For my undergraduate studies, I was faced with the problem of finding the equations of motion for a particle subject to a uniform electric field, in the relativistic case. I would like to follow the procedure presented in https://www.researchgate.net/publication/262474836_Relativistic_charged_particle_in_a_uniform_electromagnetic_field .
With that in mind, WLOG I have taken my electric field $\vec{E} = (0, 0, E),\ E>0$, for which I get the following trajectory for my charged particle $\alpha(\tau)$, $\alpha:I\subset\mathbb{R}\rightarrow M$ ($M$ being Minkowski's Space Time and $I$ an open interval that contains 0),
$$ \begin{bmatrix} (U_0^0\sinh(\omega\tau)+U_0^3(\cosh(\omega\tau)-1))/\omega \\ U_0^1\tau \\ U_0^2\tau \\ (U_0^3\sinh(\omega\tau)+U_0^0(\cosh(\omega\tau)-1))/\omega \\ \end{bmatrix} $$
Where I have defined $\omega= \frac{qE}{m}$,($q$ and $m$ are the charge and the mass of the particle), $\tau_0 = 0$ and $U(\tau_0) = U(0) = (U_0^0, U_0^1,U_0^2,U_0^3)$. (Are my calculations correct?)
Suppose I want to write this result in terms of the coordinate time $x^0$, in which the electric field is measured to be constant. Is it as straightforward as acknowledging the fact that
$$ x^0 = x^0(\tau) = (U_0^0\sinh(\omega\tau)+U_0^3(\cosh(\omega\tau)-1))/\omega? $$
In that case, what assumptions could I make WLOG regarding the initial conditions $U(\tau_0)$ as to invert this last relationship to obtain $\tau(x^0)$ (similar to taking $\vec{E}$ parallel to the z axis), and thus be able to write the trajectory in terms of $x^0$?
If I was treating simply with a three-vector, I believe I could take the initial condition to be in the YZ plane, for instance, but $U$ being a four-vector I don't know what I can assume.
