Mathematical Consistency of ECM Mass-Energy Dynamics:

March 04, 2025

Force Equation In Classical Mechanics  (Motion):

F = ma

Acceleration follows the classical inverse-mass relation:

a ∝ 1/m

​Since force is proportional to acceleration, this implies:

F ∝ a ∝ 1/m

which suggests that force arcs dynamically with acceleration.

Potential Energy and Dynamic Mass Relation:

When a system undergoes motion, the potential mass m generates kinetic energy, leading to a mass-energy equivalence in dynamic motion:

Potential Energy (PE ⇒ m), Kinetic energy (KE ⇒ 1/m)

This follows from the total energy equation:

Eₜₒₜₐₗ = PE + KE where PE ⇒ m, KE ⇒ 1/m

At rest, kinetic energy is zero, so:

Eₜₒₜₐₗ = PE, when KE = 0

As kinetic energy increases, a portion of the potential energy ΔPE converts into kinetic energy:

Eₜₒₜₐₗ = PE + KE = (PE − ΔPE) + ΔPE

Substituting mass-energy equivalence in ECM, we write:

Eₜₒₜₐₗ  = (m − Δm) + 1/Δm

​Since negative apparent mass (−Mᵃᵖᵖ) arises from the kinetic energy contribution, we identify:

−Δm ⇒ −Mᵃᵖᵖ

Thus, the negative apparent mass corresponds to the kinetic energy term in ECM, balancing the total energy equation dynamically.

Physical Coherence of −Mᵃᵖᵖ

The introduction of negative apparent mass (−Mᵃᵖᵖ) as arising from kinetic energy is consistent with ECM's premise that kinetic energy contributes to an effective mass shift.

Since −Δm represents the mass component transferred to kinetic energy, defining −Δm ⇒ −Mᵃᵖᵖ is reasonable under ECM.

Eₜₒₜₐₗ = PE + ΔPE = (PE − ΔPE) + ΔPE, where PE = (PE − ΔPE) and KE  = ΔPE

When a system (PE) undergoes energy transformation, some of its stored energy (PE − ΔPE) is converted into motion KE  = ΔPE. 

Initially, all of the system's energy is in the form of stored energy (PE). As the system moves, a portion of this stored energy is used to generate movement, reducing the amount that remains stored (PE − ΔPE). 

The part that is taken from storage becomes energy associated with motion 

KE  = ΔPE. 

However, the total energy of the system Eₜₒₜₐₗ = PE + ΔPE = (PE − ΔPE) + ΔPE does not change—only the way it is distributed between stored energy PE and motion energy KE  (=ΔPE) shifts. 

This ensures that any reduction in stored energy results in an equal increase in motion energy PE ∝ 1/KE = 1/ΔPE , maintaining balance in the system.