Self-Force on a Scalar Charge in Circular Orbit around a Schwarzschild Black Hole

In an accompanying paper, we formulate two types of regularization methods to calculate the scalar self-force on a particle of charge q moving around a black hole of mass M [Y. Mino, H. Nakano and M. Sasaki, Covariant Self-force Regularization of a Particle Orbiting a Schwarzschild Black Hole, UTAP-386, OUTAP-157 (2001)], one of which is called the “power expansion regularization”. In this paper, we analytically evaluate the self-force (which we also call the ‘reaction force’) to third post-Newtonian (3PN) order for a scalar particle in circular orbit around a Schwarzschild black hole by using the power expansion regularization. It is found that the r-component of the self-force arises at 3PN order, whereas the t- and ϕ-components, which are due to the radiation reaction, appear at 2PN and 1.5PN orders, respectively.