I am currently taking a course on electrodynamics. The last few lectures have been about relativistic electrodynamics (Lorentz transformation of four-vectors, electromagnetic field tensor, etc.). This is on the level of Griffiths' Introduction to Electrodynamics.

I've been making the exercises of "Griffiths - Introduction to Electrodynamics" but I often find myself stuck, without a clear idea of how to solve a problem.

My question is the following: what are some good sources to build intuition for problem solving in relativistic electrodynamics? And what are good things/formulas to memorise for these kind of problems?


Here is a sample problem which I find hard to solve (although it's more relativistic dynamics then electrodynamics). It is taken from Griffiths' chapter on Relativistic Electrodynamics.

Find x as function of t for motion starting from rest at the origin under the influence of a constant Minkowski force in the x direction. Leave your answer in implicit form (t as a function of x).

My approach to solving it was setting the derivative to proper time of the relativistic proper momentum equal to a constant (in the x direction, that is), but that resulted in quite a mess.

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    $\begingroup$ You don't have to memorize anything. You have to understand. $\endgroup$ – Andrei Geanta Dec 18 '17 at 13:50
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    $\begingroup$ Perhaps it would be a good idea to present some problems along with your approach of solving them. $\endgroup$ – Andrei Geanta Dec 18 '17 at 13:52
  • $\begingroup$ What does constant Minkowski force mean? The equation $dP/d\tau = F$ with F constant implies that $P(\tau) = \tau F$, assuming $P(0)=0$. But the rest mass is $m^2 = P^2 = \tau^2 F^2$, so the rest mass is increasing linearly...furthermore, the four velocity $V = P/|P| = F/|F|$ is constant. So the particle is just going in a straight line in the same direction as $F$, while linearly increasing its rest mass. Even worse, if $F$ is pointing in a spacelike direction then the particle goes faster than light. So $dP/d\tau = F$ constant is probably not the right equation... $\endgroup$ – Jules Dec 18 '17 at 15:57
  • $\begingroup$ The four velocity has constant length $|V|=1$, so what's happening in a particle trajectory is that $V$ gets rotated. Infinitesimal rotations are given by an anti-symmetric 2-tensor, so the proper generalisation of a force field to relativity is not a vector but an antisymmetric 2-tensor F, and the generalisation of the Newtonian $a = F/m$ is $A^a = dV^a/d\tau = F^a_b V^b$. $\endgroup$ – Jules Dec 18 '17 at 16:01
  • $\begingroup$ If you take $F$ constant then this equation ensures that the force is constant as felt by the particle. Note that if you take $F$ to be the electromagnetic field tensor then this is precisely the formula for the Lorentz force. $\endgroup$ – Jules Dec 18 '17 at 16:09

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