Gravitational Radiation Damping of Slowly Moving Systems Calculated Using Matched Asymptotic Expansions

Abstract

This paper treats the slow-motion approximation for radiating systems as a problem in singular perturbations. By using the method of matched asymptotic expansions, we can construct approximations valid both in the near zone and the wave zone. The outgoing-wave boundary condition applied to the wave-zone expansion leads, by matching, to a unique and easily calculable radiation resistance in the near zone. The method is developed and illustrated with model problems from mechanics and electromagnetism; these should form a useful and accessible introduction to the method of matched asymptotic expansions. The method is then applied to the general relativistic problem of gravitational radiation from gravitationally bound systems, where a significant part of the radiation can be attributed to nonlinear terms in the expansion of the metric. This analysis shows that the formulas derived from the standard linear approximation remain valid for gravitationally bound systems. In particular, it shows that, according to general relativity, bodies in free-fall motion do indeed radiate. These results do not depend upon any definition of gravitational field energy.

Parametrized post-Newtonian theory of reference frames, multipolar expansions and equations of motion in the N-body problem

“…Manasse [119] studied radial fall of a small black hole onto a massive gravitating body and calculated distortion of the shape of the black hole's horizon by making use of the matching technique. Thorne [120] and Burke [121] suggested to use the matching technique for imposing an outgoing-wave radiation condition on the post-Newtonian metric tensor for an isolated system emitting gravitational waves. This method helps to chose a causal solution of the homogeneous Einstein equations in the post-Newtonian approximation scheme and to postpone appearance of ill-defined (divergent) integrals, at least, up to the fourth PNA [122,123,124].…”

On partial spherical means formulas and radiation boundary conditions for the 3+1 wave equation

“…The t 1 → T − limit of (2.22) then yields 24) since the hemispherical means terms in (2.22) tend to 2πU z (T, 0, 0, 0) as t 1 → T − . Moreover, since U ρ will average to zero in the limit, the integral −[U t + U ρ ] (t 1 ,T −t 1 ) as a whole tends to −2πU t (T, 0, 0, 0).…”

“…After all differentiations, t 0 can be set to zero. Precisely, an order-multipole solution is given by 14) or in the Burke formalism [24],…”

Mathematical Modelling of Spray Combustion-Numerical and Analytical Analysis with application to Engineering Science

“…Perturbations, approximations and other asymptotic techniques such as the method of multiple scale, the renormalization method, the method of matched asymptotic expansion, Pade approximation, variational iterations, Lyapunov's artificial small parameter, the δ-expansion, Adomian decomposition method, Lindstedt-Poincare method are widely in science and engineering, see for examples (Burke, 1971;Abbasbandy, 2003;). All these methods are based on small/large parameters of the systems of the governing equation.…”