General Relativistic Theory of Light Propagation in the Field of Radiative Gravitational Multipoles
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The extremely high precision of current radio/optical interferometric observations and the unparalleled sensitivity of existing (LIGO) and future (LISA, ASTROD) gravitational-wave detectors demand a much better theoretical treatment of relativistic effects in the propagation of electromagnetic signals through variable gravitational fields. Especially important for future gravitational-wave observatories is the problem of propagation of light rays in the field of multipolar gravitational waves emitted by a localized source of gravitational radiation. A consistent approach giving a complete and exhaustive solution to this problem in the first post-Minkowskian approximation of General Relativity is presented in this paper. We derive a set of equations describing propagation of an electromagnetic wave in the retarded gravitational field of a time-dependent localized source emitting gravitational waves with arbitrary multipolarity and show for the first time that they can be integrated analytically in closed form. We also prove that the leading terms in observable relativistic effects depend exclusively on the values of the multipole moments of the isolated system and its time derivatives taken at the retarded instant of time on the null cone and do not depend on their integrated values. By making use of our integration technique we reproduce the known results of integration of equations of light rays both in a stationary field of a gravitational lens and in that of a plane gravitational wave, thereby establishing a relationship between our formalism and the approximations used by previous researches. The gauge freedom of our formalism is carefully studied and all gauge-dependent terms in the expressions for observable quantities are singled out and used for physically meaningful interpretation of observations. Two limiting cases of small and large values of the light-ray impact parameter, d, are elaborated in more detail. We explicitly show that in the case of small impact parameter the leading order terms for any effect of light propagation in the field of an arbitrary multipole depend neither on its radiative nor on its intermediate zone contributions. The main effect rather comes from the near zone terms. This property makes much more difficult any direct detection of gravitational waves by astronomical techniques if general relativity is correct. We also present an analytical treatment of time delay and light-ray bending in large impact parameter case corresponding to the approximation of a plane gravitational wave of arbitrary multipolarity. Explicit expressions for time delay and deflection angle are obtained in terms of the transverse-traceless (TT) part of the space-space components of the metric tensor.