Abstract

Extended Classical Mechanics (ECM) introduces a scale-sensitive and physically differentiated treatment of gravitation, antigravitation, and photon dynamics. In contrast to conventional cosmological models, ECM separates mass-dominated gravitational balance from massless kinetic manifestation, enabling a precise interpretation of the zero-gravity radius, photon kinetic termination, and the role of negative apparent mass. This work formalizes the domain of validity of the zero-gravity radius, establishes its coincidence with photon kinetic limits in localized systems, and clarifies its breakdown at the universal scale.

Keywords: Extended Classical Mechanics; Photon Dynamics; Zero-Gravity Radius; Negative Apparent Mass; Dark Energy; Kinetic Energy Manifestation; Gravitational Redshift

One Important and Non-Trivial Distinction

One important and non-trivial distinction must be emphasized. The zero-gravity radius R_ZG is defined with respect to the total gravitational mass M_G, incorporating both manifested matter mass M_M and effective dark-energy mass M_DE. As such, R_ZG is not a photon-specific construct.

In Extended Classical Mechanics (ECM), the radius r_max is instead associated with dynamic kinetic manifestation, specifically KE_ECM, and becomes relevant to photons only through kinetic evolution under gravitational influence.

Integrated ECM Treatment of R_ZG, r_max, and Scale-Dependent Gravitation

The zero-gravity radius R_ZG is meaningful only within gravitationally interacting systems such as galaxies, clusters, and superclusters. Its existence presupposes a genuine tug-of-war between inward gravitational attraction due to matter mass and outward antigravitational influence associated with effective dark-energy mass.

Physically, R_ZG functions analogously to a Lagrange-type equilibrium point, not between discrete bodies but between competing mass manifestations. This balance cannot exist in the absence of mutual gravitation.

When a photon is emitted from a source embedded within a collective galactic mass distribution, its gravitational redshift proceeds until the point where gravitational extraction of kinetic energy ceases. ECM predicts that this kinetic termination radius coincides locally with the zero-gravity radius:

r_max^local = r_max^source = R_ZG.

Photon Dynamics under Gravitational–Antigravitational Influence in Extended Classical Mechanics

ECM treats photons as massless carriers of kinetic energy rather than as rest-mass–bearing particles. Their interaction with gravitation therefore proceeds through kinetic energy conversion:

ΔKE_ECM ↔ −ΔPE_ECM.

Gravitational fields extract kinetic energy from photons via redshift only while a net gravitational influence exists. Beyond R_ZG, antigravitational dominance prevents further kinetic extraction.

Thus, photon dynamics in ECM are governed by gravitational–antigravitational influences through kinetic energy evolution, not through mass attraction.

Conceptual Correction (ECM-Consistent)

While conventional cosmology often states the dominance condition as M_DE > M_M, ECM identifies dark energy with negative apparent mass:

M^app = −ΔPE_ECM,Universe < 0.

The governing cosmic relation in ECM is therefore:

−M^app > M_M.

This expresses that unmanifested negative apparent mass dominates all manifested matter mass, producing a purely antigravitational universal regime. Consequently, no zero-gravity radius or equilibrium point exists at the universal scale.

Conclusion

Extended Classical Mechanics provides a structurally distinct and physically consistent framework for understanding photon dynamics under gravitational and antigravitational influence. By separating mass-dominated gravitational balance from massless kinetic evolution, ECM explains the coincidence of r_max and R_ZG in localized systems while rigorously excluding the concept at the universal scale. This nuanced hierarchy is not a reinterpretation of existing cosmology but a foundational extension of classical mechanics.

References

Chernin, A. D., Bisnovatyi-Kogan, G. S., Teerikorpi, P., Valtonen, M. J., Byrd, G. G., & Merafina, M. (2013). Dark energy and the structure of the Coma cluster of galaxies. Astronomy & Astrophysics, 553, A101. https://doi.org/10.1051/0004-6361/201220781

Thakur, S. N., & Bhattacharjee, D. (2023). Phase shift and infinitesimal wave energy loss equations. Zenodo, CERN. https://doi.org/10.5281/zenodo.14603601

Thakur, S. N. (2024). Extended Classical Mechanics: Vol-1 – Equivalence Principle, Mass and Gravitational Dynamics. Preprints.org. https://doi.org/10.20944/preprints202409.1190.v3

Thakur, S. N. (2024). A symmetry and conservation framework for photon energy interactions in gravitational fields. Preprints.org. https://doi.org/10.20944/preprints202411.0956.v1

Thakur, S. N. (2024). Photon Dynamics in Extended Classical Mechanics: Effective Mass, Negative Inertia, Momentum Exchange and Analogies with Dark Energy. Preprints.org. https://doi.org/10.20944/preprints202411.1797.v1

Thakur, S. N. (2025). A comparative framework for Extended Classical Mechanics’ Frequency-Governed Kinetic Energy. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.5380919

Thakur, S. N. (2025). A Unified Framework Linking Entropy, Time Distortion, Gravitational Dynamics, and Cyclic Cosmology through Extended Classical Mechanics. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.5494689

Thakur, S. N. (2025). Foundational Formulation of Extended Classical Mechanics: From Classical Force Laws to Relativistic Dynamics. Preprints.org. https://doi.org/10.20944/preprints202504.1501.v1

Thakur, S. N. (2025). A dual framework for rest and motion states: energy and mass dynamics in Extended Classical Mechanics. EasyChair Preprint. https://easychair.org/publications/preprint/cfqr