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.

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.
   PDF Link