New approach to electromagnetic wave tails on a curved spacetime

Abstract

We present an alternative method for constructing the exact and approximate solutions of electromagnetic wave equations whose source terms are arbitrary order multipoles on a curved spacetime. The developed method is based on the higher-order Green's functions for wave equations which are defined as distributions that satisfy wave equations with the corresponding order covariant derivatives of the Dirac delta function as the source terms. The constructed solution is applied to the study of various geometric effects on the generation and propagation of electromagnetic wave tails to first order in the Riemann tensor. Generally the received radiation tail occurs after a time delay which represents geometrical backscattering by the central gravitational source. It is shown that for an arbitrary weak gravitational field it is valid that the truly nonlocal wave-propagation correction (the tail term) has a universal form which is independent of multipole structure of the gravitational source. In a particular case when the electromagnetic radiation pulse is generated by the wave source during a finite time interval, the structure of the wave tail at the time after the direct pulse has passed the gravitational source is in the first approximation independent of the higher multipole moments of the source of

On the asymptotic character of electromagnetic waves in a Friedmann?Robertson?Walker universe

“…Expansions to generalized spherical harmonics have also been presented by Mankin et al [13][14][15]26], and by Laas et al [27] to construct exact and approximate solutions to electromagnetic wave equations in a curved spacetime using higher-order Green's function method.…”

“…Deng and Mannheim [12] have presented solutions to Maxwell's equations by solving these equations for spherical components of the electric and magnetic fields, separately. More recently, Mankin et al [13,14,15] have presented exact and approximate solutions to electromagnetic wave equations in a curved spacetime, based on the method proposed by Hadamard [16] and by using a higher-order Green's function for the wave equation. I adopt a very helpful method due to Skrotskii [17] who realized that electromagnetic field equations in a curved spacetime can be written in a non-covariant form formally equivalent to Maxwell's equations in a macroscopic medium in flat spacetime.…”

Electromagnetic waves in the Kerr–Schild metric

“…bounded beam of light can be described with this approximation if its wavelength is sufficiently small. Such a wave can be written in terms of a real phase (or Eikonal) function S, a real scalar amplitude α and a complex unit polarisation 1-form P, orthogonal to dS, so that g(P, P) = 1 ( 7 ) i dS P = 0 ( 8 ) where P indicates the complex conjugate of P. Equation (7) is the normalisation condition for the polarisation 1-form, while (8) ensures it is transverse. An electromagnetic 2-form given by…”

“…The presence of the Hodge map indicates that background spacetime curvature influences the behaviour of electromagnetic radiation, and this phenomenon has been widely studied ever since it was first exploited in the early tests of general relativity. Various techniques have been adopted, such as treating the gravitational field as a medium with constitutive relations derived from the metric, distilling transport properties from contractions of Maxwell's equations, exploiting Killing symmetries or using Green's function methods, fluid descriptions, 3 + 1 decompositions and geometrical optics approximations [2][3][4][5][6][7][8][9][10][11][12][13].…”

Comparison of QG-induced dispersion with standard physics effects

“…When waves propagate in a curved spacetime, they can scatter off the global curvature, a phenomenon that is usually described in terms of the formation of tails, or non-validity of the Huygens principle, rather than in terms of a modified index of refraction. It is known, for example, that a necessary condition for the validity of the Huygens principle in 4-dimensional spacetime is that the geometry be that of an Einstein space [10], and that tails generically form in the propagation of fields, both near isolated objects [11] and in a cosmological setting [12]. It would be useful therefore to analyze the latter effect in more detail in terms of modified effective dispersion relations.…”