Lorentz Transformations

Saturday, January 22, 2022

Relativistic Transformations

Due to the effects of special relativity, classical Galilean transformations do not work. Instead, a system called Lorentz transformations are used.

Lorentz transformations must be:

Linear (depending only on the first power of space and time)
Consistent with Einstein's postulates
Reducible to Galilean transformations for low velocities

The Lorentz transformations are as follows:

$x'=\frac{x-ut}{\sqrt{1-\frac{u^2}{c^2}}} \newline y'=y \newline z'=z \newline t'=\frac{t-\left(\frac{u^2}{c^2}\right)x}{\sqrt{1-\frac{u^2}{c^2}}}$

Note: the equation for $x'$ simplifies to the equation for length contraction for an observer $O$ measuring the length of an object at the one time ( $t=0$ ).

The Lorentz velocity transformations are:

$v_x'=\frac{v_x-u}{1-\frac{v_xu}{c^2}} \newline v_y'=\frac{v_y\sqrt{1-\frac{u^2}{c^2}}}{1-\frac{v_xu}{c^2}} \newline v_z'=\frac{v_z\sqrt{1-\frac{u^2}{c^2}}}{1-\frac{v_xu}{c^2}}$

For small velocities, $u$ , the equation for the x-velocity reduces to the Galilean transformation, but the y- and z- velocities do not because of how Lorentz transformations treat time.

The Lorentz time transformation is:

$dt'=\frac{dt-\left(\frac{u}{c^2}\right)dx}{\sqrt{1-\frac{u^2}{c^2}}}$

Consequences of Lorentz Transformations

Simultaneity

Events that appear to be simultaneous for one observer are not simultaneous for another observer moving relative to the other, but not due to time dilation.

The only exception would be if the events happen at the same location in space.

An example of this would be synchronized clocks, whose times drift apart once they start moving relative to one another.