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Then the quasi-local mass and Pauli-Lubanski spin are defined, respectively, ALI m1531 rev 179 the usual formulae and. As a consequence of the definitions, holds, i. Hence, is the square of the mass density of the matter fields, where hab is the spatial metric in the plane orthogonal to ta.

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If Tab satisfies the dominant energy condition i. Nevertheless, because of more advantageous properties see Section 2. Thus, even if there is a gauge-invariant and unambiguously-defined energy-momentum density of the matter fields, it is not a priori clear how the various quasi-local quantities should be introduced.

Hamiltonian introduction of the quasi-local quantities In the standard Hamiltonian formulation of the dynamics of the classical matter fields on a given ALI m1531 rev 179 necessarily flat spacetime see, e. If the matter fields have gauge freedom, then the dynamics of the system is constrained: Then the time evolution of the states in the phase space is governed by the Hamiltonian, which has the form 2. In this case, Za should be modified.

In fact, the boundary conditions are connected to the nature of the physical situations considered. For example, in electrodynamics different boundary conditions must be imposed if the boundary is to represent a conducting or an insulating surface. An identically-conserved quantity ALI m1531 rev 179 always be added to the Hamiltonian provided its functional differentiability is preserved.

The energy density and the momentum density of the matter fields can be recovered as the functional derivative of H[K] with respect to the lapse N and the shift Na, respectively. In principle, the whole analysis can be repeated quasi-locally too.

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However, apart from the promising achievements of [ 1314] for the Klein-Gordon, ALI m1531 rev 179, and the Yang-Mills-Higgs fields, as far as we know, such a systematic quasi-local Hamiltonian analysis of the matter fields is still lacking. Properties of the quasi-local quantities Suppose that the matter fields satisfy the dominant energy condition. The properties of the quasi-local quantities based on in Minkowski spacetime are, however, more interesting. Thus, in particular, the Pauli-Lubanski spin is zero.

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On the other hand, the additivity of is a slightly more delicate problem. Namely, and are elements of the dual space of the translations, and hence, we can add them and, as in the previous case, we obtain additivity. However, this additivity comes from the absolute parallelism of the Minkowski spacetime: The quasi-local energy-momenta of the different two-surfaces belong to one and the same vector space. If there were no natural connection between the Killing vectors on different two-surfaces, then the energy-momenta would belong to different vector spaces, and they could not be added. However, the finiteness of the angular momentum and center-of-mass is not ensured by the falloff. Since the typical falloff of Tab — for the electromagnetic field, for example — iswe may not impose faster than this, because otherwise we would exclude the electromagnetic field from our investigations.

Thus, in addition to the falloff, six global ALI m1531 rev 179 conditions for the leading terms of Tab must be imposed. Although quasi-locally the vanishing of the mass does not imply the vanishing of ALI m1531 rev 179 matter fields themselves the matter fields must be pure radiative field configurations with plane wave frontsthe vanishing of the total mass alone does imply the vanishing of the fields. In fact, by the vanishing of the mass, the fields must be plane waves, furthermore, bythey must be asymptotically vanishing at the same time.


However, a plane-wave ALI m1531 rev 179 can be asymptotically vanishing only if it is vanishing. Quasi-local radiative modes and a classical version of the holography for matter fields By the results of Section 2. Assuming, for the sake of simplicity, that is future and past convex in the sense of Section 4. Thus, every such spinor field can be expanded by the special radiative solutions, and hence, can also be represented by the corresponding family of holographic data.

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This fact may be a specific manifestation in classical nongravitational physics of the holographic principle see Section The difficulties The root of the difficulties: Thus, the rest mass of the source is analogous to charge in electrostatics. Following the analogy with electrostatics, we can introduce the energy density and ALI m1531 rev 179 spatial stress of the gravitational field, respectively, by 3. Thus, at this point both the gravitational energy density and the spatial stress have been made vanishing. Therefore, the gravitational energy and the spatial stress cannot be localized to a point, i. In a relativistically corrected Newtonian theory both the internal energy density u of the matter source and the energy density U of ALI m1531 rev 179 gravitational field itself contribute to the source of gravity. Thus in the traditional units, when c is the speed of light the corrected field equation could be expected to be the genuinely non-linear equation 3.

Note, however, that the theory defined by 3.

A thorough analysis of this theory, and in particular its inconsistency, is given by Giulini [ ]. Therefore, by 3. ALI m rev Free Driver Download for Windows 98SE - .


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