Lorentz transformation

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.

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.

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.

See also

• Albert Einstein
• Hendrik Lorentz
• Relativity

External links

• Deriving Lorentz transformation by Minkowski geometry