(i)–(iv) hold. Then $\Omega$ is invariant under the action of $U({\mbox{SO$_0(2,n\!-\!1)$}})$ and each uniformly accelerated observer testing $\Omega$ in finds a universal value $(1/2\pi)((\dot{\lambda}_{0 1}(0)\, x_O)^2){}^{-1/2}$ of the Unruh temperature which depends only on his particular orbit. For geodesic observers $\omega$ is a ground state; in particular, $M_{0n}$ is a positive operator.

This result is a consequence of the passivity of $\omega$, and this vacuum state is the only normal state on $\As$ which is passive for all observers. In light of Theorem \[temperature\], it is physically justified to identify the operator $M_{0n}$ with the global energy operator.

The result $\beta = 2\pi$ and [@KadRing Theorem 9.2.16] permit us to completely determine the modular unitaries corresponding to the pair $(\As(W),\Omega)$, for all $W \in \Ws$.

\[modularunitary\] Given the Standing Assumptions (i)–(iv), the modular unitaries for the pair $(\As(W_R),\Omega)$ are given by $$\Delta_{W_R}^{it} = e^{-i2\pi tM_{01}} \, , \quad t \in \RR \, .$$

Covariance and the uniqueness of the modular objects yield similar results for $\Delta_W^{it}$, for all $W \in \Ws$.

PCT symmetry
------------

Having computed the modular unitaries and the value of the inverse temperature $\beta$ seen by the uniformly accelerated observers, let us return now to the analysis 