09 April 2025
Mass–Energy Transformations in ECM: Reframing Kinetic Energy, Analysis of −Mᵃᵖᵖ, Gravitational Interaction, and the Role of Frequency in Mass–Energy Dynamics
Soumendra Nath Thakur
Abstract
This paper explores a novel reinterpretation of kinetic energy and mass–energy transformations within the Extended Classical Mechanics (ECM) framework, emphasizing the role of negative apparent mass (−Mᵃᵖᵖ) and gravitational interactions. While classical mechanics defines kinetic energy as Eᴋ = ½mv², ECM reveals that kinetic energy emerges not solely from inertial motion, but from a gravitationally-mediated redistribution of mass-energy, where −Mᵃᵖᵖ plays a central role.
For massless particles, such as photons, ECM introduces a dual-component energy structure—inherent energy (hf) and gravitational interactional energy—unified through an effective mass relation: Eₜₒₜₐₗ = hf = -Mᵉᶠᶠc² with -Mᵉᶠᶠ = f/c²
The photon’s apparent mass shifts from −2Mᵃᵖᵖ at emission to −Mᵃᵖᵖ as it escape gravitational influence, explaining gravitational redshift as a real energy loss due to field interaction, within gravitational field. This shift is also tied to ECM’s force law Fᴇᴄᴍ = −2Mᵃᵖᵖ × aᵉᶠᶠ, where effective acceleration reflects internal energy dynamics rather than changes in velocity, thereby preserving the constancy of the speed of light.
For massive particles, ECM retains the classical kinetic form but attributes the ½ coefficient to a dynamic balance between matter mass and negative apparent mass. As gravitational influence changes, −Mᵃᵖᵖ increases and effective mass decreases—mirroring a buoyant-like effect akin to Archimedes’ principle. This not only lowers the energy needed to accelerate but also provides a mechanistic explanation for why kinetic energy scales with v².
Ultimately, ECM reframes kinetic energy as a consequence of mass-energy redistribution rather than inertial motion alone. The paper concludes that the familiar ½mv² form reflects a deeper symmetry within gravitational systems, where negative apparent mass governs the dynamic interplay between motion, energy, and gravitational context—offering a consistent, physically grounded, and conceptually richer alternative to conventional interpretations.
Tagore's Electronic Lab, West
postmasterenator@gmail.com
Author declares no conflict of
interest.
Section I: Classical vs ECM Interpretation of Kinetic Energy.
The classical kinetic energy is:
Eᴋ = ½mv²
The question why the coefficients ½ m and v² appear, and ECM provides a meaningful reinterpretation.
While Classical Mechanics treats kinetic energy as a function of inertial mass and velocity, Extended Classical Mechanics (ECM) reveals that kinetic energy is not merely an inertial property but dynamically arises from gravitationally-mediated energy redistribution via negative apparent mass (−Mᵃᵖᵖ). This section unpacks how such reinterpretation leads to a more unified, dynamic understanding of mass-energy behaviour under gravitational influence.
Total Energy of Massless Particles like Photons: Manifestation through Redshift and Gravitational Dynamics
In ECM, the total energy of a photon at emission is expressed as:
Eₜₒₜₐₗ = Eᵢₙₕₑᵣₑₙₜ + Eg (ᵢₙₜₑᵣₐᴄₜᵢₒₙₐₗ)
This formulation recognizes two components:
• Eᵢₙₕₑᵣₑₙₜ: The intrinsic energy of the
photon, given by hf.
• Eg: The gravitational interactional energy, which varies with radial distance r from a gravitational source and modifies the total energy dynamically.
This interpretation connects with the mass-energy equivalence expression hf/c², which in ECM corresponds to a negative apparent mass -Mᵃᵖᵖ and, when gravitational interaction is active, to an effective mass -Mᵉᶠᶠ:
hf/c² ≡ -Mᵃᵖᵖ and -Mᵉᶠᶠc² ≡ Eₜₒₜₐₗ
Energy Shift: From −2Mᵃᵖᵖ to −Mᵃᵖᵖ
At the moment of emission within a gravitational field, the total apparent mass of the photon is:
−2Mᵃᵖᵖ = −Mᵃᵖᵖ,ᵢₙₕₑᵣₑₙₜ −Mᵃᵖᵖ,ᵢₙₜₑᵣₐᴄₜᵢₒₙₐₗ
As the photon climbs out of the gravitational well, it loses the interactional component of its apparent mass, resulting in:
Eₜₒₜₐₗ,ᵢₙᵢₜᵢₐₗ = −2Mᵃᵖᵖc² ⇒ Eₜₒₜₐₗ,ꜰᵢₙₐₗ = −Mᵃᵖᵖc²
This energy loss physically manifests as:
• Gravitational redshift: A reduction in frequency
f, since E = hf, observed when the photon moves away from a gravitational
field.
• Curvature of the photon’s path in gravitational field: As governed by the ECM force law Fᴇᴄᴍ = −2Mᵃᵖᵖaᵉᶠᶠ, where the acceleration aᵉᶠᶠ is itself a function of gravitational potential.
Thus, the drop from −2Mᵃᵖᵖ to −Mᵃᵖᵖ signifies the release of gravitational binding energy, aligning with potential energy difference ΔPE. This conversion explains the observed redshift not merely as a relativistic or metric effect, but as a mass-energy redistribution process that maintains conservation within ECM.
Consistent Frequency Energy Radius Dynamics
As r increases:
• The gravitational influence weakens,
• Eg decreases,
• aᵉᶠᶠ drops from 2c toward c,
• f decreases (redshift),
• -Mᵉᶠᶠ becomes less negative,
• And photon energy asymptotically approaches its purely inherent value.
This dynamic is fully consistent with:
Eₜₒₜₐₗ = hf = -Mᵉᶠᶠc² with -Mᵉᶠᶠ = f/c²
Hence, the photon's energy response to gravity is not due to velocity change (as speed remains c), but due to a mass-energy shift driven by the redistribution of negative apparent mass and the gravitational interaction it undergoes.
Section II: Gravitational Fields and Mass-Energy Shifts.
ECM Interpretation for Massless Particles: Redefining KE via −2Mᵃᵖᵖ (e.g. Photon):
Key Claims:
1. Massless particles have a total negative apparent mass of −2Mᵃᵖᵖ in a gravitational field.
2. This −2Mᵃᵖᵖ consists of:
• One part from inherent energy,
• One part from gravitational interactional energy.
3. The ECM force becomes:
Fᴇᴄᴍ = −2Mᵃᵖᵖ × aᵉᶠᶠ
This is an effective force, not classical, emerging from the interplay between −Mᵃᵖᵖ and gravitational acceleration in ECM. It does not imply rest inertia, but dynamic energy transfer through field interaction.
4. Effective acceleration aᵉᶠᶠ is:
• 2c within gravitational influence,
• c just after escaping the gravitational field.
• Despite effective acceleration appearing to be greater than c (e.g., aᵉᶠᶠ = 2c), this does not violate speed of light since massless particles cannot exceed c within gravitationally bound system. Instead, ECM interprets this as a redistribution of internal energy states, not a literal increase in speed.
5. The energy loss corresponds to a shift from:
• −2Mᵃᵖᵖ ⇒ −Mᵃᵖᵖ,
• This implies energy loss due to gravitational influence or redshift.
Evaluation:
• It is an elegant assignment of kinetic energy KE = ½mv²
to −2Mᵃᵖᵖ through the factor ½ (−2Mᵃᵖᵖ) = −Mᵃᵖᵖ.
• It draws symmetry between the mathematical form of
kinetic energy and the effective interactional energy dynamics of the photon
under gravity.
• The fact that energy loss of the photon corresponds to a reduction in negative apparent mass is consistent with the redshift interpretation (photon losing energy when climbing out of a gravitational well).
This supports the idea that the ½ factor is a mathematical reflection of energy sharing between interactional and inherent components of −Mᵃᵖᵖ.
Section III: Velocity and Acceleration of Massless Particles.
The assertion is that:
• Despite effective acceleration appearing to be greater
than c (e.g., aᵉᶠᶠ = 2c), this does not violate
relativity since massless particles cannot exceed c. Instead, ECM interprets
this as a redistribution of internal energy states, not a literal increase in
speed.
• The product −2Mᵃᵖᵖaᵉᶠᶠ, where aᵉᶠᶠ = 2c yields a net velocity component constrained to c, maintaining
the constancy of speed of light.
• Energy loss is reflected in the reduction of effective mass (Mᵉᶠᶠ) rather than speed.
Evaluation:
• This satisfies the observational postulate of light
speed constancy, while offering a mechanical reinterpretation of how that
constancy is maintained (via internal energy redistribution rather than
velocity change).
• Thus, ECM introduces a mass-energy redistribution mechanism instead of speed adjustment.
Section IV: For Massive Particles.
Key Statements:
1. The standard kinetic energy KE = ½mv² holds.
2. As massive particles move away from gravity wells, Mᵉᶠᶠ decreases due to increasing −Mᵃᵖᵖ.
3. Due to increased −Mᵃᵖᵖ, the effective inertia (Mᵉᶠᶠ = Mᴍ − Mᵃᵖᵖ) decreases, reducing the energy required to achieve a given acceleration.
Evaluation:
• This reflects a gravitational buoyancy-like behaviour,
analogized to Archimedes' principle (as previously proposed).
• The increase in negative apparent mass acts to dynamically reduce effective inertia, explaining why ½mv² becomes meaningful — it represents a balance between real mass and apparent reduction due to field conditions.
Section V Consistency and Answer to the Original Question: Why ½mv² and ECM's Role
ECM Interpretation:
• The ½ factor arises as a reflection of the shared
contribution between the real (positive) mass and the negative apparent mass
(which corresponds to kinetic energy) components in motion and gravitational dynamics.
• The v² term reflects the squared nature of acceleration and how energy scales with the effective acceleration product in both massless and massive scenarios.
Conclusion: ECM Consistency & Enhancement
• Mathematical consistency: This derivation and reasoning
maintain internal consistency within ECM's framework.
• Where relativity views kinetic energy as a
Lorentz-transformed rest energy, ECM frames it as an emergent product of
gravitational redistribution — offering mechanical insights into why KE ∝
v² rather than just
mathematical necessity.
• Physical consistency: ECM offers deeper insight into why
kinetic energy takes the form it does — by attributing it to energy
redistribution due to motion and gravitational influence, especially negative
apparent mass dynamics.
• Conceptual depth: ECM extends classical mechanics by
explaining how gravitational energy fields and effective mass variation shape
the apparent motion energy forms (i.e., kinetic energy expression).
• The ½ factor becomes a symmetry artifact, not just a mathematical convenience.
Conclusion:
Extended Classical Mechanics (ECM) provides a profound reinterpretation of kinetic energy, mass, and gravitational interaction by introducing the concept of negative apparent mass (−Mᵃᵖᵖ) as a dynamic agent of energy redistribution. Through this lens, kinetic energy is no longer a static property derived solely from inertial mass and velocity, but a manifestation of internal gravitational rebalancing between inherent and interactional components of energy.
For massless particles like photons, ECM demonstrates how total energy evolves from an initial state of −2Mᵃᵖᵖc² (inherent + interactional) to −Mᵃᵖᵖc² as the particle escapes a gravitational field. This drop in effective mass is observed as gravitational redshift, reinforcing that the photon's energy change arises not from velocity variation but from the loss of gravitational coupling energy. The concept of effective acceleration (aᵉᶠᶠ), peaking at 2c within a gravitational field, reflects energy transformation mechanics without violating relativistic speed constraints.
For massive particles, ECM shows that the increasing −Mᵃᵖᵖ during motion through gravitational gradients results in a decreasing effective mass (Mᵉᶠᶠ = Mᴍ − Mᵃᵖᵖ), reducing the energetic cost of acceleration. This mirrors a gravitational buoyancy effect, analogous to Archimedes' principle, where negative apparent mass offsets inertial resistance.
Ultimately, ECM not only explains the ½mv² form of kinetic energy as a dynamic interplay between matter and apparent mass components, but it also embeds mechanical meaning into abstract constants. The "½" factor arises naturally from the division of energy contributions (interactional and inherent), while the v² term emerges from the squared relationship of effective acceleration and energy scaling.
This framework resolves conceptual limitations in both classical and relativistic views, offering a logically consistent, physically grounded, and gravitationally dynamic reinterpretation of kinetic energy and mass–energy equivalence. ECM thus serves not just as a modification of classical mechanics, but as a comprehensive enhancement capable of bridging gaps in our understanding of both massive and massless particles in gravitational contexts.
References:
[1] Dark
energy and the structure of the Coma cluster of galaxies. A. D. Chernin, G. S. Bisnovatyi-Kogan, P. Teerikorpi, M. J. Valtonen, G. G. Byrd, M.
Merafina. Astronomy and Astrophysics. Vol. 553, Art. no. A101, 2013.
https://doi.org/10.1051/0004-6361/201220781
[2] A
Nuanced Perspective on Dark Energy: Extended Classical Mechanics. Thakur. S. N.
http://doi.org/10.20944/preprints202411.2325.v1
[3]
Extended Classical Mechanics: Vol-1 - Equivalence Principle, Mass and
Gravitational Dynamics. Thakur, S. N.
https://doi.org/10.20944/preprints202409.1190.v3
[4]
Classical Mechanics: Systems of Particles and Hamiltonian Dynamics by H.
Goldstein, C. Poole, and J. Safko
[5] Dark
Matter and the Dinosaurs: The Astounding Interconnectedness of the
Universe" by Lisa Randall
The Foundations of Photon Dynamics, Measurement Systems, and Temporal Constructs Beyond Relativistic Constraints:
Soumendra Nath Thakur
April 09. 2025
Mr. Vikram Zaveri’s statement, “Fundamentally, time is periodic in nature. It is not linear like in modern science,” distorts the functional and empirical interpretation of time within physical frameworks—both classical and quantum—as well as observational cosmology. Time, in its operational sense, is defined through sequential events and causality. While periodicity is an inherent characteristic of oscillatory systems (such as wave mechanics), this does not imply that time itself is periodic. Periodicity applies to physical processes within time, not time as a dimension of measurement.
1. On the Definition of Speed of Light:
The claim that “the accurate definition of speed of light is c = λ/T and not d/t” unnecessarily restricts the concept of speed to wave periodicity and overlooks the broader physical definition of speed:
c = d/t
This classical definition remains valid and fundamental. The equation c = λ/T is a specific case of d/t, when considering a periodic wave. As such, both formulations are compatible. In fact, I have shown in the ECM framework:
ΔS = Δd/Δt ⇨ c = λ/Δt
This makes your equation c = λ/T a subset of a more general and foundational expression of speed. There is no contradiction, only contextual application.
2. On the ‘Periodic Invariant’ and Null Geodesic Argument:
Your reference to the “periodic invariant” s² = λ² - c² T² and the condition s² = 0 for photons indeed mimics the relativistic null geodesic condition. However, the Extended Classical Mechanics (ECM) approach intentionally moves beyond relativistic assumptions, particularly those that involve spacetime curvature or geodesics.
ECM posits that light's constant speed can be understood non-relativistically, by considering:
• the anti-gravitational nature of photons (via negative apparent mass),
• the negligible influence of observer motion in a dominant anti-gravitational frame, and
• the physical limits imposed by Planck-scale constraints, which cap wavelength and frequency at known extremes.
The assertion that light must satisfy a periodic invariant assumes the primacy of wave periodicity over the energetic and gravitational character of photon dynamics. ECM, on the other hand, recognizes that the speed of light emerges from energy-frequency-distance consistency within a mass-energy dominated framework, not necessarily from relativistic invariants.
3. On the Role of Observer’s Mass and Motion:
Your claim that “discussion on mass of the observer is irrelevant” misses a key point. The ECM framework treats observer dynamics not merely as a mathematical abstraction but as physically significant in interpreting measurement reference frames.
In a universe where negative apparent mass (-Mᵃᵖᵖ) and effective anti-gravitational dynamics dominate, the motion of an observer (with positive matter mass) becomes negligible in contrast. This is not a metaphysical notion but a measurable consequence of mass-energy interaction and frame dominance.
Hence, dismissing the mass-energy structure of the observer as irrelevant fails to acknowledge the measurement system’s physical embedding in the universe’s gravitational architecture.
4. On the Planck Scale and the Existence of Light:
The claim “Planck scale is questionable because light did not exist at Planck time” conflates existence of photons with the validity of Planck limits.
The Planck time (≈ 5.39 × 10⁻⁴⁴ s) marks the limit of meaningful physical measurements, not the emergence of specific particles like photons. Whether or not light existed at that instant is irrelevant to the fact that wavelengths shorter than ℓₚ and frequencies higher than fₚ lose physical meaning.
Planck limits define boundaries of physical resolution, not the actual timeline of photon creation.
Furthermore, ECM accommodates the emergence of light after the Planck era—especially near the symmetry-breaking scales—while still employing Planck boundaries as fundamental constraints on physical observables, including light.
5. On the Status of Dark Matter and Dark Energy:
While you state “dark matter and dark energy are not discovered”, this overlooks strong indirect empirical evidence:
• Galaxy rotation curves (Zwicky, Rubin),
• CMB anisotropies (Planck, WMAP),
• Large-scale structure and baryon acoustic oscillations,
• Supernovae redshift observations (Riess, Perlmutter).
Although not directly detected, their gravitational and energetic effects are measurable. ECM integrates these observations within a coherent force-energy framework, utilizing negative effective mass (-Mᵉᶠᶠ) and apparent mass displacement to rationalize cosmic acceleration and photon dynamics.
Summary:
Your assertion of time’s periodicity, the absolute supremacy of relativistic invariants, and dismissal of the Planck scale or observer mass ignores the measurable, energy-based structure of the universe that ECM emphasizes.
ECM does not deny periodicity, nor does it conflict with wave mechanics. Instead, it enhances our understanding by interpreting the speed of light through:
• anti-gravitational motion of photons (via −Mᵃᵖᵖ),
• negligible gravitational motion of observers (positive Mᴍ),
• Planck constraints as limits of observable quantum motion,
• and a dominant measurement system dictated by effective mass-energy distributions, not solely geodesics or classical periodic invariants.
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