# 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 via reviewing a few user-friendly evidence relating 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 provides an in depth description of the appliance 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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13) = V*F & AnM F ED AnM, "isomorphic to", it is also natural to use the Lie derivative DX on sections p of that part PM of the space-time phase denotes for the action of 2 (p4 X x f . 9). This will, however, not be the Lie derivative as defined in standard textbooks on differential geometry, when X so constructed is not a vector field defined on Aff_. For scalar fields the only of esthetic nature; however, it becomes essential for general distinction is tensor fields. 44 4. Radiating scalar fields bundle P which lies above M.

This proves the statement. In case Of Xtrans and Xboost the argument is slightly more involved: the divergence of XA equals the two-dimensional Laplacian of the function v. 0+. 74). 73) multiplied by X'. ,O], and H(X, -':5 0 U Y + convergent. 45) vanish precisely for the same they did when X was a time translation. This finishes the proof, that every one-parameter subgroup of the Poincar6 group generates a Hamiltonian dynamical system in and _00[-1,o]. 8 "Supertranslated" hyperbolae In the previous sections we have, for the sake of simplicity, restricted ourselves hypersurfaces -9' which are hyperbolae in Minkowski space-time.

42) = restriction, . 14) "acting on the A index Of PA"' and the Lie derivative operator XX "acting on the /-t inde)e'. 42) holds with this choice of Dxp. 36) takes the following space-time form: below). time -6H j(DXPA4)jfA _ (DXfA ) JPA /"I dS1, 3. 43) by 8v)] dx1 A ... 41). For the purpose of illustration let f be a scalar field, with DXf We have from X'f,,. 41): a,, [(pl'X" - - Xt'JL) dS,, Xl'p') Jf] - Xxp"6f I dSj, . 37). in in the particular case of a scalar field theory) starting point of a definition of a Hamiltonian evolution of the field, provided the boundary integral is made to vanish by a suitable choice of boundary conditions.