Mathematical form of the Lorentz transformation
Mathematically, the form of the Lorentz transformations
illustrated pictorially above is as follows.
The units in these equations are
\(c = 1\),
following the convention for
spacetime diagrams.
Suppose that Cerulean (primed frame) is moving
relative to Vermilion (unprimed frame)
at velocity \(v\) in the \(x\) direction.
Then the coordinates
\(( t' , x' , y' , z' )\)
of an event in Cerulean's frame are related to
the coordinates
\(( t , x , y , z )\)
of the same event in Vermilion's frame by
\[
\begin{array}{l}
t' = \gamma t - \gamma v x
\\
x' = - \, \gamma v t + \gamma x
\\
y' = y
\\
z' = z
\end{array}
\ , \quad
\begin{array}{l}
t = \gamma t' + \gamma v x'
\\
x = \gamma v t' + \gamma x'
\\
y = y'
\\
z = z'
\end{array}
\]
These Lorentz transformation formulae can be written elegantly in matrix
notation:
\[
\begin{align}
\left(
\begin{array}{c}
t'
\\
x'
\\
y'
\\
z'
\end{array}
\right)
&=
\left(
\begin{array}{cccc}
\gamma & - \gamma v & 0 & 0 \\
- \gamma v & \gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}
\right)
\left(
\begin{array}{c}
t
\\
x
\\
y
\\
z
\end{array}
\right)
\ ,
\\
\left(
\begin{array}{c}
t
\\
x
\\
y
\\
z
\end{array}
\right)
&=
\left(
\begin{array}{cccc}
\gamma & \gamma v & 0 & 0 \\
\gamma v & \gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}
\right)
\left(
\begin{array}{c}
t'
\\
x'
\\
y'
\\
z'
\end{array}
\right)
\ .
\end{align}
\]
Any 4-dimensional quantity
\(( t , x , y , z )\)
that transforms according to the rules of Lorentz transformations, as above,
is called a 4-vector.
A Lorentz transformation at velocity \(v\) followed by
a Lorentz transformation at velocity \(v\) in the opposite direction,
i.e. at velocity \(-v\) yields the unit transformation, as it should:
\[
\left(
\begin{array}{cccc}
\gamma & \gamma v & 0 & 0 \\
\gamma v & \gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}
\right)
\left(
\begin{array}{cccc}
\gamma & - \gamma v & 0 & 0 \\
- \gamma v & \gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}
\right)
=
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}
\right)
\ .
\]
The determinant of the Lorentz transformation is one, as it should be:
\[
\left|
\begin{array}{cccc}
\gamma & - \gamma v & 0 & 0 \\
- \gamma v & \gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}
\right|
=
\gamma^2 ( 1 - v^2 )
=
1
\ .
\]
Indeed, requiring that the determinant be one provides another
derivation of the
formula for the Lorentz gamma factor.
|