Soumendra Nath Thakur
Correspondence :
postmasterenator@gmail.com ; postmasterenator@telitnetwork.in
March 24, 2025
Abstract
This research presents a mathematical derivation of frequency shift and phase transition within the Extended Classical Mechanics (ECM) framework, particularly in the context of the universe’s earliest moments. We establish a phase shift formula, x° = Δt × Δf × 360°, linking the frequency shift (Δf) over a time interval (Δt) to a measurable phase change. Applying this to the Planck epoch, we derive the initial frequency (f₀) at the Big Bang event as approximately 2.15 × 10⁴³ Hz, significantly higher than the Planck frequency (fᴘ). Our results indicate that the energy transition during the Big Bang was highly coherent, producing a near-complete 360° phase shift. This supports the ECM prediction that early-universe energy transformations followed a structured, deterministic process rather than arbitrary quantum fluctuations. The findings reinforce that energy-mass interactions at extreme scales maintain coherence, ensuring a smooth and continuous evolution rather than a disruptive or chaotic transition.
Keywords
1. Derivation of Phase Shift Formula:
We derived the formula for phase shift (x°) based on the relationship between frequency shift (Δf) and time interval (Δt) using:
T(deg) = (x°/f) ×⋅ (1/360) = Δt
Rearranging for x°:
x° = Δt × Δf × 360°
This formula determines the phase shift corresponding to a time delay Δt and frequency transition Δf.
Physical Consequence:
This equation represents the relationship between the frequency shift (Δf) over the Planck time interval (Δt) and the corresponding phase shift (x°). It implies that the rapid transition of frequency during the earliest moments of the universe led to a nearly complete 360° phase shift. This suggests that the energy transformation at the Planck epoch was highly coherent, reinforcing the idea that the initial Big Bang event involved a structured, non-random energy transition rather than chaotic fluctuations.
2. Derivation of Initial Frequency f₀:
We know that the Planck frequency is:
fᴘ = 2.952 × 10⁴² Hz
The total energy difference during the transition is given by Planck’s relation:
E = h f
For a photon energy of Eᴘ = 1.995 × 10⁹ J and 4.0 × 10⁻¹⁹ J, we calculate the frequency shift:
Δf = (Eᴘ − E)/h
Substituting values:
Δf = (1.995 × 10⁹ J − 4.0 × 10⁻¹⁹ J)/6.626 × 10⁻³⁴ Js
Δf = 3.01 × 10⁴³ Hz
Since Δf = f₀ − fᴘ, solving for f₀:
f₀ = Δf + fᴘ
f₀ = (3.01 × 10⁴³) + (2.952×10⁴² Hz)
f₀ ≈ 2.15 × 10⁴³ Hz
Physical Consequence:
The derivation of f₀ as the initial frequency at the Big Bang event indicates that the energy of the universe started at an extraordinarily high frequency before transitioning to lower frequencies. This frequency corresponds to an energy level significantly beyond the Planck scale, implying that the earliest state of the universe involved an ultra-high-energy phase where gravitational effects and quantum field interactions were deeply intertwined.
3. Derivation of Phase Shift x° for f₀ ⇒ fᴘ:
Using our derived formula:
x° = Δt × Δf × 360°
Given:
- Δt = 5.391247 × 10⁻⁴⁴ s
- Δf = f₀ − fᴘ = 3.01 × 10⁴³ Hz
Substituting:
x° = (5.391247 × 10⁻⁴⁴) × (3.01 × 10⁴³) × 360°
x° = 3.59.99° ≈ 360°
This confirms that the phase shift due to the frequency transition from f₀ to fᴘ is effectively a complete cycle.
Physical Consequence:
The near-complete phase transition (≈360°) confirms that the transition from f₀ to fᴘ was a highly structured and deterministic process. This supports the idea that the energy-frequency transition during the Big Bang followed a well-defined dynamical path rather than an arbitrary fluctuation. The result reinforces ECM’s prediction that energy-mass transformations in extreme conditions maintain coherence, even at superluminal speeds, ensuring a smooth and continuous energy evolution rather than a sudden collapse or discontinuous change.
4. Alphabetical listing of the mathematical terms used in the above equations:
- Δf – Frequency shift (f₀ − fᴘ)
- Δt – Time interval (Planck time, 5.391247 × 10⁻⁴⁴ s)
- E – Energy of a photon
- Eᴘ – Planck-scale energy
- f – Frequency
- f₀– Initial frequency (before transition) at the Big Bang event
- fᴘ– Planck frequency
- h – Planck’s constant
- T(deg) – Time shift in degrees
- x° – Phase shift in degrees
References:
- Thakur, S. N., & Bhattacharjee, D. (2023). Phase Shift and Infinitesimal Wave Energy Loss Equations. preprints.org (MDPI). https://doi.org/10.20944/preprints202309.1831.v1
- Thakur, S. N., & Bhattacharjee, D. (2023, October 30). Phase Shift and Infinitesimal Wave Energy Loss Equations. Longdom Publishing SL. https://www.longdom.org/open-access/phase-shift-and-infinitesimal-wave-energy-loss-equations-104719.html
