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 ![]() ![]() ![]() ![]() |
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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, e.g., [7] and references therein). | |
4 | ![]() ![]() |
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5 | As we will soon see, the leading term of the small-sphere expression of the energy-momenta in nonvacuum is of
order ![]() ![]() ![]() ![]() |
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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 ![]() ![]() |
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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 | I am grateful to Jörg Frauendiener and one of the referees for clarifying this point. | |
11 | The analogous calculations using tensor methods and the real ![]() ![]() ![]() ![]() ![]() |
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12 | 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. | |
13 | Clearly, for the Ludvigsen–Vickers energy-momentum no such ambiguity is present, because the part (8.3![]() |
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14 | In the original papers Brown and York assumed that the leaves ![]() ![]() ![]() |
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15 | The paper [184] gives a clear, readable summary of these earlier results. | |
16 | 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. | |
17 | Lau, S.R., personal communication (July 2003) | |
18 | According to this view the quasi-local energy is similar to ![]() ![]() |
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19 | This phase space is essentially ![]() ![]() ![]() ![]() ![]() ![]() |
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20 | In fact, Kijowski’s results could have been presented here, but the technique that he uses justifies their inclusion in Section 10. | |
21 | Here we concentrate only on the genuine, finite boundary of ![]() ![]() |
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22 | I am grateful to Sergio Dain for pointing this out to me. | |
23 | It could be interesting to clarify the consequences of the boost gauge choice that is based on the main
extrinsic curvature vector ![]() |
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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