To motivate the main idea behind the Brown-York definition [96, 97
], let us consider first 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 1-parameter deformation of this history,
i.e. 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 [96, 97
] 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, they considered a compact spacetime domain
with topology
such that
correspond to compact spacelike hypersurfaces
; these form a smooth
foliation of
and the 2-surfaces
(corresponding to
) form a foliation of
the timelike 3-boundary
of
. Note that this
is not a globally hyperbolic
domain14.
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
to be future pointing. The metric and extrinsic
curvature on
will be denoted, respectively, by
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, rather the ‘trace
action’) for general relativity [405, 406, 387
], and the
action for the matter fields may be included. (For the minimal, non-derivative couplings the
presence of the matter fields does not alter the subsequent expressions.) However, as Geoff
Hayward pointed out [178], to have a well-defined variational principle, the ‘trace
action’
should in fact be completed by two 2-surface integrals, one on
and the other on
.
Otherwise, as a consequence of the edges
and
, called the ‘joints’ (i.e. the non-smooth
parts of the boundary
), the variation of the metric at the points of the edges
and
could not be arbitrary. (See also [177
, 237
, 77
, 95
], 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 3 + 1 decomposition of the spacetime metric yields a 2 + 1 decomposition of the metric , too. 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
Equation (69
) and (70
) we obtain that
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 2-surface
is a leaf of the foliation of
too [247
]: 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 [245]. Here the
energy and momentum surface densities are re-expressed by the superpotential , given by
Equation (10
), in a frame adapted to the 2-surface. (Lau called the corresponding superpotential
2-form the ‘Sparling 2-form’.) However, in contrast to the usual Ashtekar variables on a spacelike
hypersurface [17
], the time gauge cannot be imposed globally on the boundary Ashtekar variables. In
fact, while every orientable 3-manifold
is parallelizable [297], and hence a globally defined
orthonormal triad can be given on
, the only parallelizable closed orientable 2-surface is
the torus. Thus, on
, we cannot impose the global time gauge condition with respect to
any spacelike 2-surface
in
unless
is a torus. Similarly, the global radial gauge
condition in the spacelike hypersurfaces
(even on a small open neighbourhood of the whole
2-surfaces
in
) can be imposed on a triad field only if the 2-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 2-surface) appear, they are valid even globally (see
also [246]). 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 [247], 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 [179], 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 2-surface
.
Booth and Mann [77] 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 [78], 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, they 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 Equation (73
), 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, non-dynamical background metric , one can construct a covariant, first order
Lagrangian
for general relativity [230
], and one can use the action
based on this
Lagrangian instead of the trace
action. Fatibene, Ferraris, Francaviglia, and Raiteri [135] 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 2-surfaces
15.
Again, the Brown-York quasi-local quantity
and
coincide if the
spacetime metrics coincide on the boundary
and
has some special form. Therefore,
although the two approaches are basically equivalent under the boundary condition above, this
boundary condition is too strong both from the points of view of the variational principle and the
quasi-local quantities. We will see in Section 10.1.8 that even the weaker boundary condition
that only the induced 3-metrics on
from
and from
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 non-referenced) action
too [97
]. 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
Equation (72
), 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 [77
]). This Hamiltonian was investigated further in [95
]. Here all the
boundary terms that appear in the variation of their Hamiltonian are determined and decomposed
with respect to the 2-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 [176
] and non-orthogonal boundaries assumptions [177
]. 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 non-compact
(for example in the asymptotically flat case) one has to impose
certain fall-off 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 which 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. non-referenced) action is
infinite. Although in the non-orthogonal boundaries case the Hamiltonian derived from the non-referenced
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 [176
], and the quasi-local energy for spheres in domains with non-orthogonal boundaries in the
Schwarzschild solution is also calculated [177
]. A similar Hamiltonian, including the ‘joints’ or ‘corner’
terms, was obtained by Francaviglia and Raiteri [141
] for the vacuum Einstein theory (and for
Einstein-Maxwell systems in [4
]), 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 [73] reexamined the general form of the Brown-York energy and angular momentum from a Hamiltonian
point of view16.
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 3-manifold
and the time coordinate
, the space
coordinates
on the 2-boundary
, and their conjugate momenta
and
,
respectively.
Their second important observation is that the Brown-York boundary conditions are too restrictive: The
2-metric, the lapse, and the shift need not to be fixed but their variations corresponding to
diffeomorphisms on the boundary must be allowed. Otherwise diffeomorphisms that are not
isometries of the 3-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 [158
, 96
]. 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 3-manifold
cannot be isometrically embedded, even locally,
into the Minkowski spacetime. (For a detailed discussion of this embeddability, see [96
] 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 in anti-de-Sitter spacetime. Assuming that
depends on the lapse
and shift
linearly, the functional derivatives
and
depend
only on the 2-metric
and on the boost-gauge that
defined on
. Therefore,
and
take the form (74
), 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 being 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 3-space, i.e. into a spacelike hyperplane of Minkowski
spacetime17.
This is the choice of Brown and York [96
, 97
]. In fact, at least for a large class of 2-surfaces
, such
an embedding exists and is unique: If
and the metric is
and has everywhere positive scalar
curvature, then there is an isometric embedding of
into the flat Euclidean 3-space [195], and
apart from rigid motions this embedding is unique [346]. The requirement that the scalar curvature of the
2-surface must be positive can be interpreted as some form of the convexity, as in the theory of surfaces in
the Euclidean space. However, there are counterexamples even to local isometric embeddability when
this convexity condition is violated [276]. A particularly interesting 2-surface that cannot be
isometrically embedded into the flat 3-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
[343
]. Thus, the construction works for a large class of
2-surfaces, but certainly not for every potentially interesting 2-surface. The convexity condition is
essential.
It is known that the (local) isometric embeddability of
into flat 3-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 non-linearity 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 3-plane, because this normal is constant. Thus, to summarize, for convex
2-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 [94, 248
, 249
]. In this approach the 2-surface
is embedded into the light cone of a point
of the Minkowski or anti-de Sitter spacetime instead of 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 [91], every locally conformally flat Riemannian
-geometry is locally isometric to an
appropriate cut of a light cone of the
dimensional Minkowski spacetime (see also [133
]). 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 [248
, 249
], 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 [94
].
Although the general, non-referenced expressions are additive, the prescription for the reference term
destroys the additivity in general. In fact, if
and
are 2-surfaces such that
is
connected and 2-dimensional (more precisely, it has a non-empty open interior for example in
),
then in general
(overline means topological closure) is not guaranteed to
be embeddable into the flat 3-space, and even if it is embeddable then the resulting reference
term
differs from the reference terms
and
determined from the individual
embeddings.
As it is noted in [77], 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 3-space reference) associated with
and the
normals of
on
is
. Similarly, the Brown-York quasi-local energy with
the light cone references in [248
] and in [94
] is also negative for such surfaces with the boosted
observers.
Update
Recently, Shi, and Tam [341
] proved interesting theorems in Riemannian 3-geometries, which can be
used to prove positivity of the Brown-York energy if the 2-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 3-manifold
with non-negative scalar curvature and smooth 2-boundary
such that each connected component
of
is homeomorphic to
and the scalar curvature of the induced 2-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 3-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 [342] by
weakening the energy condition, whenever lower estimates of the Brown-York energy can still be
given.
The energy expression for round spheres in spherically symmetric spacetimes was calculated
in [97, 77
]. In the spherically symmetric metric discussed in Section 4.2.1, on the 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 [95
].) The Newtonian limit can be derived from this 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. However, taking into account that on the Schwarzschild horizon
while at the 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 [96
, 97]. The thermodynamics of the Schwarzschild-anti-de-Sitter black
holes was investigated in terms of the quasi-local quantities in [92
]. 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 [95
].
The energy is explicitly calculated for three different kinds of 2-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 [264]. 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 2-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 we noted in the previous point, the horizon itself cannot be
isometrically embedded into a flat 3-space if the angular momentum parameter exceeds the
irreducible mass [343]. 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 [95] (see also [176]). 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 centre-of-mass is
given in [42]. It is shown that the corresponding large sphere limit is just the centre-of-mass expression of
Beig and Ó Murchadha [47
]. Here the centre-of-mass integral in terms of a charge integral of the curvature
is also given.
Although the prescription for the reference configuration by Hawking and Horowitz cannot be imposed
for a general timelike 3-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 [176
] 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 [92
].) In [93
] 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. In a slightly different formulation Booth and Creighton calculated the energy flux of
outgoing gravitational radiation [76
] (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 [249]. While in non-vacuum 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 2-sphere of radius
[as is usual in the small sphere calculations], and defines
the fleet of observers on the 2-sphere.) On the other hand, the special light cone reference used
in [94, 249] reproduces the expected result in non-vacuum, and yields
in
vacuum.
The light cone reference was shown to work in the large sphere limit near the null
and spatial infinities of asymptotically flat, and near the infinity of asymptotically anti-de-Sitter
spacetimes [248]. 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, respectively. 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 non-zero energy and for small spheres in vacuum it does not yield the Bel-Robinson
‘energy’ [250].
As we noted above, Hawking, Horowitz, and Hunter [176, 177] defined their reference configuration by
embedding the Lorentzian 3-manifold isometrically into some given Lorentzian spacetime,
e.g. into the Minkowski spacetime (see also [158]). However, for the given intrinsic 3-metric
and the
embedding 4-geometry the corresponding Gauss and Codazzi-Mainardi equations form a system of
equations for the six components of the extrinsic curvature
[96
]. 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, then even the small perturbations of the intrinsic metric
would break the conditions of embeddability. Therefore, in general this prescription for the reference
configuration can work only if the 3-surface
is ‘pushed out to infinity’ but does not work for finite
3-surfaces [96].
To rule out the possibility that the Brown-York energy can be non-zero even in Minkowski spacetime
(on 2-surfaces in the boosted flat data set), Booth and Mann [77] suggested to 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 neighbourhood
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.
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