Quantum Field Theory in Successive Rindler Spacetimes

Abstract

We study successive Rindler-like transformations in Minkowski spacetime and the corresponding sequence of vacuum states perceived by observers restricted to respective wedges. Extending the standard Rindler construction to an $n$-fold iteration, we find via Bogoliubov transformations that the vacuum of the $(n-1)^{th}$ Rindler observer appears thermal to the $n^{th}$ one. The characteristic trajectories, confined to nested wedges, exhibit characteristic accelerations and horizon shifts depending on transformation parameters ${g_1, g_2, \ldots, g_{n}}$. For the second-level transformation (\emph{Rindler Rindler} case), the late time acceleration asymptotically approaches $2g_2$ for one branch and diverges for the other. We study Minkowski, Rindler, and Rindler Rindler vacuum states from the perspective of Unruh DeWitt (UDW) detectors along inertial, Rindler, and Rindler--Rindler trajectories. The response of the UDW detector coupled to a real massless scalar field confirms the thermality: the transition rate of Rindler--Rindler observer in Minkowski vacuum matches that of a standard Rindler detector with acceleration $2g_2$, yielding a Planckian spectrum at late times. The conclusions are discussed.