Analysis of Sgr A* Environment

1. Introduction: The Galactic Center and Sgr A*

The supermassive black hole Sgr A* at the Galactic Center hosts a dynamic environment, including the S cluster (S stars and G objects) and massive young stellar objects (YSOs) like X3. The A&A study (2025) using ERIS data provides new insights into Keplerian orbits and emissions. QFunity reinterprets this as an EPT-driven system, beyond classical gravity (see EPT).

2. Classical View of the Galactic Center Environment

🌌 Classical Interpretation:

• Keplerian System: S stars, G objects, and YSOs (e.g., X3) orbit Sgr A* via its gravitational pull.
• G Objects Nature: Debated as tidally distorted stars or dust/gas clouds (A&A 2025, source).
• Forces: Gravity, hydrodynamics (pressure, shocks), and relativistic effects (precession, time dilation).

3. QFunity Approach: EPT as Fundamental Substrate

🔄 QFunity Reinterpretation: The EPT (Espace-Particule-Temps) underpins the Galactic Center dynamics.

1. Gravity as EPT-Matter Coupling

From Quantum Gravity, gravity emerges from EPT-matter interaction.

Equation:

• Explanation: Near Sgr A*, \(T_{\mu\nu}^{\text{EPT}}\) (EPT energy-momentum) modifies trajectories and vacuum properties.

2. Rotation as Organizing Principle

The pillar « Everything is Rotation » (see Rotation) drives dynamics via torsion operator \( \hat{B}_\epsilon \).

Equation:

• Explanation: Non-commutativity near Sgr A* explains G objects’ complex flows and X3’s outflows.

3. Black Hole Interior as EPT Interface

From Black Hole EPT, Sgr A*’s interior is an EPT state, with the horizon as a perceptual boundary.

• Implication: Influences accretion, gravitational fluctuations, and G object stability.

4. Refined QFunity Model for Sgr A*

1. Observed Fundamental Parameters

2. Master Equations for Sgr A*

2.1 EPT Field Equation

Ĥ_SgrA* = Ĥ_EPT + Ĥ_Kerr + Ĥ_coupling + Ĥ_matter

With:
Ĥ_EPT = ∫ d³x [½(∇Ψ)² + ½m_EPT²Ψ² + λΨ⁴]
Ĥ_Kerr = -ħc/Rₛ · (a_* · Ŝ_EPT) / (1 + √(1-a²))
Ĥ_coupling = g_EPT ∫ d³x Ψ(x) ρ_matter(x) exp(-|x|/λ_EPT)
Ĥ_matter = ∑_i (p_i²/2m_i + V_tidal(r_i) + V_disk(r_i))

Explanation: The total Hamiltonian describes the EPT scalar field (\(\Psi\)), Kerr geometry, EPT-matter coupling, and surrounding object dynamics (S stars, G objects).

2.2 Characteristic Scale \(\epsilon\) for Sgr A*

ε_SgrA* = ε_0 · (Rₛ/ℓ_P)^(D_f-3) · f(a)

Where:
ε_0 = ħ/2 ≈ 5.27 × 10⁻³⁵ m (Planck scale)
D_f ≈ 2.718 (QFunity fractal dimension)
f(a) = 1 + a²/(1+√(1-a²)) (spin correction)

Calculation:
ε_SgrA* ≈ 5.27×10⁻³⁵ · (1.27×10¹⁰/1.616×10⁻³⁵)^(-0.282) · (1 + 0.81/1.436)
        ≈ 5.27×10⁻³⁵ · (7.86×10⁴⁴)^(-0.282) · 1.564
        ≈ 5.27×10⁻³⁵ · 1.27×10⁻¹² · 1.564
        ≈ 1.05×10⁻⁴⁶ m

Significance: The tiny \(\epsilon\) indicates strong quantum dominance near Sgr A*.

2.3 EPT-Corrected Metric

g_μν^QF = g_μν^Kerr + (ℓ_P²/ε²) · h_μν^LQG + α' · g_μν^strings + δg_μν^EPT

With EPT corrections:
δg_tt^EPT = - (2GM/c²r) · [1 + β_EPT·Ψ₀²·exp(-r/λ_EPT)]
δg_φφ^EPT = (r² + a²cos²θ + 2GMa²rsin²θ/c⁴) · [1 + γ_EPT·Ψ₀²·(r/Rₛ)^(-D_f)]

Coupling parameters:
β_EPT ≈ 10⁻⁵ (EPT-time coupling)
γ_EPT ≈ 10⁻⁶ (EPT-space coupling)
λ_EPT = ħ/(m_EPT c) ≈ 1.2 mm (coherence length)
Ψ₀ = √(ρ_EPT/m_EPT) (mean EPT field)

3. Component Calculations for Sgr A*

3.1 Classical Gravitational Field

Φ_N(r) = -GM/r = - (6.67×10⁻¹¹ × 8.5×10³⁶)/r = -5.67×10²⁶/r J/kg

g(Rₛ) = GM/Rₛ² = (6.67×10⁻¹¹ × 8.5×10³⁶)/(1.27×10¹⁰)² = 3.51×10⁶ m/s²

v_esc = √(2GM/Rₛ) = √(2 × 6.67×10⁻¹¹ × 8.5×10³⁶ / 1.27×10¹⁰) = 2.98×10⁸ m/s ≈ 0.994c

3.2 EPT Rotation/Torsion Field

B̂_ε_SgrA* = ε²(∇×ω) = ε² [ (2GJ/c²r³) · (cosθ e_r - 2sinθ e_θ) ]

With J = aGM²/c

J = 0.9 × 6.67×10⁻¹¹ × (8.5×10³⁶)² / 3×10⁸ = 1.45×10⁴² kg·m²/s
B̂_ε(Rₛ) ≈ (1.05×10⁻⁴⁶)² × [2×1.45×10⁴²/(9×10¹⁶×1.27×10¹⁰)³] ≈ 10⁻¹³⁵ s⁻¹·m

Interpretation: Dominant inside \(Rₛ\) where standard geometry yields to EPT.

3.3 Central EPT Power

ρ_EPT = ρ_0 ε⁻⁴ e^(-ε/ℓ_P) = ρ_0 × (1.05×10⁻⁴⁶)⁻⁴ × e^(-6.5×10⁻¹²)

With ρ_0 = m_EPT⁴c³/ħ³, m_EPT ≈ 10⁻³ eV/c² ≈ 1.78×10⁻³⁹ kg:
ρ_0 ≈ (1.78×10⁻³⁹)⁴ × (3×10⁸)³ / (1.05×10⁻³⁴)³ ≈ 10⁻¹⁵⁰ × 2.7×10²⁵ / 1.16×10⁻¹⁰¹ ≈ 2.3×10⁻²⁴ J/m³

P_EPT = (dE_EPT/dt) = 4πRₛ² × (c/4) × ρ_EPT × f(a, D_f)
With f(a, D_f) = a²/(1-a²)^(D_f/2):
P_EPT ≈ 4π×(1.27×10¹⁰)² × (0.75×10⁸) × 2.3×10⁻²⁴ × (0.81/0.19¹.³⁵⁹)
      ≈ 1.46×10³⁴ × 10⁸ × 10⁻²⁴ × 4.7
      ≈ 6.9×10¹⁸ W ≈ 1.8×10⁻⁸ L☉

Comparison: EPT power is ~10¹² times weaker than observed luminosity, but significant for specific processes.

3.4 Accretion Disk Attraction Force

M_d ≈ 10⁻³ M☉ ≈ 8.5×10³³ kg
Φ_disk(r,z) = -2GΣ_0 ∫_0^∞ [1/√(r²+z²+r'²-2rr'cosφ)] r'dr'dφ
Σ_0 = M_d/(πR_out²), R_out ≈ 0.1 pc ≈ 3×10¹⁵ m:
Σ_0 ≈ 8.5×10³³/(π×9×10³⁰) ≈ 3×10² kg/m²

F_disk(r) ≈ 2πGΣ_0 m_* [1 - (1 + r²/z₀²)^(-1/2)]
For m_* ≈ 10 M☉ ≈ 2×10³¹ kg, z₀ ≈ 10¹⁴ m:
F_disk ≈ 2π×6.67×10⁻¹¹×3×10²×2×10³¹ × [1 - (1 + 9×10²⁸/10²⁸)^(-1/2)]
      ≈ 2.5×10²⁴ × [1 - (10)^(-1/2)] ≈ 2.5×10²⁴ × 0.684 ≈ 1.7×10²⁴ N

F_BH(r) = GMm_*/r² = 6.67×10⁻¹¹×8.5×10³⁶×2×10³¹/(9×10²⁸) ≈ 1.3×10²⁹ N
Ratio: F_disk/F_BH ≈ 1.3×10⁻⁵

4. EPT-Matter Coupling for G Objects

4.1 Modified Equation of Motion

d²r/dt² = -GM/r² · [1 + α_EPT·Ψ(r)²] e_r + β_EPT·(v × B̂_ε) + γ_EPT·∇(Ψ²)

With:
α_EPT ≈ 10⁻⁵ (gravity coupling)
β_EPT ≈ 10⁻¹⁰ m·s/kg (rotation coupling)
γ_EPT ≈ 10⁻¹⁵ m⁴/s²·kg (pressure coupling)

4.2 G Objects Evolution

ρ_EPT^G = ρ_EPT^0 · [1 + κ·(m_G/m_0)^(D_f-2)]
dm_G/dt = -Γ_acc · m_G + Γ_EPT · ρ_EPT^G · V_G

Where:
Γ_acc ≈ 10⁻¹⁰ s⁻¹ (accretion rate)
Γ_EPT ≈ 10⁻¹⁵ m³/kg·s (EPT growth rate)
V_G ≈ (10¹² m)³ = 10³⁶ m³

4.3 Anomalous Precession

Δω_EPT = (2π) · [ (3GM/c²a(1-e²)) + δ_EPT·Ψ₀²·(a/Rₛ)^(-D_f+1) ]

For a = 100 AU ≈ 1.5×10¹³ m with e ≈ 0.8:
· GR Precession: Δω_GR ≈ 2π × 3×6.67×10⁻¹¹×8.5×10³⁶/(9×10¹⁶×1.5×10¹³×0.36) ≈ 2π × 3.5×10⁻⁴ rad
· EPT Correction: Δω_EPT ≈ 2π × 10⁻⁵ × (1.5×10¹³/1.27×10¹⁰)^(-1.718) ≈ 2π × 10⁻⁵ × 3.6×10⁻⁶ ≈ 2π × 3.6×10⁻¹¹ rad

Ratio: Δω_EPT/Δω_GR ≈ 10⁻⁷

Explanation: The EPT-induced precession is subtle but detectable over decades with precise astrometric data from ERIS and GRAVITY.

5. Testable Predictions

5.1 Observable Signatures

1. Anomalies in S Star Precession:

   δω/ω ≈ 10⁻⁷ - 10⁻⁶ (dependent on a and e)

   Detectable with 30 years of GRAVITY/ERIS data.

2. Anomalous Infrared Emission from G Objects:

   L_IR^EPT/L_IR^standard ≈ 1 + η_EPT·Ψ₀²·(T/10⁴ K)^(D_f-2)

   With \(\eta_EPT \approx 10⁻³\), predicts an excess of 0.1-1% in M and N bands.

3. Positional Correlations:

   P(θ) ∝ 1 + ζ·cos(2θ - θ_0) (quadrupole EPT due to Sgr A* spin)

   With \(\zeta \approx 10⁻⁴ – 10⁻³\).

5.2 Characteristic Timescales

5.3 Parameters to Constrain by Observation

1. Effective EPT Mass:

   m_EPT = ħ/(λ_EPT c) ≈ 10⁻³⁵ - 10⁻³³ kg (10⁻³ - 10⁻¹ eV/c²)

2. Coupling Constants:
   • α_EPT (gravity): precision required 10⁻⁶
   • β_EPT (rotation): precision required 10⁻¹²
   • γ_EPT (pressure): precision required 10⁻¹⁸
3. Local Fractal Dimension:

   D_f^local = 2.718 ± δD_f with δD_f ≈ 0.01 (expected constraint)

6. Conclusion: Unified QFunity Vision of Sgr A*

Sgr A* according to QFunity is a hierarchical system:

Système_SgrA* = EPT_core ⊕ Kerr_geometry ⊕ Accretion_disk ⊕ Stellar_cluster ⊕ G_objects ⊕ EPT_halo

Synthetic Equations:

1. Global Dynamics:

   d/dt[System] = [Ĥ_EPT, System] + Flux_matter + Flux_EPT + Dissipation

2. Total Energy:

   E_total = M_BH c² + E_rot + E_disk + E_* + E_EPT + E_coupling

3. Effective Scale:

   ε_effective(r) = ε_0 · (r/Rₛ)^(3-D_f) · g(a, θ, Ψ)

Required Validations:

1. Short-term (1-3 years): Precise S-star orbit measurements with ERIS/GRAVITY to constrain α_EPT.
2. Mid-term (3-10 years): Multi-wavelength monitoring of G objects for L_IR^EPT detection.
3. Long-term (10+ years): Complete Galactic Center mapping to measure D_f^local and Ψ distribution.

QFunity posits that Sgr A*’s environment is not just an extreme gravitational system but a window into the interface between emergent spacetime and the pre-temporal EPT substrate. Observed « non-homogeneous effects » are manifestations of this dynamic interface.