Successive Rindler-Like Transformations

• Successive Rindler-like transformations define an iterative hierarchy where each nested wedge, characterized by its own acceleration, confines an observer to a progressively smaller causal region.
• A cascade of Bogoliubov transformations links the vacua of adjacent wedges, causing each observer to perceive a thermal spectrum with temperature T = g/(2π) based on their specific acceleration.
• The framework extends conventional quantum field theory concepts, offering insights into multi-horizon thermalization, quantum correlations, and applications in analogue gravity and evolving black hole systems.

Successive Rindler-like transformations define an iterative hierarchy of Lorentz-boosted—or more generally, wedge-restricted—coordinate systems in flat spacetime, where each “Rindlerization” localizes an observer to a yet more nested causal wedge, and the corresponding vacuum states form a thermally related sequence under field quantization. This framework generalizes the standard Rindler construction for uniform acceleration, yielding a multi-layer hierarchy of observer-dependent vacua and thermal phenomena, mediated by a chain of nested Bogoliubov transformations, and produces nontrivial operational signatures for quantum field theory, particle detector responses, horizon structure, and thermality.

1. Hierarchical Rindler Transformations and Nested Wedges

The standard Rindler coordinates are associated with uniformly accelerated observers, restricting Minkowski spacetime to a right (or left) wedge and inducing a vacuum that is thermally populated when seen from inertial frames. Successive Rindler-like transformations generalize this by applying additional Lorentz-boost-like maps within an existing wedge. The resulting $n$-fold transformed coordinates confine the observer to a sequence of increasingly nested wedges, each parameterized by an associated acceleration $g_1, g_2, ..., g_n$.

For $n=1$, the transformation is the standard

$$x_0 = z_1 \sinh(a z_0), \quad x_1 = z_1 \cosh(a z_0), \quad x_2 = z_2, \quad x_3 = z_3.$$

For higher $n$, the transformations are iterated, successively mapping each $(n-1)$th wedge to a subwedge via a generalized boost in the local Rindler frame, parameterized by a new $g_n$.

This iterative procedure yields a hierarchy:

• Minkowski ($M$) → Rindler ($R_1$) → Rindler–Rindler ($R_2$) → ... → $R_n$, each with its own causal wedge with a shifted horizon and associated vacuum state.

2. Bogoliubov Transformations and Thermalization Cascade

In each step, the link between adjacent vacua is formalized via a global Bogoliubov transformation relating field modes in successive coordinate patches. For the massless scalar field, quantized in each coordinate system via mode functions $u^{(n)}_{k_n}$ and ladder operators $a^{(n)}_{k_n}$, the transformation can be written

$$a^{(n-1)}_{k_{n-1}} = \int \left[\alpha(k_{n-1}, k_n)\, a^{(n)}_{k_n} + \beta(k_{n-1}, k_n)\, a^{(n)\dagger}_{k_n} \right] \, dk_n,$$

with the Bogoliubov coefficients determined by the explicit overlap of modes and possessing the property

$|\beta|^2 = \frac{1}{\exp(2\pi |k_n|/g_n) - 1},$

showing that the $(n-1)$th vacuum appears as a thermal (Planckian) state with temperature $T_n = g_n/(2\pi)$ to the $n$th observer. This process recursively “thermalizes” the vacuum, creating a “thermalization within thermalization” effect as each observer in the hierarchy sees not only the original Minkowski vacuum but also all previous Rindler vacua as thermal states, with the local temperature set by their own proper acceleration parameter.

3. Trajectories, Accelerations, and Horizon Structure

For each successive wedge, the worldlines of fiducial observers at fixed spatial coordinate in the $n$th Rindler frame—the so-called “characteristic” trajectories—are no longer simple hyperbolae. Requiring that coordinate time $t_n$ equals proper time along these worldlines induces a nonlinear equation, for example in the Rindler–Rindler ($n=2$) case: $$\dot{y}^2 = 1 - \exp\left[ -2(g/g') e^{g' y} \cosh(g' \tau) - 2g'y \right],$$ with $y(\tau)$ representing the spatial trajectory in the $R_2$ wedge. There are two branches: one where the late-time acceleration $a(\tau) \to 2g'$ and another where it diverges. The position of the causal horizon is exponentially shifted according to the accumulated transformation parameters, generalizing $x_0 - t_0 = 1/g_1$.

For higher $n$, proper acceleration and horizon location depend recursively on the set $\{g_1, ..., g_n\}$, so nested observers can experience both new causal horizons and new effective accelerations.

4. Operational Probes: Unruh–DeWitt Detector Response

To operationally probe vacuum structure, the response of Unruh–DeWitt detectors traveling along characteristic trajectories in various wedges is computed. The transition probability and transition rates are functions of the Wightman function pulled back to the detector’s worldline. In the Minkowski vacuum:

• An inertial detector yields only transient excitations that vanish with infinite interaction time.
• A Rindler detector in $R_1$ has a Planckian response at $T_1 = g_1/(2\pi)$.
• Strikingly, a detector following a Rindler–Rindler trajectory (with asymptotic acceleration $2g_2$) in the Minkowski vacuum sees, at late times and large $g_2/g_1$, a Planckian spectrum corresponding to $T_2 = 2g_2/(2\pi)$.

These results corroborate the Bogoliubov analysis and show that each successive observer in the nested hierarchy measures a thermal particle spectrum at their own effective temperature.

5. Implications for Quantum Field Theory and Vacuum Structure

The iterative Rindler construction demonstrates that the standard observer-dependence of the notion of “vacuum” in quantum field theory is not merely a dichotomy between inertial and uniformly accelerated observers, but instead extends to a hierarchy of vacua, all unitarily inequivalent, each appearing as a thermal state to its successor. This property persists at every level: to the $n$th observer, the vacua of all preceding frames (including Minkowski and lower-level Rindler wedges) appear thermal at $T_n$.

This multiplicity of inequivalent vacua and nested thermality has broader implications:

• The shifting of causal horizons and the corresponding acceleration scales suggest a multi-scale structure for quantum correlations and entanglement across horizons.
• The generalization to multiple horizons and cascaded thermalization provides a template for studying systems with layered acceleration, e.g., analogue gravity systems, evolving black holes with nontrivial causal patching, and quantum fields in dynamical settings.

6. Outlook and Theoretical Significance

Successive Rindler-like transformations introduce a robust framework to paper hierarchical vacuum structure, multi-horizon thermality, and observer dependence in quantum field theory, extending the familiar Unruh and Hawking effects. The explicit analytic forms for the coordinate transformations, Bogoliubov coefficients, detector responses, and the asymptotic acceleration limits furnish a systematic method to characterize observables and causal properties in each nested wedge.

In principle, this hierarchical thermalization structure offers a testbed for probing the robustness of thermality and information flow in quantum spacetimes, and may point toward new mechanisms for understanding thermal signatures, information distribution, and observer dependence in scenarios involving evolving, multi-layered causal horizons or nontrivial spacetime patching—all key issues in both semi-classical and quantum gravity contexts (Dubey et al., 23 Oct 2025).