Soumendra Nath Thakur | ORCiD: 0000-0003-1871-7803
October 21, 2025
Extended Classical Mechanics (ECM) proposes that gravity is not a manifestation of spacetime curvature, as described by General Relativity (GR), but a force-based interaction emerging from a mass-binding condition between mass and energy. In this framework, gravitational attraction and cosmic expansion are both understood as consequences of this binding and its natural release — rather than as geometric effects of spacetime deformation. ECM introduces the concept of negative apparent mass (−Mᵃᵖᵖ) to represent the repulsive, energy-releasing phase of this interaction, thereby offering a physical explanation for phenomena attributed to dark energy and gravitational repulsion. [1, 2, 3]
Mathematical Derivation and Observational Basis:
A detailed mathematical derivation, supported by observational data, is presented here:
Gravity Beyond GR's Spacetime Curvature: ECM Mathematical Basis of the Phase Kernel Formalism
1. Conceptual Framework
The ECM Phase Kernel Formalism reinterprets gravity as a phase modulation of waves caused by cumulative phase variations arising from mass-energy distributions. This wave-based approach reproduces the measurable predictions of GR in weak-field conditions — including Shapiro delay, gravitational lensing, and planetary perihelion precession— but attributes them to frequency and time distortions, not geometric curvature.
In ECM, gravitational influence acts as an effective refractive index that slows the phase velocity of waves and matter traversing a gravitational field. [9]
2. Gravity in ECM vs. General Relativity (GR)
General Relativity (GR):
● Interprets gravity as the curvature of spacetime produced by mass and energy. [4, 5, 6]
● Objects move along geodesics — the shortest paths — in this curved spacetime, which we perceive as gravitational motion. [7]
Extended Classical Mechanics (ECM):
● Describes gravity as a force-driven rather than geometry-driven effect. [2]
● Defines gravity as a mass-binding condition — a dynamic equilibrium that can bind or release mass-energy. [1]
● Explains cosmic expansion as the release of this bound state, independent of any spacetime expansion. [2, 8]
3. Key Concepts in ECM
• Mass-Binding Condition
Redefines gravity as a physical condition of mass-energy binding, providing a unified mechanism for subatomic, astrophysical, and cosmological phenomena. [1]
• Negative Apparent Mass (−Mᵃᵖᵖ)
Represents the repulsive or unbinding aspect of mass-energy interactions. This is not a “negative particle mass” but a derived gravitational term associated with kinetic energy, radiation, or field momentum. [2, 3]
This framework allows ECM to model cosmic acceleration and dark energy effects without invoking spacetime curvature.
• Force-Based Explanation
By maintaining a force-centric interpretation, ECM aims for a physically grounded and empirically measurable understanding of gravitational and cosmological effects. [2]
4. Implications of the ECM Model
Cosmic Expansion:
The acceleration of the universe arises from the evolving relationships among mass-energy terms — not from an expansion of spacetime. [8]
Dark Energy:
The effects attributed to dark energy emerge from the negative apparent mass (−Mᵃᵖᵖ) component inherent in radiation and kinetic energy fields. [3]
Reexplanation to GR:
ECM provides a consistent, classical-mechanical reexplanation that reproduces GR-like results under weak-field limits, while extending applicability to quantum and cosmological domains. [2]
5. Core Principles of the ECM Phase Kernel Formalism
• Gravity as Phase Modulation
Gravity manifests as a cumulative phase shift affecting all oscillatory systems — matter waves, photons, and fields — due to the presence of mass-energy.
• Time Dilation and Shapiro Delay
Gravitational time dilation corresponds to a progressive slowing of an internal beat frequency or oscillation rate as a system approaches higher gravitational potential — mathematically equivalent to the Shapiro delay observed in GR. [9]
• Unified Lensing Model
Combines geometric and Shapiro-type delays within a single phase-based framework, treating lensing as a coherent phase-variation process rather than geometric bending.
• Perihelion Precession
Interprets orbital precession as an angular phase perturbation, not as motion along a curved spacetime path.
• Testable Predictions
The model’s predictions can be verified by fitting phase-shift coefficients to precision datasets — e.g., Cassini spacecraft tracking, VLBI, and pulsar timing — to identify possible deviations from GR. [9]
References
[1] Definition of Gravity in Extended Classical Mechanics (ECM) https://www.researchgate.net/post/Definition_of_Gravity_in_Extended_Classical_Mechanics_ECM
[2] Gravitating Mass as an Emergent Polarity-Governed Quantity in ECM https://www.researchgate.net/post/Gravitating_Mass_as_an_Emergent_Polarity-Governed_Quantity_in_ECM
[3] ECM-Based Rebuttal on the Gravitational Nature and Weight of Massless Waves https://www.researchgate.net/post/ECM-Based_Rebuttal_on_the_Gravitational_Nature_and_Weight_of_Massless_Waves
[4] Einstein’s General Theory of Relativity Explained – space.com https://www.space.com/17661-theory-general-relativity.html
[5] Understanding GR: Educational Video Reference https://www.youtube.com/watch?v=c2Gv3Th6UM8
[6] Recent Study on Spacetime Curvature Dynamics – ScienceDirect https://www.sciencedirect.com/science/article/abs/pii/S0960077924012591
[7] Einstein’s Curved Spacetime Visualization https://www.youtube.com/watch?v=P4kDyWs_IuI
[8] Application of Extended Classical Mechanics (ECM) – In Brief 2 https://www.researchgate.net/post/Application_of_Extended_Classical_Mechanics_ECM-In_Brief2
[9] ECM Phase Kernel Formalism – Mathematical Framework http://www.telitnetwork.itgo.com/ExtendedClassicalMechanics/ECMPhaseKernel/
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