Single-Particle Electrodynamics - Assassination Science

correct. We must also multiply all expressions by the accelerative redshift
factor λ(r), which for each of our two charges q 1 and q 2 is simply given by
λ 1 = 1 + 1 ε(σ· ˙v),
2
λ 2 = 1 − 1 ε(σ· ˙v). (4.13)
2
For the computation of the power expression, we note that the velocities of
the charges (in the dipole rest frame) are already of order ε; the factor of
ε −1 in the values ±d/ε of the charges cancel this factor of ε; hence, there
are therefore no “spare” factors of ε −1 left which could couple to the latter
terms of (4.13). Hence, the power expression is given by the rest-frame value
of (4.10) found above:
P | v=0
= ( ˙ d ·E). (4.14)
The rigorously relativistic computation of the force expression, however,
is somewhat more subtle. Here, we find that the electric force on each charge
may couple with the second factor of (4.13):
(
F extra = ± d ) (
± 1 )
ε 2 ε(σ· ˙v) E,
where the upper (lower) signs apply to charge q 1 (q 2 ). We thus find an extra
redshift force,
F redshift = (d· ˙v)E.
Thus, the total rest-frame force on the electric dipole is in fact given by
F | v=0
= (d ·∇)E + (d· ˙v)E + ˙ d ×B.
Now, in the interests of honesty, the author must confess that this extra
“redshift force” was only discovered by him a few days before the final printing
and binding of this thesis. It has ramifications for the radiation reaction
calculations of Chapter 6; fortunately, the author was able to insert this additional
force (and its corresponding “moment-arm” torque) into the program
125