Difference between revisions of "Lorentz transformation"
(username removed) m (Added reference) |
(username removed) (Point out an interesting feature) |
||
| Line 40: | Line 40: | ||
<math> v'_y = \frac{v_y}{\gamma (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> | <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 == | == See also == | ||
Revision as of 15:58, September 15, 2016
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 to another system, , moving past this one at speed u, and with and axes colinear is:
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 x' = \gamma (x -ut) }
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 y' = y }
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 z' = z}
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' = \gamma (t - \frac{ux}{c^2} ) }
where 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 \gamma } 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:
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 v'_x = \frac{v_x -u}{1- \frac{u v_x}{c^2}} }
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 v'_y = \frac{v_y}{\gamma (1- \frac{u v_x}{c^2})} }
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 v'_z= \frac{v_z}{\gamma (1- \frac{u v_x}{c^2})} }
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.