By Piotr T. Chrusciel, Jacek Jezierski, Jerzy Kijowski
The aim of this monograph is to teach that, within the radiation regime, there exists a Hamiltonian description of the dynamics of a massless scalar box, in addition to of the dynamics of the gravitational box. The authors build this sort of framework extending the former paintings of Kijowski and Tulczyjew. they begin by means of reviewing a few uncomplicated proof referring to Hamiltonian dynamical structures after which describe the geometric Hamiltonian framework, sufficient for either the standard asymptotically flat-at-spatial-infinity regime and for the radiation regime. The textual content then supplies a close description of the applying of the recent formalism to the case of the massless scalar box. ultimately, the formalism is utilized to the case of Einstein gravity. The Hamiltonian function of the Trautman--Bondi mass is exhibited. A Hamiltonian definition of angular momentum at null infinity is derived and analysed.
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With iz: Z iz In all the cases M, -- considered below Z will be an open ball of radius one, the hyperbola. Further, it will be natural to glue image ' will OZ to a connected component of aZ_o+, so that actually be connected. will always arise from this section in Z considered of the motions 0 Moreover, the flow of a vector field X defined on M; we mention that this will not be the case for the J+-part of the dynamics in the phase space considered in of which in M will be a Sect. 5. 41) leads thus X" L) dS , .
We shall follow the philosophy adopted in Chap. 3, and use different symbols for fields defined on M, as compared to those on the model space Z x R. 7)); is proportional to the two-volume density on k is a density on B(O, 1), which S2 (in spherical coordinates (0, o) equal to sin 0), with the proportionality coefficient smooth-up-to-boundary on B(O, 1); should be thought of here c is a smooth function on (_C)O, 0] X S2 (which compactly supported; as being a subset of Y+), withOc/,9w (,Tr, 0, c) satisfy the corner conditions to all orders (cf.
2 convergence of Energy: 47 integrals corresponding Hamiltonian is the field energy. 29) the multiplication by 0 commutes with Cx because CXS-2 =- 0, which simplifies somewhat the analysis. P'+. 30) - 4 It follows that a we can safely pass to the limit finite and well defined H (X, _9',, p) when already seen - fE coming from 6P and from _PxP cancel when Xxf 6P (as X = alc')w, dangerous xxfjp Cxpjf - H(X, Y,,,, p) to obtain (1 1). 26), in the derivation of formula 0 in -4 - Cxp6f , actually have Cxp P,,).