Single-Particle Electrodynamics - Assassination Science

we define the redshift factor
λ ≡ 1 + (r· ˙v)
(G.27)
for use in the following.
G.6.5
Lab-time constituent trajectories
We now obtain the trajectory of the constituent r, without reference to the
body τ at all. Reverting t r (τ), we find
τ(t r ) = λ −1 t r − 1 2 λ−3 t 2 r(r·¨v) − 1 6 λ−4 t 3 r ˙v2 − 1 6 λ−4 t 3 r(r·... v)
− 2 3 λ−4 t 3 r ˙v2 (r· ˙v) + 1 2 λ−5 t 3 r(r·¨v) 2 − 1
24 λ−5 t 4 r(r·.... v)
− 1 8 λ−5 t 4 r( ˙v·¨v) − 1 2 λ−5 t 4 r ˙v2 (r·¨v) − 13
24 λ−5 t 4 r(r· ˙v)( ˙v·¨v)
+ 5 12 λ−6 t 4 r ˙v2 (r·¨v) + 5
12 λ−6 t 4 r(r·¨v)(r·... v) + 5 3 λ−6 t 4 r ˙v2 (r· ˙v)(r·¨v)
− 5 8 λ−7 t 4 r(r·¨v) 3 + O(t 5 r);
using (G.28) in (G.26) gives
z r (t) = r + 1 2 λ−1 t 2 ˙v + 1 6 λ−2 t 3¨v − 1 6 λ−3 t 3 (r·¨v) ˙v + 1
24 λ−3 t 4 ...
v
− 1 8 λ−4 t 4 (r·¨v)¨v − 1 24 λ−4 t 4 (r·... v) ˙v − 1 8 λ−4 t 4 ˙v 2 (r· ˙v) ˙v
+ 1 8 λ−5 t 4 (r·¨v) 2 ˙v + 1
120 λ−4 t 5 ....
v − 1
20 λ−5 t 5 (r·¨v) v
...
(G.28)
− 1 30 λ−5 t 5 (r·... v)¨v − 1
120 λ−5 t 5 (r·.... v) ˙v − 1 10 λ−5 t 5 ˙v 2 (r· ˙v)¨v
− 1 15 λ−5 t 5 ˙v 2 (r·¨v) ˙v − 1
12 λ−5 t 5 (r· ˙v)( ˙v·¨v) ˙v + 1 8 λ−6 t 5 (r·¨v) 2¨v
+ 1
12 λ−6 t 5 (r·¨v)(r·... v) ˙v + 1 4 λ−6 t 5 ˙v 2 (r· ˙v)(r·¨v) ˙v
− 1 8 λ−7 t 5 (r·¨v) 3 ˙v + O(t 6 ),
where now t is understood to be t r .
450
(G.29)