4.1 Splitting the Retarded Field
4.1.1 Conventional Method of Splitting the Retarded Field
Historically, Dirac [2] first gave the analysis of the self-force for the electromag-
netic field of a particle in flat spacetime. He was able to approach the problem in
a perturbative scheme by allowing the particle's size to remain finite and invoking
the conservation of the stress-energy tensor inside a narrow world-tube surrounding
the particle's world-line. In his analysis, the retarded field is decomposed into two
parts: (i) The first part is the ii,,, i of the advanced and retarded fields" which
is a solution of the inhomogeneous field equation resembling the Coulomb q/r
piece of the scalar potential near the particle. (ii) The second part is a i 1 h II i i1""
field which is a homogeneous solution of Maxwell's equations. Dirac describes the
self-force as the interaction of the particle with the radiation field, a well-defined
vacuum field solution.
In the analyses of the self-force in curved spacetime, first by Dewitt and
Brehme [3], and subsequently by Mino, Sasaki, and Tanaka [4], by Quinn and Wald
[5] and by Quinn [6], the Hadamard form of the Green's function is employ, .1 to
describe the retarded field of the particle. Traditionally, taking the scalar field
case for example, the retarded Green's function Gret(p,p') is divided into "direct"
and "tail" parts: (i) The first part has support only on the past null cone of the
field point p. (ii) The second part has support inside the past null cone due to the
presence of the curvature of spacetime. Accordingly, the self-force on the particle
consists of two pieces: (i) The first piece comes from the direct part of the field
and the acceleration of the world-line in the background geometry; this corresponds
to Abraham-Lorentz-Dirac (ALD) force in flat spacetime. (ii) The second piece
comes from the tail part of the field and is present in curved spacetime. Thus, the
description of the self-force in curved spacetime reduces to Dirac's result in the flat
spacetime limit.