Indeed, the basic requirement automatically yields the boundary condition that the 3-metric
should be fixed on the boundary , and that the boundary term in the Hamiltonian should be built only
from the surface stress tensor . Since the boundary conditions are given, no Legendre
transformation of the canonical variables on the 2-surface is allowed (see the derivation of Kijowski’s
expression in Section 10.2). The use of has important consequences. First, the quasi-local
quantities depend not only on the geometry of the 2-surface , but on an arbitrarily chosen boost
gauge, interpreted as a ‘fleet of observers being at rest with respect to ’, too. 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 ) unless a natural gauge choice is
prescribed^{23}.
Second, since does not contain the extrinsic curvature of in the direction , which is a part of
the 2-surface data, this extrinsic curvature is ‘lost’ from the point of view of the quasi-local quantities.
Moreover, since is a tensor only on the 3-manifold , the integral of on is not
sensitive to the component of normal to . The normal piece of the generator 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 2-surface into
the flat 3-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/anti-holomorphicity 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.) Similarly to the Dougan-Mason definitions, the general
Brown-York formulae are valid for arbitrary spacelike 2-surfaces, but solutions to the equations defining the
reference configuration exist certainly only for topological 2-spheres with strictly positive intrinsic scalar
curvature. Thus there are exceptional 2-surfaces here, too. On the other hand, the Brown-York
expressions (both for the flat 3-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 2-surface in the reference Minkowski spacetime be the same as in the physical Minkowski spacetime. In particular, if - in the physical Minkowski spacetime - does not lie in any spacelike hyperplane, then we think that it would be un-natural to require the embedding of 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 2-surface into the Minkowski spacetime by preserving its ‘universal structure’; see Section 4.1.4.)

Another objection against the embedding into flat 3-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
with the Bel-Robinson ‘energy’ as the factor of proportionality. The Brown-York
expression (even with the light cone reference ) fails to give the Bel-Robinson
‘energy’^{24}.
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 non-trapped surfaces, comes out 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
thermodynamical internal energy with the Brown-York energy, is independent of the definition of the
subtraction term.

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