Rindler-Rindler Vacuum State Analysis

• Rindler-Rindler vacuum state is a generalized vacuum obtained by iterated Rindler boosts that create nested spacetime wedges with distinct positive-frequency modes.
• Quantization via Bogoliubov transformations yields two-mode squeezed states, leading observers to detect thermal spectra with temperatures proportional to their acceleration.
• The nested construction extends to curved spacetimes with bifurcate Killing horizons, offering a framework for understanding horizon-induced thermality and its quantum gravity implications.

The Rindler-Rindler vacuum state generalizes the Fulling–Rindler vacuum construction by iterating Rindler-like boosts and quantizations within Minkowski spacetime. This hierarchy of vacua manifests in nested spacetime wedges, each associated with its distinct family of positive-frequency modes and effective temperatures, providing a powerful framework for exploring the interplay between acceleration, vacuum structure, and horizon-induced thermality—including its implications for detector theory, quantum fluctuations, and universality across bifurcate Killing horizons.

1. Hierarchy of Rindler-Type Vacua and Coordinate Construction

The Rindler-Rindler vacuum emerges naturally when applying a second, nested Rindler transformation atop the already-accelerated frame within the right Rindler wedge. The initial Rindler coordinates $(t_1, x_1)$ in Minkowski spacetime are defined via

$$T = \frac{1}{g} e^{g x_1} \sinh(g t_1), \;\;\; X = \frac{1}{g} e^{g x_1} \cosh(g t_1)$$

where $g$ is the first acceleration parameter and the right wedge $X > |T|$ is covered. Within this wedge, a second Rindler-type system $(t_2,x_2)$ is constructed by

$$t_1 = \frac{1}{g'} e^{g' x_2} \sinh(g' t_2), \;\;\; x_1 = \frac{1}{g'} e^{g' x_2} \cosh(g' t_2),$$

where $g'$ defines the "second-level" acceleration. The result is a nested, conformally flat region—dubbed the Rindler-Rindler wedge—where a further quantization is performed, leading to a new family of mode functions and associated vacuum (Kolekar et al., 2013, Dubey et al., 23 Oct 2025, Lochan et al., 2021).

This process can be iterated: one defines a chain of Rindler-type frames $R_0, R_1, R_2, \dots, R_n$, each with coordinates $\tau_n, \xi_n$, accelerations $a_n$, and spatial shifts $\ell_n$. The constructions generalize to any spacetime with a bifurcate Killing horizon, such as Schwarzschild or de Sitter (Lochan et al., 2021).

2. Quantization, Bogoliubov Transformations, and Squeezed States

Canonical quantization in the Rindler-Rindler wedge proceeds analogously to the basic Rindler construction. One defines creation and annihilation operators $a^{(2)}_m, a^{(2)\dagger}_m$ for the second-level modes and the Rindler-Rindler vacuum $|0\rangle_{RR}$ is annihilated by all $a^{(2)}_m$. The relationship between Fock spaces at each level is governed by Bogoliubov transformations: $$a^{(2)}_m = \int_{0}^{\infty} dp \; [ \alpha_{(21)}(p,m) a^{(1)}_p + \beta_{(21)}(p,m) a^{(1)\dagger}_p ]$$ with $\alpha_{(21)}, \beta_{(21)}$ coefficients mirroring the inertial–Rindler case but with acceleration parameter $g'$.

The Rindler-Rindler vacuum $|0\rangle_{RR}$ is thus realized as a two-mode squeezed state over the first-level Rindler Fock vacuum $|0\rangle_R$: $$|0\rangle_{RR} = \mathcal{N} \exp\left( \sum_{p>0} e^{-\pi p/g'} a^{(1)\dagger}_p a^{(1)\dagger}_{-p} \right) |0\rangle_R,$$ where $\mathcal{N}$ is a product of normalization factors over modes (Kolekar et al., 2013, Dubey et al., 23 Oct 2025). The process or even higher iterations produces analogous squeezed-state structures.

3. Thermality, Detector Response, and Effective Temperatures

A key structural result is that each "Rindler-Rindler" observer perceives the vacuum of the previous level as a thermal state at temperature $T' = g'/(2\pi)$. Explicit computation yields the mean particle spectrum for a level-$n+1$ observer in the vacuum of level-$n$: $$\langle N_\omega\rangle_{R_{n+1}} = \frac{1}{e^{2\pi\omega/a_{n+1}} - 1},$$ i.e., the Bose-Einstein distribution at the corresponding Unruh temperature (Lochan et al., 2021, Dubey et al., 23 Oct 2025). For the Rindler-Rindler construction ($n=2$), late-time detector response along the "physical" trajectory with asymptotic acceleration $a_\infty = 2g_2$ exhibits a Planckian transition rate at temperature $T_\infty = a_\infty/(2\pi) = g_2/\pi$ (Dubey et al., 23 Oct 2025).

Tracing over the (left) mirror-wedge degrees of freedom produces a thermal density matrix for right-wedge modes, exactly mirroring the standard Unruh effect at each level of iteration.

4. Universal and Discontinuous Turn-On of Thermality

A fundamental feature of this nested construction is the discontinuous nature of the thermalization effect: for any nonzero spatial shift $\ell_{n+1}>0$ in the nested wedge, however small, the thermal spectrum at temperature $a_{n+1}/(2\pi)$ "turns on" in full strength, independent of the size of the shift or previous acceleration parameters. When $\ell_{n+1}=0$, the Bogoliubov coefficients $\beta^{(n,n+1)}$ vanish and the vacua coincide; for any $\ell_{n+1}>0$, one immediately recovers the full Planckian spectrum (Lochan et al., 2021). This discontinuity is interpreted as an indicator of the Planckian origin of the thermality observed by Rindler-Rindler observers and persists in any locally Rindler or bifurcate Killing horizon context.

5. Generalization to Curved Spacetimes and Universal Structure

The nested Rindler-Rindler vacuum structure is not unique to flat spacetime. Any spacetime with a bifurcate Killing horizon—such as Schwarzschild black holes or de Sitter space—admits a hierarchy of Rindler-type wedges and associated vacua, each nested in the next and exhibiting the same Bogoliubov-induced thermality (Rajeev et al., 2019, Lochan et al., 2021). The analogy extends to Hawking radiation (Hartle–Hawking vacua in black holes) and the behavior of global KMS states with respect to horizon-localized time generators.

Analytic continuation and periodic-sum constructions (thermal sum/inversion) generalize to these cases and reveal a universal "thermal $\leftrightarrow$ periodic-sum $\leftrightarrow$ inversion" structure in the behavior of vacuum correlators and propagators (Rajeev et al., 2019).

6. Contrasts with Classical Field Theories and Physical Interpretation

In classical stochastic electrodynamics with zero-point radiation, no new Rindler or nested Rindler vacua arise: the field’s correlation structure is unique, determined solely by invariant geodesic separation, and is insensitive to noninertial coordinate choices or nesting (Boyer, 2012). Only quantum fields support this hierarchy of distinct, inequivalent vacua via canonical quantization in non-geodesic time, leading to the observed Bogoliubov-induced thermality and squeezed-state structure.

Quantum detectors modeled as point-like systems (e.g., Unruh–DeWitt detectors) respond thermally to the Rindler or Rindler–Rindler vacua, with the effective temperature set by the observer’s acceleration in the corresponding wedge. The phenomenon is deeply linked to horizon-induced correlations, entanglement, and the universality of horizon thermality (Dubey et al., 23 Oct 2025, Kolekar et al., 2013, Lochan et al., 2021).

7. Physical Significance and Implications

The existence of an infinite hierarchy of inequivalent Rindler-type vacua, each observer-dependent, and each displaying full thermality for arbitrarily small nested wedge shifts, demonstrates a universal relic of Planck-scale structure. This suggests deep connections to quantum gravity, the quantum equivalence principle, and horizon-shift phenomena in black hole evaporation (Lochan et al., 2021). The Rindler-Rindler vacuum and its generalizations thus provide a precise, operationally well-defined laboratory for analyzing observer dependence, particle definition, and vacuum structure in both flat and curved spacetimes.