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## Homework Statement

Question as stated: In special relativity consider the following coordinate transformation between inertial frames: first make a velocity boost [itex]v_x[/itex] in the x-direction, then make a velocity boost [itex]v_y[/itex] in the y-direction. 1) Is this a Lorentz transformation? 2) Find the matrix of this transformation. 3) Consider the boosts in inverse order - is it the same transformation?

## Homework Equations

[itex] \beta_i = \frac{v_i}{c} [/itex]

[itex] \gamma_i = \frac{1}{\sqrt{1-\frac{v_i^2}{c^2}}}[/itex]

## The Attempt at a Solution

I use the [itex] c = 1 [/itex] convention.

I wrote down the two transformations as:

x-direction:

[itex]

\begin{pmatrix}

t' \\

x'\\

y' \\

z'

\end{pmatrix} = \begin{pmatrix}

\gamma_x & -\beta_x \gamma_x & 0 & 0 \\

-\beta_x \gamma_x & \gamma_x & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1

\end{pmatrix} \begin{pmatrix}

t\\

x\\

y\\

z

\end{pmatrix}

[/itex]

and for the y-direction:

[itex]

\begin{pmatrix}

t'' \\

x''\\

y'' \\

z''

\end{pmatrix} = \begin{pmatrix}

\gamma_y & 0 & -\beta_y \gamma_y & 0 \\

0 & 1 & 0 & 0 \\

-\beta_y \gamma_y & 0 & \gamma_y & 0 \\

0 & 0 & 0 & 1

\end{pmatrix} \begin{pmatrix}

t' \\

x' \\

y' \\

z'

\end{pmatrix}

[/itex]

and combined these to get

[itex]

\begin{pmatrix}

t'' \\

x''\\

y'' \\

z''

\end{pmatrix} = \begin{pmatrix}

\gamma_y & 0 & -\beta_y \gamma_y & 0 \\

0 & 1 & 0 & 0 \\

-\beta_y \gamma_y & 0 & \gamma_y & 0 \\

0 & 0 & 0 & 1

\end{pmatrix}\begin{pmatrix}

\gamma_x & -\beta_x \gamma_x & 0 & 0 \\

-\beta_x \gamma_x & \gamma_x & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1

\end{pmatrix} \begin{pmatrix}

t\\

x\\

y\\

z

\end{pmatrix} = \begin{pmatrix}

\gamma_x \gamma_y & -\beta_x \gamma_x \gamma_y & -\beta_y \gamma_y & 0 \\

-\beta_x \gamma_x & \gamma_x & 0 & 0 \\

-\beta_y \gamma_x \gamma_y & \beta_x \beta_y \gamma_x \gamma_y & \gamma_y & 0 \\

0 & 0 & 0 & 1

\end{pmatrix} \begin{pmatrix}

t\\

x\\

y\\

z

\end{pmatrix}

[/itex]

This should be the answer to question 2).

If I inverse the order of the boosts I notice that the resulting transformation matrix is the transpose of the above matrix.

As the transformation matrix for the two cases in question are not equal the two transformations are not the same? (question 3)

I am uncertain on how to explain whether or not the resulting transformations are Lorentz transformations.

*why do the question headlines automatically appear each time I preview my post?*