\, .$$

Moreover, it is easy to see that $$\label{equation16}
e^{i\pi M_{0n}} \As(W_R) e^{-i\pi M_{0n}} = \As(\widetilde{W}_R{}') 
\quad \rm{and} \quad 
e^{i\pi M_{0n}} \As(W_R{}') e^{-i\pi M_{0n}} = \As(\widetilde{W}_R) \, .$$ The rotation in the $0$–$n$–plane by $\pi$ also reverses the orientation of the world lines $\lambda_{01}(t)x_O$ (cf. equation (\[0n01\])), so that the world lines in $\widetilde{W}_R{}'$ run in the same direction as those of $W_R{}'$, while those in $\widetilde{W}_R$ have the same orientation as those in $W_R$. Explicitly, Corollary \[modularunitary\] implies that the modular unitaries for the pair $(\As(W_R{}'),\Omega)$ are given by

$$\Delta_{W_R{}'}^{it} = e^{i2\pi tM_{01}} \, , \quad t \in \RR \, .$$

Hence, for all $t \in \RR$, one has $$\Delta_{\widetilde{W}_R}^{it} = 
   e^{i\pi M_{0n}} \Delta_{W_R{}'}^{it} e^{-i\pi M_{0n}} = 
   e^{i\pi M_{0n}} e^{i2\pi t M_{01}} e^{-i\pi M_{0n}}
  = e^{-i2\pi t M_{01}} = \Delta_{W_R}^{it} \, ,$$ by (\[0n01\]). So, $\{ e^{-i2\pi tM_{01}}\}_{t \in \RR}$ is the group of modular unitaries for $(\As(\widetilde{W}_R),\Omega)$, and we have the relation $$J_{\widetilde{W}_R} A \Omega = e^{\, -\pi M_{01}} A^* \Omega \ , \quad 
\mbox{for} \ \ A \in {{\cal A}}(\widetilde{W}_R) \, .$$ Moreover, we may appeal to (\[equation15\]) and the modular theory to find $$J_{\widetilde{W}_R} A' \Omega = e^{\, \pi M_{01}} A'{}^* \Omega \ , \quad 
