Gravitational waves in general relativity. XV: The loss-free case (Q2748064)

Using a linear approximation for loss-free gravitational radiation is studied. Outside a bounded source, a slightly azimuth-dependent stationary rotating cylindrical system is examined. The method of \textit{J. A. Wheeler} and \textit{R. P. Feynman} [Rev. Mod. Phys. 17, 157-181 (1945); ibid. 21, 425-433 (1949; Zbl 0044.27801)] which employs advanced and retarded waves to describe loss-free electromagnetic radiation is considered for its applicability to the gravitational case. NEWLINENEWLINENEWLINEThe model considered is a ``massive static axisymmetric base case, added to which there will be a small mass revolving round the static cylinder.'' As an illustrative example, the author describes a ``massive circular cylinder round which a pair of satellites (which are infinite line sources parallel to the axis) are in circular orbit on opposite sides. A small rotating quadropole (or other multipole) is therefore added to the fixed monopole.'' One expects a wave pattern, fixed in the rotating frame, basically a standing wave due to the superposition of incoming and outgoing waves. NEWLINENEWLINENEWLINETechnical analysis shows that the space is divided into an inner zone where the field equations are elliptic and an outer (wave) zone where they are hyperbolic. To first order, the angular dependence is Fourier analysed and for each Fourier component, a set of ordinary differential equations describes the field throughout empty space linking the inner and outer zones. Finally, the asymptotic behavior of the wave zone is considered and the inner zone looked at for slowly rotating sources. NEWLINENEWLINENEWLINEIn conclusion, the validity of such a linear analysis is discussed and its possible applicability to more realistic models described.