Soumendra Nath Thakur
01-10-2024
Abstract
This unified study investigates the intricate relationships among gravitational mass, matter mass, and dark matter dynamics within the framework of extended classical mechanics. By addressing the roles of dark matter mass and negative apparent mass in gravitational forces and effective mass, this research delineates the distinctions between mechanical and relativistic kinetic energy. It posits that classical mechanics is enriched by integrating these components, thus enhancing our understanding of gravitational dynamics across both macroscopic and microscopic scales.
The study redefines gravitational mass, moving beyond its traditional equivalence with matter mass to encompass contributions from dark matter and negative apparent mass. By examining the relationships between these components, it establishes a comprehensive model for gravitational interactions that includes both positive and negative mass contributions. Furthermore, the paper explores the influence of dark matter on gravitationally bound systems, highlighting its critical role in modifying effective mass and apparent mass dynamics.
In the context of cosmic dynamics, this study elucidates the transition from gravitationally bound systems to intergalactic space, where dark energy and negative mass dominate. The findings suggest that in these vast regions, negative apparent mass exerts a significant influence, leading to a repulsive force that drives the accelerated expansion of the universe.
This research ultimately provides a refined framework for understanding the complexities of gravitational dynamics and kinetic energy, laying the groundwork for future investigations into the unification of mass and energy in the cosmos.
Keywords: Dark energy, Dark matter, Effective mass, Gravitational dynamics, Gravitational mass, Matter mass, Negative apparent mass, Relativistic kinetic energy, Total effective mass,
Soumendra Nath Thakur
ORCiD: 0000-0003-1871-7803
Tagore’s Electronic Lab, WB,
Correspondence: postmasterenator@gmail.com
postmasterenator@telitnetwork.in
Declaration: No competing interest to declare,
and No funding has been received for this research.
Introduction:
This unified study explores the integration of gravitational mass, matter mass, and dark matter dynamics within extended classical mechanics, alongside a comprehensive examination of kinetic energy domains. By addressing the roles of dark matter mass (Mᴅᴍ) and negative apparent mass (Mᵃᵖᵖ) in gravitational forces and effective mass (Mᵉᶠᶠ), while distinguishing mechanical and relativistic kinetic energy, this study provides a refined framework for understanding gravitational dynamics on macroscopic and microscopic scales. Classical mechanics is enhanced by incorporating dark matter and negative mass, distinguishing mechanical energy from relativistic mass-energy equivalence.
Gravitational Mass and Matter Mass in Extended Classical
Mechanics:
Gravitational mass (Mɢ) is redefined beyond its classical equivalence with matter mass (Mᴍ), incorporating dark matter and negative apparent mass contributions [1]. Traditionally, gravitational mass (Mɢ) was considered equivalent to ordinary matter mass (Mᴏʀᴅ), but modern physics extends this view with dark matter and dark energy [2] [3].
The relationship between these components is captured by the equation:
Mᴍ = Mᴏʀᴅ + Mᴅᴍ + (-Mᵃᵖᵖ)
Here, Mᴏʀᴅ represents baryonic matter, Mᴅᴍ represents dark matter mass, and (-Mᵃᵖᵖ) accounts for negative apparent mass derived from effective acceleration. The total effective mass (Mᵉᶠᶠ) includes both dark matter and apparent mass, expressed as:
Mɢ = Mᵉᶠᶠ = Mᴍ + (-Mᵃᵖᵖ) = Mᴏʀᴅ + Mᴅᴍ + (-Mᵃᵖᵖ)
Gravitating mass (Mɢ), which governs gravitational dynamics, is the sum of matter mass (Mᴍ) and negative apparent mass (-Mᵃᵖᵖ), showing that gravitational interactions depend on both positive and negative mass contributions [1].
Dark Matter Mass and Its Role in Gravitational Dynamics:
Dark matter mass (Mᴅᴍ) significantly influences gravitationally bound systems, adding positively to the total matter mass (Mᴍ). Its contribution strengthens gravitational forces, altering the effective mass (Mᵉᶠᶠ) and apparent mass (Mᵃᵖᵖ) [1]. In gravitationally bound systems, the relationship is:
Mᵉᶠᶠ = Mᴍ + (-Mᵃᵖᵖ)
Dark matter dominates in galactic dynamics, while apparent mass reduces the overall effective mass in gravitational systems.
Modified Newtonian Laws with Dark Matter and Negative
Mass:
The study modifies
Fɢ = G·(Mᵉᶠᶠ·m₂)/r²
Where Mᵉᶠᶠ encompasses the contributions from matter mass (Mᴍ) and apparent mass (Mᵃᵖᵖ), both of which influence the gravitational force. Additionally, m₂ is defined as the sum of matter mass and apparent mass for the second mass in the interaction [1]. Similarly, the modified force equation:
F = Mᵉᶠᶠ·aᵉᶠᶠ
incorporates these components, demonstrating their effect on system dynamics.
Distinct Domains of Kinetic Energy:
Mechanical kinetic energy governs large-scale motion, while relativistic kinetic energy applies to high-energy, nuclear reactions. The equation:
Mɢ = Mᴍ + (-Mᵃᵖᵖ)
shows how mechanical kinetic energy, including the influence of dark matter, impacts macroscopic systems, while relativistic kinetic energy remains important in microscopic scales.
Negative Apparent Mass and Cosmic Dynamics:
In intergalactic space, beyond the gravitational influence of bound systems, gravitating mass (Mɢ) governs dynamics as the sum of matter mass (Mᴍ) and negative effective mass of dark energy (Mᴅᴇ), as per Chernin et al. [2]:
Mɢ = Mᴍ + Mᴅᴇ
In Vol-1, matter mass dominates, while in Vol-2, negative apparent mass (−Mᵃᵖᵖ) drives cosmic expansion. The modified equation becomes:
Mɢ = Mᴍ + Mᵉᶠᶠ
Transition from Gravitationally Bound Systems to
Intergalactic Space:
In gravitationally bound systems, such as galaxies or galaxy clusters, gravitational dynamics are primarily governed by the combined influence of matter mass (Mᴍ) and negative apparent mass (−Mᵃᵖᵖ) [1]. In these systems, the matter mass is the dominant factor, while the negative apparent mass plays a secondary role (Mᴍ>−Mᵃᵖᵖ). As a result, the gravitational mass is mostly influenced by the ordinary and dark matter, with a small reduction due to negative apparent mass.
However, in intergalactic space, beyond the reach of these gravitationally bound systems, the situation changes significantly. Dark energy and negative apparent mass become the dominant forces, with negative apparent mass contributing more heavily to the gravitational mass. This shift means that in these vast, cosmic regions, negative apparent mass exerts a much stronger influence, resulting in a repulsive force that drives the accelerated expansion of the universe. The effective mass in intergalactic space incorporates this dominant negative mass, while the matter mass plays a lesser role (Mᴍ<−Mᵃᵖᵖ).
Thus, while gravitationally bound systems are dominated by attractive forces driven by matter mass, in intergalactic space, the negative mass associated with dark energy reshapes the dynamics, leading to the expansion and evolution of cosmic structures.
Conclusion:
This study emphasizes the distinction between mechanical and relativistic kinetic energies while integrating dark matter and negative apparent mass into gravitational models. Extended Classical Mechanics provides a comprehensive framework for understanding gravitational dynamics, laying the groundwork for future exploration into mass-energy unification.
List of Mathematical Denotations:
• Mᵃᵖᵖ - Apparent mass
• Mᴅᴍ - Dark matter mass
• Mᵉᶠᶠ - Effective mass,
• Mɢ - Gravitational mass
• Mᴍ - Matter mass
• Mᴏʀᴅ - Ordinary (baryonic) matter
• −Mᵉᶠᶠ - Negative effective mass
• −Mᵃᵖᵖ - Negative apparent mass
References:
[1]
Thakur, S. N. (2024). Extended Classical Mechanics: Vol-1 - Equivalence
Principle, Mass and Gravitational Dynamics, Preprints.org (MDPI) https://doi.org/10.20944/preprints202409.1190.v2
[2]
Chernin, A. D., Bisnovatyi-Kogan, G. S., Teerikorpi, P., Valtonen, M. J., Byrd,
G. G., & Merafina, M. (2013). Dark energy and the structure of the Coma
cluster of galaxies. Astronomy and Astrophysics, 553, A101, https://doi.org/10.1051/0004-6361/201220781
[3]
Sanders, R. H., & McGaugh, S. S. (2002). Modified Newtonian dynamics as an
alternative to dark matter, Annual Review of Astronomy and Astrophysics, 40(1),
263–317, https://doi.org/10.1146/annurev.astro.40.060401.093923
[4]
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https://doi.org/10.13140/RG.2.2.28013.97763
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https://doi.org/10.36227/techrxiv.22492066.v2
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Thakur, S. N. (2024). Direct Influence of Gravitational Field on Object Motion
invalidates Spacetime Distortion. Qeios (ResearchGate).
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[7]
Thakur, S. N. (2023). Photon paths bend due to momentum exchange, not intrinsic
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[8]Thakur,
S. N. [Soumendra Nath Thakur]. (2024). Effective Mass of the Energetic Pre-
Universe: Total Mass Dynamics from Effective and Rest Mass. ResearchGate
(378298896), https://www.researchgate.net/publication/378298896.
https://doi.org/10.13140/RG.2.2.18182.18241
#Darkenergy, #Darkmatter, #Effectivemass, #Gravitationaldynamics, #Gravitationalmass, #Mattermass, #Negativeapparentmass, #Relativistickineticenergy, #Totaleffectivemass,
