Single-Particle Electrodynamics - Assassination Science

+ 1 { [1 ] .... + (r· ˙v) v + 2 [ 5 + 14(r· ˙v) ] ˙v 2¨v
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+ 5 [ 3 + 11(r· ˙v) ] ( ˙v·¨v) ˙v + 4(r·.... v) ˙v
+ 52 ˙v 2 (r·¨v) ˙v + 6(r·... v)¨v + 4(r·¨v) v
... }
τ 5 + O(τ 6 ). (3.14)
3.3.4 Accelerative redshift
The constituent r has, in the above, been treated as a part of the overall
body, and has thus had its motion described in terms of the body’s proper
time, τ. On the other hand, the constituent often needs to be considered as
an individual particle in its own right. The proper time τ r of this particle
r is defined solely by its four-trajectory z α r ; the body proper time τ is, in
this context, merely a convenient quantity that parametrises this path via
equations (3.14) and (3.13). Now, by definition,
dτ 2 r ≡ dt 2 r − dz 2 r;
thus,
d τ τ r ≡
√
(d τ t r ) 2 − (d τ z r ) 2 . (3.15)
Computing (3.15) from (3.14) and (3.13), one finds that
d τ τ r = [ 1 + (r· ˙v) ] + (r·¨v)τ + 1 2
+ 1 6
{
(r·...
v) + 3 ˙v 2 (r· ˙v) } τ 2
{
(r·....
v) + 9 ˙v 2 (r·¨v) + 10(r· ˙v)( ˙v·¨v) } τ 3 + O(τ 4 ). (3.16)
This result (3.16) is, in fact, of major importance to the considerations
of this thesis. Let us explain why. To compute the power into, force on
and torque on the relativistically rigid body, we shall need to sum up these
quantities for all of the constituents of the body. Let us, for example, consider
the power and force on the body; the torque follows in the same manner.
Now, the four-force on the body as a whole, ṗ α , is of course defined in terms
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