Effective Mass, Apparent Mass, and Acceleration of Massless Particles in Extended Classical Mechanics (ECM):

Soumendra Nath Thakur
March 19, 2025

This discussion explored the ECM framework’s interpretation of force, mass, and acceleration for massless particles, specifically photons, under gravitational influence and beyond. The key findings and mathematical formulations were verified for consistency and interpreted for their physical implications.

1. ECM Force Equation for Massive and Massless Particles

- For massive particles, the ECM force equation is:

Fₘₐₛₛ = (Mᴍ + (-Mᵃᵖᵖ)) aᵉᶠᶠ

where Mᴍ is matter mass, and -Mᵃᵖᵖ is negative apparent mass.

- For massless particles (e.g., photons), the ECM force equation is:

Fₘₐₛₛₗₑₛₛ = (-Mᵉᶠᶠ + (-Mᵃᵖᵖ)) aᵉᶠᶠ

Since Mᴍ = -Mᵉᶠᶠ and Mᴍ < 0, this equation simplifies to:

Fₘₐₛₛₗₑₛₛ = (-Mᵃᵖᵖ + -Mᵃᵖᵖ) aᵉᶠᶠ

which further resolves as:

Fₘₐₛₛₗₑₛₛ = -2 Mᵃᵖᵖ aᵉᶠᶠ

2. Effective Acceleration of Massless Particles

- The effective acceleration of a massless particle under gravitational influence is derived as:

aᵉᶠᶠ = 6 × 10⁸ m/s² ⇒ 2c

This value is obtained using the equation of motion with constant acceleration:

Δd = v₀Δt + (1/2)aᵉᶠᶠ(Δt)²

where Δd = 3 × 10⁸ m (distance travelled in 1 second), initial velocity v₀ = 0, and Δt = 1 s.

Solving for aᵉᶠᶠ gives:

aᵉᶠᶠ = 6 × 10⁸ m/s²

3. Transition of Effective Acceleration Beyond Gravitational Influence

- Beyond gravitational influence, in a zero-gravity or antigravitational field, the effective acceleration adjusts from 6 × 10⁸ m/s² to:

aᵉᶠᶠ = 3 × 10⁸ m/s² ⇒ c

Correspondingly, the force equation transitions to:

Fₘₐₛₛₗₑₛₛ = -Mᵃᵖᵖ aᵉᶠᶠ

ensuring consistency with the inherent energy E of photons beyond gravitational influence.

4. Physical Implications and Energy Flow

- The framework confirms that massless particles possess negative effective mass while retaining positive kinetic energy.

- The dynamic adjustment of negative apparent mass (-Mᵃᵖᵖ) under gravitational influence influences photon acceleration, ensuring a continuous force-energy relationship.

- The transition of effective acceleration from 2c to c when a photon leaves a gravitational field ensures conservation of its inherent energy while adjusting its force expression.

- This perspective offers an alternative explanation for energy behaviour in gravitational systems without requiring relativistic mass-energy transformations.

Conclusion

The discussion established a mathematically and physically consistent framework for the force, mass, and acceleration of massless particles under ECM. It confirmed that photons inherently experience an effective acceleration of 6 × 10⁸ m/s² within gravitational fields, which naturally transitions to 3 × 10⁸ m/s² when outside gravitational influence. These findings refine our understanding of massless particle dynamics and energy flow, contributing to ECM’s broader applications in gravitational mechanics and cosmology.