The group discussed here, also known as the Lorentz group in one dimension, describes the symmetries of special relativity with one spatial and one temporal dimension.
The metric is defined as:
\[
\eta = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
\]
The position vector is given by:
\[
x = \begin{pmatrix}
ct \\
x
\end{pmatrix}
\]
If \(x\) transforms as \(x’ = R x\), then the spacetime interval remains invariant:
\[
ds^{2} = dx^{t}\eta dx = {dx’}^{t}\eta dx’ = dx^{t}R^{t}\eta R dx
\]
This implies:
\[
R^{t}\eta R = \eta
\]
By multiplying both sides by \(R^{-1}\) and using the fact that \(\eta\eta = \mathbb{1}\), we obtain:
\[
R^{t}\eta = \eta R^{-1}
\]
Therefore:
\[
R^{t} = \eta R^{-1}\eta \quad \text{or} \quad R^{-1} = \eta R^{t}\eta
\]
Suppose \(R\) has the form:
\[
R = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\]
Then \(R^{-1}\) is given by:
\[
R^{-1} = \frac{1}{ad – cb}
\begin{pmatrix}
d & -b \\
-c & a
\end{pmatrix}
\]
From the condition \(R^{-1} = \eta R^{t} \eta\), we also find:
\[
R^{-1} = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
\begin{pmatrix}
a & c \\
b & d
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix} =
\begin{pmatrix}
a & -c \\
-b & d
\end{pmatrix}
\]
The determinant of \(R\) is:
\[
\Delta = ad – cb
\]
Using the relations \(a = \frac{d}{\Delta}\), \(c = \frac{b}{\Delta}\), \(b = \frac{c}{\Delta}\), and \(d = \frac{a}{\Delta}\), we deduce that \(\Delta^{2} = 1\), so \(\Delta = \pm 1\). For \(\Delta = 1\), we obtain \(a = d\) and \(b = c\), where \(a\) and \(b\) lie on a hyperbola:
\[
\Delta = ad – cb = a^{2} – b^{2} = 1
\]
Expressing this in hyperbolic terms:
\[
\cosh^{2}\theta – \sinh^{2}\theta = 1
\]
Thus, \(R\) can be written as:
\[
R = \begin{pmatrix}
\cosh\theta & \sinh\theta \\
\sinh\theta & \cosh\theta
\end{pmatrix}
\]
For \(\theta \to 0\):
\[
R \approx \begin{pmatrix}
1 & \theta \\
\theta & 1
\end{pmatrix} = \mathbb{1} + \theta \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}.
\]
Here, \(\varrho = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}\) is the generator of the group, and the transformation can be expressed as:
\[
R = e^{\theta\varrho}
\]
The transformed coordinates are:
\[
\begin{pmatrix}
ct’ \
x’
\end{pmatrix} =
\begin{pmatrix}
\cosh\theta & \sinh\theta \\
\sinh\theta & \cosh\theta
\end{pmatrix}
\begin{pmatrix}
ct \\
x
\end{pmatrix}.
\]
Expanding the components:
\[
ct’ = (\cosh\theta)ct + (\sinh\theta)x = \cosh\theta\left[ct + (\tanh\theta)x\right]
\]
\[
x’ = (\sinh\theta)ct + (\cosh\theta)x = \cosh\theta\left[x + (\tanh\theta)ct\right]
\]
The hyperbolic functions relate to the velocity parameter \(\beta = \tanh\theta\):
\[
\cosh\theta = \frac{e^{\theta} + e^{-\theta}}{2} \geq 1
\]
\[
\tanh\theta = \frac{\sinh\theta}{\cosh\theta} = \beta \quad \Rightarrow \quad \sinh\theta = \beta\cosh\theta
\]
From \(\cosh^{2}\theta – \beta^{2}\cosh^{2}\theta = 1\), we find:
\[
\cosh\theta = \frac{1}{\sqrt{1 – \beta^{2}}}
\]
For \(\beta = -\frac{v}{c}\), the Lorentz factor is:
\[
\gamma = \frac{1}{\sqrt{1 – \beta^{2}}} = \frac{1}{\sqrt{1 – \frac{v^{2}}{c^{2}}}}
\]
The transformations become:
\[
ct’ = \gamma\left(ct – \frac{v}{c}x\right)
\]
\[
t’ = \gamma\left(t – \frac{v}{c^{2}}x\right)
\]
\[
x’ = \gamma(x + vt)
\]