Rindler Wedges

Definition

In $(d+1)$-dimensional Minkowski spacetime with coordinates $(x^0,x^1,\dots ,x^d)$, the right Rindler wedge is

$R=\{x\in {\mathbb{R}}^{d+1}:x^1>|x^0|\},$

and the left Rindler wedge is

$L=\{x\in {\mathbb{R}}^{d+1}:x^1<-|x^0|\}.$

$R$ and $L$ are causally complementary: no causal curve connects $R$ to $L$. They are each other’s causal complement, $L=R^{\prime }$ and $R=L^{\prime }$, where the prime denotes the spacelike complement; each wedge is causally complete, $R^{\prime \prime }=R$.

Each wedge is preserved under Lorentz boosts in the $x^1$-direction: $x^0\mapsto x^0\cosh \theta +x^1\sinh \theta$, $x^1\mapsto x^0\sinh \theta +x^1\cosh \theta$. An observer uniformly accelerating in the $x^1$-direction with proper acceleration $a$ follows the trajectory

$x^{\mu }(\tau )=\frac{1}{a}(\sinh (a\tau ),\cosh (a\tau ),0,\dots ,0),$

which remains inside $R$ for all proper time $\tau$.

The Unruh effect: the Minkowski vacuum $|0⟩$, restricted to the right Rindler wedge algebra $\mathcal{M}(R)$, is a KMS (thermal) state at the Unruh temperature

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

where $a$ is the proper acceleration (in natural units $\hslash =c=k_B=1$). Algebraically, this follows from the Bisognano-Wichmann theorem: the modular flow of $\mathcal{M}(R)$ with respect to the Minkowski vacuum is precisely the boost automorphism.

Role in Research

Rindler wedges are the standard setting for the Bell-CHSH inequality program. The bipartite split is Alice = right wedge $R$, Bob = left wedge $L$. Since $R$ and $L$ are spacelike separated, the locality assumption of Bell’s theorem is automatically satisfied by microcausality.

The systematic framework (Phys. Rev. D 108, 085026; 2023) places Alice’s Weyl operators $W(f)$ with $\text{supp}(f)\subset R$ and obtains Bob’s operators via modular conjugation $J$: $JW(f)J\in \mathcal{M}(L)$. The Bisognano-Wichmann identification makes $J$ a geometric operation (CPT + boost), giving a physically transparent construction.

The Unruh-De Witt detector paper (JHEP 2024, 6 citations) bridges the abstract algebraic Rindler picture with a detector model, computing Bell correlations from the perspective of the accelerated observer.

The relative-entropy-qft line computes the Araki-Uhlmann relative entropy between the vacuum and excited states restricted to $R$, accessing information about the entanglement structure of the vacuum across the $R/L$ split.

Relations

• causal-diamonds — compact alternative to Rindler wedges; diamonds are finite-size and more physically realistic, but the same algebraic machinery applies
• tomita-takesaki-modular-theory — the Bisognano-Wichmann theorem identifies the modular flow of $\mathcal{M}(R)$ with boost automorphisms; modular conjugation $J$ maps $\mathcal{M}(R)\to \mathcal{M}(L)$
• bell-chsh-inequality — Rindler wedges provide the bipartite causal split for QFT Bell tests
• weyl-operators — Weyl operators localized in $R$ generate $\mathcal{M}(R)$; support of test function determines wedge localization
• type-iii-von-neumann-algebras — $\mathcal{M}(R)$ is a type III₁ von Neumann algebra; the vacuum is cyclic and separating for this algebra
• araki-uhlmann-relative-entropy — relative entropy between states restricted to $R$ quantifies information lost when tracing over $L$

Papers

See bell-inequalities-qft and relative-entropy-qft for full paper lists.

Notes

• The Rindler wedge is an example of a globally hyperbolic spacetime in its own right; one can quantize fields directly in Rindler coordinates (Rindler quantization), obtaining Rindler particles related to Minkowski particles by a Bogoliubov transformation.
• The thermal nature of the Minkowski vacuum restricted to $R$ is exact (not perturbative) and model-independent, following purely from the algebraic structure and the wedge geometry.
• Common confusion: the Unruh effect does not mean that Minkowski vacuum becomes a thermal state globally — it is thermal only when restricted to $R$ (or equivalently, only for observables measured by the accelerated observer).
• Rindler wedges are non-compact; this is both a mathematical advantage (clean algebraic structure) and a physical limitation (no real experiment is confined to a half-space). Causal diamonds address the compactness issue.