Gravitational reaction force on a particle in the Schwarzschild background

We formulate a new method to calculate the gravitational reaction force on a particle of mass $\mu$ orbiting a massive black hole of mass $M$. In this formalism, the tail part of the retarded Green function, which is responsible for the reaction force, is calculated at the level of the Teukolsky equation. Our method naturally allows a systematic post-Minkowskian (PM) expansion of the tail part at short distances. As a first step, we consider the case of a Schwarzschild black hole and explicitly calculate the first post-Newtonian (1PN) tail part of the Green function. There are, however, a couple of issues to be resolved before explicitly evaluating the reaction force by applying the present method. We discuss possible resolutions of these issues.Comment: 15 pages, no figure, submitted to Prog. Theor. Phy

A new analytical method for self-force regularization II. Testing the efficiency for circular orbits

In a previous paper, based on the black hole perturbation approach, we formulated a new analytical method for regularizing the self-force acting on a particle of small mass $\mu$ orbiting a Schwarzschild black hole of mass $M$, where $\mu\ll M$. In our method, we divide the self-force into the $\tilde S$-part and $\tilde R$-part. All the singular behaviors are contained in the $\tilde S$-part, and hence the $\tilde R$-part is guaranteed to be regular. In this paper, focusing on the case of a scalar-charged particle for simplicity, we investigate the precision of both the regularized $\tilde S$-part and the $\tilde R$-part required for the construction of sufficiently accurate waveforms for almost circular inspiral orbits. For the regularized $\tilde S$-part, we calculate it for circular orbits to 18 post-Newtonian (PN) order and investigate the convergence of the post-Newtonian expansion. We also study the convergence of the remaining $\tilde{R}$-part in the spherical harmonic expansion. We find that a sufficiently accurate Green function can be obtained by keeping the terms up to $\ell=13$.Comment: 21pages, 12 figure

The lifespan of classical solutions of one dimensional wave equations with semilinear terms of the spatial derivative

This paper is devoted to the lifespan estimates of small classical solutions of the initial value problems for one dimensional wave equations with semilinear terms of the spatial derivative of the unknown function. It is natural that the result is same as the one for semilinear terms of the time-derivative. But there are so many differences among their proofs. Moreover, it is meaningful to study this problem in the sense that it may help us to investigate its blow-up boundary in the near future.Comment: 10 page