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When we calculate gravitational waveforms from extreme-mass-ratio inspirals(EMRIs) by metric perturbation, it is a common strategy to use the adiabaticapproximation. Under that approximation, we first calculate the linear metricperturbation induced by geodesics orbiting a black hole, then we calculate theadiabatic evolution of the parameters of geodesics due to the radiationreaction effect through the calculation of the self-force. This procedure isconsidered to be reasonable, however, there is no direct proof that it canactually produce the correct waveform we would observe. In this paper, we studythe formal expression of the second order metric perturbation and show that itbe expressed as the linear metric perturbation modulated by the adiabaticevolution of the geodesic. This evidence supports the assumption that theadiabatic approximation can produce the correct waveform, and that theadiabatic expansion we propose in Ref.\cite{adi} is an appropriate perturbationexpansion for studying the radiation reaction effect on the gravitationalwaveform.

Modulation of the gravitational waveform by the effect of radiation reaction