Right Rindler Wedge: Spacetime & Quantum Insights

Updated 12 December 2025

Right Rindler Wedge is a spacetime region in Minkowski space defined by x > |t|, central to studying accelerating observers, causal horizons, and their thermodynamic properties.
The wedge utilizes Rindler coordinates and the boost Killing vector to construct a framework for analyzing the Unruh effect, modular theory, and thermal field quantization.
Its study provides actionable insights into quantum field dynamics, entanglement entropy, and applications in black hole thermodynamics and holographic duality.

The Right Rindler Wedge (RRW) is a fundamental spacetime region in Minkowski space, characterized by key geometric, analytic, and quantum properties. It plays a central role in the study of quantum field theory for accelerated observers, the Unruh effect, thermofield double states, horizon thermodynamics, modular theory, and the algebraic structure of local observables. The structure of the RRW serves as a prototype for causal horizons in more general settings, including black holes and AdS/CFT duality frameworks.

1. Geometric Characterization and Rindler Coordinates

The RRW in $d$ -dimensional Minkowski spacetime with metric $ds^2 = -dt^2 + dx^2 + d\mathbf{x}_\perp^2$ (where $\mathbf{x}_\perp$ denotes the $(d-2)$ transverse directions) is defined by

$\operatorname{RRW} = \{ (t, x, \mathbf{x}_\perp) \mid x > |t| \} \,.$

This domain is bounded by the null surfaces $t = \pm x$ , which act as the future and past Rindler horizons. A uniform acceleration observer at fixed Rindler spatial coordinate is confined within the RRW, never crossing the horizons.

Rindler coordinates $(\tau, \xi, \mathbf{x}_\perp)$ (with proper acceleration parameter $a>0$ ) are introduced via

$\begin{aligned} t &= \rho \sinh (a \tau),\ x &= \rho \cosh (a \tau),\ \mathbf{x}_\perp &= \text{unchanged},\ \end{aligned} \quad \rho = e^{a\xi}/a \,,\;\; \tau \in \mathbb{R},\;\; \xi \in \mathbb{R}\,.$

The line element in these coordinates reads

$ds^2 = e^{2a\xi}(-d\tau^2 + d\xi^2) + d\mathbf{x}_\perp^2 \,,$

or equivalently,

$ds^2 = - (a\rho)^2 d\tau^2 + d\rho^2 + d\mathbf{x}_\perp^2 \,.$

The boost Killing vector $\chi = x\,\partial_t + t\,\partial_x$ becomes $\partial_\tau$ in Rindler coordinates, generating isometries of the metric within the wedge (Gutti et al., 2022, Higuchi et al., 2020).

2. Quantum Field Theory in the RRW: Scalar and Spin-1 Fields

Scalar Fields

A free, massless Klein-Gordon field in RRW can be expanded as

$\phi(\tau, \xi, \mathbf{x}_\perp) = \int_0^\infty d\omega \int d^{d-2}k_\perp \left[ b^R_{\omega, k_\perp} u^R_{\omega, k_\perp}(\xi, \mathbf{x}_\perp) e^{-i\omega\tau} + \text{h.c.} \right],$

where the mode functions $u^R_{\omega, k_\perp}$ are chosen orthonormal under the Klein-Gordon inner product.

For $d=2$ , a simple choice is $u_\omega(\xi) = (4\pi\omega)^{-1/2} e^{i\omega\xi}$ . The operators $b^R_{\omega, k_\perp}$ annihilate the Rindler vacuum $|0\rangle_R$ (Gutti et al., 2022, Barman et al., 2023). For spin-1 Proca and U(1) gauge fields (both massless and massive), a canonical quantization scheme in the RRW has been established, yielding explicit mode solutions, Klein-Gordon normalization, canonical commutators, and polarization vector decompositions. The full gauge- and Lorentz-covariant structure is maintained, and the vacuum structure similarly admits a thermofield double representation (Castineiras et al., 2011, Takeuchi, 2023, Takeuchi, 25 Mar 2024).

Gravitational Waves

For linearized gravity, the RRW admits analytic mode solutions for both odd and even parity master variables, which reduce to effectively decoupled scalar fields. Canonical quantization in the RRW parallels the scalar and vector field cases, with the global Minkowski vacuum once again corresponding to a thermofield double entangled state between left and right Rindler graviton modes (Sugiyama et al., 2020).

3. Reduced Density Matrix, Unruh Effect, and Thermofield Double Structure

Restricting the Minkowski vacuum $|0\rangle_M$ to the algebra of observables in the RRW, one obtains a (Kubo-Martin-Schwinger) KMS thermal state with respect to the Rindler Hamiltonian: $\rho_R = \operatorname{Tr}_L |0\rangle_M \langle 0 | \propto \exp( - 2\pi K ),$ where $K$ is the dimensionless boost generator. The Unruh temperature is

$T_U = \frac{a}{2\pi}\,.$

The explicit form of the Minkowski vacuum is a two-mode squeezed (thermofield double) state: $|0\rangle_M = \prod_{\omega, k_\perp} \sum_{n=0}^\infty e^{-\pi \omega n/a} |n\rangle_L \otimes |n\rangle_R \,.$ Tracing over left-wedge degrees of freedom yields for each frequency a thermal (Gibbs) distribution of excitations in the RRW (Gutti et al., 2022, Higuchi et al., 2020, Gioia et al., 2020, Barman et al., 2023, Higuchi et al., 2017).

4. Modular Structure, Operator Algebras, and Symmetry

The RRW is the prototypical setting for the Bisognano–Wichmann theorem, which identifies the modular structure of the von Neumann algebra of local observables $\mathcal{A}(\mathrm{RRW})$ :

The modular group $\Delta^{it}$ acts as the one-parameter group of boosts preserving the wedge.
The modular conjugation $J$ implements CPT times a $\pi$ -rotation in the transverse plane.
$\mathcal{A}(\mathrm{RRW})$ is a Type III $_1$ factor, capturing the essential feature that the entanglement entropy across the horizon diverges as one approaches the boundary.

This structure is crucial for algebraic quantum field theory (AQFT), covariance under the Poincaré group, and causality (Asorey et al., 2017, Ju et al., 2023, Neeb et al., 2021).

For neutral fields, the full modular localization program is valid in the RRW. However, in QED or other theories with infrared dressing, the divergence of boost generators in the photon sector prevents strict modular localization of photon observables in the wedge, i.e., the Tomita–Takesaki construction breaks down for the infrared-dressed or charged sectors, resulting in spontaneous breaking of Lorentz invariance for the localized observable algebra (Asorey et al., 2017).

5. Perturbative and Algebraic Equivalence: In-in Formalism and Thermal Field Theory

All orders of perturbation theory constructed via in-in (Schwinger–Keldysh) formalism in the Minkowski vacuum for correlation functions of points within the RRW coincide precisely with a Rindler in-in formalism in a thermal (KMS) state at the Unruh temperature. The restriction of internal vertices to the RRW does not change the result for external points in the wedge (light-cone proof), reflecting that local quantum dynamics within RRW is perturbatively indistinguishable from that in a real thermal bath at $T_U$ (Higuchi et al., 2020, Barman et al., 2023).

Equivalently, the RRW canonical quantization defines a thermal quantum field theory on a static spacetime with respect to the boost Killing time. The associated path-integral or Euclidean approach yields periodicity in imaginary Rindler time, enforcing the KMS condition (Higuchi et al., 2020).

6. Entanglement, Horizon Thermodynamics, and Information-Theoretic Structure

The entanglement structure of the Minkowski vacuum across the Rindler horizon underpins the Unruh effect and redistributes information:

The global ground state is maximally entangled between right and left Rindler wedge degrees of freedom.
Restricting observables to the RRW traces out the left wedge, generating a mixed thermal state with entropy matching the area law ( $S_A = \mathrm{Area}(\partial A)/(4G_N)$ in gravitational theories).
Information-theoretic measures such as mutual information, locally accessible/inaccessible information (classical correlations and quantum discord), and entanglement of formation provide a diagnostic of how correlations are redistributed and “traded” across the causal horizon induced by acceleration (Gioia et al., 2020, Ju et al., 2023).

Mutual information between an inertial observer and a Rindler (accelerated) observer decays with increasing acceleration, while the entanglement between left and right Rindler modes grows correspondingly, with the overall entropy budget being conserved across the full system.

7. Generalizations, Nested Wedges, and Holography

The RRW structure generalizes naturally in several directions:

Nested Rindler wedges: An infinite family of nested Rindler coordinate systems within the RRW, each shifted along the inertial $x$ -axis, gives rise to inequivalent Rindler vacua. For any nonzero shift, the “daughter” RRW vacuum sees its “parent” Rindler vacuum—and all earlier vacua including the inertial vacuum—as thermal at its own Unruh temperature, a phenomenon discontinuous in the shift parameter and robust to Planck-scale uncertainties. This is a universal feature in any spacetime with a bifurcate Killing horizon (Lochan et al., 2021).
Anti-de Sitter and Holographic Duals: In asymptotically AdS spacetime, the AdS analog of the RRW—the hyperbolic Rindler wedge—is dual to a reduced density matrix (thermal state) on a single hyperbolic-space CFT. The full global AdS spacetime is reconstructed by an entangled pair of such CFTs, with entanglement across the horizon corresponding to a smooth geometric connection in the bulk; cutting the entanglement (passing to a pure state in one CFT) destroys the smooth bulk horizon, revealing the geometric significance of the RRW construction in holography (Czech et al., 2012).
Generalized Rindler wedges: The notion of “Rindler–convexity” ensures the geometric and causal compatibility of RRW–like regions in general spacetimes. In holography, entanglement wedge reconstruction and subregion/subalgebra duality find prototypical realization in the RRW (Ju et al., 2023).