Quantum Field Theory in Successive Rindler Spacetimes

TL;DR

This paper explores successive Rindler transformations in Minkowski spacetime, revealing how vacuum states appear thermal to nested observers and confirming thermality through detector responses, extending the understanding of observer-dependent vacua.

Contribution

It extends the Rindler framework to multiple iterations, analyzing the resulting vacuum states and detector responses, introducing the concept of Rindler Rindler transformations and their thermality.

Findings

Vacuum states become thermal to nested Rindler observers.

Detector responses confirm Planckian spectra consistent with thermal states.

Late-time acceleration in Rindler Rindler approaches specific limits.

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.