Reference: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Chapter 12, Problem 12.60.

The Minkowski force is the rate of change of four-momentum with respect to proper time, and allows Newton’s law to be written in its natural form

where is the proper acceleration, or second derivative of position with respect to proper time. Here we’ll investigate the behaviour of a particle subject to a constant Minkowski force in one dimension.

In terms of ordinary force, we have

The ordinary momentum is

Inserting this into 2 we get

We can integrate both sides (using software, or integral tables) to get

where is a constant of integration. If the initial conditions are at , then and we have

This is an implicit equation for the speed of the particle as a function of time. If we want the position as a function of time, we need a relation between and . Returning to 2 and 3 we have

We can use the chain rule to convert the derivative on the RHS to a derivative with respect to by multiplying both sides by

Now so and

If we call the expression in the parentheses on the RHS , then we can integrate with respect to (since is a constant):

Again, starting from rest at the origin we have when so also, and therefore , so we have

At this point we could get a relation between and by solving 14 for in terms of and then substituting this into 7. For reference, we get

so substituting will give something of a mess. To get the answer given in Griffiths requires a bit of algebra, but here is how I did it. Griffiths defines the quantity as

The quantities appearing in Griffiths’s answer are

We can rewrite 7 to get

We’ll deal with the logarithm first. Its argument is

Now we also have

Therefore

For the second term in 21, we have

Putting it all together, we have