Zero rest-mass fields and the Newman-Penrose constants on flat space

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Written by Edgar Gasperin-Garcia

Zero rest-mass fields of spin 1 (the electromagnetic field) and spin 2 propagating on flat space and their corresponding Newman-Penrose (NP) constants are studied near spatial infinity. The aim of this analysis is to clarify the correspondence between data for these fields on a spacelike hypersurface and the value of their corresponding NP constants at future and past null infinity. To do so, the framework of the cylinder at spatial infinity is employed to show that, expanding the initial data in terms spherical harmonics and powers of the geodesic spatial distance ρ to spatial infinity, the NP constants correspond to the data for the highest possible spherical harmonic at fixed order in ρ. In addition, it is shown that the NP constants at future and past null infinity, for both the Maxwell and spin-2 case, are related to each other as they arise from the same terms in the initial data. Moreover, it is shown that this observation is true for generic data (not necessarily time-symmetric). This identification is a consequence of both the evolution and constraint equations.