3. Mathematical Form of Apparent Mass (Mᵃᵖᵖ) in ECM:

Soumendra Nath Thakur

February 05, 2025

1. Fundamental Relationship in ECM

In ECM, apparent mass is defined as a function of effective mass and matter mass: 

Mᵃᵖᵖ = Mᴍ − Mᵉᶠᶠ,

where:

Mᴍ = Mᴏʀᴅ + Mᴅᴍ (matter mass, including baryonic and dark matter contributions)

Mᵉᶠᶠ = Mᴍ + (-Mᵃᵖᵖ) (effective mass in ECM)

Rearranging, we obtain: 

Mᵃᵖᵖ = −(Mᵉᶠᶠ − Mᴍ)

Thus, Mᵃᵖᵖ is directly dependent on the difference between effective mass and matter mass.

2. Variables Determining Mᵃᵖᵖ

The apparent mass is not constant; it depends on:

• Gravitational Field Strength (g): The interaction between gravitational sources affects Mᵃᵖᵖ, especially in strong fields or at large cosmic scales.

• Velocity (v): In high-velocity regimes, relativistic effects influence the contribution of Mᵃᵖᵖ.

• Distance (r): Changes in Mᵃᵖᵖ occur over different distance scales, particularly in intergalactic interactions.

• Time (t): Dynamic variations in mass distributions or cosmic expansion affect Mᵃᵖᵖ over time.

Thus, we express it as:

Mᵃᵖᵖ = f(g, v, r, t)

3. Differential Form: How Mᵃᵖᵖ Evolves Over Time and Distance

Given that ECM introduces effective acceleration (aᵉᶠᶠ) instead of classical acceleration, we relate Mᵃᵖᵖ to forces via:

F = Mᵉᶠᶠ⋅aᵉᶠᶠ 

Since: 

Mᵉᶠᶠ = Mᴍ − Mᵃᵖᵖ 

then differentiating with respect to time:

dMᵃᵖᵖ/dt = − dMᵉᶠᶠ/dt + dMᴍ/dt

For systems where matter mass is approximately conserved (dMᴍ/dt ≈ 0):

dMᵃᵖᵖ/dt = − dMᵉᶠᶠ/dt 

Thus, changes in apparent mass mirror variations in effective mass.

For spatial dependence, we express:

dMᵃᵖᵖ/dr = − dMᵉᶠᶠ/dr 

which suggests that in regions of strong gravitational influence, Mᵃᵖᵖ changes significantly.

4. Approximate Equation for Mᵃᵖᵖ

Mᵃᵖᵖ = −α(GMᴍ/c²r)

where α is a proportionality factor that depends on the local gravitational potential. This form ensures that Mᵃᵖᵖ is:

• Negligible in weak fields (e.g., near Earth)

• Significant at interstellar and intergalactic scales

Conclusion

• The mathematical form of Mᵃᵖᵖ depends on effective mass, gravitational field, velocity, and distance.

• Time evolution of Mᵃᵖᵖ is tied to changes in effective mass.

• Spatial variations suggest that Mᵃᵖᵖ increases in strong gravitational fields and at cosmic scales.

• A working equation for Mᵃᵖᵖ involves gravitational potential, ensuring consistency with ECM principles.