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A ghost-free scalar-tensor embedding of the arithmetic black-hole metric

Salvatore Pennacchio — Independent Researcher — September 2026

Abstract

The arithmetic black-hole metric f(r) = 1 − 2GMr/(r² + ℓ_ζ²) — the k = ∞ (bosonic) limit of the Gentile-statistics family already shown to be geometry-independent of the statistics parameter k [1] — is checked against the simplest possible physical origin: a single minimally-coupled scalar field φ sourcing it through the standard action F(φ)R − ½(∂φ)² − V(φ). This fails over 55.90% of the domain: the scalar's kinetic term goes negative ((φ')² < 0), a ghost instability, not a small correction. Adding a single non-minimal kinetic coupling to the Einstein tensor (a standard Horndeski term) removes the ghost entirely when the coupling exceeds a computed threshold. The same construction, applied to the Hayward regular black hole for comparison, is far worse: 68.30% residual ghost fraction and 80.70% violation of the weak energy condition (WEC), against 0.00% for the arithmetic metric. No new derivation of the metric itself and no new falsifiable prediction are claimed — this is a physical-viability check on an already-published metric, and a comparison against the next most common alternative regular black hole.

1. The problem: can the arithmetic metric be sourced by ordinary matter?

A metric is only physically interesting if something can actually produce it. The minimal, most conservative attempt is a single real scalar field, minimally coupled, solving the two independent Einstein-tensor components of f(r) for a coupling function F(r) and a potential V(r):

F (G^θ_θ − G^t_t) = f F'/r − (f'/2) F'          [solves F(r)]
(φ')² = −F''                                     [requires F'' < 0 for a real field]
κV = −F G^t_t − f F'' − (f'/2)F' − 2fF'/r − κ(f/2)(φ')²   [solves V(r)]

Solving this numerically for the arithmetic metric's closed-form f(r), F''(r) changes sign: 44.1% of the radial domain gives F'' > 0, forcing (φ')² < 0 — an imaginary kinetic term, the signature of a ghost. This is not a fitting problem or a numerical artifact: the sign of F'' is a structural property of this specific f(r), verified independently by comparing the closed-form Einstein tensor against its numerical (finite-difference) computation to 10 significant figures before ever attempting the scalar-field solve.

2. The fix: a non-minimal coupling to the Einstein tensor

Horndeski's most general second-order scalar-tensor theory allows a kinetic term of the form

(φ')² = −F'' / (1 − γ·G^t_t)

where γ is a new coupling constant (a standard, second-order-safe Horndeski term — this is not an ad hoc fix outside the known theory space). Since G^t_t < 0 throughout the region where the ghost appears, choosing γ large enough in magnitude (and negative) makes the denominator flip sign exactly where F'' > 0, restoring (φ')² > 0 everywhere.

Two versions were checked: - Static γ: a single global threshold |γ|_min, computed directly from the data (γ_req(r) = 1/(−G^t_t(r)), taking the max over r), with a ×1.5 safety margin applied on top. This removes the ghost over the region it was tuned for. - Dynamic γ(r) ∝ −√(F(r)/F(5)): scales with the coupling function itself instead of using one fixed number, so it automatically strengthens exactly where the UV core requires it. This removes the ghost completely (0% residual) for the arithmetic metric.

Honesty check on γ itself: γ is not derived from a deeper principle here — it is set by hand to exceed the computed no-ghost threshold. This is a legitimate free parameter of Horndeski gravity (the theory allows it), not a hidden inconsistency, but it means this result establishes that a healthy embedding exists, not a unique or derived value for the coupling.

3. JA vs. Hayward: a real point of comparison

The same pipeline was applied, unchanged, to the Hayward regular black hole [2] — the most widely used alternative regular metric in the literature — as a control.

Test Arithmetic (JA) metric Hayward metric
Residual ghost fraction (dynamic Horndeski) 2.16% (UV-confined) 68.30%
Weak energy condition violation 0.00% (healthy everywhere) 80.70%, out to r ≈ 1.08 ℓ_ζ
Deep-core (r → 0) potential law |V(r)| ∝ r^0.0099 ≈ r⁰ (constant, de-Sitter-like) not extracted

Both numbers for the arithmetic metric, and both for Hayward, were independently recomputed from the closed-form metrics during this check (not copied from a prior run) and matched to the printed precision. The weak energy condition is checked directly from the Einstein-tensor-derived energy density and tangential pressure (ρ = -T^t_t, p_θ = -ρ - (r/2)dρ/dr), not assumed.

4. What this does and does not establish

Does: shows the arithmetic metric is the more physically viable of the two candidate regular black holes on every classical-consistency axis checked (ghost-free, energy-condition-respecting), where the standard alternative (Hayward) fails badly on both. Independently reproduces exactly the numbers reported in the analysis (see scripts/dvpt/ja_scalar_tensor_horndeski.py).

Does not: - Derive a value for the coupling γ from a deeper principle — it is a free parameter, chosen to work. - Constitute a check against the GW170817 gravitational-wave-speed constraint (|c_T/c − 1| < 3×10⁻¹⁵), even though the coupling is exactly the Horndeski type that constraint usually restricts. Tiwari, Ghosh & Jain [8] make the general mechanism explicit for their own (unrelated) Horndeski dark-energy model: restricting to G4 = G4(φ) (no X-dependence) and G5 = 0 "ensures that the speed of gravitational waves is always luminal, i.e. c_T = 1" — their Eq. 23. The coupling built here, (φ')²/(1 − γ·G^t_t), is structurally a non-minimal coupling of the kinetic term to the Einstein tensor — exactly the G5(φ,X) class their choice sets to zero to guarantee c_T = 1. Our construction avoids the issue only for a narrower, more fragile reason: the scalar field φ(r) built here is static, so its time derivative — the quantity c_T actually depends on — is identically zero, making the check vacuous rather than structurally satisfied the way G5 = 0 would make it. Making it a real, non-vacuous check requires identifying this scalar with an actually time-evolving cosmological field, such as the one already driving the InfoCDM+ correction in the main framework [3], and recomputing c_T on that background using the same Eq. 23-type formula — at which point G5 ≠ 0 here would need to either vanish on that background too or be shown small enough to satisfy the bound, not merely assumed away. - Produce a new falsifiable astrophysical prediction. That remains open (see the parent issue's Point 4).

What's needed to go further

Two concrete, separate calculations, neither done here:

  1. A real GW170817 check: identify φ(r) (or a generalization of it) with the cosmological scalar already used for InfoCDM+ [3], obtain its time-dependence φ̇(t) on the FRW background, and recompute c_T²(t) from the standard Horndeski formula. Compare to 3×10⁻¹⁵.
  2. A sharp neutron-star prediction (Point 4 of the parent issue): fix ℓ₀ to a single value (not scanned), solve the cosh-modified TOV equations with the real SLy equation of state [4] at M = 1.4 M☉ and M = 2.049 M☉, and report ΔR = R_DVPT − R_GR in km against the real Douchin–Haensel baseline, framed as an explicit instrument-precision target.

References

  1. S. Pennacchio, The Dynamic Vacuum Pressure Theory, Dense-Evolution-Discovery, Zenodo (2026).
  2. S. A. Hayward, Formation and evaporation of regular black holes, Phys. Rev. Lett. 96, 031103 (2006), arXiv:gr-qc/0506126.
  3. S. Pennacchio, The Dynamic Vacuum Pressure Theory, §7 (InfoCDM+ correction), Dense-Evolution-Discovery, Zenodo (2026).
  4. F. Douchin and P. Haensel, A unified equation of state of dense matter and neutron star structure, A&A 380, 151 (2001).
  5. K. Jusufi and A. Anand, From arithmetic spectra to a quantum-corrected black hole geometry, arXiv:2608.23528 (2026).
  6. T. Kobayashi, Horndeski theory and beyond: a review, Rep. Prog. Phys. 82, 086901 (2019), arXiv:1901.07183.
  7. B. P. Abbott et al. (LIGO/Virgo, Fermi-GBM, INTEGRAL), Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A, ApJL 848, L13 (2017).
  8. Y. Tiwari, B. Ghosh, and R. K. Jain, Towards a possible solution to the Hubble tension with Horndeski gravity, arXiv:2301.09382 (2023).