# Special Relativity - Lorentz Transformation & Matrices

1. Nov 21, 2015

### Isaac Pepper

1. The problem statement, all variables and given/known data
There are three observers, all non accelerating. Observer B is moving at velocity vBA with respect to observer A. Observer C is moving at velocity vC B with respect to observer B. All three observers and all their relative velocities are directed along the same straight line. Calculate the matrix transforming the coordinates of an event from the reference frame of observer A to the reference frame of observer C. Comment of the form of the matrix

2. Relevant equations
Assuming normal velocities (so we can use Galilean formulae) : $$u = v + u'$$

3. The attempt at a solution
Hi, if anyone could just explain what it is I need to do in this question please - I have not done Matrices yet in First Year Physics, but have looked up and understood how to use them (I think). I've never seen Matrices used in Relativity before.
Any help would be greatly appreciated, thanks :)

EDIT :: So perhaps the coordinates of an event could be written as follows : $$\binom{t}{x}$$
In the reference frame of observer A, observer C would be going at a velocity of $$V = Vcb+Vba$$
Therefore in the reference frame of observer C, observer A would appear to going at the same speed in the opposite direction : $$V = -(Vcb+Vba)$$

EDIT2 :: So I'm guessing that would mean $$x' = x-vt$$
$$t'=t$$
and $$\binom{t'}{x'}=\binom{t}{x-vt}$$
And we're looking for a matrix that would help us move from $\binom{t'}{x'}$ to $\binom{t}{x}$
So $$\binom{t}{x-vt}=\begin{pmatrix} m&n\\ l&p \end{pmatrix} \binom{t}{x}$$
Therefore $$mt+nx=t \rightarrow m=1, n=0$$
$$lt+px=x-vt \rightarrow p=1, l=-v$$
Finally, $$\begin{pmatrix} m&n\\ l&p \end{pmatrix} = \begin{pmatrix} 1&0\\ -v&1 \end{pmatrix}$$
Is that correct?

Last edited: Nov 21, 2015
2. Nov 21, 2015

### vela

Staff Emeritus
Why are you using Galilean relativity instead of special relativity? The title of your thread suggests you should be using the latter.

3. Nov 21, 2015

### Isaac Pepper

To be honest, I got totally confused. The question that I'm given doesn't suggest that the observers are going at speeds close to the speed of light, so I'm not actually sure if I'm supposed to be using the Lorentz transformation or Galilean transformation :/

4. Nov 21, 2015

### vela

Staff Emeritus
What you did is fine for Galilean relativity. The set up in the problem sounds like it's for special relativity to me.

5. Nov 27, 2015

### Isaac Pepper

Okay so I asked my academic tutor and since they're not moving at relativistic speeds, Galilean transformations are ok for this question, thank you !