Gravity from Relative Entropy: Jackiw–Teitelboim Dynamics on the Nariai Horizon

Technical Summary: Gravity from Relative Entropy on the Nariai Horizon

Problem Statement
Recent work by Dorau and Much (2026) established that the Araki–Uhlmann relative entropy between a vacuum state and a coherent excitation of a scalar field on a local Rindler horizon equals the boost-energy flux across that horizon. Identifying this relative entropy with the Bekenstein–Hawking area variation ($S_{rel} = \delta A/4$) yields the semiclassical Einstein equations. However, this derivation faces two limitations:

1. Geometric: The local Rindler wedge possesses a bifurcation surface of infinite area. Consequently, the total entropy is formally infinite, and the identification $S_{rel} = \delta A/4$ can only be tested at the level of first-order variations ($\delta A$), lacking a finite reference state.
2. Conceptual: The derivation leaves open the question of what the entropy counts. While the relative entropy measures the distinguishability of states, the normalization $\delta A/4$ imports a statement about gravitational microstates without supplying their identity or a microscopic counting.

Methodology
To resolve these issues, the paper moves the construction from the local Rindler wedge to the Nariai spacetime ($dS_2 \times S^2$), the degenerate limit of the Schwarzschild–de Sitter family where the black hole and cosmological horizons merge.

1. Geometric Setup: The Nariai geometry possesses a global bifurcate Killing horizon with a compact bifurcation surface (a round two-sphere) of finite area $A = 4\pi/\Lambda$. This allows for a global, finite definition of the horizon entropy $S_N = \pi/\Lambda$.
2. Algebraic Framework: The authors utilize algebraic quantum field theory (AQFT) on this background. They define a preferred global vacuum state $\omega_0$ (the Hartle–Hawking-type state) via analytic continuation from the regular Euclidean section ($S^2 \times S^2$). This state is Hadamard and satisfies the Kubo–Martin–Schwinger (KMS) condition at the Bousso–Hawking temperature $T = \sqrt{\Lambda}/2\pi$.
3. Modular Theory: Using Tomita–Takesaki modular theory, the authors identify the modular flow of the horizon algebra with the geometric Killing flow. They compute the relative entropy $S_{rel}(\omega_0 \| \omega_\phi)$ for coherent excitations $\omega_\phi$ by evaluating the symplectic form on the horizon data.
4. Dimensional Reduction: The scalar field is expanded in spherical harmonics. The compactness of the transverse sphere $S^2$ discretizes the mode tower, rendering the entropy integrals finite without subtraction. The analysis focuses on the $s$-wave sector ($\ell=0$), which dominates near-horizon physics due to the exponential decoupling of higher harmonics.
5. Response Model: The paper models the backreaction of the excitation on the geometry via a linearized perturbation of the induced metric on the bifurcation sphere, sourced by the flux.

Key Contributions and Results

• Finite-Area Identification: The authors explicitly compute the relative entropy for coherent excitations on the Nariai horizon. They demonstrate that $S_{rel}$ is manifestly finite and equals the boost-energy flux $\delta Q$ (weighted by modular factors). By imposing the identification $S_{rel} = \delta A/4$, they derive the Einstein coupling $\alpha = 8\pi$ without infinite-area subtractions. The compact geometry provides a global calibration for the Bekenstein–Hawking coefficient.
• Derivation of Jackiw–Teitelboim (JT) Dynamics: The paper shows that the response model (the relation between flux and area variation) is not merely a postulate but a theorem derived from the linearized dynamics of the throat. The $s$-wave sector of the Nariai throat reduces exactly to de Sitter Jackiw–Teitelboim gravity. The dilaton equation of motion in this sector admits the response ansatz as its unique zero-mode-free solution, fixing the coupling $8\pi$ via dimensional reduction.
• Selection Rule for Nariai Instability: The Nariai spacetime is an unstable equilibrium. The paper proposes that the sign of the relative entropy flow distinguishes the two branches of instability (evaporation vs. anti-evaporation).
  • For coherent states, the boost-energy flux is pointwise non-negative, implying $\delta A \geq 0$. This selects the branch where the pierced horizon area increases (anti-evaporation for the black hole horizon, evaporation for the cosmological horizon).
  • Anti-evaporation (area decrease) is shown to require non-coherent states (e.g., squeezed states) where the flux can be locally negative.
• Squeezed States and the "Squeeze-Shape" Inequality: The paper extends the analysis to squeezed (Gaussian) excitations. It derives a "pivot identity" where the relative entropy equals the modular-weighted quantum flux. Crucially, it proves a "squeeze-shape inequality" (specifically, the vanishing of the interference functional $I_2$ for boost-positive-frequency packets). This implies that while squeezed states can generate local negative energy flux (driving anti-evaporation), the total modular-weighted flux remains positive. The negative flux is a "zero-sum" redistribution funded by positive flux elsewhere on the horizon.
• The "Running Ledger": The authors define a running ledger of entropy accumulation along the horizon. They show that for squeezed states, the interference term (source of negativity) oscillates and integrates to zero, ensuring the total relative entropy remains non-negative and thermal ($S_{rel} = \beta \Delta E_{boost}$).

Significance and Claims
The paper claims to close the gap in the "gravity from entropy" program by providing a setting where the entropy is finite, the reference state is unique, and the Einstein coupling is derived rather than assumed.

• Nature of Entropy: The authors argue that the entropy driving gravity is not a von Neumann entropy of a hidden substrate (which is undefined for type III algebras) but a relative entropy: a measure of the information-theoretic distinguishability between the realized state (with an event) and the counterfactual vacuum (without the event). Gravity is the mechanism that records this distinguishability in the geometry ($\delta A = 4 S_{rel}$).
• Global Calibration: Unlike the Rindler wedge, the Nariai horizon allows for a global calibration of the entropy-area relation against a finite total capacity ($A/4$).
• Limitations and Scope: The paper is modest regarding the full dynamical implications. The selection rule for the instability branches is presented as a derived rule within the linear-response, quasi-equilibrium regime. The extension to fully dynamical, teleological horizons outside this window is identified as an open problem. The derivation of the sum rule for squeezed states relies on a conjecture regarding the convergence of the product state to the natural cone in the type III setting (Shale's theorem), though the paper provides strong evidence for its validity.

In summary, the paper establishes that on the Nariai horizon, the semiclassical Einstein equations emerge as a bookkeeping identity for the relative entropy of coherent and squeezed excitations, with the coupling $8\pi$ fixed by the dimensional reduction to Jackiw–Teitelboim gravity.