[University Special Relativity] Lorentz Transformation and Boosts

[Athenian]

Homework Statement

Find the matrix for the Lorentz transformation consisting of a boost of speed $v$ in the $x$-direction followed by a boost of speed $w$ in the $y'$ direction. Show that the boosts performed in the reverse order would give a different transformation.

Relevant Equations

When it comes to Lorentz transformations, I feel like there are countless equations available for it. However, for this particular problem, I used the following equation for the Lorentz transformation and boost.

Lorentz Transformation:
$$S(P, P) = \eta_{\alpha \beta} x^{\alpha} x^{\beta} = x^{\alpha} \eta_{\alpha \beta} x^{\beta} = -c^2 t^2 + x^2 + y^2 + z^2$$

Boost:
$ct' = Act + Bx$ $\Rightarrow$ $(ct') = (A, B, 0, 0) (ct)$
$x' = Cct + Dx$ $\Rightarrow$ $(x') = (C, D, 0, 0) (x)$
$y' = y$ $\Rightarrow$ $(y') = (0, 0, 1, 0) (y)$
$z' = z$ $\Rightarrow$ $(z') = (0, 0, 0, 1) (z)$

Where A is ultimately equal to 1, B = 0, C = 0, and D =1.
*Please note that the above equation for boost is supposed to be a matrix. However, typing it out on the forum didn't work very well for me. Perhaps there's a way to type out a matrix here that I don't know yet.

Unfortunately, I am not entirely confident of the above equations being able to do the trick and ultimately solve for the question. However, my guess is that using the equation written above for "boost", I could perhaps use $v$ and insert it into the $x$-direction part of the matrix (somehow) and do the same for the speed $w$ in the $y'$-direction which is on the other (left) side of the matrix equation for boost.

Furthermore, when it comes to Lorentz transformations, I believe the question would involve the Lorentz transformation identity $I$ during the calculation process since "something times its inverse is identity $I$" (e.g. $L \cdot L^{-1} = I$).

Other than these small ideas, I have a difficult time knowing how to proceed with solving the question. Does anyone in the community perhaps have any helpful suggestions to help me move forward in solving the problem? Any help would be sincerely appreciated. Thank you very much for reading through this!

 

[mitochan]

I recently participated in the thread:
https://www.physicsforums.com/threads/lorentz-transformation-for-3-frames-2-dimensions.978937/that might show you a way. Two successive perpendicular boosts equals a rotation after a boost.

 

[Athenian]

Thank you very much for sharing the thread, mitochan!

First off, my apologies for replying this late despite your kind assistance. I had a tough math exam a day and a half ago and I have also taken the liberty to study more in-depth your thread as well as other online sources regarding (Lorentz) boosts.

With that in mind, here are a couple of things I learned (and understood) through my research - which, unfortunately, doesn't amount to as much as I would like.

To begin with, a Lorentz boost (a Lorentz transformation which does not involve rotation) in the $x$ direction would look like the following:
$$\begin{bmatrix} \gamma & -\beta \gamma & 0 & 0 \\ -\beta \gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 &0 & 0 & 1 \end{bmatrix}$$
The coordinates are written as $(t, x, y, z)$.

Please note that $\beta = \frac{v}{c}$ and $\gamma = \frac{1}{\sqrt{1- \frac{v^2}{c^2}}}$.

The following is a Lorentz boost in the $y$ direction:
$$\begin{bmatrix} \gamma & 0 & -\beta \gamma & 0 \\ 0 & 1 & 0 & 0 \\ -\beta \gamma & 0 & \gamma & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$$

However, despite knowing that this matrix of the Lorentz boost in the $y$ direction does not affect the $x$ and $z$ directions - rather only affects both time and the $y$ direction - I still don't quite get how people are able to get this calculation.

*Source: I learned this from StackExchange - https://physics.stackexchange.com/questions/30166/what-is-a-lorentz-boost-and-how-to-calculate-it

"Attempted Solution"
I heard from someone on StackExchange that once I "find the correct values for my boost of speed $v$ in the $x$ direction, I should do the same with the second boost in the $y'$ direction with a speed of $w$ - except that the matrix should have $B, C,$ and $D$ in the second row and/or column. Finally, I am supposed to multiply the matrics in one order, and then in the opposite order afterward".
What does everybody in the community think of such a solution process?
In addition, is the Lorentz boost in the $x$-direction I have written above consist speed $v$ in the matrix like the question requires?
Beyond that, how should I go about finding the boost of speed $w$ in the $y'$ direction?

I apologize for the load of questions. But, perhaps not surprisingly, I am terribly confused by this problem and I sincerely hope to understand how to solve this question in an orderly fashion.

Regardless, I sincerely appreciate your help as well as sharing that thread with me. While I was able to learn a few points in the thread, I do admit a lot of its calculations escaped my understanding even though I tried to search on the web for assistance. I'll still continue to try to understand your thread, however, any help from you - or anybody else from the community - to push me in the right direction in solving the problem would be much appreciated. Thank you!