14.4 On the Brown–York–type expressions

The idea of Brown and York that the quasi-local conserved quantities should be introduced via the canonical formulation of the theory is quite natural. In fact, as we saw, one could arrive at their general formulae from different points of departure (functional differentiability of the Hamiltonian two-surface observables). If the a priori requirement that we should have a well-defined action principle for the trace-χ-action yielded undoubtedly well behaving quasi-local expressions, then the results would a posteriori justify this basic requirement (like the holomorphicity or antiholomorphicity of the spinor fields in the Dougan–Mason definitions). However, if not, then that might be considered as an unnecessarily restrictive assumption, and the question arises as to whether the present framework is wide enough to construct reasonable quasi-local energy-momenta and angular momenta.

Indeed, the basic requirement automatically yields the boundary condition that the three-metric γab should be fixed on the boundary 𝒮, and that the boundary term in the Hamiltonian should be built only from the surface stress tensor τ ab. Since the boundary conditions are given, no Legendre transformation of the canonical variables on the two-surface is allowed (see the derivation of Kijowski’s expression in Section 10.2). The use of τab has important consequences. First, the quasi-local quantities depend not only on the geometry of the two-surface 𝒮, but on an arbitrarily chosen boost gauge, interpreted as a ‘fleet of observers ta being at rest with respect to 𝒮, as well. This leaves a huge ambiguity in the Brown–York energy (three arbitrary functions of two variables, corresponding to the three boost parameters at each point of 𝒮) unless a natural gauge choice is prescribed.23 Second, since τab does not contain the extrinsic curvature of 𝒮 in the direction a t, which is a part of the two-surface data, this extrinsic curvature is ‘lost’ from the point of view of the quasi-local quantities. Moreover, since τab is a tensor only on the three-manifold 3B, the integral of Ka τ tb ab on 𝒮 is not sensitive to the component of Ka normal to 3B. The normal piece a b v vbK of the generator a K is ‘lost’ from the point of view of the quasi-local quantities.

The other important ingredient of the Brown–York construction is the prescription of the subtraction term. Considering the Gauss–Codazzi–Mainardi equations of the isometric embedding of the two-surface into the flat three-space (or rather into a spacelike hyperplane of Minkowski spacetime) only as a system of differential equations for the reference extrinsic curvature, this prescription – contrary to frequently appearing opinions – is as explicit as the condition of the holomorphicity/antiholomorphicity of the spinor fields in the Dougan–Mason definition. (One essential, and, from pragmatic points of view, important, difference is that the Gauss–Codazzi–Mainardi equations form an underdetermined elliptic system constrained by a nonlinear algebraic equation.) Similar to the Dougan–Mason definitions, the general Brown–York formulae are valid for arbitrary spacelike two-surfaces, but solutions to the equations defining the reference configuration exist certainly only for topological two-spheres with strictly positive intrinsic scalar curvature. Thus, there are exceptional two-surfaces here, too. On the other hand, the Brown–York expressions (both for the flat three-space and the light cone references) work properly for large spheres.

At first sight, this choice for the definition of the subtraction term seems quite natural. However, we do not share this view. If the physical spacetime is the Minkowski one, then we expect that the geometry of the two-surface in the reference Minkowski spacetime would be the same as in the physical Minkowski spacetime. In particular, if 𝒮 – in the physical Minkowski spacetime – does not lie in any spacelike hyperplane, then we think that it would be unnatural to require the embedding of 𝒮 into a hyperplane of the reference Minkowski spacetime. Since in the two Minkowski spacetimes the extrinsic curvatures can be quite different, the quasi-local energy expressions based on this prescription of the reference term can be expected to yield a nonzero value even in flat spacetime. Indeed, there are explicit examples showing this defect. (Epp’s definition is free of this difficulty, because he embeds the two-surface into the Minkowski spacetime by preserving its ‘universal structure’; see Section 4.1.4.)

Another objection against the embedding into flat three-space is that it is not Lorentz covariant. As we discussed in Section 4.2.2, Lorentz covariance (together with the positivity requirement) was used to show that the quasi-local energy expression for small spheres in vacuum is of order r5 with the Bel–Robinson ‘energy’ as the factor of proportionality. The Brown–York expression (even with the light cone reference k0 = √2-𝒮-R-) fails to give the Bel–Robinson ‘energy’.

Finally, in contrast to the Dougan–Mason definitions, the Brown–York type expressions are well defined on marginally trapped surfaces. However, they yield just twice the expected irreducible mass, and they do not reproduce the standard round sphere expression, which, for nontrapped surfaces, arises from all the other expressions discussed in the present section (including Kijowski’s definition). It is remarkable that the derivation of the first law of black hole thermodynamics, based on the identification of the thermodynamic internal energy with the Brown–York energy, is independent of the definition of the subtraction term.

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