Nariai Relative Entropy

Companion repository for the paper

Relative Entropy on the Nariai Horizon: A Finite-Area Completion of the Entropic Derivation of the Semiclassical Einstein Equations (B. S. Hartshorn, 2026, draft)

which extends Dorau & Much, From Quantum Relative Entropy to the Semiclassical Einstein Equations, Phys. Rev. Lett. 136, 091602 (2026) [arXiv:2510.24491], from the local Rindler wedge to the Nariai spacetime dS2 x S2 -- the degenerate limit of Schwarzschild-de Sitter in which the black-hole and cosmological horizons merge.

Why Nariai

• Finiteness. The bifurcation surface is a compact two-sphere of area 4*pi/Lambda, so the relative entropy, the area, and the identification S_rel = dA/4 all involve finite quantities. The infinite-area caveat of the Rindler construction disappears, and the ledger equation S = pi/Lambda + S_rel relates finite numbers.
• Unambiguous temperature. The Killing normalization is fixed by the geometry (Bousso-Hawking), removing the Rindler boost-rescaling ambiguity.
• Dynamics. Positivity of relative entropy becomes a selection rule on the branches of the Nariai (anti-)evaporation instability: within the coherent sector only the pierced horizon cut can grow; anti-evaporation is a certificate of non-coherent horizon flux.

Key exact result (Appendix B)

With U = -exp(-kappa*u)/kappa, the modular weight is exactly the boost Jacobian: (-U) dU = du / kappa. Hence the interference functional

I2 = Int (-U) (d_U chi)^2 dU dvol

is the zero-total-boost-frequency Fourier component of (d_u chi)^2 and vanishes identically for boost-positive-frequency packets. Consequences:

1. The squeezed-state relative entropy takes the thermal first-law form S_rel = beta * dE_boost, beta = 2*pi/sqrt(Lambda).
2. Local flux-negativity windows (the engine of anti-evaporation) integrate to zero against the modular weight: negativity is a zero-sum reallocation.
3. |I2|/I1 is a binary wedge-locality detector: 0 for boost-adapted data, 1 for boost-blind data.

Contents

File	Purpose
supplementary.tex	Supplementary material: TikZ diagrams + numerical figures
appendix_b_verification.py	Numerical checks for Appendix B (I2 vanishing, negativity windows, one-mode sum rule)
generate_figures.py	Generates Figures S1-S3 (matplotlib) into figures/
Makefile	Build system

Quickstart

make verify        # run the numerical checks (numpy)
make figures       # regenerate Figures S1-S3 (matplotlib)
make               # build supplementary.pdf (needs pdflatex + TikZ)

Requirements: Python >= 3.10 with numpy and matplotlib; a TeX distribution with TikZ (TeX Live recommended).

Verified numbers

From make verify (kappa = 1):

• |I2|/I1 = 1.3e-15 for a boost-positive-frequency packet (prediction: 0), with the boost/affine cross-check agreeing to 2e-16.
• |I2|/I1 = 1.000000 for a real (boost-blind) packet: no wedge-local squeeze exists at any finite r.
• Squeezed flux at (r, theta) = (0.5, pi): negative on 51% of the packet support, funded depth 21% of the positive part, total positive.
• One-mode sum rule S_rel = beta * dE verified to <= 2e-15 at three parameter points, matching beta * w * sinh(r)^2 * coth(beta*w/2).

Status and open problems

• (i) Squeeze-shape inequality: closed (exact vanishing theorem).
• (ii) dS = 0 for the quadratic generator: verified at the one-mode level; the type-III domain-theoretic lift remains open.
• (iii) Backreaction with sign-indefinite flux (teleological horizon): the open dynamical frontier, constrained by the zero-sum theorem.

Papers

• https://doi.org/10.5281/zenodo.21303604
• https://doi.org/10.5281/zenodo.21382744