A new analytical method for self-force regularization I. scalar charged particle in Schwarzschild spacetime

摘要

We formulate a new analytical method for regularizing the self-force actingon a particle of small mass $\mu$ orbiting a black hole of mass $M$, where$\mu\ll M$. At first order in $\mu$, the geometry is perturbed and the motionof the particle is affected by its self-force. The self-force, however,diverges at the location of the particle, and hence should be regularized. Itis known that the properly regularized self-force is given by the tail part (orthe $R$-part) of the self-field, obtained by subtracting the direct part (orthe $S$-part) from the full self-field. The most successful method ofregularization proposed so far relies on the spherical harmonic decompositionof the self-force, the so-called mode-sum regularization or mode decompositionregularization. However, except for some special orbits, no systematicanalytical method for computing the regularized self-force has been given. Inthis paper, utilizing a new decomposition of the retarded Green function in thefrequency domain, we formulate a systematic method for the computation of theself-force. Our method relies on the post-Newtonian (PN) expansion but theorder of the expansion can be arbitrarily high. To demonstrate the essence ofour method, in this paper, we focus on a scalar charged particle on theSchwarzschild background. The generalization to the gravitational case isstraightforward, except for some subtle issues related with the choice of gauge(which exists irrespective of regularization methods).