Mass and Effective Mass: Matter, Gravitating Mass, and Dark Energy Impacts:

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

07-06-2024

Abstract:

This study serves as a supplementary resource for researchers investigating length deformation in classical and relativistic mechanics. It complements existing studies such as the 'Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics' series, 'Dynamics between Classical Mechanics and Relativistic Insights', and 'Advancing Understanding of External Forces and Frequency Distortion: Part -1.'

The focus of this resource is on the intricate dynamics of mass and effective mass, exploring fundamental concepts like matter mass and gravitating mass. The relationship between effective mass and forms of energy is examined in detail, elucidating how effective mass accounts for resistance to acceleration or changes in velocity under gravitational forces, akin to the gravitational potential energy of inertial mass.

Through a detailed analysis and illustrative examples, including the study 'Dark Energy and the Structure of the Coma Cluster of Galaxies,' the interplay between matter and dark energy is deciphered. This offers profound insights into the complex dynamics of the universe, shedding light on the interaction between matter and dark energy.

This resource serves as a supplementary guide for researchers investigating length deformation in classical and relativistic mechanics. It complements studies such as the 'Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics' series,[₁,¹, ₂,²] 'Dynamics between Classical Mechanics and Relativistic Insights'[₃,³], and 'Advancing Understanding of External Forces and Frequency Distortion: Part -1.'[₄,⁴] Delve into nuanced discussions on mass and effective mass, covering fundamental concepts like matter mass and gravitating mass. Explore the intricate relationship between effective mass and forms of energy, understanding how it accounts for resistance to acceleration or changes in velocity under gravitational forces, akin to the gravitational potential energy of inertial mass. Through examples like the study 'Dark Energy and the Structure of the Coma Cluster of Galaxies,'[₅,⁵] witness how this resource deciphers the interplay between matter and dark energy, offering profound insights into their intricate dynamics.

Keywords: Matter Mass, Gravitating Mass, Effective mass, Dark energy.

The Concepts of Mass:

1. Matter Mass (Mᴍ):

The matter mass (Mᴍ) is a measure of the amount of matter in a substance or object. Anything that has volume and mass is classified as matter. Although mass itself cannot be seen, it is quantifiable and can be measured. The basic SI unit for mass is the kilogram (kg). The only entities that are not matter are forms of energy.

Mass vs. Weight: Mass is often confused with weight, but they measure different things. Mass measures the amount of matter in an object, while weight measures the force of gravity acting on that object. The force of gravity on an object depends on its mass and the strength of gravity. If the strength of gravity is held constant (as it is across the Earth), an object’s mass is directly proportional to its weight, meaning greater mass corresponds to greater weight.

Volume measures the amount of space that a substance or object occupies. The basic SI unit for volume is the cubic meter (m³), but smaller volumes may be measured in cubic centimeters (cm³), and liquids may be measured in liters (L) or milliliters (mL). The method used to measure the volume of matter depends on its state.

2. Gravitating Mass (Mɢ):

The gravitating mass (Mɢ) is the mass of an object as measured in a gravitational field. This requires a gravitational field, a scale, and various known masses.

Inertial Mass vs. Gravitational Mass: Inertial mass and gravitational mass of an object are identical in value but differ in their measurement methods. Inertial mass is measured by assessing an object's resistance to changes in velocity, while gravitational mass describes the force on an object within a gravitational field. Inertial mass is derived from the concept of inertia—the tendency of objects to remain motionless or in uniform motion unless acted upon by a force. Gravitational mass is measured using a scale.

Since the motion of an object on a spring involves constant changes in velocity, and the behavior of the spring system is related to the mass on the spring, inertial mass is often measured by attaching a mass to a spring. This involves using a spring with a known spring constant (k), which describes the 'stiffness' of the spring.

3. Effective Mass (mᵉᶠᶠ):

Energy, while not matter, is equivalent to mass. The concept of effective mass (mᵉᶠᶠ) refers to the mass equivalent of the gravitational effect of forms of energy.

Effective mass accounts for the resistance to acceleration or changes in velocity of a particle when responding to a gravitational force. This is similar to how the gravitational potential energy of any inertial mass contributes to resistance to acceleration or changes in velocity.

In this context, the effective mass represents the mass equivalent of the gravitational effects of forms of energy, accounting for resistance to acceleration or changes in velocity. Thus, effective mass is the representation of the mass equivalent of energy's resistance to acceleration under the influence of gravitational force.

Therefore, the effective mass is the mass equivalent of the gravitational effect of forms of energy, representing their resistance to acceleration or changes in velocity when responding to specific forces. This concept is similar to the gravitational potential energy of any inertial mass, which also accounts for resistance to acceleration or changes in velocity.

Example:

The study "Dark Energy and the Structure of the Coma Cluster of Galaxies" by A. D. Chernin et al. explores the mass distribution within the Coma cluster, introducing the concept of gravitating mass (Mɢ), which represents the cluster's total mass, and effective mass (Mᴅᴇ or mᵉᶠᶠ), indicating the mass equivalent of dark energy's gravitational effect. The effective mass is defined as the difference between the gravitating mass and the matter mass (Mᴅᴇ = Mɢ − Mᴍ). This equation highlights the contribution of dark energy within a specific radius. By distinguishing between matter mass (Mᴍ), dark-energy effective mass (Mᴅᴇ <0), and gravitating mass (Mɢ = Mᴍ + Mᴅᴇ), the study provides a detailed understanding of the interaction between matter and dark energy in the cluster's structure and dynamics.

References:

1. Thakur, S. N. Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics. Preprints.org. https://doi.org/10.20944/preprints202405.1271.v1
2. Thakur, S. N. Comparative Analysis of Length Deformation in Classical and Relativistic Mechanics: Part-2. Preprints.org. https://doi.org/10.20944/preprints202405.1332.v1
3. Thakur, S. N. Dynamics between Classical Mechanics and Relativistic Insights: ResearchGate. https://doi.org/10.13140/RG.2.2.21005.96481/1
4. Thakur, S. N. Advancing Understanding of External Forces and Frequency Distortion: Part -1. ResearchGate. https://doi.org/10.13140/RG.2.2.35236.28809
5. 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 & Astrophysics, 553, A101. https://doi.org/10.1051/0004-6361/201220781

The Influence of Force Vectors on Resultant Vector:

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

05-06-2024

A force vector represents a force with both magnitude and direction, acting on a body in motion. The resultant vector (R) and its magnitude depend on the combined magnitudes of two force vectors.

Key Concepts:

Force Vectors:

• Force vector = Magnitude × Direction

• Gravitational force vector (g) acts vertically downward.

• Velocity vector (v) acts horizontally.

Resultant Vector:

• Resultant vector (R) is determined by combining g and v.

• R's magnitude depends on g and v.

Influence of v and g:

• When ∥v∥≫∥g∥, g becomes negligible.

• R is predominantly determined by v.

• R's magnitude can be approximated as ∥v∥.

Vertical Component g:

• In a scenario where v≫g, the vertical component g is insignificant.

• R is primarily influenced by v.

A force vector represents a force with both magnitude and direction, acting on a body in motion. The resultant vector (R) and its magnitude depend on the combined magnitudes of two force vectors.

When the magnitude of the velocity vector (v) exceeds that of the gravitational force vector (g), the gravitational force's influence becomes negligible. Consequently, the resultant vector (R) is predominantly determined by the horizontal velocity vector (v), leading to the effective velocity of the body being approximated as v, with the gravitational force having an insignificant impact on R.

In a scenario where a 10g body accelerates to a desired velocity, with v≫g, the resultant vector R is primarily determined by the horizontal velocity vector v, rendering the vertical component g negligible.

In this context, the vertical component (g) is insignificant, allowing the effective velocity to be approximated as v and rendering the gravitational force's influence insignificant

Description of the Perception of Effective Mass:

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

05-06-2024

Inertial mass under the influence of gravity relates to the perception of weight, as the weight of a given mass can vary depending on the distance between the centres of gravity of two massive bodies and their respective masses, assuming no external forces are acting on the bodies.

The perception of weight is generally based on the amount of mass a body has. In a fixed gravitational potential where the body is at rest, a heavier body will have more weight than a smaller one. Additionally, this perception of weight may feel heavier or lighter without any change in the amount of mass of the bodies involved when the distance between the centres of gravity of the bodies changes. This can occur when an external force is applied to one body relative to the other, resulting in either vertical separation or horizontal movement. The applied forces, whether vertical or horizontal, between the massive bodies lead to a perceived change in the weight of the moving body in any relative direction. The change is more significant in the vertical direction, but a higher speed in the horizontal direction can also cause a perceived change in weight when the horizontal motion component significantly affects the vertical component of gravitational attraction between the bodies.

This perception of changing weight corresponds to changes in the total energy of the moving body relative to the other. These changes in energy, whether potential or kinetic, cause a perceived change in weight, which is referred to as effective mass.

Application of Effective Mass through Newtonian Mechanics:

Soumendra Nath Thakur
ORCiD: 0000-0003-1871-7803

05-06-2024

Effective mass is a fundamental concept that arises from the equivalence of energy and mass, but it does not represent actual mass by mass equivalent energy that has intractability with the force. In various physical contexts, effective mass serves as a vital tool for understanding the behaviour of massive manifestation of energetic state under the influence of force or dynamics. It allows us to simplify complex systems and predict their behaviour accurately.

Effective mass manifests not only in the traditional sense of mass but also in the energetic state of particles. This manifestation is rooted in the fundamental relationship between mass and energy. Even in situations where mass is not converted into other forms of energy, the concept of effective mass remains applicable, aiding in describing the behaviour of particles and systems.

Whether in classical mechanics, nuclear physics, semiconductor physics, cosmology, or other fields, effective mass provides a valuable framework for understanding and predicting the behaviour of particles and systems. It bridges the gap between the mass-like manifestations of energy and traditional mass, offering insights into the dynamics of physical phenomena.

The concept of effective mass is crucial in understanding the gravitational dynamics of the Coma cluster. Effective mass incorporates both the traditional matter mass and the impact of dark energy, providing a comprehensive understanding of the cluster's gravitational behaviour. This research highlights the importance of considering dark energy in mass estimations of galaxy clusters, as it significantly influences their structure and dynamics.

The research on "Dark energy and the structure of the Coma cluster of galaxies" by employing Newtonian mechanics, describes the local dynamical effects of dark energy and defines three masses to characterize the structure of a regular cluster: the matter mass, the effective mass, and the gravitating mass. It reveals that the effective mass, which considers the influence of dark energy on the cluster's dynamics, is negative, indicating an antigravity effect.

Analysis of Newton's Second Law and Effective Mass using Energy Conservation Principles:

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

05-06-2024

This study delves into Newton's second law of motion and the concept of effective mass in the context of classical mechanics, considering the variability of effective mass depending on the object's velocity and kinetic energy.

In classical mechanics, effective mass (mᵉᶠᶠ) embodies the mass equivalent of an object's kinetic energy in motion. While it is often comparable to or less than inertial mass (m), scenarios exist where effective mass (mᵉᶠᶠ) surpasses inertial mass (m), notably at high velocities or with significant kinetic energies. Thus, the equivalence between effective mass (mᵉᶠᶠ) and inertial mass (m) is contingent upon the specific circumstances of motion.

Considering Newton's second law (F = m⋅a), it becomes apparent that an increase in force (F) directly yields a rise in acceleration (a), showcasing a direct proportionality (F ∝ a). Furthermore, for a constant force, acceleration (a) inversely relates to the mass (m) of the object (a ∝ 1/m). Hence, alterations in effective mass (mᵉᶠᶠ) due to changes in velocity or kinetic energy reflect directly in the object's acceleration.

In the context of energy conservation principles, effective mass (mᵉᶠᶠ) corresponds to the kinetic energy (KE) of the object, delineating a scenario where mᵉᶠᶠ = KE. However, it's crucial to discern that this equivalence does not necessitate the equality of gravitational potential energy (PE) and kinetic energy (KE). Typically, PE and KE denote distinct forms of energy within the system and are seldom equal unless under specific conditions.

While the approximation that effective mass (mᵉᶠᶠ) cannot exceed inertial mass (m) aligns with classical mechanics' typical assumptions, it's imperative to acknowledge its potential deviations in certain circumstances. Particularly in systems subjected to extreme conditions or non-conservative forces, the relationship between effective mass and inertial mass may diverge from conventional expectations, emphasizing the need for meticulous analysis and consideration of the specific motion dynamics involved.

The general form of the total energy (Eᴛᴏᴛ) equation is expressed as:

Eᴛᴏᴛ = PE + KE

where Eᴛᴏᴛ is the total energy of the system, KE is the kinetic energy, and PE is the potential energy.

For an object of inertial mass m moving at velocity v and under the influence of gravitational potential energy, the total energy equation can be written as:

Eᴛᴏᴛ = (1/2)mv² + mgh

where (1/2)mv² is the kinetic energy, and mgh is the gravitational potential energy, with g being the acceleration due to gravity and h the height above a reference level.

In relativistic mechanics, the total energy equation includes rest mass energy and is expressed as:

Eᴛᴏᴛ = γmc²

where γ is the Lorentz factor {= 1/√(1-v²/c²)}, m is the rest mass of the object, and c is the speed of light.

In classical mechanics, elastic deformations occur when the atomic or molecular structure of materials rearranges under the influence of external forces. The speeds involved in these deformations are significantly less than the speed of light, assuming no nuclear reactions or changes within the materials. Piezoelectric materials, including accelerometer devices, exhibit clear empirical evidence of these occurrences when subjected to mechanical forces and gravity. Therefore, the relativistic energy-mass equivalence equation E=mc²  does not apply at speeds significantly less than the speed of light without nuclear reactions or changes within the object materials.

The concept of effective mass (mᵉᶠᶠ) represents the mass of a particle considering not only its inertial properties but also the influence of external forces, such as gravitational or electromagnetic fields, and kinetic energy. It is particularly useful in scenarios where the particle's behaviour is influenced by motion and the surrounding environment, where interactions between particles alter the apparent mass, without nuclear reactions or changes within the object materials.

The concept of effective mass (mᵉᶠᶠ) is introduced in the context of classical mechanics and kinetic energy. This idea finds support in the research paper titled "Dark energy and the structure of the Coma cluster of galaxies" by A. D. Chernin, et al. The paper explores the implications of dark energy on the foundational principles of Newtonian mechanics within galaxy clusters, investigating the behaviour of celestial entities. The findings suggest that dark energy influences the dynamics of galaxy clusters, challenging and expanding our understanding of classical mechanics. In this context, it is valid to interpret effective mass in terms of kinetic energy, particularly when considering the influence of external forces and the motion of celestial bodies within these clusters. Therefore, it is accurate to assert that the concept of effective mass, as used in the study of galaxy clusters and dark energy, is closely related to kinetic energy. 

Analysing Newton's second law of motion reveals that when the potential energy of inertial mass (m) decreases due to the application of force and corresponding acceleration, an equivalent kinetic energy is generated, which can be represented as effective mass (mᵉᶠᶠ). This means the inertial mass (m) can be viewed in terms of the object's kinetic energy (KE), which is represented as effective mass (mᵉᶠᶠ). Thus, the expression of total energy (Eᴛᴏᴛ) becomes the sum of inertial mass (m) and effective mass (mᵉᶠᶠ), expressed as:

Eᴛᴏᴛ = m + mᵉᶠᶠ

where the inertial mass (m) and effective mass (mᵉᶠᶠ) represent the potential energy (PE) of the inertial mass (m) and the kinetic energy (KE) due to the motion of the effective mass (mᵉᶠᶠ), respectively.

Effective mass (mᵉᶠᶠ)

Definitive Description:

Effective mass (mᵉᶠᶠ) is a concept in physics that represents the mass of a particle when taking into account not only its inertial properties but also the influence of external forces, such as gravitational or electromagnetic fields, as well as kinetic energy. It is particularly useful in scenarios where the behaviour of the particle is affected by its motion and the surrounding environment, such as in relativistic mechanics or within certain materials where interactions between particles alter the apparent mass.

Example 1:  1% of the Speed of Light

Consider a 10-gram object accelerating to 1% of the speed of light (approximately 2997924.58 m/s). The object's effective mass can be determined by accounting for the kinetic energy as per the respective speed, finding kinetic energy using KE = (1/2)mv² in Jules, and then determining the effective mass mᵉᶠᶠ based on the values of KE in joules.

Given Values:

• Inertial mass (m): 10 grams = 0.01 kg

• Velocity (v): 2997924.58 m/s (0.01c)

• Time (t): 10000 seconds

Calculation Steps:

Calculate Kinetic Energy:

KE = (1/2) × 0.01 kg × (2997924.58 m/s)²

KE = 4.494 × 10¹¹ J 

Calculate Effective Mass:

mᵉᶠᶠ = (2 × 4.494 × 10¹¹ J)/(2997924.58 m/s)²

mᵉᶠᶠ ≈ 14.97 kg

In this example, the effective mass (mᵉᶠᶠ) is not necessarily equal to the inertial mass (m) due to the significant velocity of 1% of the speed of light. At such speeds, the relativistic effects become non-negligible, causing the effective mass to deviate from the inertial mass. Therefore, mᵉᶠᶠ differs from 0.01 kg when considering the kinetic energy (KE) associated with the given velocity.

Example 2: 10% of the Speed of Light

Given Values:

• Inertial mass (m): 10 grams = 0.01 kg

• Velocity (v): 29979245.8 m/s (0.1c)

• Time (t): 10000 seconds

Calculate Kinetic Energy:

KE = (1/2) × 0.01 kg × (29979245.8 m/s)²

KE = 4.494 × 10¹³ J 

Calculate Effective Mass:

mᵉᶠᶠ = (2 × 4.494 × 10¹³ J)/(29979245.8 m/s)²

mᵉᶠᶠ ≈ 149.97 kg

In this example, the effective mass (mᵉᶠᶠ) is not necessarily equal to the inertial mass (m) due to the significant velocity of 10% of the speed of light.

Example 3: 50% of the Speed of Light

Given Values:

Inertial mass (m) = 10 grams = 0.01 kg

Velocity (v) = 149896229 m/s (0.5c)

Time (t) = 10000 seconds

Calculate Kinetic Energy:

KE = (1/2) × 0.01 kg × (149896229 m/s)²

KE = 1.124 × 10¹⁵ J 

Calculate Effective Mass:

mᵉᶠᶠ = (2 × 1.124 × 10¹⁵ J)/(149896229 m/s)²

mᵉᶠᶠ ≈ 374.92 kg

In this example, the effective mass (mᵉᶠᶠ) is not necessarily equal to the inertial mass (m) due to the significant velocity of 50% of the speed of light.

Summary of Effective mass:

1. 1% of the Speed of Light:

mᵉᶠᶠ ≈ 14.97 kg

2. 10% of the Speed of Light:

mᵉᶠᶠ ≈ 149.97 kg

3. 50% of the Speed of Light:

mᵉᶠᶠ ≈374.91 kg

These examples illustrate the variability of effective mass with increasing velocities and highlight the importance of considering relativistic effects in such scenarios.

Interpretation:

The effective mass concept helps us understand that the mass of an object can appear different when influenced by external forces or when moving at significant velocities. In these examples, the effective mass differs from the inertial mass due to relativistic effects at higher velocities. This deviation illustrates the dynamic nature of mass in various physical contexts and emphasizes the need to account for relativistic effects in accurate predictions and analyses.