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