Single-Particle Electrodynamics - Assassination Science

and reverting t(τ) from (2.91), one finds
+ 1 { ....
v + 4 ˙v 2¨v + 3( ˙v·¨v) ˙v } τ 4 + O(τ 5 ), (2.95)
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τ(t) = t − 1 6 ˙v2 t 3 − 1 8 ( ˙v·¨v)t4 − 1
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{
3 ˙v 4 + 3¨v 2 + 4( ˙v·... v) } t 5
− 1 {
( ˙v·....
v) + 2(¨v·... v) + 6 ˙v 2 ( ˙v·¨v) } t 6 + O(t 7 ); (2.96)
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substituting (2.96) into (2.95), we are returned to the definition (2.87); this
is the second check. Finally, we note that γ(τ), computed in (2.93) as d τ t(τ),
may alternatively be computed via
γ(τ) ≡
1
√1 − v 2 (τ) ; (2.97)
using (2.95), we are returned to the expression (2.93).
Thus, even without seeing the explicit derivation of (2.91) and (2.92),
one knows that they are, in fact, a correct parametrisation of the path of the
point particle, around its instantaneous-rest event.
2.8.6 Spin degrees of freedom
We now consider the case in which the point particle possesses three internal
degrees of freedom constituting a spin vector, σ, in its rest frame. As
described in Section 2.6.7, this three-vector generalises to the four-vector
Σ when the particle is in arbitrary relativistic motion. Clearly, we could
consider a manifestly-covariant parametrisation of Σ (τ) à la that of z(τ),
namely,
Σ (τ) = c ′ 0 + c ′ 1τ + c ′ 2τ 2 + c ′ 3τ 3 + c ′ 4τ 4 + O(τ 5 ), (2.98)
but from the discussion of Section 2.8.2, we already know that this will
introduce a redundant parameter in each order, since the four-spin Σ must
satisfy the constraint
(Σ ·U) = 0.
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