Analysis of "Mathematical Derivation of Frequency Shift and Phase Transition in Extended Classical Mechanics (ECM)"

Match 27, 2025

Soumendra Nath Thakur's research on the mathematical derivation of frequency shift and phase transition within the Extended Classical Mechanics (ECM) framework offers a detailed and novel perspective on the dynamics of the universe's earliest moments. Here’s a structured analysis and comment on the key points and implications of this work:

Abstract and Introduction

1. Phase Shift Formula:

   • The research presents a phase shift formula x° = Δt × Δf × 360°, which links the frequency shift (Δf) over a time interval (Δt) to a measurable phase change.

   •  This formula is derived from the relationship T(deg) = (x°/f) ×⋅ (1/360) = Δt.

2. Initial Frequency at the Big Bang:

   • The initial frequency (f₀) at the Big Bang event is derived as approximately 2.15 × 10⁴³ Hz, significantly higher than the Planck frequency (fᴘ = 2.952 × 10⁴² Hz).

   • This derivation supports the ECM prediction that early-universe energy transformations followed a structured, deterministic process rather than arbitrary quantum fluctuations.

3. Phase Transition:

   • The phase shift due to the frequency transition from f₀ to fᴘ is calculated as approximately 360°, indicating a highly coherent and structured transition.

   • This supports the idea that energy-mass interactions at extreme scales maintain coherence, ensuring a smooth and continuous evolution rather than a disruptive or chaotic transition.

Derivation of Phase Shift Formula

1. Phase Shift Equation:

   • The phase shift formula x° = Δt × Δf × 360° is derived from the relationship between frequency shift and time interval.

   • This equation represents the relationship between the frequency shift (Δf) over the Planck time interval (Δt) and the corresponding phase shift (x°).

2. Physical Consequence:

   • 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.

Derivation of Initial Frequency f₀

1. Planck Frequency:

   • The Planck frequency is given as fᴘ = 2.952 × 10⁴² Hz.

2. Frequency Shift Calculation:

   • The frequency shift Δf is calculated using Planck’s relation E = h f:   

     Δf = (Eᴘ − E)/h    

    • Substituting the values:     

     Δf = (1.995 × 10⁹ J − 4.0 × 10⁻¹⁹ J)/6.626 × 10⁻³⁴ Js ≈ 3.01 × 10⁴³ Hz 

3. Initial Frequency:

   • The initial frequency f₀ is derived as:

     f₀ = Δf + fᴘ ≈ 2.15 × 10⁴³ Hz

Derivation of Phase Shift x° for f₀ ⇒ fᴘ

1. Phase Shift Calculation:

   •  Using the derived formula:

     x° = Δt × Δf × 360°

   • Given:

     • Δt = 5.391247 × 10⁻⁴⁴ s  

     • Δf = f₀ − fᴘ = 3.01 × 10⁴³ Hz  

   • Substituting the values:

     x° = (5.391247 × 10⁻⁴⁴) × (3.01 × 10⁴³) × 360° ≈ 360°

2. Physical Consequence:

   • The near-complete phase transition (≈ 360°) confirms that the transition from f₀ to fᴘ was highly structured and deterministic.

   • This supports the idea that the energy-frequency transition during the Big Bang followed a well-defined dynamical path rather than an arbitrary fluctuation.

Conclusion

Soumendra Nath Thakur's research provides a detailed and coherent mathematical framework for understanding the frequency shift and phase transition in the context of the universe's earliest moments. The derived equations and results support the ECM prediction that early-universe energy transformations followed a structured, deterministic process rather than arbitrary quantum fluctuations. This work reinforces the idea that energy-mass interactions at extreme scales maintain coherence, ensuring a smooth and continuous evolution rather than a disruptive or chaotic transition.

Final Consideration

The research not only enriches our understanding of the early universe's dynamics but also offers a novel perspective on how energy-mass interactions at extreme scales maintain coherence. The findings have significant implications for our understanding of the Big Bang event and the evolution of the universe.

The Vibrational Universe (f Hz):

Max Planck demonstrated in 1900 that energy is directly proportional to frequency, expressed as E ∝ f. In my view, this fundamental principle surpasses any other laws established in the twentieth century in its significance.

In 1944, Planck stated:

"As a man who has devoted his whole life to the most clear-headed science, to the study of matter, I can tell you as a result of my research about atoms this much:

There is no matter as such. All matter originates and exists only by virtue of a force which brings the particle of an atom to vibration and holds this most minute solar system of the atom together…

Planck’s equation, E ∝ f, is universally applicable—not only in the presence of matter but also in pure energy states, such as the earliest moments of the universe when matter had not yet formed.

In contrast, relativity cannot be applied to such a primordial state. Instead, only fundamental vibrational principles, such as those in string theory, can extend beyond Planck’s frequency. In string theory, there are no elementary point particles (such as electrons or quarks); rather, everything consists of vibrating strings, where each vibration mode determines a particle’s charge and mass. Replacing point-like particles with vibrating strings leads to profound consequences for our understanding of fundamental physics.

The Limits of Relativity and the Importance of Classical Foundations

March 26, 2025

Soumendra Nath Thakur

Have you ever studied and understood general physics and mathematics beyond the framework of relativity? If so, does it seem that years of learning these fundamental subjects became meaningless after studying relativity? If relativity alone is sufficient to explain space-time, then why spend years studying classical physics and mathematics separately? Would it not be more logical to focus solely on relativity from the outset?

The truth is that gravity is a force, not a curvature of spacetime as Einstein postulated. Space and time are abstract extensions, not physical entities, and thus cannot behave as relativistic interpretations suggest. What appears as an expanding spacetime is not a physical expansion but an indefinite extension of spatial and temporal measurements due to ever-changing existential events.

To truly understand the physical world, one must respect the foundational principles of general physics and classical mechanics rather than accept flawed relativistic interpretations uncritically. Science thrives on objective reasoning, not consensus or ideological influence.

Mathematical Derivation of Frequency Shift and Phase Transition in Extended Classical Mechanics (ECM)

Soumendra Nath Thakur

Correspondence : 

postmasterenator@gmail.com ; postmasterenator@telitnetwork.in

DOI: http://dx.doi.org/10.13140/RG.2.2.36663.02721

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

Phase Shift, Frequency Transition, Extended Classical Mechanics (ECM), Big Bang Energy Evolution, Planck-Scale Dynamics,

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:

1. 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
2. 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

Clarifying ECM’s Energy-Mass Perspective: Addressing Misconceptions and Reaffirming Core Principles

March 23, 2025

Dear Mr. Gary Stephens,

I appreciate your engagement in this discussion. However, your reference to relativistic simultaneity and the associated "Relativity of Simultaneity wiki (c - v).png" is misaligned with the core objective of this discussion, which is focused on the dynamics of massless particles in Extended Classical Mechanics (ECM).

As I have already outlined in my previous response to Ms. Larissa Borissova , ECM refines classical mechanics without relying on relativistic space-time constructs. Unlike relativity, which attributes gravitational effects to space-time curvature and geodesic motion, ECM establishes that mass arises as a consequence of motion and gravitational dynamics. This results in a distinct mass-energy relationship where massless particles, such as photons, experience effective acceleration under gravitational influence, governed by the interplay between negative apparent mass (-Mᵃᵖᵖ) and effective mass (Mᵉᶠᶠ).

Your reference to relativistic simultaneity disregards these principles by reintroducing relativistic velocity transformations, which are not applicable within ECM’s framework. Moreover, the relativistic interpretation of the speed of light (c) in relation to an observer’s speed (S) does not engage with the fundamental and precise understanding that ECM provides regarding light’s speed beyond relativistic constraints.

ECM rigorously incorporates the role of negative apparent mass and gravitational interactions to explain how photons dynamically behave in various energy-mass frameworks. This perspective naturally accounts for observational phenomena, including redshift, without invoking relativistic postulates. Furthermore, the Planck scale imposes fundamental limits on measurements, ensuring that beyond these limits, conventional descriptions—including relativistic simultaneity—lose physical significance.

Therefore, I encourage discussions to remain aligned with ECM’s principles rather than reverting to relativistic constructs that do not directly engage with the established framework presented here. If you wish to engage in a focused dialogue on the mathematical and physical consistency of ECM, I welcome it. However, introducing relativistic simultaneity into this discussion is neither relevant nor necessary to address the dynamics of massless particles within ECM.

Best Regards,

Soumendra Nath Thakur