The causal structure of the Green’s functions is
richer in curved spacetime: While in flat spacetime the retarded
Green’s function has support only on the future light cone of , in curved spacetime its support extends inside the light cone as well;
is therefore nonzero when
, which
denotes the chronological future of
. This property
reflects the fact that in curved spacetime, electromagnetic waves
propagate not just at the speed of light, but at all speeds smaller than or equal to the speed
of light; the delay is caused
by an interaction between the radiation and the spacetime
curvature. A direct implication of this property is that the
retarded potential at
is now generated by the point charge
during its entire history prior to the retarded time
associated with
: The potential depends on the particle’s
state of motion for all times
(see
Figure 2
).
The physically relevant solution to
Equation (13) is obviously the
retarded potential
, and as in flat spacetime,
this diverges on the world line. The cause of this singular
behaviour is still the pointlike nature of the source, and the
presence of spacetime curvature does not change the fact that the
potential diverges at the position of the particle. Once more this
behaviour makes it difficult to figure out how the retarded field
is supposed to act on the particle and determine its motion. As in
flat spacetime we shall attempt to decompose the retarded solution
into a singular part that exerts no force, and a smooth radiative
part that produces the entire self-force.
To decompose the retarded Green’s function into
singular and radiative parts is not a straightforward task in
curved spacetime. The flat-spacetime definition for the singular
Green’s function, Equation (9), cannot be adopted
without modification: While the combination half-retarded plus
half-advanced Green’s functions does have the property of being
symmetric, and while the resulting vector potential would be a
solution to Equation (13
), this candidate for
the singular Green’s function would produce a self-force with an
unacceptable dependence on the particle’s future history. For
suppose that we made this choice. Then the radiative Green’s
function would be given by the combination half-retarded minus
half-advanced Green’s functions, just as in flat spacetime. The
resulting radiative potential would satisfy the homogeneous wave
equation, and it would be smooth on the world line, but it would
also depend on the particle’s entire history, both past (through
the retarded Green’s function) and future (through the advanced
Green’s function). More precisely stated, we would find that the
radiative potential at
depends on the particle’s
state of motion at all times
outside the interval
; in the limit where
approaches the world
line, this interval shrinks to nothing, and we would find that the
radiative potential is generated by the complete history of the
particle. A self-force constructed from this potential would be
highly noncausal, and we are compelled to reject these definitions
for the singular and radiative Green’s functions.
The proper definitions were identified by
Detweiler and Whiting [23], who proposed the
following generalization to Equation (9
):
The potential constructed
from the singular Green’s function can now be seen to depend on the
particle’s state of motion at times
restricted to the
interval
(see Figure 3
). Because this
potential satisfies Equation (13
), it is just as
singular as the retarded potential in the vicinity of the world
line. And because the singular Green’s function is symmetric in its
arguments, the singular potential can be shown to exert no force on
the charged particle. (This requires a lengthy analysis that will
be presented in the bulk of the paper.)
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From the radiative potential we form an
electromagnetic field tensor , and the curved-spacetime generalization to
Equation (4
) is