# 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 this type of framework extending the former paintings of Kijowski and Tulczyjew. they begin by way of reviewing a few straightforward 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 offers a close description of the applying of the hot formalism to the case of the massless scalar box. eventually, the formalism is utilized to the case of Einstein gravity. The Hamiltonian position of the Trautman--Bondi mass is exhibited. A Hamiltonian definition of angular momentum at null infinity is derived and analysed.

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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. 39) after integration. 41) 6X sin 0 dO dW Y(-00,01 Throughout the discussion above we have assumed that solutions of the wave equation, which are smooth on 1 1, exist, this question is discussed in detail in Appendix B.

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.

61) 1 101. Hamiltonian role of the Trautman-Bondi energy for scalar fields In physics one energy which would like to be able to determine the maximal amount of can be extracted out of frameworks have often been used a given field configuration. Hamiltonian devices for producing candidates for the appropriate energy expressions. Now, a Hamiltonian approach can only give a convincing result if the expressions so obtained are unique. As already pointed out at the end of Sect. 1, Hamiltonians are indeed unique (modulo an additive constant), when the form S?