10.1 The Brown–York expression

10.1.1 The main idea

To motivate the main idea behind the Brown–York definition [120Jump To The Next Citation Point, 121Jump To The Next Citation Point], let us first consider 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, (qa,t1) 1 and (qa ,t2) 2, respectively, the corresponding action functional is I1[q(t)] := ∫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 (qa(u,t(u)),t(u)) be a smooth one-parameter deformation of this history, for which (qa(0,t(0)),t(0)) = (qa(t),t), and u ∈ (− 𝜖,𝜖) for some 𝜖 > 0. Then, denoting the derivative with respect to the deformation parameter u at u = 0 by δ, one has the well known expression

( ) ( ) 1 ∫ t2 ∂L d ∂L a a ∂L a t ∂L a t δI [q (t)] = --a-− -- --a- (δq − q˙δt)dt + ---aδq |2t1 − --aq˙ − L δt|t21. (10.1 ) t1 ∂q dt ∂ ˙q ∂ ˙q ∂ ˙q
Therefore, introducing the Hamilton–Jacobi principal function S1(qa1,t1;qa2,t2) as the value of the action on the solution qa(t) of the equations of motion from (qa,t) 1 1 to (qa,t ) 2 2, the derivative of S1 with respect to a q2 gives the canonical momenta 1 a p a := (∂L ∕∂q˙), while its derivative with respect to t2 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 ∫ t2(dh ∕dt)dt t1 with arbitrary smooth function h = h(qa(t),t) is an equally good action for the same dynamics. Clearly, the subtraction term 0 I [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 [120Jump To The Next Citation Point, 121Jump 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, Brown and York considered a compact spacetime domain D with topology Σ × [t1,t2] such that Σ × {t} correspond to compact spacelike hypersurfaces Σt; these form a smooth foliation of D and the two-surfaces 𝒮 := ∂Σ t t (corresponding to ∂Σ × {t}) form a foliation of the timelike three-boundary 3 B of D. Note that this D is not a globally hyperbolic domain.14 To ensure the compatibility of the dynamics with this boundary, the shift vector is usually chosen to be tangent to 𝒮t on 3B. The orientation of 3B is chosen to be outward pointing, while the normals, both of Σ1 := Σt1 and of Σ2 := Σt2, are chosen to be future pointing. The metric and extrinsic curvature on Σ t will be denoted, respectively, by h ab and χ ab, and those on 3B by γ ab and Θab.

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, the ‘trace χ action’) for general relativity [572, 573, 534Jump To The Next Citation Point], and the action for the matter fields may be included. (For minimal, nonderivative couplings, the presence of the matter fields does not alter the subsequent expressions.) However, as Geoff Hayward pointed out [243], to have a well-defined variational principle, the ‘trace χ action’ should in fact be completed by two two-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 nonsmooth 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 [242Jump To The Next Citation Point, 315Jump To The Next Citation Point, 100Jump To The Next Citation Point, 119Jump To The Next Citation Point], where the ‘orthogonal boundaries assumption’ is also relaxed.) Let η 1 and η 2 be the scalar product of the outward-pointing normal of 3B and the future-pointing normal of Σ1 and of Σ2, respectively. Then, varying the spacetime metric (for the variation of the corresponding principal function S1) they obtained the following:

∫ 1 ∘ ---( ) δS1 = ------ |h| χab − χhab δhabd3x − Σ2∫ 16πG --1--∘ ---( ab ab) 3 − 16πG |h | χ − χh δhabd x − ∫Σ1 ∘ --- − --1--- |γ |(Θab − Θ γab)δγ d3x − 3B 16πG ab 1 ∮ ∘ --- 1 ∮ ∘ --- − ----- tanh− 1η2δ |q|d2x + ----- tanh −1η1δ |q |d2x. (10.2 ) 8πG 𝒮2 8πG 𝒮1
The first two terms together correspond to the term p1δqa|t2 a t1 of Eq. (10.1View Equation), and, in fact, the familiar ADM expression for the canonical momentum ab &tidle;p is just -1-- 16πG ∘ --- ab ab |h|(χ − χh ). The last two terms give the effect of the presence of the nondifferentiable ‘joints’. Therefore, it is the third term that should be analogous to the third term of Eq. (10.1View Equation). In fact, roughly, this is proportional to the proper time separation of the ‘instants’ Σ1 and Σ2, 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 S := S1 − S0, where S0 is assumed to be an arbitrary function of the three-metric on the boundary ∂D = Σ2 ∪ 3B ∪ Σ1. Then
ab 2 δS 1 ( ab ab) 2 δS0 τ := − ∘-------- = -----Θ − Θγ + ∘--------- (10.3 ) |γ|δγab 8πG |γ| δγab
defines a symmetric tensor field on the timelike boundary 3B, and is called the surface stress-energy tensor. (Since our signature for γ ab on 3B is (+, − ,− ) rather than (− ,+, + ), we should define τab with the extra minus sign, according to Eq. (2.1View Equation).) Its divergence with respect to the connection 3 De on 3B determined by γab is proportional to the part γabTbcvc of the energy-momentum tensor, and hence, in particular, τab is divergence-free in vacuum. Therefore, if (3B,γab) admits a Killing vector, say Ka, then, in vacuum
∮ Q 𝒮 [K ] := Ka τab¯tbd 𝒮, (10.4 ) 𝒮
the flux integral of ab τ 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.)

Clearly, the trace-χ 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 χ, and the latter depends on the location of the boundary itself. Such a Lagrangian was found by Pons [431]. This depends on the coordinate system adapted to the boundary of the domain 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 Γ Γ Lagrangian of Einstein, rather than to that of Hilbert.

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 γab, as well. Let N and a N be the lapse and the shift of this decomposition on 3 B. Then the corresponding decomposition of τab defines the energy, momentum, and spatial-stress surface densities according to

1 1 δS0 𝜀 := tatbτab = −----k + ∘--------, (10.5 ) 8 πG |q|δN 1 1 δS0 ja := − qabtcτbc =-----Aa + ∘-------a-, (10.6 ) 8πG |q|δN 1 [ ] 2 δS0 sab := ΠacΠbdτ cd = -----kab − kqab + qabte (∇etf )vf + ∘--------, (10.7 ) 8πG |q|δqab
where q ab is the spacelike two-metric, A e is the SO (1,1 ) vector potential on 𝒮 t, Πa b is the projection to 𝒮t introduced in Section 4.1.2, 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 two-surfaces 𝒮t (which coincides with that determined by the foliation Σt in the ‘orthogonal boundaries’ case). The gauge potential Ae is taken in this gauge. Thus, although 𝜀 and j a on 𝒮 t are built from the two-surface data (in a particular boost-gauge), the spatial surface stress depends on the part a b t (∇atb )v of the acceleration of the foliation Σt as well. Let ξa be any vector field on 3B tangent to 3B, and ξa = nta + na its 2 + 1 decomposition. Then we can form the charge integral (10.4View Equation) for the leaves 𝒮t of the foliation of 3B
∮ ∮ a a ab a Et [ξ ,t ] := ξaτ tbd𝒮t = (n𝜀 − n ja) d𝒮t. (10.8 ) 𝒮t 𝒮t
Obviously, in general Et [ξa,ta] is not conserved, and depends not only on the vector field ξa and the two-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 := 𝒮 t by
a a a EBY (𝒮, t ) := Et [t,t ], (10.9 )
i.e., they link the ‘quasi-time-translation’ (i.e., the ‘generator of the energy’) to the preferred unit normal ta of 𝒮 t. Since the preferred unit normals ta 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 E Σ[ta] given by Eq. (2.6View Equation) for the matter fields or Eq. (3.17View Equation) for the gravitational ‘field’ rather than to the charges Q 𝒮[K ]. For vector fields ξa = na with closed integral curved in 𝒮t the quantity Et [ξa,ta] might be interpreted as angular momentum corresponding to a ξ.

The quasi-local energy is still not completely determined, because the ‘subtraction term’ 0 S 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 E [ξa,ta] t (in particular zero quasi-local energy), and identify 0 S with the 1 S of the reference spacetime. Thus, by Eq. (10.5View Equation) and (10.6View Equation) we obtain that

-1---( 0) -1---( 0) 𝜀 = − 8πG k − k , ja = 8πG Aa − A a , (10.10 )
where 0 k and 0 A a 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 A0a 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.

For a definition of the Brown–York energy as a quasi-local energy oparator in loop quantum gravity, see [565Jump To The Next Citation Point].

10.1.4 Further properties of the general expressions

As we noted, 𝜀, 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 two-surface 𝒮t is a leaf of the foliation of 3B ′ as well [333Jump To The Next Citation Point]. If {¯Σt} is another foliation of D such that ∂¯Σ = 𝒮 t t and ¯Σ t is orthogonal to 3B, then the new 𝜀′, j′ a, and ′ sab are built from the old 𝜀, ja, and sab and the 2 + 1 pieces on 𝒮t of the canonical momentum ab ¯&tidle;p, defined on ¯Σt. Apart from the contribution of 0 S, these latter quantities are

j⊢ := ∘-2--vavb¯&tidle;pab = --1--l, (10.11 ) |h| 8πG 2 1 ˆja := ∘----qabvc¯&tidle;pbc = ----Aa, (10.12 ) |h| 8πG 2 cd 1 e e tab := ∘----qacqbdp¯&tidle; = -----[lab − qab(l + v (∇evf )t )], (10.13 ) |h| 8 πG
where l ab is the extrinsic curvature of 𝒮 t corresponding to its normal ta (we denote this by τ ab in Section 4.1.2), and l is its trace. (By Eq. (10.12View Equation) ˆ 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 two-metric of the radial foliation of Σt [333Jump To The Next Citation Point, 119Jump To The Next Citation Point], which are, in fact, essentially the components of the canonical momentum.) Thus, the required transformation formulae for 𝜀, ja, and sab follow from the definitions and those for the extrinsic curvature and the SO (1,1) gauge potential of Section 4.1.2. The various boost-gauge invariant quantities that can be built from 𝜀, ja, sab, j⊢, and tab are also discussed in [333Jump To The Next Citation Point, 119Jump 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 [331]. Here the energy and momentum surface densities are re-expressed by the superpotential ∨bae, given by Eq. (3.6View Equation), in a frame adapted to the two-surface. (Lau called the corresponding superpotential 2-form the ‘Sparling 2-form’.) However, in contrast to the usual Ashtekar variables on a spacelike hypersurface [30Jump To The Next Citation Point], the time gauge cannot be imposed globally on the boundary Ashtekar variables. In fact, while every orientable three-manifold Σ is parallelizable [410], and hence, a globally-defined orthonormal triad can be given on Σ, the only parallelizable, closed, orientable two-surface is the torus. Thus, on 3B, we cannot impose the global time gauge condition with respect to any spacelike two-surface 𝒮 in 3 B unless 𝒮 is a torus. Similarly, the global radial gauge condition in the spacelike hypersurfaces Σt (even in a small open neighborhood of the whole two-surfaces 𝒮t in Σt) can be imposed on a triad field only if the two-boundaries 𝒮t = ∂Σt are all tori. Obviously, these gauge conditions can be imposed on every local trivialization domain of the tangent bundle T𝒮 t of 𝒮 t. However, since in Lau’s local expressions only geometrical objects (like the extrinsic curvature of the two-surface) appear, they are valid even globally (see also [332]). On the other hand, further investigations are needed to clarify whether or not the quasi-local Hamiltonian, using the Ashtekar variables in the radial–time gauge [333], is globally well defined.

In general, the Brown–York quasi-local energy does not have any positivity property even if the matter fields satisfy the dominant energy conditions. However, as G. Hayward pointed out [244], for the variations of the metric around the vacuum solutions that extremalize the Hamiltonian, called the ‘ground states’, the quasi-local energy cannot decrease. On the other hand, the interpretation of this result as a ‘quasi-local dominant energy condition’ depends on the choice of the time gauge above, which does not exist globally on the whole two-surface 𝒮.

Booth and Mann [100Jump To The Next Citation Point] shifted the emphasis from the foliation of the domain D to the foliation of the boundary 3 B. (These investigations were extended to include charged black holes in [101], where the gauge dependence of the quasi-local quantities is also examined.) In fact, from the point of view of the quasi-local quantities defined with respect to the observers with world lines in 3B 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 Σt of the foliation of D are orthogonal to 3B. As a result, Booth and Mann recovered the quasi-local charge and energy expressions of Brown and York derived in the ‘orthogonal boundary’ case. However, they suggested a new prescription for the definition of the reference configuration (see Section 10.1.8). Also, they calculated the quasi-local energy for round spheres in the spherically-symmetric spacetimes with respect to several moving observers, i.e., in contrast to Eq. (10.9View Equation), they did not link the generator vector field ξa to the normal ta of 𝒮t. 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, nondynamic background metric g0ab, one can construct a covariant first-order Lagrangian L(gab,g0ab) for general relativity [306Jump To The Next Citation Point], and one can use the action ID [gab,g0ab] based on this Lagrangian instead of the trace χ action. Fatibene, Ferraris, Francaviglia, and Raiteri [184Jump To The Next Citation Point] clarified the relationship between the two actions, ID [gab] and 0 ID [gab,gab], and the corresponding quasi-local quantities. Considering the reference term 0 S in the Brown–York expression as the action of the background metric g0ab (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 g0 ab coincide on the boundary ∂D. Using L (g ,g0) ab ab, they construct the conserved Noether current for any vector field a ξ and, by taking its flux integral, define charge integrals a 0 Q𝒮 [ξ ,gab,gab] on two-surfaces 𝒮.15 Again, the Brown–York quasi-local quantity Et[ξa,ta] and Q 𝒮t[ξa,gab,g0ab] coincide if the spacetime metrics coincide on the boundary ∂D and if ξa has some special form. Therefore, although the two approaches are basically equivalent under the boundary condition above, this boundary condition is too strong from both the point of view of the variational principle and that of the quasi-local quantities. We will see in Section 10.1.8 that even the weaker boundary condition, that requires only the induced three-metrics on 3B fromgab and from g0ab to 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 ∫ 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 χ action as the action for general relativity, it is natural to derive the corresponding Hamiltonian in the analogous way. Following this route Brown and York derived the Hamiltonian, corresponding to the ‘basic’ (or nonreferenced) action I1 as well [121Jump 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 Eq. (10.8View 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 [100Jump To The Next Citation Point]). This Hamiltonian was investigated further in [119Jump 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 two-surface ∂Σ. It is shown that the change of the Hamiltonian under a boost of Σ yields precisely the boosts of the energy and momentum surface density discussed above.

Hawking, Horowitz, and Hunter also derived the Hamiltonian from the trace χ action 1 ID [gab] both with the orthogonal [241Jump To The Next Citation Point] and nonorthogonal boundary assumptions [242Jump To The Next Citation Point]. They allowed matter fields ΦN, whose dynamics is governed by a first-order action I1 [gab,ΦN ] mD, to be present. However, they treated the reference configuration in a different way. In the traditional canonical analysis of the fields and the geometry based on a noncompact Σ (for example in the asymptotically flat case) one has to impose certain falloff conditions that ensure the finiteness of the action, the Hamiltonian, etc. This finiteness requirement excludes several potentially interesting field + gravity configurations from our investigations. In fact, in the asymptotically flat case we compare the actual matter + gravity configurations with the flat spacetime + vanishing matter fields configuration. Hawking and Horowitz generalized this picture by choosing a static, but otherwise arbitrary, solution 0 gab, 0 ΦN of the field equations, considered the timelike boundary 3B of D to be a timelike cylinder ‘near the infinity’, and considered the action

I [g ,Φ ] := I1 [g ] + I1 [g ,Φ ] − I1 [g0 ] − I1 [g0 ,Φ0 ] D ab N D ab mD ab N D ab mD ab N

and those matter + gravity configurations that induce the same value on 3B as Φ0N and g0ab. Its limit as 3B is ‘pushed out to infinity’ can be finite, even if the limit of the original (i.e., nonreferenced) action is infinite. Although in the nonorthogonal boundaries case the Hamiltonian derived from the nonreferenced action contains terms coming from the ‘joints’, by the boundary conditions at 3 B 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 [241Jump To The Next Citation Point], and the quasi-local energy for spheres in domains with nonorthogonal boundaries in the Schwarzschild solution is also calculated [242Jump To The Next Citation Point]. A similar Hamiltonian, including the ‘joints’ or ‘corner’ terms, was obtained by Francaviglia and Raiteri [191Jump To The Next Citation Point] for the vacuum Einstein theory (and for Einstein–Maxwell systems in [9Jump 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 [95Jump To The Next Citation Point] reexamined the general form of the Brown–York energy and angular momentum from a Hamiltonian point of view.16 Their starting point is the observation that the domain 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 ab (hab, &tidle;p ) on the typical three-manifold Σ and the time coordinate t, the space coordinates xA on the two-boundary 𝒮 = ∂Σ, and their conjugate momenta π and π A.

The second important observation of Booth and Fairhurst is that the Brown–York boundary conditions are too restrictive. The two-metric, lapse, and shift need not be fixed, but their variations corresponding to diffeomorphisms on the boundary must be allowed. Otherwise diffeomorphisms that are not isometries of the three-metric γ ab 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 π 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 γab on the timelike boundary 3B. To specify S0, it seems natural to expect that the principal function S be zero in Minkowski spacetime [216Jump To The Next Citation Point, 120Jump To The Next Citation Point]. Then S0 would be the integral of the trace Θ0 of the extrinsic curvature of 3B, if it were embedded in Minkowski spacetime with the given intrinsic metric γ ab. However, a general Lorentzian three-manifold 3 ( B, γab) cannot be isometrically embedded, even locally, into the Minkowski spacetime. (For a detailed discussion of this embedability, see [120Jump 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 𝜀, in some reference spacetime, e.g., in Minkowski or anti-de Sitter spacetime. Assuming that 0 S depends on the lapse N and shift N a linearly, the functional derivatives (δS0 ∕δN ) and (δS0 ∕δN a) depend only on the two-metric qab and on the boost-gauge that 3B defined on 𝒮t. Therefore, 𝜀 and ja take the form (10.10View Equation), and, by the requirement of the vanishing of 𝜀 in the reference spacetime it follows that k0 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 unique (since, in particular, there are two independent normals of 𝒮t in the spacetime and it would not be fixed which normal should be used to calculate k0), and hence the construction would be ambiguous. On the other hand, one could require (𝒮t,qab) to be embedded into flat Euclidean three-space, i.e., into a spacelike hyperplane of Minkowski spacetime. This is the choice of Brown and York [120Jump To The Next Citation Point, 121Jump To The Next Citation Point]. In fact, as we already noted in Section 4.1.3, for two-surfaces with everywhere positive scalar curvature, such an embedding exists and is unique. (The order of the differentiability of the metric is reduced in [261] to C2.) A particularly interesting two-surface that cannot be isometrically embedded into the flat three-space is the event horizon of the Kerr black hole, if the angular momentum parameter a exceeds the irreducible mass (but is still not greater than the mass parameter m), i.e., if √ -- 3m < 2 |a| < 2m [463Jump To The Next Citation Point]. (On the other hand, for its global isometric embedding into 4 ℝ, see [203].) Thus, the construction works for a large class of two-surfaces, but certainly not for every potentially interesting two-surface. The convexity condition is essential.

It is known that the (local) isometric embedability of (𝒮, qab) into flat three-space with extrinsic curvature k0ab is equivalent to the Gauss–Codazzi–Mainardi equations δa(k0ab − δabk0) = 0 and 𝒮R − (k0)2 + k0abk0ab = 0. Here δa is the intrinsic Levi-Civita covariant derivative and 𝒮R 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 0 kab, and hence, if the embedding exists, qab determines 0 k. Therefore, the subtraction term 0 k can also be interpreted as a solution of an under-determined elliptic system, which is constrained by a nonlinear algebraic equation. In this form the definition of the reference term is technically analogous to the definition of those in Sections 7, 8, and 9, but, by the nonlinearity of the equations, in practice it is much more difficult to find the reference term 0 k 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 A0 a will be zero in the boost-gauge in which the timelike normal of 𝒮 t in the reference Minkowski spacetime is orthogonal to the spacelike three-plane, because this normal is constant. Thus, to summarize, for convex two-surfaces, the flat space reference of Brown and York is uniquely determined, k0 is determined by this embedding, and A0a = 0. Then ∫ 8πGS0 = − 𝒮 N k0d𝒮t 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 [118Jump To The Next Citation Point, 334Jump To The Next Citation Point, 335Jump To The Next Citation Point]. In this approach the two-surface (𝒮t,qab) is embedded into the light cone of a point of the Minkowski or anti-de Sitter spacetime instead of into a spacelike hypersurface of constant curvature. The essential difference between the new (‘light cone reference’) and the previous (‘flat space reference’) prescriptions is that the embedding into the light cone is not unique, but the reference term 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 [115], 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, [178Jump 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 [334Jump To The Next Citation Point, 335Jump To The Next Citation Point], yielding ∘ ------------ k0 = 2𝒮R + 4 ∕λ2, where 𝒮R is the Ricci scalar of (𝒮, qab) and λ 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 [118Jump To The Next Citation Point].

10.1.7 Further properties and the various limits

Although the general, nonreferenced expressions are additive, the prescription for the reference term 0 k destroys the additivity in general. In fact, if 𝒮 ′ and 𝒮 ′′ are two-surfaces such that 𝒮 ′ ∩ 𝒮′′ is connected and two-dimensional (more precisely, it has a nonempty open interior, for example, in 𝒮 ′), then in general ------------------ 𝒮′ ∪ 𝒮 ′′ − 𝒮′ ∩ 𝒮 ′′ (overline means topological closure) is not guaranteed to be embeddable, the flat three-space, and even if it is embeddable then the resulting reference term 0 k differs from the reference terms ′0 k and ′′0 k determined from the individual embeddings.

As noted in [100Jump 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 Σ 1 be the spacelike hyperboloid ∘ -2----2 t = − ρ + r, Σ0 the hyperplane t = − T = const. < − ρ < 0 and 𝒮 := Σ0 ∩ Σ1, the sphere of radius ∘ -------- T 2 − ρ2 in the t = − T hyperplane. Then the trace of the extrinsic curvature of 𝒮 in Σ0 and in Σ1 is ∘ -------- 2∕ T2 − ρ2 and ∘ -------- 2T ∕ρ T2 − ρ2, respectively. Therefore, the Brown–York quasi-local energy (with the flat three-space reference) associated with 𝒮 and the normals of Σ1 on 𝒮 is ∘ ---------------3 − (T + ρ)(T − ρ) ∕(ρG ). Similarly, the Brown–York quasi-local energy with the light cone references in [334Jump To The Next Citation Point] and in [118Jump To The Next Citation Point] is also negative for such surfaces with the boosted observers.

Recently, Shi and Tam [458Jump To The Next Citation Point] have proven interesting theorems in Riemannian three-geometries, which can be used to prove positivity of the Brown–York energy if the two-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 Σ be a compact, smooth Riemannian three-manifold with non-negative scalar curvature and smooth two-boundary 𝒮 such that each connected component 𝒮i of 𝒮 is homeomorphic to 2 S and the scalar curvature of the induced two-metric on 𝒮i is strictly positive. Then, for each component ∮ ∮ 𝒮i kd 𝒮i ≤ 𝒮i k0d𝒮i holds, where k is the trace of the extrinsic curvature of 𝒮 in Σ with respect to the outward-directed normal, and k0 is the trace of the extrinsic curvature of 𝒮i in the flat Euclidean three-space when 𝒮i is isometrically embedded. Furthermore, if in these inequalities the equality holds for at least one 𝒮i, then 𝒮 itself is connected and Σ is flat. This result is generalized in [459] by weakening the energy condition, in which case lower estimates of the Brown–York energy can still be given. For some rigidity theorems connected with this positivity result, see [461]; and for their generalization for higher dimensional spin manifolds, see [329].

The energy expression for round spheres was calculated in [121Jump To The Next Citation Point, 100Jump To The Next Citation Point]. In the spherically-symmetric metric discussed in Section 4.2.1, on round spheres the Brown–York energy with the flat space reference and fleet of observers ∂∕∂t on 𝒮 is a GEBY [𝒮r,(∂∕∂t) ] = r(1 − exp(− α)). In particular, it is ∘ ----------- 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 [119Jump To The Next Citation Point].) Still in the spherically-symmetric context the definition of the Brown–York energy is extended to spherical two-surfaces beyond the event horizon in [347] (see also [443]). A remarkable result is that while the total energy of the electrostatic field of a point charge in any finite three-volume surrounding the point charge in Minkowski spacetime is always infinite, the negative gravitational binding energy compensates the electrostatic energy so that the quasi-local energy is negative within a certain radius under the event horizon in the Reissner–Nordström spacetime and tends to − |e | as r → 0. The Brown–York energy is discussed from the point of view of observers in spherically-symmetric spacetimes (e.g., the connection between this energy and the effective energy in the geodesic equation for radial geodesics) in [90, 576]. The explicit calculation of the Brown–York energy with the (implicitly assumed) flat-space reference in Friedmann–Robertson–Walker spacetimes (as particular examples for the general round sphere case) is given in [6Jump To The Next Citation Point].

The Newtonian limit can be derived from the round sphere expression 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. (For the calculation of the Newtonian limit in the covariant Newtonian spacetime, see [564].) However, taking into account that on the Schwarzschild horizon GEBY = 2m, while at 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 [120Jump To The Next Citation Point, 121Jump To The Next Citation Point]. The thermodynamics of the Schwarzschild–anti-de Sitter black holes was investigated in terms of the quasi-local quantities in [116Jump 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 [119Jump To The Next Citation Point].

The energy is explicitly calculated for three different kinds of two-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 [356]. 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 2 4πR, 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 two-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 √ --2----2 r+ := m + m − a is 2 2 4 4 2m [1 − a ∕(8m ) + 𝒪 (a ∕r+)], which is twice the irreducible mass of the black hole. An interesting feature of this calculation is that the energy cannot be calculated for the horizon directly, because, as previously noted, the horizon itself cannot be isometrically embedded into a flat three-space if the angular momentum parameter exceeds the irreducible mass [463]. The energy for the ergosurface is positive, as for the other two kinds of surfaces.

The spacelike infinity limit of the charges interpreted as the energy, spatial momentum, and spatial angular momentum are calculated in [119] (see also [241Jump 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 center-of-mass is given in [57]. It is shown that the corresponding large sphere limit is just the center-of-mass expression of Beig and Ó Murchadha [64]. Here the center-of-mass integral is also given in terms of a charge integral of the curvature. The large sphere limit of the energy for metrics with the weakest possible falloff conditions is calculated in [181Jump To The Next Citation Point, 462]. A further demonstration that the spatial infinity limit of the Brown–York energy in an asymptotically Schwarzschild spacetime is the ADM energy is given in [180].

Although the prescription for the reference configuration by Hawking and Horowitz cannot be imposed for a general timelike three-boundary 3B (see Section 10.1.8), asymptotically, when 3B is pushed out to infinity, this prescription can be used, and coincides with the prescription of Brown and York. Choosing the background metric g0 ab to be the anti-de Sitter one, Hawking and Horowitz [241Jump 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 [116Jump To The Next Citation Point].) In [117Jump To The Next Citation Point] the null infinity limit of the integral of N (k0 − k)∕(8πG ) 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. (The limit of the Brown–York energy on general asymptotically hyperboloidal hypersurfaces is calculated in [330].) In a slightly different formulation Booth and Creighton calculated the energy flux of outgoing gravitational radiation [94Jump 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 [335Jump To The Next Citation Point]. While in nonvacuum the quasi-local energy is the expected (4π∕3 )r3T tatb ab, in vacuum it is proportional to ab ab 4EabE + HabH, instead of the Bel–Robinson ‘energy’ a bc d Tabcdt tt t. (Here Eab and Hab are, respectively, the conformal electric and conformal magnetic curvatures, and a t plays a double role. It defines the two-sphere of radius r [as is usual in the small sphere calculations], and defines the fleet of observers on the two-sphere.) On the other hand, the special light cone reference used in [118, 335] reproduces the expected result in nonvacuum, and yields [1 ∕(90G )]r5T tatbtctd abcd in vacuum. The small sphere limit was also calculated in [181] for small geodesic spheres in a time symmetric spacelike hypersurface.

The light cone reference ∘ ------------ k0 = 2𝒮R + 4∕λ2 was shown to work in the large sphere limit near the null and spatial infinities of asymptotically flat spacetimes and near the infinity of asymptotically anti-de Sitter spacetimes [334]. Namely, the Brown–York quasi-local energy expression with this null-cone reference term tends to the Bondi–Sachs, the ADM, and Abbott–Deser energies. The supermomenta of Geroch at null infinity can also be recovered in this way. The proof is simply a demonstration of the fact that this light cone and the flat space prescriptions for the subtraction term have the same asymptotic structure up to order 𝒪 (r− 3). This choice seems to work properly only in the asymptotics, because for small ellipsoids in the Minkowski spacetime this definition yields nonzero energy and for small spheres in vacuum it does not yield the Bel–Robinson ‘energy’.17

A formulation and a proof of a version of Thorne’s hoop conjecture for spherically symmetric configuarations in terms of EBY are given in [402Jump To The Next Citation Point], and will be discussed in Section 13.2.2.

10.1.8 Other prescriptions for the reference configuration

As previously noted, Hawking, Horowitz, and Hunter [241, 242] defined their reference configuration by embedding the Lorentzian three-manifold 3 ( B, γab) isometrically into some given Lorentzian spacetime, e.g., into the Minkowski spacetime (see also [216]). However, for the given intrinsic three-metric γab and the embedding four-geometry the corresponding Gauss and Codazzi–Mainardi equations form a system of 6 + 8 = 14 equations for the six components of the extrinsic curvature Θ ab [120Jump 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, even the small perturbations of the intrinsic metric hab would break the conditions of embedability. Therefore, in general, this prescription for the reference configuration can work only if the three-surface 3B is ‘pushed out to infinity’, but does not work for finite three-surfaces [120].

To rule out the possibility that the Brown–York energy can be nonzero even in Minkowski spacetime (on two-surfaces in the boosted flat data set), Booth and Mann [100] suggested that one embed (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 ξa = N ta + N a of the dynamics, tangent to 3 B, to a vector field 0a ξ in 0 M such that ∗ 0 0 xξqab = ϕ (xξ qab) and a ∗ 0a 0 ξ ξa = ϕ (ξ ξa). Here ϕ is a diffeomorphism mapping an open neighborhood U of 𝒮 in M into 0 M such that ϕ |𝒮, the restriction of ϕ to 𝒮, is an isometry, and xξqab denotes the Lie derivative of qab along ξa. 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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