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 expressionon the solution of the equations of motion from to , the derivative of with respect to gives the canonical momenta , while its derivative with respect to gives minus the energy, , at . Obviously, neither the action nor the principal function are unique: for any of the form with arbitrary smooth function is an equally good action for the same dynamics. Clearly, the subtraction term alters both the canonical momenta and the energy according to and , respectively.
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 , 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 obtainedmass, while for spacelike with closed orbits in the charge is called angular momentum. (Here is not necessarily an element of the foliation of , and is the unit normal to tangent to .) Update 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 . 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’s.
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 tothe leaves of the foliation of they link the ‘quasi-time-translation’ (i.e. the ‘generator of the energy’) to the preferred unit normal of . Since the preferred unit normals are usually interpreted as a fleet of observers who are at rest with respect to , in their spirit the Brown-York-type quasi-local energy expressions are similar to given by Equation (6) for the matter fields or Equation (18) for the gravitational ‘field’ rather than to the charges . For vector fields with closed integral curved in the quantity might be interpreted as angular momentum corresponding 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 : 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 areradial foliation of [247, 95], which are, in fact, essentially the components of the canonical momentum.) Thus the required transformation formulae for , , and follow from the definitions and those for the extrinsic curvatures and the gauge potential of Section 4.1.2. The various boost-gauge invariant quantities that can be built from , , , , and are also discussed in [247, 95].
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 . 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 , the time gauge cannot be imposed globally on the boundary Ashtekar variables. In fact, while every orientable 3-manifold is parallelizable , 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 ). 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 , 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 , 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  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 , 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 , and one can use the action based on this Lagrangian instead of the trace action. Fatibene, Ferraris, Francaviglia, and Raiteri  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 . 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 ). This Hamiltonian was investigated further in . 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  and non-orthogonal boundaries assumptions . 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 , and the quasi-local energy for spheres in domains with non-orthogonal boundaries in the Schwarzschild solution is also calculated . A similar Hamiltonian, including the ‘joints’ or ‘corner’ terms, was obtained by Francaviglia and Raiteri  for the vacuum Einstein theory (and for Einstein-Maxwell systems in ), 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  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  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 , and apart from rigid motions this embedding is unique . 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 . 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 . 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 , every locally conformally flat Riemannian -geometry is locally isometric to an appropriate cut of a light cone of the dimensional Minkowski spacetime (see also ). 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 .
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 , 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  and in  is also negative for such surfaces with the boosted observers. Update Recently, Shi, and Tam  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  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 .) 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 . 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 .
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 . 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 . 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  (see also ). 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 . It is shown that the corresponding large sphere limit is just the centre-of-mass expression of Beig and Ó Murchadha . 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  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 .) In  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  (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 . 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 . 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’ .
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 ). 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 . 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 .
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  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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