Poincare group

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

These transformations are:

Spatial translations:
Failed to parse (syntax error): {\displaystyle 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:
$L_{x}:(x,y,z,ct)=(x',y,z,ct')$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_y : (x, y, z, ct) = (x, y', z, ct')}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z : (x, y, z, ct) = (x, y, z', ct')}