Lorentz Transformation

A 4-D transformation satisfied by all Four-Vectors $a^\nu$ ,

$\begin{displaymath} {a'}^{\mu} =\Lambda_\nu^\mu a^\nu. \end{displaymath}$

(1)

In the theory of special relativity,

the Lorentz transformation replaces the Galilean Transformation as the valid transformation law between reference frames moving with respect to one another at constant Velocity. Let $x^\nu$ be the Position Four-Vector with

, and let the relative motion be along the

axis with Velocity

. Then (1) becomes

$\begin{displaymath} {x'}^{\mu} =\Lambda_\nu^\mu x^\nu, \end{displaymath}$

(2)

where the Lorentz Tensor is given by

$\begin{displaymath} {\hbox{\sf L}} =\left[{\matrix{ \Lambda_0^0 & \Lambda_1^0 & ... ... \gamma & 0 & 0\cr 0 & 0 & 1 & 0\cr 0 & 0 & 0 & 1\cr}}\right]. \end{displaymath}$

(3)

Here,

$\displaystyle \beta$	$\textstyle \equiv$	$\displaystyle {v\over c}$	(4)
$\displaystyle \gamma$	$\textstyle \equiv$	$\displaystyle {1\over\sqrt{1-\beta^2}}.$	(5)

Written explicitly, the transformation between $x^\nu$ and ${x^\nu}'$ coordinate is

$\displaystyle {x^0}'$	$\textstyle =$	$\displaystyle \gamma(x^0-\beta x^1)$	(6)
$\displaystyle {x^1}'$	$\textstyle =$	$\displaystyle \gamma(x^1-\beta x^0)$	(7)
$\displaystyle {x^2}'$	$\textstyle =$	$\displaystyle x^2$	(8)
$\displaystyle {x^3}'$	$\textstyle =$	$\displaystyle x^3.$	(9)

The Determinant of the upper left $2\times 2$ Matrix in (3) is

$\begin{displaymath} D=(\gamma)^2-(-\gamma\beta)^2=\gamma^2(1-\beta^2) = {\gamma^2\over \gamma^2}=1, \end{displaymath}$

(10)

$\displaystyle {\hbox{\sf L}}^{-1}$	$\textstyle =$	$\displaystyle \left[\begin{array}{ccccccccccccc} (\Lambda^{-1})_0^0 & (\Lambda^... ...^{-1})_1^3 & (\Lambda^{-1})_2^3 & (\Lambda^{-1})_3^3\end{array}\right]\nonumber$
	$\textstyle \equiv$	$\displaystyle \left[\begin{array}{ccccccccccccc} \gamma & \gamma\beta & 0 & 0\\... ... & 0 & 0\nonumber\\ 0 & 0 & 1 & 0\nonumber\\ 0 & 0 & 0 & 1\end{array}\right].$

A Lorentz transformation along the -axis can also be written

$\begin{displaymath} \left[{\matrix{{x^0}'\cr {x^1}'\cr {x^2}'\cr {x^3}'\cr}}\rig... ...] \left[{\matrix{{x^0}\cr {x^1}\cr {x^2}\cr {x^3}\cr}}\right], \end{displaymath}$

(11)

where $\theta$ is called the rapidity,

$\begin{displaymath} x^0 \equiv ct, \end{displaymath}$

(12)

and

$\displaystyle \tanh\theta$	$\textstyle \equiv$	$\displaystyle \beta\equiv {v\over c}$	(13)
$\displaystyle \cosh\theta$	$\textstyle \equiv$	$\displaystyle \gamma\equiv {1\over\sqrt{1-\beta^2}}$	(14)
$\displaystyle \sinh\theta$	$\textstyle =$	$\displaystyle \gamma\beta.$	(15)

References

Fraundorf, P. ``Accel-1D: Frame-Dependent Relativity at UM-StL.'' http://www.umsl.edu/~fraundor/a1toc.html.

Griffiths, D. J. Introduction to Electrodynamics. Englewood Cliffs, NJ: Prentice-Hall, pp. 412-414, 1981.

Morse, P. M. and Feshbach, H. ``The Lorentz Transformation, Four-Vectors, Spinors.'' §1.7 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 93-107, 1953.