Changes

The Poincaré Group is the full set of transformations which leave the spacetime interval invariant.  Just as in 3-dimensional space distance is preserved by translations and rotations, so in Minkowski space the transformations of the Poincaré Group preserve the spacetime interval.

The Poincaré Group is the full set of transformations which leave the spacetime interval invariant.  Just as in 3-dimensional space distance is preserved by translations and rotations, so in Minkowski space the transformations of the Poincaré Group preserve the spacetime interval.

There are ten basic transformations in the Poincaré Group, and all other transformations may be written as combinations of these (for example, a translation followed by a rotation).  In the language of [[group theory]], these ten elements are said to be [[generator|generators]] of the Poincaré Group, which thus has the structure of a ten-dimensional [[Lie group]].  Six of these are the familiar translations and rotations of space.  In additional to translations of time, the Poincaré Group has three additional generators, corresponding to the Lorentz transformations.

There are ten basic transformations in the Poincaré Group, and all other transformations may be written as combinations of these (for example, a translation followed by a rotation).  In the language of [[group theory]], these ten elements are said to be [[generator]]s of the Poincaré Group, which thus has the structure of a ten-dimensional [[Lie group]].  Six of these are the familiar translations and rotations of space.  In additional to translations of time, the Poincaré Group has three additional generators, corresponding to the Lorentz transformations.

===Spatial translations===

==Spatial translations==

*<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>

*<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>

*<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==

*<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==

*<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>

*<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>

*<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==

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

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