Difference between revisions of "Poincare group"
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| − | 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. | ||
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These transformations are: | These transformations are: | ||
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| − | Spatial translations | + | ===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 | + | ===Time Translations=== |
| − | + | *<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_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 | + | ===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> | |
[[Category:Physics]] | [[Category:Physics]] | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
Revision as of 06:39, August 1, 2009
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
- 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 T_y : (x, y, z, ct) = (x, y + \epsilon , 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 T_z : (x, y, z, ct) = (x, y, z + \epsilon , ct)}
Time Translations
- 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 T_t : (x, y, z, ct) = (x, y, z, ct + \epsilon )}
Spatial Rotations
- 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 R_x : (x, y, z, ct) = (x, cos \theta y + sin \theta z, - sin \theta y + cos \theta y, 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 R_y : (x, y, z, ct) = (cos \theta x + sin \theta z, y, - sin \theta x + cos \theta 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 R_z : (x, y, z, ct) = (cos \theta x + sin \theta y, - sin \theta x + sin \theta y, z, ct)}
Lorentzian Transformations
- 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_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')}