Lorentz Transformation in New Coordinates

[soupdejour]

I know the Lorentz transformation in rectangular coordinates

$$ct' = \gamma (ct - \beta x)$$
$$x' = \gamma (x - \beta ct)$$
$$y' = y$$
$$z' = z$$

I want to do this same transformation, but from cylindrical coordinates $$(r,\theta,z)$$ to $$(r',\theta',z')$$.

Any ideas?

 

[soupdejour]

To make things straightforward, I just want the new frame to be moving along one dimension, say z. My guess is

$$ct' = \gamma (ct - \beta z)$$
$$r' = r$$
$$\theta' = \theta$$
$$z' = \gamma (z - \beta ct)$$

 

[DrGreg]

To make things straightforward, I just want the new frame to be moving along one dimension, say z. My guess is

$$ct' = \gamma (ct - \beta z)$$
$$r' = r$$
$$\theta' = \theta$$
$$z' = \gamma (z - \beta ct)$$

Yes, that is exactly right. If the motion is in the z direction, the x and y coordinates do not change, and the r and θ coordinates depend only on x and y (and vice-versa).

 

[soupdejour]

Yes, that is exactly right. If the motion is in the z direction, the x and y coordinates do not change, and the r and θ coordinates depend only on x and y (and vice-versa).

Awesome, I understand, Thanks!