Nested Rindler Vacuum States

• The nested Rindler vacuum states extend conventional Rindler transformation by iteratively defining observer-dependent vacua that each thermalize the prior state.
• Bogoliubov transformations between successive mode expansions reveal a Planckian distribution with temperatures determined by the corresponding acceleration parameters.
• This framework has practical implications for quantum information and gravitational studies, highlighting altered entanglement structures and horizon phenomena.

The Rindler Rindler vacuum state is a hierarchy of vacuum constructions in quantum field theory that extends the concept of observer-dependence in the definition of vacuum. Standard Rindler coordinates, introduced to analyze quantum fields from the perspective of a uniformly accelerated observer, yield the celebrated Unruh effect: the inertial (Minkowski) vacuum appears as a thermal bath to the accelerated observer. The Rindler Rindler framework generalizes this by performing a further Rindler-like transformation inside the Rindler wedge, yielding a nested or iterative structure of vacua. Each iteration defines a new class of observers whose associated vacuum state is inequivalent to the previous one and, crucially, the vacuum of each previous (n–1)th level appears as a thermal state to the nth-level observer. This hierarchy has deep implications for the structure of field theory, the concept of particle, and the interplay between quantum information and horizon thermality.

1. Coordinate Hierarchy and Rindler–Rindler Transformations

The nested Rindler construction begins by applying a standard Rindler transformation to Minkowski spacetime:

• Minkowski coordinates $(T, X)$ are mapped to Rindler coordinates $(t_1, x_1)$ via:

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

for some proper acceleration parameter $g$.

The Rindler–Rindler (or “second-order Rindler”) frame is obtained by performing a similar transformation inside the Rindler wedge:

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

with another acceleration-like parameter $g'$. After n such transformations (general n-fold iteration), the resulting observers (“nth-level Rindler observers”) inhabit increasingly nested wedges with associated vacuum states.

The line element in Rindler–Rindler coordinates is:

$ds^2 = e^{2g x_1} e^{2g' x_2} (-dt_2^2 + dx_2^2)$

Each level of the hierarchy is associated with its own vacuum state defined via positive-frequency modes with respect to its time coordinate.

2. Bogoliubov Structure and Thermalization of Nested Vacua

The central mechanism by which the vacuum from one level appears thermal to the next arises from Bogoliubov transformations between the mode expansions associated with successive frames. Explicitly, the scalar field admits expansions both in $(n-1)$th and $n$th-level coordinates:

$$\phi(t, x) = \int \frac{dk_{n-1}}{\sqrt{2|k_{n-1}|}} [a_{k_{n-1}} e^{i(k_{n-1}x_{n-1} - |k_{n-1}| t_{n-1})} + \text{h.c.}]$$

$$= \int \frac{dk_n}{\sqrt{2|k_n|}} [b_{k_n} e^{i(k_n x_n - |k_n| t_n)} + \text{h.c.}]$$

The two operator sets are related via Bogoliubov coefficients. The key result is that the vacuum $|0\rangle_{n-1}$ for the $(n-1)$th observer has nonzero “beta” Bogoliubov coefficients when expressed in the $n$th observer's mode basis, yielding a non-zero expectation value for the $n$th observer's particle number operator at $t_{n-1}=0$:

$$\langle N_{n, n-1}(|k_n|) \rangle = \frac{\delta(0)}{e^{2\pi|k_n|/g_n} - 1}$$

This is a Planckian (thermal) distribution with temperature $T_n = g_n / (2\pi)$. Each new wedge thus “thermalizes” the previous vacuum with a temperature controlled by its own acceleration parameter.

3. Physical Trajectories, Proper Acceleration, and Horizons

After the second Rindler transformation, the time coordinate $t_2$ does not generically coincide with any observer’s proper time. The physical interpretation requires constructing specific worldlines for which $dt_2$ is the proper time differential. Solving for such a trajectory $y(\tau)$ yields a nontrivial nonlinear equation, due to conformal factors. In the Rindler–Rindler case, one branch of physically relevant trajectories exhibits proper acceleration asymptotically approaching $a(\tau) \to 2g'$ as $\tau \to \infty$, while the causal horizon is shifted to $x_0 - t_0 \to 1/g$ (see (Dubey et al., 23 Oct 2025)).

These features mean that a Rindler–Rindler observer ultimately experiences an effective acceleration that is double the parameter $g'$, and a horizon distinct from that of the parent Rindler observer. This encapsulates a shift in both the observer’s accessible spacetime region and in the structure of the accessible quantum state.

4. Detector Response and Planckian Spectrum

Operationally, the thermality of the nested vacuum is confirmed by analyzing the response of Unruh–DeWitt (UDW) detectors along nested Rindler trajectories. The detector's excitation probability and transition rate, computed via standard first-order perturbation theory, reproduce the same thermal response predicted by the Bogoliubov analysis:

$R \propto \frac{1}{e^{\omega / T_n} - 1}$

For the Rindler–Rindler observer in the Minkowski vacuum, the transition rate matches that of a Rindler observer (first-level) with acceleration $2g'$, confirming the Planckian emission with $T = 2g'/2\pi$ at late times (Dubey et al., 23 Oct 2025, Kolekar et al., 2013).

This reiterative thermality is also confirmed by alternate methods such as tracing mode function overlaps, saddle-point approximations, or direct computation of the two-point function.

5. Observer-Dependence, Nesting Hierarchy, and Discontinuity

The nested Rindler construction demonstrates a profound observer-dependence: each nth-level observer defines its own vacuum and horizon, and interprets both the Minkowski and any previous Rindler or nested-Rindler vacuum as a thermal state at temperature $g_n/2\pi$(see (Lochan et al., 2021, Dubey et al., 23 Oct 2025)). Remarkably, for wedge shifts even as small as a Planck length, the onset of thermality is discontinuous; no intermediate regime exists where the spectrum is non-thermal. For any nonzero spatial shift, the full thermal Planckian spectrum is observed (see the discontinuity discussion in (Lochan et al., 2021)).

This suggests an intrinsic “Planckian origin” to the universality of the observed thermality, potentially encoding quantum gravity effects even in flat spacetime.

6. Implications: Quantum Information, Gravity, and Horizon Physics

The Rindler Rindler formalism supplies a framework for the paper of quantum field theory in successive causal domains, each with its own observer-dependent vacuum structure. The direct application to relativistic quantum information includes the explicit construction of channels whose noise and capacity are set by successive Unruh-like effects (Bradler et al., 2010).

In gravitational contexts, analogous nested wedge constructions can be introduced around black hole or cosmological horizons (bifurcate Killing horizons), with the horizon position undergoing a characteristic shift for each level. The universal property that any small shift triggers full thermality is potentially a powerful diagnostic for quantum gravitational structure (Lochan et al., 2021). The observer-dependence and non-transitivity in the definition of reduced states for nested regions have been shown to yield crucial differences in entanglement structure and entropy calculations, with possible impact on the black hole information problem (Gutti et al., 2022).

7. Summary Table: Hierarchical Rindler Vacua and Thermal Perceptions

Vacuum Level	Observer Coordinates	Parent Vacuum Perceived as	Observed Temperature
Minkowski	$(T, X)$	Pure vacuum	0
Rindler (1st)	$(t_1, x_1)$	Thermal state	$g_1/2\pi$
Rindler–Rindler (2nd)	$(t_2, x_2)$	Thermal state	$g_2/2\pi$ (Minkowski and Rindler vacua)
...	...	Thermal state	$g_n/2\pi$ (for all previous vacua)

The table encodes that every higher-level observer perceives all prior vacua—including the Minkowski and previous Rindler ones—as strictly thermal with their own local temperature.

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The Rindler Rindler vacuum paradigm establishes a robust, mathematically precise structure in which quantum field theory’s observer-dependence, horizon thermality, and information-theoretic properties can be examined. The nested construction reveals the universality and rigidity of the thermality induced by acceleration, points to Planck-scale sensitivity in horizon structure, and offers a controlled setting for exploring operational definitions and quantum gravitational implications of horizon-induced entropy (Kolekar et al., 2013, Lochan et al., 2021, Dubey et al., 23 Oct 2025).