From

Transformation

Result

(3-8)

$\begin{matrix} Φ^{μ ν} = [\begin{matrix} C^{00} \exp (i (\mp E t \pm P r)) & 0 \\ 0 & C^{11} \exp (i (\mp E t \pm P r)) \end{matrix}] \\ = [\begin{matrix} C^{00} \exp (i (\mp E t \pm P r + 2 π)) & 0 \\ 0 & C^{11} \exp (i (\mp E t \pm P r + 2 π)) \end{matrix}] \end{matrix}$

$2 π$ symmetry

Table 5

$h_{μ ν}^{3} = (\begin{matrix} a & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & b \end{matrix}) = (\begin{matrix} - 1 & 0 & 0 & 0 \\ 0 & \cos ϕ & - \sin ϕ & 0 \\ 0 & \sin ϕ & \cos ϕ & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) (\begin{matrix} a & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & b \end{matrix}) (\begin{matrix} - 1 & 0 & 0 & 0 \\ 0 & \cos ϕ & \sin ϕ & 0 \\ 0 & - \sin ϕ & \cos ϕ & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$

where $a = C^{00} \exp (i (\mp E t \pm P r))$

where $b = C^{11} \exp (i (\mp E t \pm P r))$

$ϕ$ independent