Frequency-Governed Mass, Apparent Mass Gradients, and the Emergence of Gravitation in Extended Classical Mechanics

Abstract

This work presents a unified Extended Classical Mechanics (ECM) formulation of gravitational and cosmic redshift grounded in a foundational principle of frequency-governed mass variation. Photon energy is treated as negative apparent mass (NAM), whose radial redistribution under inverse-square field structure produces phase shift and time distortion. A closed-form ECM redshift function f(r) is derived directly from apparent mass gradients, unifying local gravitational redshift and cosmological redshift without invoking spacetime curvature, metric expansion, or intrinsic photon energy dissipation.

Keywords

Extended Classical Mechanics, Frequency-Governed Mass, Negative Apparent Mass, Phase Shift, Time Distortion, Inverse-Square Law, Gravitational Redshift, Cosmic Redshift

Foundational Axiom of Extended Classical Mechanics (ECM)

Extended Classical Mechanics (ECM) is founded on the principle that gravity and antigravity are not fundamental forces, but emergent manifestations of frequency-governed mass variation. All gravitational phenomena arise from the spatial and temporal redistribution of apparent mass, particularly negative apparent mass (NAM), encoded through frequency.

ΔKE_ECM = hf = −ΔM^appc²

A radial concentration of frequency-bound apparent mass produces gravitational attraction, while radial dilution through cumulative phase shift produces antigravitational behaviour. Gravitational and cosmological redshift are therefore manifestations of phase-induced time distortion, not consequences of spacetime curvature or intrinsic photon energy loss.

All subsequent formulations in this work—photon phase shift, time distortion, inverse-square apparent mass gradients, and the closed-form ECM redshift law—follow directly from this axiom.

Foundational Axioms: Time in Extended Classical Mechanics (ECM)

Axiom T1 – Reciprocal Definition of Time:
Time is defined through the reciprocal relation between frequency and temporal interval:

f = 1 / Δt,     Δt = 1 / f = T

where f is the oscillatory frequency, Δt the fundamental temporal interval, and T the full oscillatory period of the process.

Axiom T2 – Phase-Dependent Temporal Intervals:
Observable time intervals may correspond to only a fraction of a complete oscillatory cycle:

Δt = T (x° / 360)

where x°/360 represents the phase fraction of the cycle. This fraction may vary due to external influences such as motion, gravitational interaction, thermal processes, or chemical reactions.

Axiom T3 – Standard Clock Time:
Standard clock time is defined as the reference condition in which f>0 remains fixed and stable. In this state, the phase increment approaches zero (x → 0), ensuring that temporal intervals are constant and independent of external factors.

Axiom T4 – Cosmic Time and Phase Deviation:
Cosmic time corresponds to the effective temporal progression of physical processes. Deviations from clock time arise when the phase fraction varies under entropic or environmental influences:

Δt = t_cos − t_cl = T (x° / 360)

where t_cos is cosmic time, t_cl is the clock-time reference, and the right-hand side represents phase-evolved temporal increments.

Interpretation: Time in ECM emerges from oscillatory phase progression, with clock time providing a stable reference, while cosmic time reflects the influence of environmental and entropic factors.

Photon Phase Shift and Time Distortion

In ECM, a photon’s frequency is a direct measure of its negative apparent mass. Interaction with a gravitational or antigravitational field induces a phase shift x° in the photon’s oscillation, resulting in time distortion relative to the emission clock:

Δt(x) = (x° / 360°)(1 / f_{photon, emission})

This time distortion alters the observed frequency while preserving electromagnetic coherence. The process accumulates continuously as the photon propagates radially outward.

Inverse-Square Apparent Mass Gradient

ECM fields obey inverse-square structure. Consequently, the magnitude of photon negative apparent mass varies radially according to:

dM^app_ph/dr ∝ −1/r²

Introducing a source-associated apparent mass strength M^app_src, the radial NAM gradient is written as:

dM^app_ph/dr = −κ M^app_src / r²

Integrating from the emission radius r₀ to r yields:

ΔM^app_ph(r) = −κ M^app_src (1/r − 1/r₀)

For r ≫ r₀, this reduces to:

ΔM^app_ph(r) ≈ −κ M^app_src/r

Closed-Form ECM Redshift Function

Since photon frequency is linearly proportional to the magnitude of negative apparent mass, the radial frequency evolution follows:

f(r)/f₀ = |M^app(r)| / |M^app₀|

Defining the ECM characteristic radial scale:

r_s ≡ κ M^app_src / |M^app₀|

the closed-form ECM redshift law becomes:

f(r) = f₀(1 − r_s/r)

Conclusion

By grounding gravitation in frequency-governed mass variation, ECM provides a unified physical explanation of gravitational and cosmological redshift. Gravity and antigravity emerge as complementary manifestations of radial redistribution of negative apparent mass governed by inverse-square field structure.

Frequency—not spacetime geometry—thus serves as the fundamental mediator of mass, energy, gravitation, and cosmic evolution within Extended Classical Mechanics.

References

1. S. N. Thakur, Phase Shift and Infinitesimal Wave Energy Loss Equations, Longdom Publishing.
   https://www.longdom.org/open-access/phase-shift-and-infinitesimal-wave-energy-loss-equations-104719.html
2. S. N. Thakur, A Nuanced Perspective on Dark Energy: Extended Classical Mechanics, Magnivel International.
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