Quantum Field Theory in Successive Rindler Spacetimes

TL;DR

The paper develops an $n$-fold hierarchy of Rindler-like transformations in Minkowski spacetime, showing that the vacuum seen by the $(n-1)^{\text{th}}$ observer appears thermal to the $n^{\text{th}}$ observer via Bogoliubov transformations. It constructs the characteristic trajectories confined to nested wedges, analyzes their horizon shifts and late-time accelerations, and validates the thermality of these vacua using Unruh–DeWitt detectors across Minkowski, Rindler, and Rindler–Rindler states. In the two-level RR case, the detector response confirms an effective acceleration of $2g_2$ at late times, with a Planckian spectrum, and the results extend to higher $n$ with comparable behavior in appropriate branches. The work connects wedge-restriction thermality to multi-horizon structures, offering a precise flat-spacetime framework for multi-scale thermality with potential relevance to analogue gravity and time-dependent horizon scenarios.

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.

Figures (9)

• Figure 1: The left panel of the above plot depicts the trajectory $t_2 = \tau$ obtained from solving Eq.\ref{['eq:10']} with $g=g'=0.01$, while the right panel shows the proper acceleration in the fourth quadrant of the left panel. Dashed lines indicate the approximate analytical solutions discussed in Sections \ref{['subsec:negydot']} and \ref{['subsec:posydot']} with $C=0$, while the solid curves correspond to the exact numerical solutions. The red colour represents the positive root of $\dot{y}$, while the blue colour represents the negative root.
• Figure 2: The plots above illustrate the Rindler Rindler trajectories in the Minkowski plane, as introduced in Eq. \ref{['eq:10']} and further discussed in Section \ref{['sec:numtraj']}. The red curves correspond to the positive root of $\dot{y}$, while the blue curves represent the negative root. The label RRR denotes trajectories in the right Rindler Rindler wedge, and RRL denotes those in the left Rindler Rindler wedge. The parameters used are $g = g' = 0.01$.
• Figure 3: The above plots illustrate the transition probability of a UDW detector interacting with a real massless scalar field in the Minkowski vacuum in 1+1 spacetime dimensions, as discussed in Section \ref{['tranprobMink']}.
• Figure 4: The above plots illustrate the transition probability of a UDW detector interacting with a real massless scalar field in the Rindler vacuum in 1+1 spacetime dimensions, as discussed in Section \ref{['tranprobrind']}.
• Figure 5: The above plots illustrate the transition probability of a UDW detector interacting with a real massless scalar field in the Rindler Rindler vacuum in 1+1 spacetime dimensions, as discussed in Section \ref{['tranprobRrind']}.