# Transformation of three-force

1. Jul 22, 2007

### jostpuur

Consider a following problem. A particle is moving with a velocity $\boldsymbol{v}$, and is experiencing a force $\boldsymbol{F}$. By the force I mean time rate change of the momentum. If we boost into a different frame, which is moving with a velocity $\boldsymbol{u}$ in the original frame, then what is the force that this particle experiences in this new frame?

A correct answer, I believe, is
$$\boldsymbol{F}' = \frac{\boldsymbol{F}\sqrt{1-|u|^2/c^2} - \boldsymbol{F}\cdot(\boldsymbol{v}/c^2 - (1-\sqrt{1-|u|^2/c^2})\boldsymbol{u} / |u|^2)\boldsymbol{u}}{1-\boldsymbol{v}\cdot\boldsymbol{u}/c^2}$$

My question to you is, that have you seen this equation anywhere?

I have not, except in my own notes. I derived this myself, and used it to derive an expression of an electromagnetic force that one moving particle exerts on another one, so that the result agreed (in the special case of no acceleration) with the one that is usually obtained using retarted potentials (the Green's function stuff and the Lienard-Wiechert potentials). So I believe I made no mistake in this.

My derivation went through first solving transformations of location, velocity and momentum in a similar three-vector formalism, which to my understanding doesn't seem to be very popular. Usually texts about relativity always proceed straight into the tensor formalism after the first basic equations (dilation and contraction stuff).

This is related to an earlier post of mine in the thread https://www.physicsforums.com/showthread.php?t=175438

pervect, since the underlying principles in this calculation are quite different than those in the potential approach, which is actually based on finding solutions of some PDE (Maxwell's equations) instead of transforming a time derivative of momentum, I'm not yet fully convinced that this was a perfectly standard calculation. Or maybe it was? I don't know. That's why I'm asking about this

Last edited: Jul 22, 2007
2. Jul 23, 2007

### Meir Achuz

A simpler formula for the transformation of "force" (if defined as dp/dt)
is given in arXiv:physics/0603110 for the case v=0.

Last edited: Jul 23, 2007
3. Jul 23, 2007

### robphy

4. Jul 23, 2007

### jostpuur

Very nice robphy! I wish I had had this source when I was accused of coming up with my "own theories", when I was deriving these three-vector equations. How unfortunate, that the pdf seems to be of horrible quality