\left(
\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}
\right).\end{aligned}$$ In the Appendix \[A2\] we derive the solutions to the linearized vacuum field equations (\[eq:ein:lin\]), (\[eq:BD:lin\]) directly in the gauge $h_{\mu0}=0$.

Detector response in the conformal gauge {#s:2}
========================================

In this section we investigate the motion of a free test mass in the background of the plane, massive scalar gravitational wave in the conformal gauge. The result will enable us to obtain the response of the laser interferometer. In what follows we make use of the derivation given in [@Shibata94], [@Nakao01] for the massless scalar field.

To this end we first consider an arbitrary conformal wave moving in the direction $-z$ with the velocity ${\rm v}$, ${\rm h}_{s}(t,z)=h_{s}(t+z/{\rm v})$ (so e.g. it can be one of a Fourier modes of the Eq. (\[eq:BD:lin:sol2\])), in which case the background geometry has the form: $$\begin{aligned}
\label{eq:conf}
ds^2 & = & \left[1 + h_{s}\left(t + \frac{z}{{\rm v}}\right)\right]\eta_{\mu\nu}dx^{\mu}dx^{\nu}.\end{aligned}$$ Free motion of a test body can be obtained from the Lagrangian $$L({\bf x},\dot{\bf x})= \frac12\left[1 + h_{s}\left(t + \frac{z}{{\rm v}}\right)\right]\left( 