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

Reaffirming ECM’s Foundations: A Response to Misinterpretations of Mass-Energy Dynamics:

March 22, 2025                                  ResearchGate Discussion Link

Dear Ms. Larissa Borissova ,

I appreciate your perspective on the expansion of mathematical frameworks in theoretical physics. However, your assertion that such expansions do not fit within the framework of existing theories seems to overlook the fact that the expansion of the universe, as understood in modern cosmology, is based on extensive observational evidence and remains the most consistent theory describing cosmic evolution.

Scientific theories evolve precisely because of the need to reconcile observational data with theoretical models, and ECM follows this principle by refining classical mechanics rather than relying on relativistic space-time constructs.

Your discussion of space-time in GTR, the role of singular surfaces, and the hypothetical connection to dark matter is certainly intriguing. However, it does not directly engage with the core objective of this discussion, which focuses on the dynamics of massless particles in ECM. The framework I have outlined does not rely on vacuum energy concepts associated with dark energy. Instead, ECM establishes that mass arises as a consequence of motion and gravitational dynamics, offering a distinct perspective from relativistic mass-energy interpretations.

ECM provides an alternative formulation where massless particles, such as photons, experience effective acceleration under gravitational influence, governed by the interplay between negative apparent mass and effective mass. This framework allows for a coherent force-energy relationship without invoking relativistic geodesics or space-time curvature. Unlike GTR, which attributes gravitational effects to the curvature of space-time, ECM derives mass and energy interactions from fundamental mechanical principles.

If you wish to engage in a discussion on mass-energy dynamics within ECM, I would welcome a focused dialogue on the mathematical and physical consistency of the framework rather than a broader discourse on relativistic space-time theories.

Best Regards, 

Soumendra Nath Thakur

The Coma Cluster as a Testbed for Extended Classical Mechanics (ECM):

March 21, 2025

The Coma Cluster, a large galaxy cluster, provides crucial observational data for testing and refining theories about dark energy and gravity. Specifically, it reveals the presence of dark matter and the influence of dark energy's antigravity, particularly at its outer edges.

ECM's Explanation of Dark Energy's Effects:

Extended Classical Mechanics (ECM) offers an alternative to the standard cosmological constant model of dark energy. It proposes that dark energy's influence can be explained by refining classical mechanics, particularly through the concept of "effective mass" (Mᵉᶠᶠ<0). This effective mass accounts for the interplay between matter mass (Mᴍ) and dark energy's antigravity.

Alignment with A.D. Chernin's Research:

A.D. Chernin's studies on the Coma Cluster and dark energy strongly support ECM's principles. His research demonstrates that:

• Antigravity dominates at large radii (R ≳ 14 Mpc).
• The concept of effective mass, which can become negative, aligns with observed antigravity effects.
• Local antigravity effects are also observed.
• The idea of a zero gravity radius, where objects are no longer gravitationally bound, is also supported.

ECM provides a framework that naturally explains these observations without requiring exotic vacuum energy interpretations. It derives antigravity effects from mass-energy interactions, offering a self-consistent model that aligns with observational data from the Coma Cluster.

Critique of Viscous Time Theory (VTT) in "The Informational Precipitation of the Universe: Beyond Hawking’s Fine-Tuning Paradox"

My comment of the post, 'The Informational Precipitation of the Universe: Beyond Hawking’s Fine-Tuning Paradox' https://www.facebook.com/share/p/1Edh9TQ8Q7/

1. VTT’s Attempt to Describe Expansion Without Established Fundamental Interactions  

From the quoted text, Viscous Time Theory (VTT) attempts to describe the expansion of the universe, a phenomenon that has been traditionally explained using the four fundamental interactions—primarily gravity. However, VTT introduces an extra-fundamental interaction (informational precipitation and density gradients) without properly defining or establishing it.  

This leads to several logical problems:

  

- Expansion is an observed phenomenon explained by fundamental interactions (mainly gravity). Any new theory that replaces gravity’s role must provide a more rigorous explanation, not just a conceptual alternative.  

- VTT does not properly establish how its extra-fundamental interaction functions physically. Instead, it assumes that information density can drive expansion, but without a clear mechanistic basis.  

- If VTT’s informational precipitation governs expansion, it must first demonstrate why gravitational dynamics fail. Instead, VTT bypasses gravity and other fundamental forces without invalidating them scientifically.

Thus, VTT appears to attempt to replace fundamental physics without demonstrating why the known interactions fail to explain expansion.

2. Overruling Known Fundamental Interactions Without Justification 

A scientifically consistent theory must **either integrate existing fundamental principles or logically replace them by proving they are insufficient. However, VTT does not do this. Instead, it:  

- Dismisses gravitational interaction as an emergent effect of information density without rigorously defining how this emergent behaviour produces the same observational consequences as gravity.  

- Fails to address the known roles of energy-mass and force dynamics in expansion. Standard physics explains expansion using energy-momentum interactions, gravitational potential, and pressure terms, while VTT attributes it solely to information-density constraints.  

- Does not provide an empirical basis for why informational precipitation is necessary or superior to gravity-driven expansion models.  

Without a proper framework proving why gravity, dark energy, or even inflation fields fail, VTT’s dismissal of fundamental interactions is unfounded and speculative.

3. Can a Theory Be Considered Rational When It Ignores Fundamental Physics Without Invalidating It? 

A rational scientific theory must meet the following criteria:  

1. It must be based on empirical evidence.  

2. It must address known observations better than existing theories.

3. It must be falsifiable, making predictions that can be tested.

4. It must integrate with or improve upon established physics rather than arbitrarily replacing it.

VTT does not meet these criteria because:  

- It does not provide a falsifiable mechanism to distinguish informational precipitation from gravitational interaction.  

- It does not offer testable predictions that are distinct from existing physics.  

- It replaces fundamental interactions arbitrarily without showing where they fail.  

- It lacks empirical grounding, as "informational density gradients" are not well-defined physical quantities within tested physics frameworks.  

Thus, VTT is not a scientifically consistent theory in its current form because it replaces well-established physics with an ill-defined concept without proving the need for such a replacement.

4. Is VTT Scientifically Consistent?  

No, VTT is not scientifically consistent for the following reasons:  

It does not provide a mechanism for how informational interactions replace gravity. 

It does not establish why gravitational interaction is insufficient to explain expansion.  

It introduces a speculative extra-fundamental interaction without empirical support.  

It disregards fundamental physics without proving its necessity.

Final Verdict:  

- VTT is speculative and lacks scientific rigor.

- It is not a rational replacement for fundamental interactions unless it provides empirical validation and testable predictions.  

- It currently stands as a philosophical interpretation rather than a physical theory.

- Soumendra Nath Thakur

 March 21, 2025