Deriving the Lorentz transformation

[silmaril89]

Homework Statement

Derive the Lorentz transformation by assuming that the transformation is linear, and does not change the perpendicular coordinates. Write the transformation as

x' = A1 (x - vt), y' = y, z' = z, t' = A2 t + A3 x,

Determine A1, A2, A3 by requiring that a flash of light produces an outgoing spherical wave, with velocity c, in either frame F or F'

Homework Equations

I believe we can use x'^2 + y'^2 + z'^2 = c^2 * t'^2 to prove it, but I'd like to see if we don't need that

Maybe the Galilean Transformation may also be of use

x' = x - vt, y' = y, z' = z, t' = t

The Attempt at a Solution

All attempts I've made have just been manipulating the two supplied equations above. My biggest problem, is that I can't seem to understand exactly what a linear transformation really is, or what any transformation really is, (at least as far as deriving it myself, or putting it into a matrix). I think once I can understand that, the rest should be trivial. Any help and clarifications on what a linear transformation is and how to apply it to deriving the Lorentz transformation would be greatly appreciated.

 

[vela]

A linear transformation L is a mapping between two vector spaces that has the property

L(rx+sy) = rL(x)+sL(y)

where r and s are scalars and x and y are vectors.

In this problem, the Lorentz transformation maps the coordinates of an event in one inertial frame F to the coordinates of the event in another inertial frame F'. In other words, knowing the coordinates (t, x, y, z) in F, you can calculate the coordinates (t', x', y', z') in F' of the same event.

Requiring the transformation to be linear means that it can be represented by a matrix multiplication:

$$\begin{pmatrix}t' \\ x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix}<br /> \Lambda_{00} & \Lambda_{01} & \Lambda_{02} & \Lambda_{03} \\<br /> \Lambda_{10} & \Lambda_{11} & \Lambda_{12} & \Lambda_{13} \\<br /> \Lambda_{20} & \Lambda_{21} & \Lambda_{22} & \Lambda_{23} \\<br /> \Lambda_{30} & \Lambda_{31} & \Lambda_{32} & \Lambda_{33}<br /> \end{pmatrix} \begin{pmatrix}t \\ x \\ y \\ z \end{pmatrix}$$

Hopefully, you can see that the equations you have can be written in the form above with the appropriate choices for Λ_μν, so the equations you were given already embody the assumption that the Lorentz transformation is linear.

 

[silmaril89]

So, as my answer, I wrote my transformation as this

$$\begin{pmatrix}t' \\ x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix}<br /> A2 & A3 & 0 & 0 \\<br /> -A1 v & A1 x & 0 & 0 \\<br /> 0 & 0 & 1 & 0 \\<br /> 0 & 0 & 0 & 1<br /> \end{pmatrix} \begin{pmatrix}t \\ x \\ y \\ z \end{pmatrix}$$

And then from there, I should be able to solve for A1 A2 and A3, by using x'^2 + y'^2 + z'^2 = c^2 * t'^2, is that correct?