The natural relativistic generalization of the Abraham-Lorentz formula (Eq. 11.80) would...

The natural relativistic generalization of the Abraham-Lorentz formula (Eq. 11.80) would seem to be

This is certainly a 4-vector, and it reduces to the Abraham-Lorentz formula in the nonrelativistic limit v ≪ c.

(a) Show, nevertheless, that this is not a possible Minkowski force. [Hint: See Prob. 12.39d.]

(b) Find a correction term that, when added to the right side, removes the objection you raised in (a), without affecting the 4-vector character of the formula or its nonrelativistic limit.

Reference prob 12.39d

Define proper acceleration in the obvious way:

(d) Write the Minkowski version of Newton’s second law, Eq. 12.68, in terms of