# Find lorentz transformation for arbitrary velocity (v) relat

1. Feb 15, 2015

### Pual Black

Hello
i have to find the Lorentz transformation for arbitrary velocity (v) relative to (O)

the information's i have:
1-i have to use all 3 components of velocity $(V_x, V_y, V_z )$
2- $x'=\frac{x-vt}{\sqrt{1-\frac{v^2}{c^2}}}$

$y'=y$

$z'=z$

3- $V_x'=\frac{V_x-V}{1-\frac{VV_x}{c^2}}$

$V_y'=\frac{V_y\sqrt{1-\frac{v^2}{c^2}}}{1-\frac{VV_x}{c^2}}$

$V_z'=\frac{V_z\sqrt{1-\frac{v^2}{c^2}}}{1-\frac{VV_x}{c^2}}$

i searched the internet and this forum and found that i have to use a matrix to solve this question but i don't know how to do that.

2. Feb 15, 2015

### Staff: Mentor

Do you mean that you can't find the general matrix form of the Lorentz transformation, or that you don't know how to use that matrix?

3. Feb 15, 2015

### Pual Black

I can't find a way to get a solution of the problem above. I just have the information but don't know how to use them. Of course i also don't know how to use the matrix and even don't know if i have to use a matrix to get the solution of lorentz transformation.
So i need your help to solve the whole question and get the final equation.
Please give me a hint how i have to start.
Thank you

4. Feb 15, 2015

### A.T.

Last edited: Feb 15, 2015
5. Feb 15, 2015

### Pual Black

Thank you for the link but Wikipedia was the first page i visited and from this site i know that i shall use the matrix. But I don't understand it. I can't just copy the solution without understandig it.
Therefore i asked for your help so i can solve this solution step by step.

6. Feb 15, 2015

### Staff: Mentor

The Wikipedia page doesn't say you have to use a matrix. There's a whole section on the vector form before the one on the matrix form. Have you tried looking at that?

7. Feb 15, 2015

### Staff: Mentor

This is not correct for the general case you are trying to solve; it's only correct for the special case in which only $V_x$ is nonzero.

8. Feb 15, 2015

### Pual Black

Do you mean "
Boost in the x-direction and in y - z direction "

Yes i read it but the question says in arbitrary direction. Therefore x,y,z and not just in one direction.
Correct me please if im wrong

9. Feb 15, 2015

### Staff: Mentor

Can you say more specifically about what you don't understand about it? You don't understand how to use it, or you don't understand where it comes from? Specifically, this:

10. Feb 15, 2015

### Staff: Mentor

I mean the section entitled "Vector form" under "Boost in any direction".

A boost in "any" direction is the same thing as a boost in an "arbitrary" direction. That's why A.T. directed you to that portion of the Wikipedia page.

11. Feb 15, 2015

### Pual Black

I don't know where they come from.

12. Feb 15, 2015

### Pual Black

yes i searched a little and found this. is it right now?

$x'=\frac{x-vt}{\sqrt{1-\frac{v^2}{c^2}}}$

$y'=\frac{y-vt}{\sqrt{1-\frac{v^2}{c^2}}}$

$z'=\frac{z-vt}{\sqrt{1-\frac{v^2}{c^2}}}$

13. Feb 15, 2015

### pervect

Staff Emeritus
It seems to me that the easiest conceptual approach is to find some purely spatial rotation (which can be represented by a matrix R) that aligns the boost in some specified direction of your choice (x,y,z).

Then the total boost should be $R^{-1} B R$ where R is the purely spatial rotation matrix (padded out from the usual 3x3 matrix to a 4x4 matrix), $R^{-1}$ is its inverse, and B is the boost along the axis of your choice (x,y,z).

I.e to boost in an arbitrary direction, you rotate your frame of reference so that one of the x,y,or z axis (your choice) points in the boost direction, then you preform the boost along your chosen axis, then you undo the rotation.

This also strikes me as more work than I care to do. But you should eventually be able to get the right answer, I think.

14. Feb 15, 2015

### Staff: Mentor

No. I'm not sure I understand why you keep searching for solutions to copy, since you said this in an earlier post:

Searching isn't going to help you understand; you need to take some time and look at the information we've already linked you to, and try to figure out what it's telling you, and ask questions about it.

15. Feb 15, 2015

### Staff: Mentor

They come from writing the vector equations in the "vector form" section out component by component, and then putting all that information in matrix form. Once again, have you looked at the "vector form" section? Do you have questions about what it says there? If you want to understand how a boost in an arbitrary direction works, that section is a good place to start.

16. Feb 16, 2015

### Pual Black

And as i said i just have a few informations which are not enough. Therefore i must keep searching and learning.

Now i want your help to understand why we used the matrix. What is the reason ? Why not a determined or something else

17. Feb 16, 2015

### Staff: Mentor

Sigh. You aren't even reading my posts. Twice now I have pointed you at the vector form of the transformation. That is not the matrix form. You do not have to use the matrix.

And even if you end up using the matrix for calculations, understanding the vector form and how it works will help you to understand the matrix form. So, for a third time, have you looked at the vector form of the transformation?

18. Feb 16, 2015

### PWiz

Just curious, would you generally recommend using the vector form over the matrix form for boosts in any direction? I must admit that the matrix complicates matters a little more than I'd like it to.

19. Feb 16, 2015

### Staff: Mentor

For understanding what's going on, yes. For actually doing computations, it depends; for many computations, the matrix form will probably be easier to use.

20. Feb 16, 2015

### pervect

Staff Emeritus
Matrices represent linear trasnformations, a tool of linear algebra. I would suggest finding an introductory textbook on the topic on matrices (and/or linear algebra, though the later might be a bit broader search term) to learn more about them.

I don't have any specific textbook recommendations, but you should be able to find some books at the library.

If you are struggling to understand the Lorentz transformation and its representation via matrices, it might be helpful to first study rotations (which I hope you are familiar with conceptually) and how matrices are used to represent rotations.