thus, in general, this
is a highly overdetermined system, and hence it may be expected to have a solution only in exceptional
cases. However, even if such an embedding existed, even the small perturbations of the intrinsic metric
would break the conditions of embedability. Therefore, in general, this prescription for the reference
configuration can work only if the three-surface
is ‘pushed out to infinity’, but does not work for finite
three-surfaces [120].
To rule out the possibility that the Brown–York energy can be nonzero even in Minkowski spacetime (on
two-surfaces in the boosted flat data set), Booth and Mann [100] suggested that one embed
isometrically into a reference spacetime
(mostly into the Minkowski spacetime) instead of a
spacelike slice of it, and to map the evolution vector field
of the dynamics, tangent to
, to a vector field
in
such that
and
. Here
is a
diffeomorphism mapping an open neighborhood
of
in
into
such that
, the
restriction of
to
, is an isometry, and
denotes the Lie derivative of
along
.
This condition might be interpreted as some local version of that of Hawking, Horowitz, and
Hunter. However, Booth and Mann did not investigate the existence or the uniqueness of this
choice.
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
![]() This work is licensed under a Creative Commons License. E-mail us: |