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The relativity group of Minkowski spacetime is the subgroup $P < Aff(4,\mathbb{R})$ which preserves the proper time $c^2 (x_4-y_4)^2 - \|\mathbf{x}-\mathbf{y}\|^2$ between two events $X=(\mathbf{x},x_4,1)$ and $Y=(\mathbf{y},y_4,1)$, where $c$ is a positive constant called the speed of light.

Basically I need to find out what $P$ is. i tried taking a matrix $a$ in the affine group and finding out $a(X-Y)$, then trying to put the entries of this column vector into the proper time equation and choosing matrix entries so that it works, and after a lot of expanding brackets etc i came up with

$P=\left\{\begin{pmatrix} A & 0 & u \\ 0 & 1 & x \\ 0 & 0 & 1 \end{pmatrix} \ : \ A\in O(3), u\in\mathbb{R}^3, x\in\mathbb{R}\right\}~$

but then I found out this is wrong. but I'm not sure what is right and how to derive it.

I then need to determine its Lie algebra. which i could do if I knew $P$ i think.

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    @blib: Have you derived the generators of Poincare group in matrix form? – 2014-02-19

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