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Lorentz transformation

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The Lorentz transformations is a set of equations that describe a linear transformation between a stationary reference frame and a reference frame in constant velocity. The equations are given by:

[math]x'=\frac{x-vt}{\sqrt{1-\frac{v^2}{c^2}}}[/math] , [math]y'=y[/math] , [math]z'=z[/math] , [math]t'=\frac{t-\frac{vx}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}[/math]

where [math]x'[/math]represents the new x co-ordinate, [math]v[/math] represents the velocity of the other reference frame, [math]t[/math] representing time, and [math]c[/math] the speed of light.

On a Cartesian coordinate system, with the vertical axis being time (t), the horizontal axis being position in space along one axis (x), the gradients represent velocity (shallower gradient resulting in a greater velocity). If the speed of light is set as a 45° or 1:1 gradient, Lorentz transformations can rotate and squeeze other gradients while keeping certain gradients, like a 1:1 gradient constant. Points undergoing a Lorentz transformations on such a plane will be transformed along lines corresponding to [math]t^2-x^2=n^2[/math] where n is some number

Points undergoing a Lorentz transformation follow the green, conjugate hyperbola, where the vertical axis represents time, [math]y^2-x^2=n^2[/math]