Appendix 6: Angular-Time Correspondence in Extended Classical Mechanics — ∏ᵈᵉᵍ as a Physical Angular Object and Phase-Time Displacement Δt.

Soumendra Nath Thakur

Tagore’s Electronic Lab, WB, India

Email: postmasterenator@gmail.com| postmasterenator@telitnetwork.in

Date: June 11, 2025

Quantized Angular Objects and Time Displacement: Formalization of ∏ᵈᵉᵍ and T(θ°) = Tₓ° in Extended Classical Mechanics

Abstract:

Extended Classical Mechanics (ECM) requires all mathematical entities to correspond to real, measurable, physical structures. The abstract constant ∏, commonly regarded as a dimensionless scalar, is instead formalized here as an angular object ∏ᵈᵉᵍ, representing the measurable degree-equivalent of one radian. Simultaneously, the derived relation:

                   T(θ°) = Tₓ° = θ°/360f = Δt

Interprets angular phase shifts in real systems as measurable temporal displacements. These formulations extend ECM's core principle: every mathematical transformation reflects a physical redistribution—whether of mass, energy, or time—and all quantities must preserve dimensional identity.

1. Formalization of ∏ᵈᵉᵍ as a Physical Angular Object

In ECM, circular and rotational motion must reflect real physical angular displacements, not abstract ratios. Traditionally, ∏ represents the ratio of a circle’s circumference to its diameter, used across trigonometric and rotational contexts. However, ECM interprets this ratio in terms of real, countable angular units, resulting in the definition:

∏ᵈᵉᵍ = 180°/∏ ≈ 57.2948°

This value corresponds to the physical angular span subtended by one radian in a circle when expressed in degrees. Rather than treating ∏ as dimensionless, ECM treats ∏ᵈᵉᵍ as an angular object with measurable identity. The number of such angular units required to span a half-circle becomes:

180°/∏ᵈᵉᵍ ≈ 3.14158

So we express:

180° = 3.14158 × ∏ᵈᵉᵍ 

This formulation matches ECM’s unit consistency protocol and parallels ECM's other physicalised constructs, such as phase-time shifts and energy-based deformation.

2. Angular Phase Shift as Temporal Displacement in ECM

Just as angular constants are converted into physical angular displacements, ECM requires phase shifts to represent real temporal displacement. When a system oscillates at a frequency f, and undergoes an angular shift θ° or ₓ°, the corresponding time shift Δt is given by:

T(θ°) = Tₓ° = θ°/360f  = Δt

Where:

• θ° or ₓ°: Angular shift in degrees

• f: Oscillatory frequency (Hz)

• Δt: Actual physical time delay due to angular offset

This is consistent with earlier ECM derivations of time modulation due to angular displacement in rotating or oscillating systems. It physically represents the temporal redistribution required to generate a phase delay in systems such as waveforms, rotating fields, or piezoelectric deformations.

Illustrative Example:

For a 90° phase shift at 50 Hz:

T(90°) = 90/(360 × 50) = 1/(4 × 50) = 0.005 sec

This quantifies how an angular rotation of 90° corresponds to a real delay of 0.005 seconds in the waveform or rotating field, causing time distortion in the waveform.

3. Physical Implications in ECM Modelling

The dual application of ∏ᵈᵉᵍ and T(θ°) or Tₓ° supports ECM’s unified treatment of geometry and time:

• In circular or rotational geometry, ∏ is no longer abstract but counts as ∏ᵈᵉᵍ units of angular displacement.

• In periodic systems, angular displacements translate into real temporal redistributions, measurable as Δt.

These relations find application in:

• Rotor and gyroscopic dynamics

• Phase-shifted electrical signals

• Electromechanical resonance

• Polarized wave front modulation

• Photon delay or advancement due to angular phase in ECM field theory

Conclusion:

The reinterpretation of ∏ as ∏ᵈᵉᵍ and the derivation of:

T(θ°) or Tₓ° = θ°/360f or ₓ°/360f  

Collectively advance ECM’s central thesis: all observable effects—geometric, temporal, or energetic—must be grounded in real, quantifiable displacements. These constructs replace dimensionless scalars with physically representative units, aligning ECM’s language with observable structure and enforcing continuity across mass, space, and time.

A Unified Interpretation of Gravitational Dynamics and Quantum Consistency: Negative Apparent Mass and Mass Continuity in ECM

June 11, 2025

In Extended Classical Mechanics (ECM), "negative apparent mass" refers to a dynamic, not an intrinsic, property of matter that arises due to gravitational interactions. This concept is crucial for understanding mass continuity in ECM, where a portion of the matter's mass is redistributed under gravity, with the negative apparent mass opposing the gravitational confinement. 

Explanation:

1. Negative Apparent Mass:

ECM introduces the idea that a portion of a body's matter mass can be displaced as negative apparent mass when subjected to gravitational interaction. This displacement is not a fundamental property of the matter but rather an emergent effect related to the gravitational field. 

2. Effective Mass:

The effective mass in ECM is defined as the sum of matter mass and negative apparent mass. This means that the total force experienced by a body is determined by the combined effect of its matter mass and the opposing negative apparent mass. 

3. Mass Continuity:

ECM maintains the principle of mass continuity by suggesting that the apparent mass redistribution under gravity doesn't violate the conservation of mass. Instead, it highlights a dynamic process where matter mass is temporarily displaced or "converted" into apparent mass under the influence of gravity, which then opposes the gravitational field. 

4. Connection to Buoyant Forces:

The concept of negative apparent mass in ECM is sometimes compared to the buoyant force in fluids. Just as an object submerged in a fluid experiences an upward force due to the displaced fluid, a body under a gravitational field experiences an apparent "buoyant" force in the form of negative apparent mass. 

5. Implications for Gravitational Dynamics:

ECM's understanding of negative apparent mass and mass continuity helps explain various phenomena, including: 

Cosmic Expansion: The concept of negative apparent mass and its interaction with gravity can contribute to understanding the acceleration of the universe. 

Dark Energy: ECM suggests that dark energy might be related to the energy-equivalent mass of negative apparent mass, rather than a separate entity. 

Gravitational Repulsion: ECM provides a framework for understanding how negative apparent mass can contribute to gravitational repulsion at large scales. 

Motion of Massless Particles (Photons): The negative apparent mass can also influence the motion of massless particles like photons, impacting their trajectory and energy. 

6. Consistency with Quantum Principles:

The ECM framework aims to be consistent with established quantum principles, and the concept of negative apparent mass is linked to quantum mechanical frequency relations, ensuring that it aligns with quantum mechanics.

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