Difference between revisions of "Lorentz transformation"
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| − | '''Lorentz transformations''' form the group of linear | + | '''Lorentz transformations''' form the group of linear isometries of [[Minkowski space]]. |
| − | In [[physics]], a Lorentz transformation is the conversion of space and time between two different inertial frames of reference. | + | In [[physics]], a Lorentz transformation is the conversion of space and time between two different [[Inertial reference frame|inertial frames of reference]]. |
| − | ==External | + | According to [[Einstein]] the principles of [[Special Relativity]] are mathematically expressed by Lorentz transformations owing to which it is possible to transform equations for mechanical and electromechanical phenomena between inertial systems.<ref>{{cite book |
| + | |author=Michal Andrle | ||
| + | |title=Whitheadova Filosofie Přírody | ||
| + | |publisher=Charles University | ||
| + | |location=Prague | ||
| + | |isbn=978-80-87378-22-9 | ||
| + | |pages=155 | ||
| + | |quote= | ||
| + | |language=Czech}}</ref> | ||
| + | |||
| + | == Mathematics of the Transforms == | ||
| + | === Displacement and Time === | ||
| + | |||
| + | The transformation from one coordinate system <math> (x, y, z, t) </math> to another system, <math> (x', y', z', t') </math>, moving past this one at speed u, and with <math>x </math> and <math> x' </math> axes colinear is: | ||
| + | |||
| + | <math> x' = \gamma (x -ut) </math><br/> | ||
| + | <math> y' = y </math><br/> | ||
| + | <math> z' = z</math><br/> | ||
| + | <math> t' = \gamma (t - \frac{ux}{c^2} ) </math> | ||
| + | |||
| + | where <math> \gamma </math> is the [[Lorentz factor]]<ref>{{cite book | ||
| + | |author=Bradley W. Carroll, Dale A Ostlie | ||
| + | |title=An Introduction to Modern Astrophysics | ||
| + | |publisher=Pearson | ||
| + | |location=London | ||
| + | |isbn= | ||
| + | |pages= | ||
| + | |quote= | ||
| + | |language=English}}</ref>. | ||
| + | |||
| + | === Velocity Transforms === | ||
| + | |||
| + | The velocity transforms can be found by [[Derivative (calculus)|differentiating]] the displacement transforms with respect to [[time]]. For the above coordinate systems, we find: | ||
| + | |||
| + | <math> v'_x = \frac{v_x -u}{1- \frac{u v_x}{c^2}} </math><br/> | ||
| + | <math> v'_y = \frac{v_y}{\gamma (1- \frac{u v_x}{c^2})} </math><br/> | ||
| + | <math> v'_z= \frac{v_z}{\gamma (1- \frac{u v_x}{c^2})} </math> | ||
| + | |||
| + | Note how the transformations for [[velocity]] in the y and z directions also depend on the velocity of the particle in the x direction, not just the relative speed between the [[inertial frame of reference|frames of reference]]. This is different to the displacement transformations, in which y and z are independent of x. | ||
| + | |||
| + | == See also == | ||
| + | *[[Albert Einstein]] | ||
| + | *[[Hendrik Lorentz]] | ||
| + | *[[Relativity]] | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
| + | |||
| + | ==External links== | ||
*[http://www.relativitycalculator.com/Minkowski_special_relativity_geometry.shtml Deriving Lorentz transformation by Minkowski geometry ] | *[http://www.relativitycalculator.com/Minkowski_special_relativity_geometry.shtml Deriving Lorentz transformation by Minkowski geometry ] | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
| − | [[Category: | + | [[Category:Relativity]] |
Latest revision as of 21:21, September 8, 2020
Lorentz transformations form the group of linear isometries of Minkowski space.
In physics, a Lorentz transformation is the conversion of space and time between two different inertial frames of reference.
According to Einstein the principles of Special Relativity are mathematically expressed by Lorentz transformations owing to which it is possible to transform equations for mechanical and electromechanical phenomena between inertial systems.[1]
Mathematics of the Transforms
Displacement and Time
The transformation from one coordinate system <math> (x, y, z, t) </math> to another system, <math> (x', y', z', t') </math>, moving past this one at speed u, and with <math>x </math> and <math> x' </math> axes colinear is:
<math> x' = \gamma (x -ut) </math>
<math> y' = y </math>
<math> z' = z</math>
<math> t' = \gamma (t - \frac{ux}{c^2} ) </math>
where <math> \gamma </math> is the Lorentz factor[2].
Velocity Transforms
The velocity transforms can be found by differentiating the displacement transforms with respect to time. For the above coordinate systems, we find:
<math> v'_x = \frac{v_x -u}{1- \frac{u v_x}{c^2}} </math>
<math> v'_y = \frac{v_y}{\gamma (1- \frac{u v_x}{c^2})} </math>
<math> v'_z= \frac{v_z}{\gamma (1- \frac{u v_x}{c^2})} </math>
Note how the transformations for velocity in the y and z directions also depend on the velocity of the particle in the x direction, not just the relative speed between the frames of reference. This is different to the displacement transformations, in which y and z are independent of x.