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 non-dynamical background, just in accordance with the
principle of equivalence. If there were some non-dynamical background metric ![]() ![]() ![]() ![]() |
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3 | Since Einstein’s Lagrangian is only weakly diffeomorphism invariant, the situation would even be worse if we used Einstein’s Lagrangian. The corresponding canonical quantities would still be coordinate dependent, though in certain ‘natural’ coordinate system they yield reasonable results (see for example [2] and references therein). | |
4 | ![]() ![]() |
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5 | If, in addition, the spinor constituent ![]() ![]() ![]() |
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6 | As we will see soon, the leading term of the small sphere expression of the energy-momenta in non-vacuum is of order
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7 | Because of the fall-off, no essential ambiguity in the definition of the large spheres arises from the use of the coordinate radius instead of the physical radial distance. | |
8 | In the so-called Bondi coordinate system the radial coordinate is the luminosity distance ![]() ![]() |
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9 | 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. | |
10 | I thank Paul Tod for pointing this out to me. | |
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 2-surface twistor spaces, discussed in Section 7.2.1, on different 2-surfaces. | |
13 | Clearly, for the Ludvigsen-Vickers energy-momentum no such ambiguity is present, because the part (59![]() |
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14 | In the original papers Brown and York assumed that the leaves ![]() ![]() ![]() |
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15 | The paper gives a clear, well 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 | The problem to characterize this embeddability is known as the Weyl problem of differential geometry. | |
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 may justify their inclusion in the previous 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 out this 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 ![]() |
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24 | It might be interesting to see the small sphere expansion of the Kijowski and Kijowski-Liu-Yau expressions in vacuum. |
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