# What are the Lorentz transformations for polar coordinates?

Is there a reason I can't find anywhere the Lorentz transformation for polar coordinates? I can only find the lorentz transformations for cartesian coordinates.

Can anyone guide me on how to transform polar coordinates, or is the only way to convert them to cartesian coordinates first, and then transform those and then convert those back to polar coordinates?

• There's some info at physics.stackexchange.com/a/603032/123208 – PM 2Ring May 16 at 4:32
• @PM2Ring i have seen that question, and i believe it is different and does not address what i am asking – silverrahul May 16 at 4:34
• Did you look at the linked article arxiv.org/abs/2008.08780 which discusses both cylindrical & spherical coordinates? – PM 2Ring May 16 at 4:38
• What difficulty do you have in taking the ordinary representation of the Lorentz transformations (either in terms of abstract vectors or in Cartesian coordinates) and just doing the transformation from Cartesian to polar coordinates? – ACuriousMind May 16 at 13:07
• The coordinates i want to transform are in polar coordinates . I know i can convert them to cartesian coords and then transform those cartesian coords . My question is whether there was any formula for directly transforming polar coordinates without first converting them – silverrahul May 16 at 14:41

## 1 Answer

Start with coordinate free:

$$t' =\gamma\Big( t-\frac{\vec v\cdot\vec r}{c^2} \Big)$$

$$\vec r'=\vec r +(\gamma-1)(\vec r\cdot\hat v)\hat v-\gamma ct \vec v$$

and use polar coordinates to evaluate the dot products.

• Can you tell me where you got this from ? Book , reference , links etc ? My vector maths is not the strongest and i want to see if there is any more explanation of this – silverrahul May 16 at 4:36
• Wikipedia – G. Smith May 16 at 4:44
• @silverrahul See Goldstein's Classical Mechanics – mithusengupta123 May 16 at 5:40
• @mithusengupta123 thank you. That was very helpful – silverrahul May 16 at 6:08
• The best book on Special Relativity I have seen so far is the one by Stepanov, which I recommend very strongly. In that source, Section 6.5 provides the simplest explanation of SR in rotating frames. (In fact, each section in that textbook is a little masterpiece.) amazon.com/… – Michael_1812 May 16 at 13:05