the trajectories of the emitter and detector were synchronized in the absence of the wave. Then the [*time–of–flight*]{} of the light signal from $E$ to $D$, $\Delta \tau_{ED}(t) \equiv \tau_{D}(t) - \tau_{E}\left[t-\delta t(t)\right]$, where $t$ and $t-\delta t(t)$ are the coordinate times of the emission and the detection respectively, is a coordinate–independent quantity, in fact it enters the detector response of all the gravitational wave laser interferometers. Namely, if we compare a laser signal $A_{E}=A_{L}e^{i\omega_{L}\tau_{E}}$ with angular frequency $\omega_{L}$ sent from the emitter to the detector with the identical template laser signal $A_{D}=A_{L}e^{i\omega_{L}\tau_{D}}$ at the detector the change in the phase will be proportional to $\Delta\tau_{ED}$: $$\begin{aligned}
\label{eq:laser_1}
A_{D}(\tau_{D}(t)) - A_{E}\left[\tau_{E}(t-\delta t(t))\right] \simeq A_{L}\,i\omega_{L}\left[ \tau_{D}(t) - \tau_{E}[t-\delta t(t)] \right].\end{aligned}$$ In the background given by Eq. (\[eq:conf\]) light travels along the null lines of the Minkowski metric $\eta_{\mu\nu}$ thus to the leading order $$\begin{aligned}
\delta t(t) & = & |{\bf x}_{D}(t) - {\bf x}_{E}(t-L)| =
|\,{\bf x}_{D0} - {\bf x}_{E0} + \frac{{{\boldsymbol \Omega}}}{2 {\rm v}}
\int\limits_{t-L-\frac{{{\boldsymbol \Omega}}\cdot{\bf x}_{E0}}{{\rm v}}}^{t-\frac{{{\boldsymbol \Omega}}\cdot{\bf x}_{D0}}{{\rm v}}}h_{s}(v)\,dv\,| \nonumber \\
& \simeq &
L + \frac{{{\boldsymbol \Omega}}\cdot{\bf n}_{ED}}{2 {\rm v}}
\int\limits_{t-L-\frac{{{\boldsymbol \Omega}}\cdot{\bf x}_{E0}}{{\rm v}}}^{t-\frac{{{\boldsymbol \Omega}}\cdot{\bf x}_{D0}}{{\rm v}}}h_{s}(v)\,dv \nonumber\end{aligned}$$ and using (\[eq:solt\]) $$\begin{aligned}
\label{eq:DT1}
\Delta \tau_{ED}(t)
\label{eq:tau}
& 