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10.1 The Brown-York expression

10.1.1 The main idea

To motivate the main idea behind the Brown-York definition [96Jump To The Next Citation Point97Jump To The Next Citation Point], let us consider first a classical mechanical system of n degrees of freedom with configuration manifold Q and Lagrangian L : TQ × R --> R (i.e. the Lagrangian is assumed to be first order and may depend on time explicitly). For given initial and final configurations, (qa1,t1) and (qa2 ,t2), respectively, the corresponding action functional is I1[q(t)] := integral t2L(qa(t),qa(t),t)dt t1, where qa(t) is a smooth curve in Q from qa(t ) = qa 1 1 to a a q (t2) = q2 with tangent a q (t) at t. (The pair a (q (t),t) may be called a history or world line in the ‘spacetime’ Q × R.) Let a (q (u,t(u)),t(u)) be a smooth 1-parameter deformation of this history, i.e. for which (qa(0,t(0)),t(0)) = (qa(t),t), and u (- (- e,e) for some e > 0. Then, denoting the derivative with respect to the deformation parameter u at u = 0 by d, one has the well known expression

integral t2 ( ) ( ) dI1[q(t)] = @L--- -d @L-- (dqa - qadt) dt +-@L-dqa|t2t - @L--qa- L dt| tt2. (65) t1 @qa dt @qa @qa 1 @qa 1
Therefore, introducing the Hamilton-Jacobi principal function 1 a a S (q1,t1;q2,t2) as the value of the action on the solution qa(t) of the equations of motion from (qa,t1) 1 to (qa,t2) 2, the derivative of S1 with respect to qa 2 gives the canonical momenta p1 := (@L/@qa) a, while its derivative with respect to t 2 gives minus the energy, 1 1 a - E = -(paq - L), at t2. Obviously, neither the action 1 I nor the principal function S1 are unique: I[q(t)] := I1[q(t)] - I0[q(t)] for any I0[q(t)] of the form integral tt2(dh/dt) dt 1 with arbitrary smooth function h = h(qa(t),t) is an equally good action for the same dynamics. Clearly, the subtraction term I0[q(t)] alters both the canonical momenta and the energy according to 1 1 a pa '--> pa = pa - (@h/@q ) and 1 1 E '--> E = E + (@h/@t), respectively.

10.1.2 The variation of the action and the surface stress-energy tensor

The main idea of Brown and York [96Jump To The Next Citation Point97Jump To The Next Citation Point] 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 E above. To formulate this idea mathematically, they considered a compact spacetime domain D with topology S × [t1,t2] such that S × {t} correspond to compact spacelike hypersurfaces St; these form a smooth foliation of D and the 2-surfaces St := @St (corresponding to @S × {t}) form a foliation of the timelike 3-boundary 3 B of D. Note that this D 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 St on 3B. The orientation of 3B is chosen to be outward pointing, while the normals both of S1 := St1 and S2 := St2 to be future pointing. The metric and extrinsic curvature on St will be denoted, respectively, by hab and xab, those on 3B by gab and Qab. 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 I1 the ‘trace K action’ (or, in the present notation, rather the ‘trace x action’) for general relativity [405406387Jump To The Next Citation Point], 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 x action’ should in fact be completed by two 2-surface integrals, one on S1 and the other on S2. Otherwise, as a consequence of the edges S1 and S2, called the ‘joints’ (i.e. the non-smooth parts of the boundary @D), the variation of the metric at the points of the edges S1 and S2 could not be arbitrary. (See also [177Jump To The Next Citation Point237Jump To The Next Citation Point77Jump To The Next Citation Point95Jump To The Next Citation Point], where the ‘orthogonal boundaries assumption’ is also relaxed.) Let j1 and j2 be the scalar product of the outward pointing normal of 3B and the future pointing normal of S1 and of S2, respectively. Then, varying the spacetime metric, for the variation of the corresponding principal function S1 they obtained

integral 1 V~ ---( ) dS1 = ------ |h| xab - xhab dhabd3x - S2 integral 16pG --1-- V~ ---( ab ab) 3 - 16pG |h| x - xh dhabd x - integral S1 V~ --( ) - --1--- |g| Qab- Qgab dgabd3x - 3B 16pG 1 gf - 1 V~ --- 2 1 gf -1 V~ --- 2 - ----- tanh j2d |q|d x + ----- tanh j1d |q|d x. (66) 8pG S2 8pG S1
The first two terms together correspond to the term p1adqa|t2t1 of Equation (65View Equation), and, in fact, the familiar ADM expression for the canonical momentum ab p~ is just --1- 16pG V~ --- ab ab |h |(x - xh ). The last two terms give the effect of the presence of the non-differentiable ‘joints’. Therefore, it is the third term that should be analogous to the third term of Equation (65View Equation). In fact, roughly, this is proportional to the proper time separation of the ‘instants’ S1 and S2, and it is reasonable to identify its coefficient as some (quasi-local) analog of the energy. However, just as in the case of the mechanical system, the action (and the corresponding principal function) is not unique, and the principal function should be written as 1 0 S := S - S, where 0 S is assumed to be an arbitrary function of the 3-metric on the boundary @D = S2 U 3B U S1. Then
ab --2---dS- -1--( ab ab) --2-- dS0- t := - V~ ---dg = 8pG Q - Qg + V~ ---dg (67) |g| ab |g| ab
defines a symmetric tensor field on the timelike boundary 3B, and is called the surface stress-energy tensor. (Since our signature for gab on 3B is (+, -,- ) rather than (-,+, +), we should define tab with the extra minus sign, just according to Equation (1View Equation).) Its divergence with respect to the connection 3De on 3B determined by gab is proportional to the part gabTbcvc of the energy-momentum tensor, and hence, in particular, tab is divergence-free in vacuum. Therefore, if (3B, g ) ab admits a Killing vector, say a K, then in vacuum
gf Q [K] := K tabt dS, (68) S S a b
the flux integral of ab t Kb on any spacelike cross section S of 3 B, is independent of the cross section itself, and hence defines a conserved charge. If a K is timelike, then the corresponding charge is called a conserved mass, while for spacelike Ka with closed orbits in S the charge is called angular momentum. (Here S is not necessarily an element of the foliation St of 3B, and ta is the unit normal to S tangent to 3B.) UpdateJump To The Next Update Information Clearly, the trace-x 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 x, and the latter depends on the location of the boundary itself. Such a Lagrangian was found by Pons [317]. This depends on the coordinate system adapted to the boundary of the domain D 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 GG Lagrangian of Einstein, rather than to that of Hilbert’s.

10.1.3 The general form of the Brown-York quasi-local energy

The 3 + 1 decomposition of the spacetime metric yields a 2 + 1 decomposition of the metric gab, too. Let N and N a be the lapse and the shift of this decomposition on 3B. Then the corresponding decomposition of tab defines the energy, momentum, and spatial stress surface densities according to

0 e := t t tab = --1--k + V~ -1-dS--, (69) a b 8pG |q|dN 0 ja := - qabtctbc =--1--Aa + V~ -1--dS--, (70) 8pG |q|dN a [ ] 0 sab := TTacTTbdt cd = -1---kab - kqab + qabte ( \~/ etf )vf + V~ -2--dS-, (71) 8pG |q|dqab
where qab is the spacelike 2-metric, Ae is the SO(1, 1) vector potential on St and a TTb is the projection to St introduced in Section 4.1.2, and kab is the extrinsic curvature of St corresponding to the normal va orthogonal to 3B, and k is its trace. The timelike boundary 3B defines a boost-gauge on the 2-surfaces St (which coincides with that determined by the foliation St in the ‘orthogonal boundaries’ case). The gauge potential Ae is taken in this gauge. Thus, although e and ja on St are built from the 2-surface data (in a particular boost-gauge), the spatial surface stress depends on the part ta( \~/ atb)vb of the acceleration of the foliation St too. Let qa be any vector field on 3B tangent to 3B, and qa = nta + na its 2 + 1 decomposition. Then we can form the charge integral (68View Equation) for the leaves St of the foliation of 3B
gf gf a a ab a Et [q ,t ] := qat tb dSt = (ne - n ja) dSt. (72) St St
Obviously, in general Et[qa,ta] is not conserved, and depends not only on the vector field qa and the 2-surface data on the particular St, but on the boost-gauge that 3B defines on St, i.e. the timelike normal ta as well. Brown and York define the general form of their quasi-local energy on S := St by
EBY (S, ta) := Et [ta,ta], (73)
i.e. they link the ‘quasi-time-translation’ (i.e. the ‘generator of the energy’) to the preferred unit normal a t of St. Since the preferred unit normals a t are usually interpreted as a fleet of observers who are at rest with respect to St, in their spirit the Brown-York-type quasi-local energy expressions are similar to ES[ta] given by Equation (6View Equation) for the matter fields or Equation (18View Equation) for the gravitational ‘field’ rather than to the charges Q [K] S. For vector fields qa = na with closed integral curved in St the quantity a a Et[q ,t ] might be interpreted as angular momentum corresponding to a q.

The quasi-local energy is still not completely determined, because the ‘subtraction term’ S0 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 a a Et[q ,t ] (in particular zero quasi-local energy), and identify S0 with the S1 of the reference spacetime. Thus by Equation (69View Equation) and (70View Equation) we obtain that

( ) ( ) e = - -1--- k- k0 , ja = -1--- Aa - A0 , (74) 8pG 8pG a
where k0 and A0a are the reference values of the trace of the extrinsic curvature and SO(1, 1)-gauge potential, respectively. Note that to ensure that k0 and A0 a really be the trace of the extrinsic curvature and SO(1, 1)-gauge potential, respectively, in the reference spacetime, they cannot depend on the lapse N and the shift N a. This can be ensured by requiring that S0 be a linear functional of them. We return to the discussion of the reference term in the various specific constructions below.

10.1.4 Further properties of the general expressions

As we noted, e, ja, and sab depend on the boost-gauge that the timelike boundary defines on St. Lau clarified how these quantities change under a boost gauge transformation, where the new boost-gauge is defined by the timelike boundary 3B' of another domain D' such that the particular 2-surface St is a leaf of the foliation of 3 ' B too [247Jump To The Next Citation Point]: If {St} is another foliation of D such that @St = St and St is orthogonal to 3 B, then the new ' e, ' ja, and s'ab are built from the old e, ja, and sab and the 2 + 1 pieces on St of the canonical momentum ~pab, defined on St. Apart from the contribution of S0, these latter quantities are

2 ab 1 j |- := V~ ---vavb~p = -----l, (75) |h| 8pG ^ --2-- bc -1--- ja := V~ |h|qabvc~p = 8pG Aa, (76) t := V~ -2-q q p~cd = --1--[l - q (l + ve( \~/ v )te)], (77) ab |h| ac bd 8pG ab ab e f
where lab is the extrinsic curvature of St corresponding to its normal a t (we denoted this by tab in Section 4.1.2), and l is its trace. (By Equation (76View Equation) ^ja is not an independent quantity, that is just ja. These quantities were originally introduced as the variational derivatives of the principal function with respect to the lapse, the shift and the 2-metric of the radial foliation of S t [247Jump To The Next Citation Point95Jump To The Next Citation Point], which are, in fact, essentially the components of the canonical momentum.) Thus the required transformation formulae for e, ja, and sab follow from the definitions and those for the extrinsic curvatures and the SO(1, 1) gauge potential of Section 4.1.2. The various boost-gauge invariant quantities that can be built from e, ja, sab, j |-, and tab are also discussed in [247Jump To The Next Citation Point95Jump To The Next Citation Point].

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 \/ bae, given by Equation (10View Equation), 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 [17Jump To The Next Citation Point], the time gauge cannot be imposed globally on the boundary Ashtekar variables. In fact, while every orientable 3-manifold S is parallelizable [297], and hence a globally defined orthonormal triad can be given on S, the only parallelizable closed orientable 2-surface is the torus. Thus, on 3B, we cannot impose the global time gauge condition with respect to any spacelike 2-surface S in 3B unless S is a torus. Similarly, the global radial gauge condition in the spacelike hypersurfaces St (even on a small open neighbourhood of the whole 2-surfaces St in St) can be imposed on a triad field only if the 2-boundaries St = @St are all tori. Obviously, these gauge conditions can be imposed on every local trivialization domain of the tangent bundle TSt of St. 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 S.

Booth and Mann [77Jump To The Next Citation Point] shifted the emphasis from the foliation of the domain D to the foliation of the boundary 3B. (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 3 B and orthogonal to S it is irrelevant how the spacetime domain D is foliated. In particular, the quasi-local quantities cannot depend on whether or not the leaves St of the foliation of D are orthogonal to 3B. 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 (73View Equation), they did not link the generator vector field qa to the normal ta of S t. In particular, the world lines of the observers are not integral curves of (@/@t) in the coordinate basis given in Section 4.2.1 on the round spheres.

Using an explicit, non-dynamical background metric g0 ab, one can construct a covariant, first order Lagrangian 0 L(gab,gab) for general relativity [230Jump To The Next Citation Point], and one can use the action 0 ID[gab,gab] based on this Lagrangian instead of the trace x action. Fatibene, Ferraris, Francaviglia, and Raiteri [135] clarified the relationship between the two actions, ID[gab] and ID[gab,g0ab], and the corresponding quasi-local quantities: Considering the reference term S0 in the Brown-York expression as the action of the background metric g0 ab (which is assumed to be a solution of the field equations), they found that the two first order actions coincide if the spacetime metrics gab and 0 gab coincide on the boundary @D. Using L(gab, g0) ab, they construct the conserved Noether current for any vector field qa and, by taking its flux integral, define charge integrals Q [qa,g ,g0] S ab ab on 2-surfaces S 15. Again, the Brown-York quasi-local quantity Et[qa,ta] and QSt[qa,gab,g0ab] coincide if the spacetime metrics coincide on the boundary @D and qa 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 3B from gab and from g0ab be the same is still too strong.

10.1.5 The Hamiltonians

If we can write the action I[q(t)] of our mechanical system into the canonical form integral t2[p qa- H(qa, p ,t)]dt t1 a a, then it is straightforward to read off the Hamiltonian of the system. Thus, having accepted the trace x 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 I1 too [97Jump To The Next Citation Point]. They obtained the familiar integral of the sum of the Hamiltonian and the momentum constraints, weighted by the lapse N and the shift a N, respectively, plus a a a Et[N t + N ,t ], given by Equation (72View Equation), 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 [77Jump To The Next Citation Point]). This Hamiltonian was investigated further in [95Jump To The Next Citation Point]. Here all the boundary terms that appear in the variation of their Hamiltonian are determined and decomposed with respect to the 2-surface @S. It is shown that the change of the Hamiltonian under a boost of S yields precisely the boosts of the energy and momentum surface density discussed above.

Hawking, Horowitz, and Hunter also derived the Hamiltonian from the trace x action I1 [gab] D both with the orthogonal [176Jump To The Next Citation Point] and non-orthogonal boundaries assumptions [177Jump To The Next Citation Point]. They allowed matter fields PN, whose dynamics is governed by a first order action 1 ImD[gab,PN ], 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 S (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 g0ab, P0N of the field equations, considered the timelike boundary 3B of D to be a timelike cylinder ‘near the infinity’, and considered the action

[ ] [ ] ID [gab,PN ] := I1D [gab] + I1mD [gab,PN ]- I1D g0ab - I1mD g0ab,P0N

and those matter + gravity configurations which induce the same value on 3 B as 0 P N and 0 gab. Its limit as 3B 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 3B 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 [176Jump To The Next Citation Point], and the quasi-local energy for spheres in domains with non-orthogonal boundaries in the Schwarzschild solution is also calculated [177Jump To The Next Citation Point]. A similar Hamiltonian, including the ‘joints’ or ‘corner’ terms, was obtained by Francaviglia and Raiteri [141Jump To The Next Citation Point] for the vacuum Einstein theory (and for Einstein-Maxwell systems in [4Jump To The Next Citation Point]), 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 D 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 (h ,p~ab) ab on the typical 3-manifold S and the time coordinate t, the space coordinates A x on the 2-boundary S = @S, and their conjugate momenta p and pA, 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 gab on 3B 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 p conjugate to the time coordinate t. 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.

10.1.6 The flat space and light cone references

The quasi-local quantities introduced above become well-defined only if the subtraction term S0 in the principal function is specified. The usual interpretation of a choice for S0 is the calibration of the quasi-local quantities, i.e. fixing where to take their zero value.

The only restriction on 0 S that we had is that it must be a functional of the metric gab on the timelike boundary 3B. To specify S0, it seems natural to expect that the principal function S be zero in Minkowski spacetime [158Jump To The Next Citation Point96Jump To The Next Citation Point]. Then S0 would be the integral of the trace Q0 of the extrinsic curvature of 3B if it were embedded in Minkowski spacetime with the given intrinsic metric g ab. However, a general Lorentzian 3-manifold 3 (B, gab) cannot be isometrically embedded, even locally, into the Minkowski spacetime. (For a detailed discussion of this embeddability, see [96Jump To The Next Citation Point] and Section 10.1.8.)

Another assumption on S0 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 e, in some reference spacetime, e.g. in Minkowski or in anti-de-Sitter spacetime. Assuming that 0 S depends on the lapse N and shift N a linearly, the functional derivatives (dS0/dN ) and (dS0/dN a) depend only on the 2-metric qab and on the boost-gauge that 3B defined on St. Therefore, e and j a take the form (74View Equation), and by the requirement of the vanishing of e in the reference spacetime it follows that 0 k should be the trace of the extrinsic curvature of St in the reference spacetime. Thus it would be natural to fix 0 k as the trace of the extrinsic curvature of St when (St,qab) is embedded isometrically into the reference spacetime. However, this embedding is far from being unique (since, in particular, there are two independent normals of S t in the spacetime and it would not be fixed which normal should be used to calculate 0 k), and hence the construction would be ambiguous. On the other hand, one could require (St,qab) 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 [96Jump To The Next Citation Point97Jump To The Next Citation Point]. In fact, at least for a large class of 2-surfaces (St,qab), such an embedding exists and is unique: If S ~~ S2 t and the metric is C2 and has everywhere positive scalar curvature, then there is an isometric embedding of (St,qab) 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 a exceeds the irreducible mass (but is still not greater than the mass parameter m), i.e. if V~ -- 3m < 2 |a|< 2m [343Jump To The Next Citation Point]. 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 (S,qab) into flat 3-space with extrinsic curvature k0 ab is equivalent to the Gauss-Codazzi-Mainardi equations d (k0a - dak0) = 0 a b b and S 0 2 0 0ab R - (k ) + kabk = 0. Here da is the intrinsic Levi-Civita covariant derivative and S R is the corresponding curvature scalar on S determined by qab. Thus, for given qab and (actually the flat) embedding geometry, these are three equations for the three components of k0ab, and hence, if the embedding exists, qab determines k0. Therefore, the subtraction term k0 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 k0 than the spinor fields in the constructions of Sections 7, 8, and 9.

Accepting this choice for the reference configuration, the reference SO(1, 1) gauge potential 0 Aa will be zero in the boost-gauge in which the timelike normal of St 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, k0 is determined by this embedding, and A0 = 0 a. Then 8pGS0 = - integral N k0dS St t, from which sab 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 [94Jump To The Next Citation Point248Jump To The Next Citation Point249Jump To The Next Citation Point]. In this approach the 2-surface (St,qab) 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 0 k may be given explicitly, in a closed form. The positivity of the Gauss curvature of the intrinsic geometry of (S,qab) is not needed. In fact, by a result of Brinkmann [91], every locally conformally flat Riemannian n-geometry is locally isometric to an appropriate cut of a light cone of the n + 2 dimensional Minkowski spacetime (see also [133Jump To The Next Citation Point]). To achieve uniqueness some extra condition must be imposed. This may be the requirement of the vanishing of the ‘normal momentum density’ 0 j |- in the reference spacetime [248Jump To The Next Citation Point249Jump To The Next Citation Point], yielding V~ ------------ k0 = 2SR + 4/c2, where SR is the Ricci scalar of (S, qab) and c is the cosmological constant of the reference spacetime. The condition j0 |- = 0 defines something like a ‘rest frame’ in the reference spacetime. Another, considerably more complicated choice for the light cone reference term is used in [94Jump To The Next Citation Point].

10.1.7 Further properties and the various limits

Although the general, non-referenced expressions are additive, the prescription for the reference term 0 k destroys the additivity in general. In fact, if S' and S'' are 2-surfaces such that S'/ ~\ S'' is connected and 2-dimensional (more precisely, it has a non-empty open interior for example in S'), then in general -'----''----'----''- S U S - S /~\ S (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 k0 differs from the reference terms k'0 and k''0 determined from the individual embeddings.

As it is noted in [77Jump To The Next Citation Point], 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 S1 be the spacelike hyperboloid V~ ------- t = - r2 + r2, S0 the hyperplane t = - T = const. < - r < 0 and S := S /~\ S 0 1, the sphere of radius V~ T-2--r2- in the t = -T hyperplane. Then the trace of the extrinsic curvature of S in S0 and in S1 is V~ -2----2- 2/ T - r and V~ -2----2- 2T /r T - r, respectively. Therefore, the Brown-York quasi-local energy (with the flat 3-space reference) associated with S and the normals of S1 on S is V~ ---------------- - (T + r)(T - r)3/(rG). Similarly, the Brown-York quasi-local energy with the light cone references in [248Jump To The Next Citation Point] and in [94Jump To The Next Citation Point] is also negative for such surfaces with the boosted observers. UpdateJump To The Next Update Information Recently, Shi, and Tam [341Jump To The Next Citation Point] proved interesting theorems in Riemannian 3-geometries, which can be used to prove positivity of the Brown-York energy if the 2-surface S 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 S be a compact, smooth Riemannian 3-manifold with non-negative scalar curvature and smooth 2-boundary S such that each connected component Si of S is homeomorphic to 2 S and the scalar curvature of the induced 2-metric on Si is strictly positive. Then for each component gf gf Si k dSi < Si k0 dSi holds, where k is the trace of the extrinsic curvature of S in S with respect to the outward directed normal, and k0 is the trace of the extrinsic curvature of S i in the flat Euclidean 3-space when S i is isometrically embedded. Furthermore, if in these inequalities the equality holds for at least one Si, then S itself is connected and S 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 [97Jump To The Next Citation Point77Jump To The Next Citation Point]. 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 @/@t on S is GEBY[Sr, (@/@t)a] = r(1- exp(- a)). In particular, it is V~ ----------- r[1 - 1- (2m/r)] for the Schwarzschild solution. This deviates from the standard round sphere expression, and, for the horizon of the Schwarzschild black hole it is 2m (instead of the expected m). (The energy has also been calculated explicitly for boosted foliations of the Schwarzschild solution and for round spheres in isotropic cosmological models [95Jump To The Next Citation Point].) The Newtonian limit can be derived from this by assuming that m is the mass of a fluid ball of radius r and m/r is small: It is GEBY = m + (m2/2r) + O(r -2). 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 m and radius r from individual particles, bringing them together from infinity. However, taking into account that on the Schwarzschild horizon GE = 2m BY while at the spatial infinity it is just m, the Brown-York energy is monotonically decreasing with r. Also, the first law of black hole mechanics for spherically symmetric black holes can be recovered by identifying EBY with the internal energy [96Jump To The Next Citation Point97]. The thermodynamics of the Schwarzschild-anti-de-Sitter black holes was investigated in terms of the quasi-local quantities in [92Jump To The Next Citation Point]. Still considering EBY 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 [95Jump To The Next Citation Point].

The energy is explicitly calculated for three different kinds of 2-spheres in the t = const. 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 r = const. surfaces (such as the outer horizon), spheres whose intrinsic metric (in the given slow rotation approximation) is of a metric sphere of radius R with surface area 4pR2, and the ergosurface (i.e. the outer boundary of the ergosphere). The slow rotation approximation is defined such that |a |/R « 1, where R is the typical spatial measure of the 2-surface. In the first two cases the angular momentum parameter a enters the energy expression only in the m2a2/R3 order. In particular, the energy for the outer horizon V~ -------- r+ := m + m2 - a2 is 2m[1 - a2/(8m2) + O(a4/r4+)], 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 [176Jump To The Next Citation Point]). 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 [47Jump To The Next Citation Point]. 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 3 B (see Section 10.1.8), asymptotically, when 3 B is pushed out to infinity, this prescription can be used, and coincides with the prescription of Brown and York. Choosing the background metric g0ab to be the anti-de-Sitter one, Hawking and Horowitz [176Jump To The Next Citation Point] 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 [92Jump To The Next Citation Point].) In [93Jump To The Next Citation Point] the null infinity limit of the integral of 0 N (k - k)/(8pG) was calculated both for the lapses N 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 [76Jump To The Next Citation Point] (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 [249Jump To The Next Citation Point]. While in non-vacuum the quasi-local energy is the expected (4p/3)r3Tabtatb, in vacuum it is proportional to 4E Eab + H Hab ab ab instead of the Bel-Robinson ‘energy’ T tatbtctd abcd. (Here Eab and Hab are, respectively, the conformal electric and conformal magnetic curvatures, and a t plays a double role: It defines the 2-sphere of radius r [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 [94249] reproduces the expected result in non-vacuum, and yields [1/(90G)]r5Tabcdtatbtctd in vacuum.

The light cone reference 0 V~ --S--------2 k = 2 R + 4/c 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 O(r -3). 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].

10.1.8 Other prescriptions for the reference configuration

As we noted above, Hawking, Horowitz, and Hunter [176177] defined their reference configuration by embedding the Lorentzian 3-manifold 3 (B, gab) isometrically into some given Lorentzian spacetime, e.g. into the Minkowski spacetime (see also [158]). However, for the given intrinsic 3-metric gab and the embedding 4-geometry the corresponding Gauss and Codazzi-Mainardi equations form a system of 6 + 8 = 14 equations for the six components of the extrinsic curvature Q ab [96Jump To The Next Citation Point]. 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 hab would break the conditions of embeddability. Therefore, in general this prescription for the reference configuration can work only if the 3-surface 3B 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 (S,qab) isometrically into a reference spacetime (M 0,g0ab) (mostly into the Minkowski spacetime) instead of a spacelike slice of it, and to map the evolution vector field qa = N ta + N a of the dynamics, tangent to 3 B, to a vector field 0a q in 0 M such that * 0 ´Lqqab = f (´Lq0qab) and a * 0a 0 q qa = f (q qa). Here f is a diffeomorphism mapping an open neighbourhood U of S in M into M 0 such that f |S, the restriction of f to S, is an isometry, and ´L q q ab denotes the Lie derivative of q ab along qa. 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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