-\dot{t}^2 + \dot{x}^2 + \dot{y}^2 + \dot{z}^2 \right),$$ where the dot stands for the proper time derivative, $\dot{x}^{\mu}(\tau)\equiv \frac{d x^{\mu}(\tau)}{d \tau}$. The equations of motion read $$\begin{aligned}
\label{eq:eomt}
\frac{d}{d\tau}\left[ (1 + h_{s})\dot{t} \right] & = & -\frac12 {\rm h}_{s,t}\left( -\dot{t}^2 + \dot{x}^2 + \dot{y}^2 + \dot{z}^2 \right)\\
\label{eq:eomi}
\frac{d}{d\tau}\left[ (1 + h_{s})\dot{x}^{i} \right] & = & \frac12 {\rm h}_{s,i}\left( -\dot{t}^2 + \dot{x}^2 + \dot{y}^2 + \dot{z}^2 \right),\qquad i=1,2,3.\end{aligned}$$ We assume that in the absence of gravitational wave the test body was at rest with respect to the coordinates and hence we seek for the leading order solution having the form $\dot{t}=1 + A_{t}h$, $\dot{x}^i= A_{i}h$. From Eqs. (\[eq:eomi\]) it follows that, as a first step, we can specify $A_x=A_y=0$. Using the (leading order) identities ${\rm h}_{s,t}=\dot{h}_{s}$ and ${\rm h}_{s,z}=\dot{h}_{s}/{\rm v}$ from Eqs. (\[eq:eomt\]), (\[eq:eomi\]) it then follows that $A_{t} = -\frac12$, $A_{z}=-\frac{1}{2 {\rm v}}$ and we get $$\begin{aligned}
\label{eq:sol1x}
x(t) & = & x_{0} \\
\label{eq:sol1y}
y(t) & = & y_{0} \\
\label{eq:sol1z}
z(t) & = & z_{0} - \frac{1}{2 {\rm v}}\int\limits_{\infty}^{t}h_{s}[t'+ z(t')/{\rm v}]\,dt' = z_{0} - \frac{1}{2 {\rm v}}\int\limits_{\infty}^{t+\frac{z_{0}}{{\rm v}}}h_{s}(v)\,dv \\
\label{eq:sol1t}
\tau(t) & = & t + \frac12\int\limits_{\infty}^{t+\frac{z_{0}}{{\rm v}}}h_{s}(v)\,dv,\end{aligned}$$ where $v(t')=t'+z(t')/{\rm v}$ and $x_0$, $y_0$, $z_0$ are the 