6.4. Equivalence of Apparent Mass and Dark Energy's Negative Effective Mass:

The equivalence between Apparent Mass (−Mᵃᵖᵖ) and Dark Energy's Negative Effective Mass (Mᴅᴇ) is crucial for understanding gravitational dynamics within the framework of extended classical mechanics. Both concepts are characterized by their negative mass properties, which influence gravitational interactions across cosmic scales. Their negative effective mass leads to repulsive forces that counteract the traditional attractive forces of ordinary matter and dark matter, shaping the structural formation and evolution of cosmic entities.

The alignment of these two concepts can be expressed as Mɢ = Mᴍ + (−Mᵃᵖᵖ), which mirrors the Chernin et al. framework. The presence of Dark Energy's negative effective mass (Mᴅᴇ) is conceptually equivalent to the apparent mass (−Mᵃᵖᵖ). This equivalence underscores the importance of both negative mass characteristics in determining the overall gravitational dynamics.

The implications of this equivalence become particularly pronounced under extreme conditions, such as high velocities and strong gravitational fields. The repulsive nature of both Apparent Mass and Dark Energy Effective Mass can significantly alter expected gravitational behaviours, particularly within galaxy clusters where the interplay between these masses dictates the formation and behaviour of large-scale structures.

Incorporating these concepts into gravitational models allows for better understanding of cosmic phenomena, such as galaxy formation and the universe's expansion rate.