is present, giving the same implication as in the other cases, and is the only
contribution to the self-force for the gravitational field.
3.2 Analytical Calculations of Self-force
3.2.1 Dewitt and Dewitt: Falling Charges
Dewitt and Dewitt [7] computed the self-force on the electric charge falling
freely in the non-relativistic limit of small velocities in a static weak gravitational
field which is characterized by (in harmonic coordinates)
hb 2GM b, (3-17)
where G is the gravitational constant and M is the total mass contained in the
spacetime. By simplifying the bi-vector retarded Green's function in this limit,
they were able to evaluate the I l! integral" in Eq. (3-3) directly. The result of
calculation shows that, in this limit, the force separates naturally into the following
two parts (looking into the spatial components of the force):
F = F + FNC
CMr + 22 ( ) (GM) (3-18)
4 3 r
where Fc is a conservative force which arises from the fact that the mass of the
particle is not concentrated at a point but is partly distributed as electric field
energy in the space surrounding the particle, and FNC is a non-conservative force
which gives rise to radiation la,,n',:,j having a linear dependence on both the
velocity of the charge and the curvature of the background geometry.
FC corresponds to a repulsive inverse square potential
e2GM
c 2GM (3-19)
2r2