V2.638: Schwarzschild-de Sitter Two-Horizon Interpolation

Motivation

V2.635 showed that γ_BH = -4a - 2c/3 is topologically protected against spin. The cosmological prediction uses δ = -4a with no Weyl contribution (-2c/3 term absent). Why? The standard answer is “de Sitter is conformally flat, so the Weyl tensor vanishes.” But this explanation doesn’t show HOW the Weyl contribution turns off as we go from a BH to a cosmological context.

Schwarzschild-de Sitter spacetime has BOTH a BH horizon and a cosmological horizon, providing the natural arena to understand the relationship.

Key Result

The Weyl integral on any SdS horizon is W = -8πM/r_H. This is a single formula that unifies the BH and cosmological predictions:

Horizon	r_H	W	γ	Weyl coefficient
BH (Λ→0)	2M	-4π	-4a - 2c/3	2/3
Nariai	3M	-8π/3	-4a - 4c/9	4/9
Cosmo (M→0)	∞	0	-4a	0

The master formula:

γ(r_H) = -4a · [1 + (c/a) · (2M)/(3r_H)]

where c/a = 0.985 for SM+graviton. The Weyl contribution scales as 1/r_H and smoothly interpolates between the two limits.

Why It Works

For SdS spacetime, the Weyl scalar Ψ₂ = -M/r³ is independent of Λ. The cosmological constant enters ONLY through the Ricci tensor, not the Weyl tensor. This is because the Weyl tensor measures tidal distortion (from the BH mass M), while Λ is a uniform background.

On a horizon at radius r_H:

• C_{⊥⊥} = -2M/r_H³ (constant on the S² since SdS is spherically symmetric)
• dA = 4πr_H² (horizon area)
• W = C_{⊥⊥} × A_H = -8πM/r_H

For the cosmological horizon (r_C → √(3/Λ) as M → 0):

• W_C = -8πM/r_C → 0 as M → 0
• This recovers δ = -4a (no Weyl term) from the SdS geometry

The conformal flatness of de Sitter is a CONSEQUENCE: when M = 0, the Weyl tensor Ψ₂ = 0 everywhere, hence the spacetime is conformally flat.

The Two-Horizon Scan

At fixed Λ, varying the dimensionless mass μ = 3M√Λ from 0 (de Sitter) to 1 (Nariai limit):

μ	r_BH/r_C	γ_BH	γ_C	Δγ
0.001	0.0004	-14.456	-12.417	-2.038
0.3	0.12	-14.427	-12.669	-1.758
0.5	0.23	-14.370	-12.876	-1.494
0.7	0.38	-14.262	-13.124	-1.138
0.9	0.58	-14.098	-13.393	-0.705
0.999	0.95	-13.811	-13.741	-0.070

At the Nariai limit (μ → 1), both horizons converge: γ_BH = γ_C = -4a - 4c/9.

Astrophysical Corrections

For real BHs in our universe (Λ_obs ≈ 10⁻⁵² m⁻²):

BH	M/M_☉	Δγ/γ_BH
Stellar	10	3 × 10⁻⁴⁴
Sgr A*	4 × 10⁶	5 × 10⁻³³
M87*	6.5 × 10⁹	10⁻²⁶
TON 618	6.6 × 10¹⁰	10⁻²⁴

The Λ correction to γ_BH is negligible to absurd precision. V2.635’s topological protection holds for all astrophysical BHs.

Structural Insight: Why a Appears Alone in Λ

The cosmological prediction Λ ∝ |δ| = 4a uses ONLY the Euler anomaly coefficient ‘a’. The BH prediction γ_BH = -4a - 2c/3 uses BOTH a and c. This experiment reveals why:

1. a multiplies ∫R^Σ = 8π, which is purely topological (Gauss-Bonnet)
2. c multiplies ∫C_{⊥⊥} = -8πM/r_H, which is geometrical

On the cosmological horizon, r_H → ∞, so the c contribution vanishes. The cosmological constant sees ONLY the topology of the horizon (S²), not its detailed geometry. This is why Λ is determined by the Euler density coefficient ‘a’ alone.

The ratio γ_BH/δ = 1 + (c/a)(2M)/(3r_H) is controlled by c/a = 0.985, a pure number fixed by the SM field content. This ratio connects the cosmological and BH predictions in a parameter-free way.

Honest Assessment

What’s New

1. The formula W(r_H) = -8πM/r_H provides a continuous interpolation between BH and cosmological predictions in a single spacetime
2. Explains WHY the cosmological prediction has no Weyl contribution (not just “de Sitter is conformally flat” but “1/r_H → 0”)
3. The Nariai limit gives a specific intermediate value γ_N = -4a - 4c/9
4. Quantifies the Λ correction to γ_BH (negligible for all astrophysical BHs)

What’s Strong

1. The formula is exact (follows from Ψ₂ = -M/r³ for SdS, a standard result)
2. Cleanly connects V2.628 (BH), V2.635 (spin protection), and the cosmological prediction (δ = -4a) in a unified framework
3. The master formula γ = -4a[1 + (c/a)(2M)/(3r_H)] is elegant and predictive

What’s Weak

1. SdS is an idealization — real universe has matter, radiation, not just Λ
2. The Nariai limit (both horizons equal) is unphysical for our universe
3. The key formula Ψ₂ = -M/r³ is a standard GR result, not new physics
4. Neither γ_BH nor the interpolation is currently measurable

What It Means for the Framework

The experiment reveals a structural principle: the cosmological constant knows only about horizon topology (via ‘a’), while BH entropy also knows about local curvature (via ‘c’). The two predictions are related by the ratio c/a, which is fixed by the SM field content. This deepens the connection between the BH and cosmological sectors of the framework.

Files

• src/sds_horizons.py — SdS geometry, Weyl integral, gamma computation
• tests/test_sds.py — 8 verification tests (all pass)
• run_experiment.py — Full 6-phase experiment
• results.json — Machine-readable results