A Hamiltonian field theory in the radiating regime by Piotr T. Chrusciel, Jacek Jezierski, Jerzy Kijowski

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 any such framework extending the former paintings of Kijowski and Tulczyjew. they begin by way of reviewing a few uncomplicated evidence pertaining to Hamiltonian dynamical platforms 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 offers an in depth 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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We will have aat(W, yA)IaW 0 = on = _ at will in general not be constant in w on (-oo, -11, correspond to a function in 9[-,,0] any more. Thus Tt is only defined for an interval of positive times depending upon the point (,Tr, 0, c) of 9[-1,0], hence is only a local dynamical system. 41) with [-r-, -r] so = (such as Einstein equations) that this kind of behaviour has to be [- 1, 01 shows that the dynamics is where blow up in accepted. Formula Hamiltonian, with 52 4. 51) and S? 41) vanishes. 52) - 1,0] = S,_ over (61X7 62X) D,, 1,2, have been identified with the OraiOalca)'s 0.

From the following, modified I 2 Lagrangian: 2 - L+ "momentum as a + 12 2 V. 25). 0+ by continuity, which is not the case for L. 54) and the discussion below). CX Q . 13). 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).

34) 48 4. y, ((XxP, Xx f), (JP, Jf)) --- + Of f OW a y, a J1 sin 0 dO do. 35) analysis may be performed for that part of -3 , which is mapped boundary ,0+ of M. 34) = - Iim 6-+0 lim I we X: jX(W'YA)=jj(W'P=j'YA). 36) (Hamiltonian) con- phase (61X, 62X) 0 denote we space of the field data on ,0+. 33). The OW J, X) 1P=1-6 sin 0 dO d sin 0 dO dV Hamiltonian corresponding . + ,], x) = (-fiP-CXhL;;-+ the = X. 39) HP (X, X) dO d o S2 a rather general formalism, in which the or lack thereof was hidden in the formalism.

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