Comparative Study: Classical and Relativistic Mechanics - Principles, Examples, and Discrepancies in Length Contraction Predictions:

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

08-06-2024

Abstract:

This study investigates the principles of classical and relativistic mechanics, exploring their applications through examples and analysing the differences in length contraction predictions. Classical mechanics concepts such as Newton's second law and Hooke's Law are discussed, alongside relativistic mechanics principles including the Lorentz factor. The study provides examples of both classical deformation and relativistic length contraction, demonstrating how these phenomena manifest at different velocities. By comparing the predicted length contractions, significant discrepancies are identified, particularly at higher velocities. Factors contributing to these differences, such as acceleration-related length deformation and the limitations of the Lorentz factor, are examined. 

Keywords: classical mechanics, relativistic mechanics, length contraction, Lorentz factor, Newton's second law, Hooke's Law, velocity, acceleration,

Lorentz factor (γ):

The Lorentz factor (γ), introduced by Albert Einstein, quantifies the effects of special relativity on time, length, and relativistic mass for objects moving relative to an observer. It is defined by the equation γ = 1/√(1-v²/c²), where v is the velocity of the moving object and c is the speed of light. At rest (v=0), γ=1, indicating no time dilation or length contraction.

As velocities increase, the Lorentz factor approaches 1, indicating negligible relativistic effects. For example, at v=100 m/s, γ≈1.0000000000000556, and at v=1,000,000 m/s, γ≈1.0000055556. The Lorentz factor becomes significant when it exceeds 1.1, corresponding to velocities approaching 0.413 times the speed of light (41.3% of c), such as v=123,900,000 m/s.

Comparison between classical deformation and relativistic length contraction reveals significant discrepancies in relativistic predictions for length changes under similar conditions. Relativistic length contractions are notably smaller than their classical counterparts, as evidenced by the differences measured.

In classical mechanics:

1. The equation v = u + at relates the initial velocity (u), acceleration (a), time (t), and final velocity (v).

2. Newton's second law states F = ma, where force (F) is directly proportional to acceleration (a) and inversely proportional to mass (m).

3. Hooke's Law expresses the relationship between length deformation (ΔL) and applied force (F): F = k⋅ΔL, where k is the spring constant.

4. The spring constant (k) is calculated as k = F/ΔL, where F is the applied force and ΔL is the displacement.

Given an object with mass m = 10g and a spring constant k = 29979.2458N/m, initial velocity u = 0 m/s:

For velocities v=100m/s, v=1,000,000m/s, and v=123900000m/s, time (t) can be determined from (v−u) = 100m/s, yielding t = 1 second.

Using F = ma, forces (F) are calculated for each velocity. Then, ΔL is determined using ΔL = F/k.

The respective deformations (ΔL) for the given velocities are approximately:

• ΔL ≈ 3.34 × 10⁻⁵ m for v = 100 m/s

• ΔL ≈ 0.333 m for v = 1,000,000 m/s

• ΔL ≈ 41.32 m for v = 123,900,000 m/s

To find the total lengths after deformation, add these values to the original length of 1 meter:

• For v = 100 m/s: Total length ≈ 1 meter + 3.34 × 10⁻⁵ m

• For v = 1,000,000 m/s: Total length ≈ 1 meter + 0.333 m

• For v = 123,900,000 m/s: Total length ≈ 1 meter + 41.32 m

These total lengths represent the final lengths of the object after deformation.

Relativistic Mechanics Examples:

Given an object with a rest length (L⁰) of 1 meter and a rest mass of 10 grams (0.01 kg), initially at rest (v = 0 m/s), the object separates from the reference frame at t = 0 s and achieves velocities of 100 m/s, 1,000,000 m/s, and 123,900,000 m/s respectively.

To calculate the length contraction (ΔL) for each velocity using the Lorentz factor (γ), we employ the formula ΔL = γ⋅L⁰.

Using the speed of light (c ≈ 3 × 10⁸ m/s), we find:

1. For v = 100 m/s:

γ ≈ 1.0000000000000556

ΔL ≈ 1.0000000000000556 m

2. For v = 100,000 m/s:

γ ≈ 1.0000055556

ΔL ≈ 1.0000055556 m

3. For v = 123,900,000 m/s:

γ ≈ 1.1

ΔL ≈ 1.1 m

These values represent the change in length (contraction) relative to the observer due to the object's motion, with the original length L⁰ being 1 meter.

Comparison between Classical deformation and relativistic length contraction:

Comparison between classical deformation and relativistic length contraction reveals significant discrepancies in their predictions for length changes under similar conditions. Relativistic length contractions are notably smaller than their classical counterparts, as evidenced by the following differences:

At v = 100 m/s, the relativistic contraction is 0.000033399999444m smaller.

At v = 1,000,000 m/s, the relativistic contraction is 0.3329944444m smaller.

At v = 123,900,000 m/s, the relativistic contraction is 41.22 m smaller.

These differences arise due to several factors. Firstly, the relativistic Lorentz factor (γ) does not account for acceleration-related length deformation as the moving frame transitions from rest to the desired velocity. Additionally, it does not consider the stiffness of the material or the spring constant (k), leading to unaccounted changes in length.

Moreover, the applicability of the Lorentz factor is limited in everyday scenarios where speeds are well below the speed of light. Furthermore, the traditional application of γ·mc² is not suitable for processes involving speeds equal to or exceeding the speed of light, such as nuclear conversions of mass into energy. Therefore, the relativistic Lorentz factor is flawed and inferior to classical mechanics' interpretations of material deformation (ΔL).

#classicalmechanics, #relativisticmechanics, #lengthcontraction, #lorentzfactor, #NewtonsSecondLaw, #hookeslaw, #velocity, #acceleration,

The products of nuclear processes in brief:

Nuclear reactions like fission and fusion split large nuclei, releasing heat and gamma rays, and merge light nuclei, releasing energy, while radioactive decay loses energy through radiation to unstable nuclei.

Nuclear fission products are atomic fragments left after a large nucleus splits into smaller nuclei, releasing heat energy and gamma rays. Nuclear fusion involves merging two light nuclei to form a single heavier nucleus, releasing energy as the resulting mass is less than the original nuclei's. Radioactive decay is the process by which an unstable atomic nucleus loses energy through radiation.

Nuclear Fission:

• In nuclear fission, large nuclei split into smaller fragments, releasing heat energy and gamma rays.

• The products of nuclear fission are atomic fragments (such as isotopes of different elements) resulting from the splitting process.

Nuclear Fusion:

• Nuclear fusion involves merging two light nuclei to form a single, heavier nucleus.

• This process releases energy because the resulting mass is slightly less than the sum of the original nuclei’s masses.

Radioactive Decay:

• Radioactive decay occurs when an unstable atomic nucleus spontaneously emits radiation.

• During decay, the unstable nucleus loses energy, leading to the transformation of the nucleus into a more stable state.

The nuclear reactions encompass both fission and fusion, each with distinct outcomes.

Keywords: nuclear reactions,  fission, fusion, radioactive decay,

#NuclearReactions #fission #fusion #RadioactiveDecay

The paper "Dark energy and the structure of the Coma cluster of galaxies" by Chernin et al.

The paper "Dark energy and the structure of the Coma cluster of galaxies" by Chernin et al. explores the influence of dark energy on the structure and mass of the Coma cluster of galaxies. The authors approach the Coma cluster as a gravitationally bound system embedded in the dark energy background described by the ΛCDM cosmology. They aim to determine whether the dark energy density is significant enough to affect the structure of such a large cluster.

Key Concepts and Methods:

1. Three Characteristic Masses:

• Matter Mass (Mᴍ): The total mass of matter (dark matter and baryons) within a given radius.

• Dark Energy Effective Mass (Mᴅᴇ or mᵉᶠᶠ): The mass equivalent of the dark energy's gravitational effect within the same radius. It is negative due to dark energy's repulsive nature. It is negative due to dark energy's repulsive nature and is defined as the difference between the gravitating mass and the matter mass, calculated as:

Mᴅᴇ (or mᵉᶠᶠ) = Mɢ - Mᴍ

• Gravitating Mass (Mɢ): The total mass that includes the effect of dark energy, defined as

Mɢ = Mᴍ + Mᴅᴇ (or mᵉᶠᶠ).

2. Zero-Gravity Radius (Rᴢɢ):

• The radius where the gravitational pull due to matter is exactly balanced by the repulsive effect of dark energy.

• Calculated as 

Rᴢɢ = [Mᴍ/(8π/3)ρᴅᴇ]^1/3, 

where Mᴍ is the matter mass, and ρᴅᴇ is the density of dark energy.

3. Matter Density Profile:

• The authors propose a new density profile that accounts for dark energy effects, improving the fit to observational data from 1.4 Mpc to 14 Mpc.

Findings:

1. Impact of Dark Energy:

• At small radii (R ≤ 14 Mpc), dark energy has a negligible effect on the mass distribution, so

Mɢ ≃ Mᴍ. 

• In this context: Mɢ is the gravitational mass. Mᴍ is the matter mass.

• At larger radii (R ≥ 14 Mpc), dark energy significantly affects the structure, and its repulsive effect becomes comparable to or greater than the gravitational attraction of matter.

2. Mass Estimates:

• The upper limit for the total size of the Coma cluster is approximately 20 Mpc, beyond which the cluster cannot remain gravitationally bound due to the repulsive force of dark energy.

• The total matter mass within this radius is estimated to be Mᴍ ≲ 6.2 × 10¹⁵ M⊙.

3. Comparison with Traditional Profiles:

• The NFW (Navarro-Frenk-White) and Hernquist profiles are traditional models used to describe the matter density of clusters but do not adequately account for dark energy.

• The new profile proposed by the authors provides a better fit to observational data, especially at larger radii where dark energy effects are significant.

Conclusion:

The study concludes that dark energy significantly affects the structure of the Coma cluster at large distances from the centre. The proposed matter density profile, which incorporates the effects of dark energy, provides a more accurate representation of the cluster's mass distribution. The findings suggest that the Coma cluster's total matter mass is capped at around 6.2 × 10¹⁵ M⊙, and its size is limited to about 20 Mpc due to the influence of dark energy.

Implications:

• Large-Scale Structure: The results highlight the importance of considering dark energy when studying the structure and mass of galaxy clusters.

• Observational Verification: Future observations can test these predictions, particularly the proposed upper limits on the Coma cluster's size and mass.

• Cosmological Models: The findings reinforce the ΛCDM model and its implications for the role of dark energy in the Universe.

• By integrating dark energy into the analysis, this paper provides a more comprehensive understanding of the Coma cluster's structure, offering insights into the broader effects of dark energy on cosmic structures.

Reference: 

Chernin, A. D., Bisnovatyi-Kogan, G. S.,Teerikorpi, P., Valtonen, M. J., Byrd, G.G., & Merafina, M. (2013). Dark energyand the structure of the Coma cluster ofgalaxies. Astronomy & Astrophysics, 553,A101. https://doi.org/10.1051/0004-6361/201220781 

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