WKB-like approach to the Unruh temperature for arbitrary acceleration

In this work we study the Unruh temperature as arising from tunneling through a barrier for an observer in flat Minkowski spacetime with arbitrary acceleration $a(t)$. For the defining case of constant acceleration $a(t) = a_0$, the Unruh temperature (W. Unruh, Phys. Rev. D 14, 870 (1976)) is given by $k_b\,T_U =\tfrac{\hbar\,a_0}{2\,\pi\,c}$. Extending the work of de Gill et al. (A. de Gill, D. Singleton, V. Akhmedova and T. Pilling, Am. J. Phys. 78, 685 (2010)) we generalize the gravitational WKB approach to derive the Unruh temperature for arbitrary acceleration. We show that the often employed Schwarzschild-like form of the flat metric is not appropriate for the WKB calculation with an arbitrary $a(t)$, and instead derive a generalized Unruh temperature for the generalized Rindler metric where $a_0\to a(t)$. We derive a generalization of the Rindler coordinates appropriate for arbitrary $a(t)$, and stress the importance of the role of the integrated acceleration $\chi(t) = \int^t dt'\,a(t')$, which can also act as a temporal coordinate. We explore several non-trivial examples of $a(t)$ and their generalized Unruh temperatures. We additionally develop an approximation to the Unruh temperature for small deviations away from constant acceleration by the standard approach of considering the negative frequency content of a purely positive frequency plane wave of an inertial observer, as measured by the co-moving arbitrarily accelerated observer. Lastly, we develop and explicit coordinate transformation between the arbitrarily accelerated observer and conformal coordinates, where the plane wave structure of the solutions of the wave equation is readily transparent, and analogous to the form for the inertial observer.