The first evaluation of the electromagnetic
self-force was carried out by DeWitt and DeWitt [41] for a charge moving
freely in a weakly-curved spacetime characterized by a Newtonian
potential . (This condition must be imposed
globally, and requires the spacetime to contain a matter
distribution.) In this context the right-hand side of
Equation (33
) reduces to the tail
integral, since there is no external force acting on the charge.
They found the spatial components of the self-force to be given
by
A similar expression was obtained by Pfenning and Poisson [46] for the case of a scalar charge. Here
whereThe force required to hold an electric charge in place in a Schwarzschild spacetime was computed, without approximations, by Smith and Will [54]. As measured by a free-falling observer momentarily at rest at the position of the charge, the total force is
and it is directed in the radial direction. Here,The intriguing phenomenon of mass loss by a
scalar charge was studied by Burko, Harte, and Poisson [15] in the simple
context of a particle at rest in an expanding universe. For the
special cases of a de Sitter cosmology, or a spatially-flat
matter-dominated universe, the retarded Green’s function could be
computed, and the action of the scalar field on the particle
determined, without approximations. In de Sitter spacetime the
particle is found to radiate all of its rest mass into monopole
scalar waves. In the matter-dominated cosmology this happens only
if the charge of the particle is sufficiently large; for smaller
charges the particle first loses a fraction of its mass, but then
regains it eventually.
In recent years a large effort has been devoted
to the elaboration of a practical method to compute the (scalar,
electromagnetic, and gravitational) self-force in the Schwarzschild
spacetime. This work originated with Barack and Ori [7] and was pursued by
Barack [2, 3] until it was put
in its definitive form by Barack, Mino, Nakano, Ori, and
Sasaki [6
, 9
, 11
, 38
]. The idea is to
take advantage of the spherical symmetry of the Schwarzschild
solution by decomposing the retarded Green’s function
into spherical-harmonic modes which can be computed
individually. (To be concrete I refer here to the scalar case, but
the method works just as well for the electromagnetic and
gravitational cases.) From the mode-decomposition of the Green’s
function one obtains a mode-decomposition of the field gradient
, and from this subtracts a mode-decomposition of the
singular field
, for which a local expression is known.
This results in the radiative field
decomposed
into modes, and since this field is well behaved on the world line,
it can be directly evaluated at the position of the particle by
summing over all modes. (This sum converges because the radiative
field is smooth; the mode sums for the retarded or singular fields,
on the other hand, do not converge.) An extension of this method to
the Kerr spacetime has recently been presented [44
, 34
, 10], and Mino [37] has devised a
surprisingly simple prescription to calculate the time-averaged
evolution of a generic orbit around a Kerr black hole.
The mode-sum method was applied to a number of
different situations. Burko computed the self-force acting on an
electric charge in circular motion in flat spacetime [12], as well as on a
scalar and electric charge kept stationary in a Schwarzschild
spacetime [14], in a spacetime that
contains a spherical matter shell (Burko, Liu, and Soren [17]), and in a Kerr spacetime (Burko and
Liu [16]). Burko also
computed the scalar self-force acting on a particle in circular
motion around a Schwarzschild black hole [13], a calculation that
was recently revisited by Detweiler, Messaritaki, and
Whiting [21]. Barack and Burko
considered the case of a particle falling radially into a
Schwarzschild black hole, and evaluated the scalar self-force
acting on such a particle [4];
Lousto [33] and Barack and
Lousto [5
], on the other hand,
calculated the gravitational self-force.