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Created page with '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 inte...'
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:
<math>T_x : (x, y, z, ct) = (x + /epsilon, y, z, ct)</math>
<math>T_y : (x, y, z, ct) = (x, y + /epsilon, z, ct)</math>
<math>T_z : (x, y, z, ct) = (x, y, z + /epsilon, ct)</math>
Time Translations:
<math>T_t : (x, y, z, ct) = (x, y, z, ct + /epsilon)</math>
Spatial Rotations:
<math>R_x : (x, y, z, ct) = (x, cos /theta y + sin /theta z, - sin /theta y + cos /theta y, ct')</math>
<math>R_y : (x, y, z, ct) = (cos /theta x + sin /theta z, y, - sin /theta x + cos /theta z, ct')</math>
<math>R_z : (x, y, z, ct) = (cos /theta x + sin /theta y, - sin /theta x + sin /theta y, z, ct')</math>
Lorentzian Transformations:
<math>L_x : (x, y, z, ct) = (x', y, z, ct')</math>
<math>L_y : (x, y, z, ct) = (x, y', z, ct')</math>
<math>L_z : (x, y, z, ct) = (x, y, z', ct')</math>
These transformations are:
Spatial translations:
<math>T_x : (x, y, z, ct) = (x + /epsilon, y, z, ct)</math>
<math>T_y : (x, y, z, ct) = (x, y + /epsilon, z, ct)</math>
<math>T_z : (x, y, z, ct) = (x, y, z + /epsilon, ct)</math>
Time Translations:
<math>T_t : (x, y, z, ct) = (x, y, z, ct + /epsilon)</math>
Spatial Rotations:
<math>R_x : (x, y, z, ct) = (x, cos /theta y + sin /theta z, - sin /theta y + cos /theta y, ct')</math>
<math>R_y : (x, y, z, ct) = (cos /theta x + sin /theta z, y, - sin /theta x + cos /theta z, ct')</math>
<math>R_z : (x, y, z, ct) = (cos /theta x + sin /theta y, - sin /theta x + sin /theta y, z, ct')</math>
Lorentzian Transformations:
<math>L_x : (x, y, z, ct) = (x', y, z, ct')</math>
<math>L_y : (x, y, z, ct) = (x, y', z, ct')</math>
<math>L_z : (x, y, z, ct) = (x, y, z', ct')</math>