Poincare group

The Poincaré Group consists of 10 mathematical transformations that can be applied to different coordinates of the space-time graph (x, y, z, ct), but do not affect the "spacetime interval". The "spacetime interval" is roughly the 4-D equivalent of 3-D "distance".

The Poincaré Group is a "symmetry group" because the spacetime interval remains invariant.

These transformations are:

Spatial translations:
$T_{x}:(x,y,z,ct)=(x+\epsilon ,y,z,ct)$
$T_{y}:(x,y,z,ct)=(x,y+\epsilon ,z,ct)$
$T_{z}:(x,y,z,ct)=(x,y,z+\epsilon ,ct)$

Time Translations:
$T_{t}:(x,y,z,ct)=(x,y,z,ct+\epsilon )$

Spatial Rotations:
$R_{x}:(x,y,z,ct)=(x,cos\theta y+sin\theta z,-sin\theta y+cos\theta y,ct)$
$R_{y}:(x,y,z,ct)=(cos\theta x+sin\theta z,y,-sin\theta x+cos\theta z,ct)$
$R_{z}:(x,y,z,ct)=(cos\theta x+sin\theta y,-sin\theta x+sin\theta y,z,ct)$

Lorentzian Transformations:
$L_{x}:(x,y,z,ct)=(x',y,z,ct')$
$L_{y}:(x,y,z,ct)=(x,y',z,ct')$
$L_{z}:(x,y,z,ct)=(x,y,z',ct')$