Single-Particle Electrodynamics - Assassination Science

this brings a vanishing integrand r-dependence up to a finite r-dependence.
The dipole torque avoids this problem because the angular momentum has
the vector r explicitly crossed into its definition, and hence the integrand
in question only diverges to the same order as the corresponding integrand
for the force on a monopole field, i.e., it vanishes over the small volume
surrounding the particle.)
Let us now make the above analyses relativistically rigorous, by again
employing the rigid body formalism described in Chapter 3. We need to add
in the effects of the accelerative redshift factor,
λ(r) ≡ 1 + (r· ˙v).
At any position in our circular loop, we use (4.22) to find
λ(θ) = 1 + ε { ˙v x cos θ + ˙v y sin θ } . (4.44)
If one examines the expressions carefully, one finds that the redshift correction
terms in (4.44), of order ε, will not contribute to the torque integral,
because that integral already has z(θ) crossed into it, which is itself of order
ε. On the other hand, we do find contributions to the power and force
integrals. For the former, one finds
P redshift = µ· ˙v×E.
Now, if one compares this to our original result (4.33), and then looks back
at the electric dipole redshift force found in Section 4.2.1, one finds that we
seem to be establishing a “redshift prescription”,
∇ −→ ∇ + ˙v. (4.45)
Again, since this effect was only discovered by the author a few days before
the printing of this thesis, only a small amount of contemplation has followed
139