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Calculations



You advised to substitute sqrt(ax^2 + ay^2 + az^2) for a, I did it the other
way around thus substitute (ax^2 + ay^2 + az^2) by a^2 and ...

Wow, that really simplifies (after some other tricks):

S|S + 2 a^2*(1+A|S)*(Cosh[a*T]-1) - 2*a^3*t*Sinh[a*T] = 0

A=[0 ax ay az]
S=[t  x  y  z]

Solving gives:

t' = 1/a * Ln[(d+Sqrt[4 f g a^4 + d^2])/(2g a^2)]
                ^ or a minus-sign
With:
 d=2 a^2 (1+A|S) - S|S
 f=-1 - A|S - a t
 g= 1 + A|S - a t


It looks much like your solution, but isn't the same:
t' = 1/a * ln((-q - sqrt(q^2 - 4 * p * r)) / (2 * p))

 > k = 1 - A|S
 > p = k - a * t = 1 - A|S - a t
 > q = a^2 * S^2 - 2 * k = a^2 S|S - 2 - 2 A|S
 > r = k + a * t = 1 - A|S + a t 

Timothy