Self-Force in the Radiation Reaction Formula Adiabatic Approximation of a Metric Perturbation and an Orbit

We investigate a calculation method for the gravitational evolution of an extreme mass ratio binary, i.e. a binary constituting of a galactic black hole and a stellar mass compact object. The inspiralling stage of this system is considered to be a possible source of detectable gravitational waves. Because of the extreme mass ratio, one may approximate such a system by a black hole geometry (a Kerr black hole) plus a linear metric perturbation induced by a point particle. With this approximation, a self-force calculation was proposed for a practical calculation of the orbital evolution, including the effect of the gravitational radiation reaction, which is now known as the MiSaTaQuWa self-force . In addition, a radiation reaction formula was proposed 2) as an extension of the well-known balance formula of Press and Teukolsky . The radiation reaction formula provides us a convenient method to calculate an infinite time averaged loss of “the constants of motion” (i.e. the orbital energy, the z component of the angular momentum, and the Carter constant) through the gravitational radiation reaction, with which one may approximately calculate the orbital evolution. Because these methods are approximately equivalent, we investigate the consequence of the orbital evolution using the radiation reaction formula. To this time, we have used the so-called adiabatic approximation of the orbital evolution and considered a method to evaluate the MiSaTaQuWa self-force by use of a linear metric perturbation. In this approach, we point out that there is a theoretical question concerning the choice of the gauge condition in the calculation of the MiSaTaQuWa self-force. Because of this gauge ambiguity, there is a case in which the MiSaTaQuWa self-force might not predict the orbital evolution in a physically expected manner, and this forces us to calculate a waveform only in the so-called dephasing time. We discuss the reason that such an unexpected thing happens and find that it is primarily because we consider the linear metric perturbation separately from the orbital evolution due to the self-force. We propose a new metric perturbation scheme under a possible constraint of the gauge conditions in which we obtain a physically expected prediction of the orbital evolution caused by the MiSaTaQuWa self-force. In this new scheme of a metric perturbation, an adiabatic approximation is applied to both the metric perturbation and the orbit. As a result, we are able to predict the gravitational evolution of the system in the so-called radiation reaction time scale, which is longer than the dephasing time scale. However, for gravitational wave detection by LISA, this may still be insufficient. We further consider a gauge transformation in this new metric perturbation scheme, and find a special gauge condition with which we can calculate the gravitational waveform of a time scale long enough for gravitational wave detection by LISA.