Equivalence of Inertial and Gravitational Mass in Classical Mechanics.

15-09-2024

The equivalence principle in classical mechanics posits that inertial mass (Mᴍ) is equal to gravitational mass (Mɢ), i.e., 

Mᴍ = Mɢ.

Given that:

Inertial mass is related to acceleration by:

Mᴍ = F/a

Gravitational mass is expressed as:

Mɢ = Fɢ·r²/G·Mᴍ

Thus, the equivalence principle (Mᴍ = Mɢ) leads to:

F/a = Fɢ·r²/G·Mᴍ

This equation shows that the force (F) causing acceleration (a) due to an object’s inertial mass is equivalent to the gravitational force (Fɢ) exerted by mass (Mᴍ) at a distance (r), scaled by the gravitational constant (G).

Interpretation:

The equivalence principle demonstrates that an object's resistance to acceleration (inertial mass) is indistinguishable from its gravitational interaction (gravitational mass), highlighting the fundamental relationship between gravity and inertia in classical mechanics.

This principle is key to understanding both classical and relativistic physics, illustrating the balance between gravitational and inertial forces through r, a, and G.

#EquivalencePrinciple #ClassicalMechanics

Clarifying the Inapplicability of Time Dilation to Light and the Role of Planck Length

15 September 2024

ResearchGate Discussion Link

Dear Robert A. Phillips,

Thank you for your thoughtful question. My paper maintains that light is not subject to time dilation, and this is a consistent stance given that time dilation applies to objects or systems experiencing either relative velocity or differences in gravitational potential, both of which apply to masses moving at speeds less than the speed of light. Time dilation, as understood in relativistic terms, does not apply to light itself.

For time dilation to occur, one needs two clocks—one stationary and one in motion. When the moving clock reaches the speed of light, it ceases to function as a clock since time, in that frame, would no longer progress in a measurable way. Hence, time dilation is not applicable to light, which always travels at a constant speed, unaffected by these considerations.

The consideration of light's redshift or blueshift due to gravitational effects is important, but it's critical to differentiate between these phenomena and time dilation. Redshift or blueshift causes a change in the frequency and wavelength of light, which results in time delay or distortion, not dilation. A standard clock would measure time distortion or delay (a deviation in the measured time due to the change in wavelength) but not the enlargement of time associated with time dilation.

Regarding your point on the Planck length, this unit is derived from fundamental constants such as c, G, ħ, and kB and thus naturally includes gravitational consequences. The Planck length, defined long before the introduction of general relativity and the concept of spacetime warpage, remains consistent within the relativistic framework, although it represents a scale at which classical interpretations of spacetime break down, and quantum gravitational effects must be considered.

Thus, while Planck length is a vital concept, it does not directly tie into the observable warpage of spacetime in the way time dilation is often described.

In Summary:

In my paper, I maintain that light is not subject to time dilation, as time dilation arises from relative motion or gravitational potential differences between two clocks, which are constrained by velocities below the speed of light. Since light always travels at the speed of light, it cannot experience time dilation like matter does. If a clock were to reach the speed of light, it would no longer function, as it would lose its capacity to measure time.

Redshift and blueshift result from changes in wavelength and frequency, which I define as time distortions, not time dilation. The equation c = f⋅λ ensures that frequency and wavelength changes are inversely related, and these variations cause time distortions Δt = 1/T, distinct from time dilation (t' > t), which involves the relative expansion of time.

Regarding Planck length, it belongs to the Planck unit system, based on constants c, G, ħ, and kB. Planck length includes gravitational effects and was formulated before the concept of spacetime warpage in general relativity. While gravitational lensing results from spacetime warping, it does not alter the Planck units themselves. Near the Planck scale, quantum gravity effects dominate, rendering classical relativity inapplicable.

Best regards,

Soumendra Nath Thakur

Clarifying Dark Matter, Dark Energy, and Gravitational Dynamics: A Response to Robert A. Phillips

15 September 2024

ResearchGate Discussion Link

Dear Mr. Robert A. Phillips,

Thank you for sharing your thought-provoking question. I appreciate the depth of your inquiry, but I would like to offer a few clarifications and alternative perspectives based on my research.

Firstly, I would differ slightly with the beginning of your statement, "Based on the descriptions and observations of dark matter and dark energy." The current scientific understanding does not actually provide direct descriptions or observations of dark matter and dark energy themselves. Instead, we observe their gravitational effects on the universe. The term "dark" is used to signify that these entities do not emit or reflect light, as opposed to "illuminating" baryonic matter. Dark matter, especially baryonic dark matter, is hypothesized to be made up of baryons, manifesting in forms like diffuse gas clouds, low-luminosity stars, and planets. However, this baryonic dark matter makes up only a small fraction of the overall dark matter content in the universe.

Moreover, when discussing dark matter in the context of your model, my focus lies not on dark matter as a medium but on its observable gravitational effects. My research addresses how matter mass (Mᴍ), which includes both baryonic and non-baryonic dark matter, affects classical mechanics, especially in terms of gravitational dynamics and mass measurements within an extended framework.

As for your interpretation of dark energy as "propagating gravitational waves caused by the gravitational acceleration of matter," this seems to suggest treating gravitational waves as a form of substance. However, my research frames dark energy not as a substance, but as a phenomenon emerging from cosmic motion and gravitational dynamics. Dark energy manifests through the effects of motion and the gravitational interactions of matter rather than being a material entity. Specifically, the apparent negative mass (−Mᵃᵖᵖ) and the negative effective mass of dark energy (Mᴅᴇ) are better understood as consequences of motion and gravitational dynamics rather than being tangible substances.

This view aligns with the extended classical mechanics framework, offering a coherent explanation of gravitational interactions, particularly in systems impacted by dark energy and apparent negative mass. My work emphasizes the role of cosmic motion and its relationship with gravitational forces in shaping our understanding of dark energy.

I hope this response clarifies my perspective and contributes to our shared understanding of these cosmic phenomena. I appreciate your engagement with these complex ideas.

Best regards,

Soumendra Nath Thakur

Redefining Mass Profiles in Extended Classical Mechanics:

Date: 15 September 2024

The research titled "Extended Classical Mechanics: Vol-1 - Equivalence Principle, Mass and Gravitational Dynamics" builds on the concept of dark energy's antigravity, aligning with the work of A.D. Chernin et al. on dark energy and the structure of the Coma Cluster of galaxies. It redefines mass profiles by introducing the concept of Apparent Mass (Mᵃᵖᵖ), a negative mass component that influences Effective Mass (Mᵉᶠᶠ). By incorporating the antigravity effects of dark energy and revising mass profiles, this study enhances the applicability of classical mechanics to cosmic structures, offering new insights into the universe's gravitational dynamics. The integration of these concepts helps to refine our understanding of cosmic phenomena and supports the development of a more comprehensive model in extended classical mechanics.

Dark Energy and the Structure of the Coma Cluster of Galaxies: Redefining Mass and Gravitational Dynamics in Extended Classical Mechanics:

The study explores the implications of dark energy on classical mechanics, focusing on its influence on gravitational dynamics and mass measurements within the framework of extended classical mechanics. Building on the ΛCDM cosmology, which models dark energy as a uniform, vacuum-like fluid, this research re-examines the structure of galaxy clusters, particularly the Coma cluster, using a theoretical framework that incorporates both dark energy and matter mass.

Recent advancements highlight the need to integrate dark energy's effects into gravitational dynamics, revealing that dark energy contributes significantly to the effective mass and influences the structure of galaxy clusters at large scales. The analysis, based on observational data and theoretical models, identifies three critical masses: matter mass Mᴍ and total gravitating mass Mɢ = Mᴍ + Mᴅᴇ , dark-energy effective mass  Mᴅᴇ (negative). This framework leads to a new matter density profile that aligns with observed data for the Coma cluster and addresses the limitations of traditional models.

The study proposes that the effective gravitating density of dark energy is negative, producing antigravity that can surpass gravitational effects on scales up to 20 Mpc. This finding challenges existing models and suggests that galaxy clusters' total size and mass must account for dark energy's influence. The analysis also revises the matter mass profiles of clusters, using a modified Hernquist profile that better matches observational data and provides more accurate estimates of cluster mass and size.

By incorporating dark energy's antigravity effects and redefining mass profiles, this work enhances classical mechanics' applicability to cosmic structures and provides new insights into the universe's gravitational dynamics. The integration of these concepts helps refine our understanding of cosmic phenomena and supports the development of a more comprehensive model of extended classical mechanics.

Keywords: dark energy, classical mechanics, galaxy clusters, Coma cluster, effective mass, gravitational dynamics, ΛCDM cosmology, antigravity, mass profiles,

#darkenergy, #classicalmechanics, #galaxyclusters, #Comacluster, #effectivemass, #gravitationaldynamics, #ΛCDMcosmology, #antigravity, #massprofiles,

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