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I'm confused about the difference between a Galilean transformation and boost with reference to their matrices. I was given four statements (listed below) but I'm not sure what I should be looking for to determine their validity. Apologies if this has been answered before I could only see boost vs translation but wasn't sure if this was the same.

  1. $ \begin{bmatrix} 1 & 0 & 0 & 0 \\ v & 1 & 0 & 1 \\ -v & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ \end{bmatrix} $ is a Galilean transformation

  2. $ \begin{bmatrix} 1 & 0 & 0 & 0 \\ v & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 2v & -1 & 0 & 0 \\ \end{bmatrix} $ is a Galilean transformation

  3. $ \begin{bmatrix} 1 & 0 & 0 & 0 \\ v & 1 & 0 & 0 \\ v & 0 & 1 & 0 \\ -v & 0 & 0 & 1 \\ \end{bmatrix} $ is a Galilean boost

  4. $ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 2v & 0 & -1 & 0 \\ v & 1 & 0 & 0 \\ 2v & 0 & 0 & 1 \\ \end{bmatrix} $ is a Galilean transformation but not a Galilean boost

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  • $\begingroup$ Do you understand how to write the linear transformations between $t,x,y,z$ and $t’,x’,y’,z’$ that these matrices represent? $\endgroup$
    – Ghoster
    Commented Apr 24 at 21:15
  • $\begingroup$ @Ghoster I think so, you just multiply by [t, x, y, z] transpose. I know from the solution I was given that the issue is with the first matrix so I'm assuming it is something to do with x' = vt + x + z but I dont really know how to interpret this or why it is an issue $\endgroup$
    – rose
    Commented Apr 24 at 21:19

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