Changes

The Poincaré Group consists of 10 [[Mathematics|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''' consists of 10 [[Mathematics|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.

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

 

These transformations are:

These transformations are:

 

Spatial translations:

===Spatial translations===

<br> <math>T_x : (x, y, z, ct) = (x + \epsilon , y, z, ct)</math>

*<math>T_x : (x, y, z, ct) = (x + \epsilon , y, z, ct)</math>

<br> <math>T_y : (x, y, z, ct) = (x, y + \epsilon , z, ct)</math>

*<math>T_y : (x, y, z, ct) = (x, y + \epsilon , z, ct)</math>

<br> <math>T_z : (x, y, z, ct) = (x, y, z + \epsilon , ct)</math>

*<math>T_z : (x, y, z, ct) = (x, y, z + \epsilon , ct)</math>

 

Time Translations:

===Time Translations===

<br> <math>T_t : (x, y, z, ct) = (x, y, z, ct + \epsilon )</math>

*<math>T_t : (x, y, z, ct) = (x, y, z, ct + \epsilon )</math>

 

Spatial Rotations:

===Spatial Rotations===

<br> <math>R_x : (x, y, z, ct) = (x, cos \theta y + sin \theta z, - sin \theta y + cos \theta y, ct)</math>

*<math>R_x : (x, y, z, ct) = (x, cos \theta y + sin \theta z, - sin \theta y + cos \theta y, ct)</math>

<br> <math>R_y : (x, y, z, ct) = (cos \theta x + sin \theta z, y, - sin \theta x + cos \theta z, ct)</math>

*<math>R_y : (x, y, z, ct) = (cos \theta x + sin \theta z, y, - sin \theta x + cos \theta z, ct)</math>

<br> <math>R_z : (x, y, z, ct) = (cos \theta x + sin \theta y, - sin \theta x + sin \theta y, 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:

===Lorentzian Transformations===

<br> <math>L_x : (x, y, z, ct) = (x', y, z, ct')</math>

*<math>L_x : (x, y, z, ct) = (x', y, z, ct')</math>

<br> <math>L_y : (x, y, z, ct) = (x, y', z, ct')</math>

*<math>L_y : (x, y, z, ct) = (x, y', z, ct')</math>

<br> <math>L_z : (x, y, z, ct) = (x, y, z', ct')</math>

*<math>L_z : (x, y, z, ct) = (x, y, z', ct')</math>

[[Category:Physics]]

[[Category:Physics]]

[[Category:Mathematics]]

[[Category:Mathematics]]