Consistency of Effective Mass and Gravitating Mass in an Extended Classical Mechanics System:

Soumendra Nath Thakur

March 04, 2025

In a system:

The effective mass (Mᵉᶠᶠ) is defined as the sum of the matter mass (Mᴍ) and the negative apparent mass (−Mᵃᵖᵖ). The matter mass itself consists of the ordinary matter mass (Mᴏʀᴅ) and the mass of dark matter (Mᴅᴍ). Consequently, the effective mass is equivalent to the gravitating mass (Mɢ).

The effective mass remains positive (Mᵉᶠᶠ>0) when the absolute magnitude of the matter mass |Mᴍ| exceeds the absolute magnitude of the negative apparent mass |−Mᵃᵖᵖ|. Conversely, the effective mass becomes negative (Mᵉᶠᶠ<0) when the absolute magnitude of the matter mass is less than the absolute magnitude of the negative apparent mass.

Similarly, the gravitating mass follows the same conditions as the effective mass, remaining positive (Mɢ>0) when the absolute magnitude of the matter mass is greater than the absolute magnitude of the negative apparent mass and becoming negative (Mɢ<0) when the absolute magnitude of the matter mass is smaller than the absolute magnitude of the negative apparent mass.

Additionally, the negative apparent mass can be expressed as the difference between the matter mass and the effective mass. Since the effective mass is equivalent to the gravitating mass, the negative apparent mass can also be described as the difference between the matter mass and the gravitating mass.

Mathemetical Presentation:

In a system:

The effective mass (Mᵉᶠᶠ) is defined as the sum of the matter mass (Mᴍ) and the negative apparent mass (−Mᵃᵖᵖ). The matter mass consists of the ordinary matter mass (Mᴏʀᴅ) and the mass of dark matter (Mᴅᴍ), so that:

Mᴍ = Mᴏʀᴅ + Mᴅᴍ

Since the effective mass is derived from the matter mass and negative apparent mass, it is equivalent to the gravitating mass (Mɢ), meaning:

Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ) = Mɢ

The sign of the effective mass depends on the relative magnitudes of the matter mass and the negative apparent mass. Specifically:

• The effective mass remains positive (Mᵉᶠᶠ > 0) when the absolute magnitude of the matter mass |Mᴍ| is greater than the absolute magnitude of the negative apparent mass |−Mᵃᵖᵖ|.
• Conversely, the effective mass becomes negative (Mᵉᶠᶠ < 0) when the absolute magnitude of the matter mass is smaller than the absolute magnitude of the negative apparent mass.

Since the gravitating mass follows the same fundamental equation as the effective mass, it exhibits the same conditions:

• The gravitating mass remains positive (Mɢ > 0) when |Mᴍ| > |−Mᵃᵖᵖ|.
• The gravitating mass becomes negative (Mɢ < 0) when |Mᴍ| < |−Mᵃᵖᵖ|.

Additionally, the negative apparent mass (−Mᵃᵖᵖ) can be expressed as the difference between the matter mass and the effective mass:

Mᵃᵖᵖ = Mᴍ − Mᵉᶠᶠ

Since the effective mass is equal to the gravitating mass, this also means:

Mᵃᵖᵖ = Mᴍ − Mɢ