Summary of Extended Classical Mechanics: Vol-1 - Equivalence Principle, Mass and Gravitational Dynamics

The research paper "Extended Classical Mechanics: Vol-1 - Equivalence Principle, Mass and Gravitational Dynamics," by Soumendra Nath Thakur, provides a comprehensive re-evaluation of classical mechanics by incorporating modern concepts from astrophysics and cosmology. The paper aims to extend the traditional framework of classical mechanics to address new phenomena related to gravitational dynamics, dark matter, and dark energy.

Part 1: Introduction and Overview

The first part of the paper introduces the motivation behind extending classical mechanics to include concepts like dark matter and dark energy. It outlines the need to reconcile classical mechanics with observational evidence from astrophysics, particularly in relation to the behaviour of gravitational systems on large scales.

Key Points:

• The traditional framework of classical mechanics is well-established but limited in its ability to address phenomena related to dark matter and dark energy.

• The paper proposes an extension of classical mechanics to incorporate these concepts, aiming to provide a unified framework for understanding gravitational dynamics.

Part 2: Equivalence Principle and Mass

This section discusses the equivalence principle in classical mechanics, which states that inertial mass (related to acceleration) and gravitational mass (related to gravitational interaction) are equivalent. The paper extends this principle to systems involving both normal matter and dark matter.

Key Points:

• The equivalence principle is reaffirmed, with the paper proposing that the effective gravitational mass (Mɢ) of a system reflects the combined inertial mass of normal matter and dark matter.

• The concept of matter mass (Mᴍ) is defined as the sum of baryonic matter and dark matter.

• The paper explores how gravitational dynamics can be influenced by both matter mass and the negative apparent mass associated with dark energy.

Part 3: Mathematical Presentation

This section provides a detailed mathematical treatment of the concepts introduced. It discusses the relationship between apparent mass and effective mass, including the role of negative apparent mass in gravitational dynamics.

Key Points:

• The paper redefines gravitational mass (Mɢ) to include the negative apparent mass (−Mᵃᵖᵖ), providing a revised framework for understanding gravitational interactions.

• Newton's second law and Newton's law of universal gravitation are reformulated to incorporate the effects of apparent mass.

• The discussion includes the implications of apparent mass for kinetic energy, object deformation, and relativistic effects.

Part 4: Future Directions and References

The final part outlines future research directions and provides a list of references for further reading.

Key Points:

• Future research will explore the relationship between apparent mass and kinetic energy, its impact on object deformation, and connections with relativistic Lorentz transformations.

• References include key works on dark energy, classical mechanics, and cosmology, providing a foundation for further study.

Overall Summary

The research paper represents an ambitious effort to extend classical mechanics by incorporating modern concepts from astrophysics. The main contributions of the paper include:

1. Extension of the Equivalence Principle:

• The paper extends the classical equivalence principle to systems involving both normal matter and dark matter, proposing that the effective gravitational mass of such systems is equivalent to the combined inertial mass.

2. Integration of Dark Matter and Dark Energy:

• The paper introduces the concept of negative apparent mass and integrates it with gravitational dynamics. This extension provides a framework for understanding phenomena related to dark energy and cosmic acceleration.

3. Reformulation of Gravitational Dynamics:

• Traditional equations of motion and gravitational forces are modified to include the effects of apparent mass, offering a revised approach to gravitational interactions.

4. Future Research Directions:

• The paper outlines potential areas for future research, including the impact of apparent mass on kinetic energy and its relation to relativistic effects.

Overall, the paper successfully bridges classical mechanics with modern astrophysical concepts, providing a comprehensive framework for understanding gravitational dynamics and cosmic phenomena. The proposed extensions offer valuable insights and suggest avenues for further exploration and refinement in the field of classical mechanics and cosmology.

#ExtendedClassicalMechanics

Dark Energy Effective Mass (Mᴅᴇ):

In the research paper "Dark energy and the structure of the Coma cluster of galaxies" by Chernin et al. (2013), the concept of Dark Energy Effective Mass ($M_{de}$) is introduced as part of the analysis of the Coma cluster's structure. The paper explores how dark energy, characterized by its antigravitational effects, influences the structure of galaxy clusters.

Description of Dark Energy Effective Mass:

1. Definition and Role:

   • Dark Energy Effective Mass ($M_{de}$) is defined as the effective mass of dark energy that contributes to the gravitational dynamics of a galaxy cluster. Unlike traditional matter, dark energy has a negative effective mass ($M_{de}<0$) due to its repulsive, antigravitational properties. This negative mass affects the total gravitating mass of the cluster.​

2. Mathematical Formulation:

   • The effective mass of dark energy within a spherical volume of radius $R$ is given by: $$M_{de}(R) = \frac{8 \pi}{3} \rho_{de} R^3$$where $\rho_{de}$ is the density of dark energy. For instance, at different radii:
     • At $R = 1.4$ Mpc: $M_{de} = -2.3 \times 10^{12} M_{\odot}$
     • At $R = 4.8$ Mpc: $M_{de}=-9.4\times 10^{13}{M}_{\odot }$
     • At $R = 14$ Mpc: $M_{de} = -2.3 \times 10^{15} M_{\odot}$​

3. Equation for Total Gravitating Mass:

   • The total gravitating mass ($M_g$) within a radius $R$ of a galaxy cluster is the sum of the matter mass ($M_m$) and the dark energy effective mass ($M_{de}$): $$M_g = M_m + M_{de}$$This equation allows us to calculate the total gravitating mass of the cluster by adding the matter mass to the dark energy effective mass. For example, at $R = 14$  Mpc, the total gravitating mass $M_g$ approximates $4.7 \times 10^{15} M_{\odot}$.

4. Implications:

   • The negative effective mass of dark energy implies that, at large distances from the cluster center, the dark energy's repulsive force can exceed the gravitational attraction of the matter within the cluster. This antigravitational effect becomes significant at distances beyond the zero-gravity radius ($R_{zg}$), beyond which the dark energy's influence dominates.

The study by Chernin et al. highlights the substantial impact of dark energy on the structure and mass estimation of galaxy clusters, underscoring its role in shaping cosmic structures and influencing their dynamics.