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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 $x^0=ct$, and let the relative motion be along the $x^1$ axis with Velocity $v$. 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)

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


A Lorentz transformation along the $x^1$-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)

See also Hyperbolic Rotation, Lorentz Group, Lorentz Tensor


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.



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© 1996-9 Eric W. Weisstein
1999-05-25