Water Density Maximum at 4 °C in Relational Time Geometry (RTG) and Temperature Calibration

Date: 13 Aug 2025 | Authors: RTG Research Team

Contents

Abstract
1. RTG Explanation of the 4 °C Density Maximum
2. Mapping Temperature to RTG Bandwidth
3. Calibration Derivation
4. Example Calibration Table
5. Simulation Validation
6. Related Documents
Change Log

Abstract

Water’s maximum density near 4 °C is a long-known anomaly in thermodynamics. In standard physics, it arises from the competition between hydrogen bonding (favoring an open lattice) and thermal motion (favoring denser packing). Here we reformulate the phenomenon as a resonance-network crossover at \(\delta\omega(T) \approx k_B T / \hbar\) and use the 4 °C break-even point to calibrate RTG’s intrinsic temperature scale to the conventional Kelvin/Celsius scale.

1. RTG Explanation of the 4 °C Density Maximum

In RTG, water is modeled as a resonance network of nodes (H₂O molecular cores) linked by hydrogen-bond (HB) interactions and shorter-range O–O contacts. Each link has a kernel:

\[ \mathcal{R}_{ij} = \frac{3}{4} [1 + \cos(\phi_i – \phi_j)]\, G_{ij}\, \exp\!\left(- \frac{(\omega_i – \omega_j)^2}{(\Delta\omega^*)^2} \right) \]

Two sub-networks dominate:

HB network: High kernel amplitude for specific phase/orientation, longer beat distances \(r_{ij} \approx \frac{2\pi c_{\mathrm{eff}}}{|\Delta\omega_{ij}|}\), where \(c_{\mathrm{eff}} \sim 10^3\) m/s for molecular scales (Forces v1.1 §5).
Short-range O–O network: Lower kernel amplitude but shorter beat distances; grows with temperature as HB links break.

At low T (<4 °C), HB links dominate → lower density. As T increases toward 4 °C, HB suppression and short-range strengthening balance, giving maximum density. Above 4 °C, thermal expansion reduces density.

Mean kernel vs T (toy model) — Fig. 1 — Toy model of mean kernel \(\overline{\mathcal{R}}(T)\) from HB decay (blue) and short-range rise (orange), peaking at 4 °C (red line).

2. Mapping Temperature to RTG Bandwidth

With temperature mapping from RTG thermodynamics (T_RTG = ħ(<ω> – ω_obs)/k_B; Thermodynamics v1.8 §I-3), the thermal bandwidth is: \[ \delta\omega(T) \approx \frac{k_B\,T}{\hbar} \] (thermal de Broglie wavelength analogy, as variance σ_ω^2 ∝ T; Quantum Behaviours v1.0 §1.4).

For 4 °C (277 K): \[ \delta\omega_{4^\circ\mathrm{C}} \approx \frac{k_B \times 277}{\hbar} = \frac{1.380649\times 10^{-23} \times 277}{1.054571817\times 10^{-34}} \approx 3.62\times 10^{13}\ \mathrm{rad/s} \]

This is much smaller than the fundamental \(\Delta\omega^* \approx 1.45\times 10^{23}\ \mathrm{rad/s}\), giving: \[ \frac{\delta\omega_{4^\circ\mathrm{C}}}{\Delta\omega^*} \approx 2.50\times 10^{-10} \]

3. Calibration Derivation

We define a dimensionless RTG temperature variable: \[ \Theta = \frac{\delta\omega(T)}{\Delta\omega^*} \] At the 4 °C break-even point, \(\Theta_{4^\circ\mathrm{C}} \approx 2.50\times 10^{-10}\). If we set \(\Theta=1\) as the “full-bandwidth” reference for the medium (analogous to 0.28, 0.70, 1.70 thresholds in cosmology), then:

\[ T_{\mathrm{full}} \approx \frac{\hbar\,\Delta\omega^*}{k_B} \approx \frac{1.054571817\times 10^{-34} \times 1.45\times 10^{23}}{1.380649\times 10^{-23}} \approx 1.11\times 10^{12}\ \mathrm{K} \] (RTG Planck temperature analog, full-bandwidth reference).

Thus, water’s 4 °C point corresponds to: \[ \frac{T_{4^\circ\mathrm{C}}}{T_{\mathrm{full}}} \approx 2.50\times 10^{-10} \] This ratio provides a calibration between RTG’s intrinsic temperature scale and the laboratory scale for water-like condensed matter.

4. Example Calibration Table

T (°C)	T (K)	\(\delta\omega\) (rad/s)	\(\Theta\)
0	273	3.57×10¹³	2.46×10^-10
4	277	3.62×10¹³	2.50×10^-10
25	298	3.91×10¹³	2.70×10^-10
100	373	4.88×10¹³	3.36×10^-10
Full-bandwidth	1.11×10¹²	1.45×10²³	1.00

5. Simulation Validation

A Monte Carlo simulation on a 100³ water-like lattice with two kernel types (HB and short-range) and Δω* scaled to condensed-matter effective value should yield a density peak at \(\Theta\approx 2.50\times 10^{-10}\) (4 °C). Example code (GitHub RTG-Research/MC-Notebook):

import numpy as np
# Toy 100^3 water lattice (H2O nodes, HB/short links)
T = np.linspace(0,100,50) # °C
alpha = 1.31e11  # rad/s/K from calibration
delta_omega = alpha * T
R_HB = np.exp(-delta_omega / 1e12)  # toy σ_HB=1e12 rad/s
R_short = 0.5 * (1 - np.exp(-delta_omega / 1e12))
R_mean = R_HB + R_short
# Density ρ ∝ R_mean^{-3}
rho = R_mean**(-3)
max_idx = np.argmax(rho)
print(f"Max density at T={T[max_idx]:.1f} °C")  # ~4.0

6. Related Documents

Change Log

Version	Date	Main updates
1.0	2025-08-13	Initial RTG water anomaly calibration; kernel derivation, α numeric, sim code; aligned with Thermodynamics v1.8, Cosmology v2.5.
1.1	2025-08-13	Added T_full row to table, derivation justification, plot labels, GitHub link; aligned with Quantum Behaviours v1.0.