Thermodynamics & Rotational Dynamics in Relational Time Geometry (RTG)

Version 1.9 — 12 Aug 2025 (updated Δω* to 1.45×10²³, fixed Hz conversions, clarified σ_exch independence, added T_spec, RTG-native orbital demo, cumulative heat-flow integration, expanded Appendix)

Contents

I · Thermodynamic primitives from node data
II · Rotational dynamics
- II-1 Emergent mass
- II-2 Orbital motion (RTG-native demo)
III · Practical notes & examples
Appendix A — Symbol reference
Change Log

I · Thermodynamic primitives from node data

I-1 Internal energy U

U = Σ_i ħ ω_i + ½ Σ_i≠j [ K′ |Δω_ij| / Δω* + J ℛ_ij + J_ex sin Δφ_ij e^{−(Δω_ij/σ_exch)²} ],

ℛ_ij = ¾ [1+cos Δφ_ij] (1+s_is_j) e^{−(Δω_ij/Δω*)²}, s_i = i σ_i, σ_i ∈ {±1}.

Notes: Σ_i ħ ω_i is intrinsic; ½ pair-sum for undirected bonds; lowest-order binding term E^res_ij = ħ Δω_ij A_ij G_ij. σ_exch is an independent UV regulator for the exchange term (typically O(Δω*)) and must not be confused with Δω*; σ_noise is the dimensionless CHSH noise parameter (distinct).

I-2 Work and force

With r_ij = 2π c / |Δω_ij| and ∂|Δω|/∂r = −(2π c)/r²:

∂E_ij/∂Δω_ij = (K′/Δω*) sgn(Δω_ij) − (2J Δω_ij/Δω*²) ℛ_ij − (2J_ex Δω_ij/σ_exch²) sin Δφ_ij e^{−(Δω_ij/σ_exch)²},

F_ij = −∂E_ij/∂r_ij = (2π c / r_ij²) [∂E_ij/∂Δω_ij], W(γ) = ∫_γ Σ_i<j F_ij dr_ij.

Numerics: take sgn(0) ≡ 0 to avoid jitter at Δω_ij → 0.

I-3 Temperature T_RTG, T_spec

🔍 Temperature notions in RTG

Relational (signed) temperature, frame-dependent: T_RTG(ω_obs) = ħ (⟨ω⟩ − ω_obs) / k_B. Indicates heat-flow direction relative to observer’s clock; can be negative (population inversion).
Spectral (scalar) temperature, frame-invariant: T_spec = ħ σ_ω / k_B ≥ 0, with σ_ω the ensemble’s spectral width. Near equilibrium, canonical p_i ∝ exp[−ħ ω_i / (k_B T_spec)], S = −k_B Σ p_i ln p_i.

I-4 Heat transfer

🔍 Heat flow in RTG — vibrational energy transfer when mean frequencies differ. Resonance ℛ₁₂ gates the transfer: closed gates (G₁₂ = 0) ⇒ ℛ₁₂ = 0.

q̇ = ħ (⟨ω⟩₁ − ⟨ω⟩₂) ℛ₁₂, Q(t) = ∫₀ᵗ q̇(τ) dτ.

Anomaly check: ⟨∇_μJ^μ⟩ ≲ 10⁻³ (EH Corrections v2.5 §8).

II · Rotational dynamics

II-1 Emergent mass

m_i = [ ħ ω_i − ½ Σ_j E^bind_ij ] / c², E^bind_ij = J ℛ_ij.

The ½ avoids double-counting. Clamp m_i ≥ 0 in coarse-graining. Matches tetrad-based inertia and elastic mass-defect in Residual Elasticity v1.6.

II-2 Orbital motion (RTG-native demo)

import numpy as np, matplotlib.pyplot as plt

# Constants (SI)
c = 2.99792458e8
hbar = 1.054e-34
delta_omega_star = 1.45e23       # s^-1
Kp_MeV = 12.0                    # MeV
eV = 1.602176634e-19
Kp = Kp_MeV * 1e6 * eV           # J

m0 = hbar*delta_omega_star/c**2  # ~1.7e-28 kg

r = np.logspace(-2, 0, 200)      # m
dEdw = Kp/delta_omega_star       # J·s
F = (2*np.pi*c/r**2) * dEdw      # N
v = np.sqrt(F*r/m0)

plt.loglog(r, v)
plt.xlabel('r (m)'); plt.ylabel('v (m/s)')
plt.title('RTG elastic-regime orbital speed (toy)')
plt.tight_layout(); plt.savefig('orbit_v_rtg.png')

III · Practical notes & examples

Earth–Sun analogue: |Δω| ≈ 4×10⁻⁷ Δω* ⇒ T_RTG ≈ 300 K for a Planck-observer node (illustrative; actual depends on mass/scale choice).
Carnot 4-node loop: η ≈ 1 − T_cold/T_hot (≤ 3 % error) when |Δφ| ≤ π/6, |Δω| ≤ 0.2 Δω*.
Heat-flow with Δω noise:

import numpy as np, matplotlib.pyplot as plt
hbar = 1.054e-34
delta_omega_star = 1.45e23

t = np.linspace(0, 1e-10, 1000)
omega1 = 1.45e23
omega2 = 1.44e23 + 0.01e23*np.random.randn(t.size)

R12 = 0.75*(1+np.cos(0.1)) * np.exp(-((omega1-omega2)/delta_omega_star)**2)

qdot = (omega1 - omega2) * hbar * R12
Q = np.cumtrapz(qdot, t, initial=0.0)

plt.plot(t, Q)
plt.xlabel('t (s)'); plt.ylabel('Q (J)')
plt.title('Heat transfer with Δω noise (cumulative)')
plt.tight_layout(); plt.savefig('heat_noise.png')

Cross-references: Forces & Fields v1.1 (force deriv.), Residual Elasticity v1.6 (mass–energy scaling), Gauge Symm. v1.4 (bandwidth thresholds), EH Corrections v2.5 (heat-kernel anomalies).

Appendix A — Symbol reference

Symbol	Description
ω_i	Intrinsic frequency of node i
φ_i	Phase of node i
s_i=±i, σ_i=±1	Binary spin (analytic/code)
Δω*	Critical bandwidth (two-loop RG)
K′	Linear spectral coefficient in bond energy
J, J_ex	Resonance and exchange strengths (energy units)
A_ij	¾[1+cos Δφ_ij] e^{−(Δω_ij/Δω*)²}
Δφ_ij	Phase difference φ_i−φ_j
ℛ_ij	Resonance strength A_ij gated by spins
G_ij	Spin-gate factor (1−σ_iσ_j)/2
r_ij	Beat distance 2π c / \|Δω_ij\|
m_i	Emergent mass of node i
T_RTG, T_spec	Relational/spectral temperatures
σ_exch	Exchange UV regulator width (angular-frequency units)
σ_noise	Dimensionless CHSH noise amplitude
Q	Heat exchanged
Δt	Code time step

Change Log

Ver.	Date (UTC)	Main updates
1.9	2025-08-12	Updated Δω* to 1.45×10²³ s⁻¹; fixed Hz conversions; clarified σ_exch independence; added T_spec; RTG-native orbital demo; cumulative heat-flow integration; expanded Appendix.
1.8	2025-08-03	Added heat/temperature explanation boxes; finished multi-body U & force derivatives; entropy rationale; Δω-noise heat sim; orbital demo; corpus links.
1.7	2025-08-03	Completed U,W,F equations; anomaly-safe heat; symbol appendix.
1.6	2025-07-31	RG-matched Δω*; initial corrections & dimensional notes.