List of Footnotes

1		Sometimes in the literature this requirement is introduced as some new principle in the Hamiltonian formulation of the fields, but its real content is not more than to ensure that the Hamilton equations coincide with the field equations.
2		Since we do not have a third kind of device to specify the spatio-temporal location of the devices measuring the spacetime geometry, we do not have any further operationally defined, maybe nondynamic background, just in accordance with the principle of equivalence. If there were some nondynamic background metric $g0 ab$ on $M$ , then, by requiring $g0 = ϕ∗g0 ab ab$ we could reduce the almost arbitrary diffeomorphism $ϕ$ (essentially four arbitrary functions of four variables) to a transformation depending on at most ten parameters.
3		Since Einstein’s Lagrangian is only weakly diffeomorphism invariant, the situation would be even worse if we used Einstein’s Lagrangian. The corresponding canonical quantities would still be coordinate dependent, though in certain ‘natural’ coordinate systems they yield reasonable results (see, for example, [2 ] and references therein).
4		$E(𝒮)$ can be thought of as the 0-component of some quasi-local energy-momentum four-vector, but, because of the spherical symmetry, its spatial parts are vanishing. Thus, $E(𝒮)$ can also be interpreted as the mass, the length of this energy-momentum four-vector.
5		As we will soon see, the leading term of the small-sphere expression of the energy-momenta in nonvacuum is of order $r3$ , in vacuum it is of order $r5$ , while those of the relativistic angular momentum are $r4$ and $r6$ , respectively.
6		Because of the falloff, no essential ambiguity in the definition of the large spheres arises from the use of the coordinate radius instead of the physical radial distance.
7		In the Bondi coordinate system the radial coordinate is the luminosity distance $rD := − 1∕ρ$ , which tends to the affine parameter $r$ asymptotically.
8		Since we take the infimum, we could equally take the ADM masses, which are the minimum values of the zero-th component of the energy-momentum four-vectors in the different Lorentz frames, instead of the energies.
9		I thank Paul Tod for pointing this out to me.
10		The analogous calculations using tensor methods and the real $ab ω$ instead of spinors and the anti-self-dual (a.s.d.) part of $ab ω$ would be technically more complicated [386 , 387 , 392 , 203 ].
11		Recall that, similarly, we did not have any natural isomorphism between the two-surface twistor spaces, discussed in Section 7.2.1, on different two-surfaces.
12		Clearly, for the Ludvigsen–Vickers energy-momentum no such ambiguity is present, because the part (8.3 ) of their propagation law defines a natural isomorphism between the space of the Ludvigsen–Vickers spinors on the different two-surfaces.
13		In the original papers Brown and York assumed that the leaves $Σt$ of the foliation of $D$ were orthogonal to $3 B$ (‘orthogonal boundaries assumption’).
14		The paper [168 ] gives a clear, readable summary of these earlier results.
15		Thus, in principle, we would have to report on their investigations in the next Section 11. Nevertheless, since essentially they re-derive and justify the results of Brown and York following only a different route, we discuss their results here.
16		According to this view the quasi-local energy is similar to $E Σ$ of Equation (2.6 ), rather than to the charges, which are connected somehow to some ‘absolute’ element of the spacetime structure.
17		This phase space is essentially $∗ T TQ$ , the cotangent bundle of the tangent bundle of the configuration manifold $Q$ , endowed with the natural symplectic structure, and can be interpreted as the collection of quadruples $a a (q , ˙q ,pa, ˙pa)$ . The usual Lagrangian (or velocity) phase space $TQ$ and the Hamiltonian (or momentum) phase space $∗ T Q$ are special submanifolds of $∗ T T Q$ .
18		In fact, Kijowski’s results could have been presented here, but the technique that he uses justifies their inclusion in the Section 10.
19		Here we concentrate only on the genuine, finite boundary of $Σ$ . The analysis is straightforward even in the presence of ‘boundaries at infinity’ at the asymptotic ‘ends’ of asymptotically flat $Σ$ .
20		I am grateful to Sergio Dain for pointing this out to me.
21		It could be interesting to clarify the consequences of the boost gauge choice that is based on the main extrinsic curvature vector $Qa$ , discussed in Section 4.1.2. This would rule out the arbitrary element of the construction.