Check invariance under rotation group in spacetime

[mcas]

Homework Statement

Check invariance under rotation group in spacetime of a relativistic Newton equation of a charged particle in e-m field with Lorentz force.

Relevant Equations

$\frac{dp^\mu}{ds}=\frac{e}{c}F^{\mu \nu} u_\nu$

I started by inserting $ds=\sqrt{dx'^{\mu} dx'_{\mu}}$ and $p'^{\mu}=mc \frac{dx'^{\mu}}{ds}$.
So we have:
$$\frac{dp'^{\mu}}{ds}=mc \frac{d}{dx'^{\mu}} \frac{d}{dx'_{\mu}} (x'^{\mu})$$
Now I know that
$dx'^{\mu}=C_\beta \ ^\mu dx^\beta$
and
$dx'_{\mu}=C^\gamma \ _\mu dx_\gamma$
where $C$ is the transformation and $C_\beta \ ^\mu C^\gamma \ _\mu = \delta^\gamma _\beta$.
Taking this, we have
$$\frac{dp'^{\mu}}{ds}=mc \frac{d}{C_\beta \ ^\mu dx^\beta C^\gamma \ _\mu dx_\gamma} (x'^{\mu})=mc \frac{d}{dx^\beta dx_\beta} (x'^{\mu})=mc \frac{d}{ds} (x'^{\mu})$$

And now if I were to write $x'^{\mu}=C_\delta \ ^\mu dx^\delta$, this equation wouldn't be an invariant but I think it should be.

 

[vanhees71]

I'm a bit puzzled, what is really asked? You have an equation which is manifestly covariant under Lorentz transformations. Since the rotations form a subgroup of the Lorentz group, it's also covariant under rotations.

NB: I guess "invariance" here should read "covariance" or "invariance" is in fact "form invariance" which is synonymous with "covariance" ;-))

 

[mcas]

I'm a bit puzzled, what is really asked? You have an equation which is manifestly covariant under Lorentz transformations. Since the rotations form a subgroup of the Lorentz group, it's also covariant under rotations.

NB: I guess "invariance" here should read "covariance" or "invariance" is in fact "form invariance" which is synonymous with "covariance" ;-))

I'm sorry, English is not my first language so the translation might be a little bit clanky and I assume you must be right.
So let me try rephrasing it - I have to show that this equation is covariant under Lorentz transformations using the following properties:
$A_\alpha=C_\alpha \ ^\beta A'_\beta$
$A^\alpha=C^\alpha \ _\beta A'^\beta$
$A'_\alpha=C^\beta \ _\alpha A_\beta$
$A'^\alpha=C_\beta \ ^\alpha A^\beta$
$C^\gamma \ _\alpha C_\gamma \ ^\beta=\delta_\alpha ^\beta$
$C_\alpha \ ^\gamma C^\beta \ _\gamma=\delta_\alpha ^\beta$
$\frac{\partial}{\partial x_\alpha}=C^\alpha \ _\beta \frac{\partial}{\partial x'_\beta}$
$\frac{\partial}{\partial x'_\alpha}=C _\beta \ ^\alpha \frac{\partial}{\partial x_\beta}$
(I don't have to use all of them, obviously.)

I've tried to go from $\frac{dp'^{\mu}}{ds}$ to $\frac{dp^{\mu}}{ds}$ but that didn't work as seen in my attempt above.

 

[vanhees71]

Well, for the left-hand side you first need to deduce, how $p^{\mu}$ and $s$ transform under Lorentz transformations, and on the right-hand side $q$ and $F^{\mu \nu}$ as well as $u_{\nu}$. So just think about the definition of the various quantities and their transformation properties under Lorentz transformations.