in this section is to investigate the response of an spatially extended detector, working in its co-moving frame and coupled to the Minkowski vacuum state of the scalar field, following De Bievre and Merkli [@DeBievre:2006pys]. We consider the corresponding centre of mass, with co-ordinates $(x_0(\tau))$, of the detector to follow the Rindler trajectory with uniform acceleration $g$. We work in Rindler co-ordinates with the following form of the metric $$ds^2 = \exp{(2 g z)} \left( - dt^2 + dz^2 \right) + d x^2_{\perp}
\label{rindlermetric}$$

We further assume the usual monopole interaction Hamiltonian term proportional to the value of the field on the trajectory, but now the field is replaced by the smeared field $\phi(\tau)$ obtained through $$\phi(\tau) = \int dz d^2 x_{\perp} e^{g z} f \left(z, x_{\perp}, z_0(\tau), x_{\perp 0}(\tau) \right) \phi(x)
\label{smeared}$$ where $dz d^2 x_{\perp} e^{g z}$ is the 3-spatial volume of the $t = $ constant hypersurface and $f \left(z, x_{\perp}, z_0(\tau), x_{\perp 0}(\tau) \right)$ is the profile function which encodes the spatial geometry of the extended detector itself. For the particular detector considered, $z_0(\tau) =0 = x_{\perp 0}(\tau) $, the detector is centred, with its centre of mass at the origin.

The pullback of the Wightman function relevant for calculating the detector response function is $$\begin{aligned}
W(\tau,\tau^\prime) = \langle 0_M | \phi(\tau) \phi(\tau^\prime) | 0_M \rangle
\label{whitmannfunction}\end{aligned}$$ with the transition rate being $${\dot {\cal F}}(E) =  2 \operatorname{Re}\int_0^\infty ds \,e^{-i E s} \, W(\tau,\tau - s)$$

The quantised scalar field in terms of the mode solutions for the metric in Eq.(\[rindlermetric\]) is $$\phi(x) = \int d\omega \int d^2 k_{\perp} \left[ {\hat a}_{\omega, k_{\perp}}  v_{\omega, k_{\perp}}(x) + {\hat a}^{\dagger}_{\omega, k_{\perp}}  v^{\star}_{\omega, k_{\perp}}(x) \right] 
\label{field}$$ where the mode solutions are given in terms of the modified Bessel function as $$v_{\omega, k_{\perp}