A charged particle is accelerated from rest in a uniform electric field in the x

Question

A charged particle is accelerated from rest in a uniform electric field in the x axis. The electromagnetic field tensor, F, has components F ^mu _nu = {E mu = 1, nu =0 E nu = 1, mu = 0 otherwise where the electric field strength is E. i) Show that the electric field has the same strength in the particle rest frame as in the observer's frame, at all stages in the motion. ii) The equation of motion of the particle is m dv ^mu/d pi = qF ^mu _nu v ^nu where V is the velocity four-vector and the particle has mass m and charge q. What is the quantity pi? solve this system of equations to show that an observer sees the Lorentz factor of the particle change with time as gamma = (1 + q^2E^2t^2/m^2)^1/2 What electric field is needed for an electron to reach an energy of 1 GeV in 1 ms?

Explanation / Answer

Clearly we arrive at only two coupled differential equations in of the form

mdv0d=qEv1mdv0d=qEv1

mdv1d=qEv0mdv1d=qEv0

but I haven't been able to arrive at the relation indicated at all. I believe that is the proper time, in which case I think it is given by

dt=ddt=d

but I'll admit that that I am not too confident on this. I can also solve the coupled equations to get v0()v0() and v1()v1() and I get a linear combination of exponentials with ±Eq/m±Eq/m as their exponents, but I can't see how to get out of this.

Any help would be wonderful as I'm tearing my hair out a bit

EDIT: Going into more detail about what i've done so far, rather than converting from to ttstraight away we start with the two differential equations (1) and (2). Rearranging (1) to get v1v1 on its own and plugging into (2) we get, after a little bit of work:

d2v0d2=q2E2m2d2v0d2=q2E2m2

which has as a solution

v0()=AeqE/m+BeqE/m.v0()=AeqE/m+BeqE/m.

For A and B some constants of integration. Plugging this into our first equation, we find that

v1()=AeqE/mBeqE/m.v1()=AeqE/mBeqE/m.

This is as far as I'm fairly sure on my answer. I've tried one or two other things but they all quickly led me to a dead end, in particular I tried using dt=ddt=d where is the standard lorentz factor in the differential equations, where I would then arrive at things like

mdv0d=qEv1mdv0d=qEv1

but solving these equations became much more tricky since (I believe) depends on time as it is a function of velocity which is changing.

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