the maximum.

Let us look at the combination $\epsilon |\cos\Theta_0|$. As mentioned earlier, $\epsilon$ and $|\cos\Theta_0|$ always appear as a product in the expression. Let us assume that for $\Theta_0 \neq \pi/2$, the product $\epsilon |\cos\Theta_0| \gg 1$ by assuming $\epsilon \gg 1$ for a finite $|\cos\theta_0|$. In this case, the term with pre-factor $b^4 \epsilon^4$ dominates over rest of the terms in the braces and ${\dot {\cal F}}_{\Theta_0}(\omega)$ is KMS thermal with the inverse of the temperature being equal to $2 \pi/ g - (4/g)\tan^{-1} \left( |\cos\Theta_{0}|g\epsilon \right)$. The $\epsilon$ parameter represents the length scale of the spatial extension of the detector. A large $\epsilon$ would signify a sufficiently extended detector. Whereas, as mentioned earlier, in the point-like limit of the detector $\epsilon \rightarrow 0$, one recovers the usual isotropic Unruh temperature. This suggests that the features mentioned above are especially due to the spatial extension of the detector.

We comment on this spurious result in the discussion section after analysing the spatially extended detector from the Rindler co-moving frame of reference in the next section.

One might suspect that for the direction dependence feature of the spatially extended detector to vanish in the point-like limit $\epsilon \rightarrow 0^+$, could possibly be a feature of the transition rate of the detector wherein one has formally subtracted an infinite constant term by taking the difference of the transition probability of the detector at $\tau$ and $\tau + d\tau$ to arrive at the transition rate expression. The constant infinite term may, perhaps, contain the information about the direction dependence $\Theta_0$ even in the point-like limit $\epsilon \rightarrow 0^+$.

One can check for the above suspicion by explicitly computing the transition probability for the extended detector when the detector is switched ON and switch OFF smoothly *around* the finite proper time $\tau_0$ and $\tau_f$ respectively. The general 