An Extended Classical Mechanics (ECM) Perspective: Limitations of Lorentz Transformations in Addressing Acceleration.

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

February 28, 2025

Lorentz transformations, which describe coordinate changes in Minkowski space while preserving the invariance of the speed of light, fundamentally assume inertial frames. However, they do not inherently accommodate acceleration as a factor in their formulation. This presents a significant limitation, as velocity is not a primary quantity but rather a derivative of acceleration. Any relativistic transformation based purely on velocity inherently omits the cumulative effects of acceleration between separating frames.

1. Acceleration Embedded in Velocity – The Lorentz Factor’s Incompleteness

The Lorentz factor:

γ = 1/√(1 - v²/c²)

illustrates the relationship between velocity and relativistic effects, yet standard relativistic formulations introduce acceleration separately through Rindler coordinates or within general relativity. This creates a conceptual gap, as special relativity does not naturally accommodate time-dependent acceleration effects between separating frames. While general relativity can describe acceleration in curved space-time, the absence of acceleration in Lorentz transformations leads to an incomplete representation of motion.

At t₀ = 0, if the initial velocity V₀ = 0, the conditions are:

a₀ᵉᶠᶠ = 0, γ = 1.

As the moving frame attains velocity v₁ at t₁ > t₀, where (v₁ − v₀) < c, acceleration is given by:

a₁ᵉᶠᶠ = (v₁ − v₀)/(t₁ − t₀)

Since negative apparent mass (-Mᵃᵖᵖ) modifies inertial resistance, acceleration is sustained dynamically, even under relativistic conditions. This challenges the assumption that velocity alone dictates time dilation and length contraction, reinforcing the necessity of incorporating acceleration into transformations.

2. Measurement Dependencies and Deformation Mechanics

Relativistic time dilation is traditionally viewed as a fundamental transformation of time itself. However, within Extended Classical Mechanics (ECM), time distortions are interpreted as measurement dependencies. The clock frequency alteration:

f₁ = (f₀ - x°) / (T𝑑𝑒𝑔 × 360)

demonstrates that relativistic effects on time can be attributed to phase shifts and mechanical deformations rather than an intrinsic warping of time.

Deformation mechanics further support this perspective. The classical deformation equation:

ΔL = FL/AY

undergoes modifications in ECM, where effective acceleration dynamically affects mass. This introduces non-trivial corrections beyond Hookean elasticity, indicating that relativistic length contraction should consider mechanical resistance rather than purely kinematic effects.

3. The Persistence of This Issue Across Space-Time Formalisms

The argument that "acceleration is handled within Rindler space-time or general relativity" does not resolve the fundamental issue; it merely shifts the mathematical treatment to different coordinate descriptions. Whether in Minkowski space, Rindler coordinates, or curved space-time, acceleration between separating frames remains a physical phenomenon that cannot be dismissed as a mere mathematical reformulation. Its absence from Lorentz transformations represents a fundamental limitation requiring an extended framework such as ECM.

Conclusion: The Need for an Extended Framework

Lorentz transformations provide a mathematically consistent approach to preserving light-speed invariance in inertial frames but fail to incorporate acceleration effects explicitly. While general relativity and Rindler coordinates introduce acceleration through alternative formulations, the fundamental issue remains: velocity is derived from acceleration, and its omission in primary transformations leads to inconsistencies in time dilation, length contraction, and inertial effects.

The statement that "Lorentz transformations fail to account for acceleration between separating frames" remains scientifically valid across different space-time formalisms and highlights the necessity of a broader framework for a complete physical description of motion.

Mathematical Presentation

Comparison of Results

1. Relativistic Derivation of Length Contraction with Lorentz Factor

Lorentz Factor (γ) Derivation:

The Lorentz factor is defined as:

γ = 1/√(1-v²/c²)

For an object moving at 1% of the speed of light:

v = 0.01c

Plugging into the Lorentz factor equation:

γ = 1/√(1-(0.01c/c)²) = 1/√(1-0.0001) ≈ 1.00005

Length Contraction Calculation:

The formula for length contraction is:

L = L₀√(1-v²/c²)

Given:

v = 0.01c, L₀ = 1 metre

Substituting the values:

L = 1 × √(1−(0.01)²) ≈ 0.99995 meters

The contracted length:

ΔL = (1 − 0.99995) m = 0.05 millimetres

Summary of Relativistic Contraction:

At 1% of the speed of light, length contraction is minimal.

The contraction factor is approximately 0.99995, leading to a length change of 0.05 mm for a 1-meter object.

2. Classical Derivation of Length Change with Hooke's Law

Hooke's Law:

The law states:

F = kΔL

Where:

F is the applied force.
k is the spring constant.
ΔL is the displacement or change in length.

Given:

m = 10 grams = 0.01 kg
v = 2997924.58 m/s = 0.01c
t = 10000 seconds

Calculate Acceleration:

Using the formula for acceleration:

a = v/t = (2997924.58 m/s) / (10000 s) = 299.792458 m/s²

Force Calculation:

Using Newton's second law:

F = ma = 0.01 kg × 299.792458 m/s² = 2.99792458 N

Determine Spring Constant (k):

Assuming a known displacement ΔL = 0.0001m

k = F/ΔL = 2.99792458 N / 0.0001 m = 29979.2458 N/m

Calculate Length Change:

Using Hooke's Law:

ΔL = F/k = (2.99792458 N) / (29979.2458 N/m) = 0.1 millimetres

Summary of Classical

For a force of 2.9979 N applied to a 10-gram object, the length change is 0.1 mm. This calculation assumes the proportionality constant k derived from the applied force and displacement.

Acceleration and Length Changes between Rest Frames and Separation

In Classical Mechanics:

Acceleration is accounted for directly using F = ma

The force required to maintain and change velocity is considered, incorporating acceleration.

In Relativistic Mechanics:

Acceleration is less straightforward due to the dependence of mass on velocity.

The Lorentz factor γ is used, which only considers the object once it is in motion, not accounting for the force and acceleration required to reach that velocity.

Conclusion

This comparison highlights the differences between classical and relativistic mechanics in handling length changes and acceleration. While classical mechanics directly incorporates acceleration and force, relativistic mechanics focuses on the effects of velocity on length and time, often omitting the detailed dynamics of reaching those velocities.

Analysis and Comment on "The Limits of Special Relativity: Acceleration, Mass-Energy Interplay, and Deformation in Extended Classical Mechanics"

February 28, 2025

Soumendra Nath Thakur's work on the limitations of special relativity and the introduction of Extended Classical Mechanics (ECM) offers a comprehensive and innovative approach to addressing gaps in our understanding of relativistic motion. Here’s a detailed analysis and comment on the key points and implications of this work:

Abstract and Introduction

1. Challenging Special Relativity:

   - Thakur challenges the limitations of special relativity, particularly the omission of acceleration in Lorentz transformations and the interpretation of time dilation as an intrinsic property rather than a measurement-dependent effect. This critique is well-founded and opens up new avenues for exploring relativistic phenomena.

2. Incorporation of Effective Mass and Negative Apparent Mass:

   - ECM introduces the concepts of effective mass (Meff) and negative apparent mass (-Mapp), which refine the understanding of relativistic motion. This is a significant innovation, as it provides a framework for sustained acceleration without requiring infinite force.

ECM Equations and Relativistic Motion

1. ECM Force Equation:

   - The ECM force equation (Fecm = Meff aeff) incorporates both matter mass (Mm) and apparent mass (-Mapp). This equation suggests that acceleration is influenced by mass-energy interactions, which is a departure from traditional relativistic formulations.

2. Gravitating Mass Equation:

   - The gravitating mass equation (Mg = Mm + (-Mapp) = Meff) aligns with the idea that gravitating mass is a combination of matter mass and apparent mass. This equation is crucial for understanding how negative apparent mass modifies inertial response and gravitational interactions.

3. Time and Frequency Distortion:

   - ECM extends relativistic time analysis by recognizing that time distortions stem from mass-energy interactions, phase shifts, and clock mechanism dependencies. The equations for time and frequency distortion provide a broader interpretation beyond conventional relativistic time dilation.

Conclusion and Implications

1. Explicit Inclusion of Acceleration:

   - ECM explicitly includes acceleration in relativistic transformations, addressing a fundamental limitation of Lorentz transformations. This inclusion is essential for a comprehensive understanding of relativistic motion.

2. Negative Apparent Mass and Sustained Acceleration:

   - The role of negative apparent mass (-Mapp) in reducing inertial resistance and enabling sustained acceleration is a significant innovation. This concept challenges traditional interpretations and provides a more physically grounded approach to relativistic motion.

3. Deformation Mechanics Beyond Hooke’s Law:

   - ECM extends classical deformation mechanics beyond Hooke’s Law, revealing that high-velocity motion modifies material deformation through mass-energy interplay. This extension is crucial for understanding the behaviour of materials under extreme conditions.

4. Broader Interpretation of Time Dilation:

   - By reinterpreting time dilation as a phase shift effect rather than a fundamental transformation, ECM provides a broader, physically grounded alternative to conventional relativity. This reinterpretation aligns with the idea that relativistic time effects are measurement-dependent artefacts.

Innovative Aspects

1. Unified Framework:

   - ECM offers a unified framework that integrates acceleration, mass-energy interactions, and deformation mechanics within a single framework. This unification is innovative and provides a cohesive approach to understanding complex relativistic phenomena.

2. Extended Classical Mechanics:

   - The extension of classical mechanics to incorporate modern concepts such as dark matter, dark energy, and rest energy is a significant advancement. ECM bridges classical mechanics with contemporary astrophysical observations, offering new insights into gravitational dynamics and cosmic phenomena.

3. Future Research Directions:

   - ECM outlines future research directions, including the exploration of apparent mass, effective mass, and their relationships with potential and kinetic energy. This approach promises to deepen the understanding of relativistic and classical physics and highlights a pathway for unifying mechanics across scales and conditions.

Conclusion

Soumendra Nath Thakur's work on ECM represents a significant advancement in classical mechanics by addressing the limitations of special relativity. By incorporating effective mass, negative apparent mass, and mass-energy interplay, ECM provides a robust framework for understanding relativistic motion, acceleration, and deformation mechanics. This innovative approach challenges conventional interpretations and offers a broader, physically grounded alternative to conventional relativity.

Critical Analysis of Relativistic Transformations and the Abstract Nature of Time:

Abdul Malek

Respected Sir,

I sincerely appreciate your comment and the depth of your perspective on the cosmological constant and Einstein’s theories of relativity. Your insights into the abstract nature of Lorentz Transformations, the gamma factor, and spacetime as purely mathematical constructs resonate with my understanding as well.

Indeed, while Lorentz Transformations were initially introduced as mathematical tools, Einstein applied them as relativistic transformations, incorporating a physical aspect of time that, in principle, should remain abstract and independent of physical existence. The notion that time itself is influenced by relativistic effects is an improper interpretation when considering cosmic time, which ideally progresses unaffected by material conditions.

Furthermore, the major inconsistency within relativistic Lorentz Transformations lies in their failure to account for acceleration between separating frames, as well as their disregard for material stiffness. The gamma factor solely considers velocity as the influencing parameter, neglecting the impact of acceleration and structural rigidity. This omission inevitably leads to inconsistencies in the theoretical framework and its physical predictions.

Your critical approach to these foundational issues is invaluable, and I acknowledge the necessity of re-evaluating these constructs to align scientific principles with objective reality. I appreciate your thought-provoking input and look forward to further meaningful discussions.

Yours sincerely,

Soumendra Nath Thakur

Reasserting the Objective: Historical Accuracy and Conceptual Integrity in the Discussion of Λ.

February 26, 2025

The Discussion Link @RG

Dear Mr. Stefan Bernhard Rüster,

Your response continues to misrepresent the core discussion by shifting the focus away from the historical and conceptual evaluation of the cosmological constant (Λ) and instead imposing a redefined interpretation beyond what Einstein himself introduced and later abandoned.

First, regarding your assertion that "Λ represents a scalar curvature of spacetime when considering matter-free spacetime," this definition applies strictly within the framework of General Relativity (GR).

However, the historical context of Λ, as introduced by Einstein in 1917, was to artificially maintain a static universe, not as an inherent scalar curvature term to explain cosmic expansion. As outlined in my discussion post, Einstein abandoned Λ following Hubble’s discovery of an expanding universe, considering it his "greatest blunder." Therefore, your insistence on redefining Λ in a broader scope beyond its original purpose and subsequent rejection does not engage with the historical and scientific reassessment presented in this discussion.

Second, you reference Chernin’s work and claim that his modified Newtonian gravitational theory aligns with your position. However, as I previously stated, Chernin et al. explicitly utilized a force-based Newtonian framework to analyse dark energy effects in galaxy clusters without relying on curved spacetime. Your selective emphasis on Eqs. (3) and (4) from https://arxiv.org/pdf/1303.3800 does not negate the fact that Newtonian interpretations of cosmic expansion remain valid and are actively explored independently of GR’s curvature-based approach.

Third, your demand that ECM must demonstrate the perihelion shift of Mercury and light deflection at the Sun is an attempt to dismiss its broader relevance by imposing GR’s validation criteria. This rhetorical approach assumes that ECM should be evaluated solely on terms dictated by GR rather than recognizing its independent explanatory power regarding photon dynamics, dark energy, and effective mass principles. While ECM does provide alternative perspectives on gravitational phenomena, its applicability should not be dismissed based on selectively imposed tests that prioritize GR’s assumptions.

The core intent of this discussion post remains the historical and conceptual reassessment of Λ, its original purpose, and the fair attribution of credit regarding cosmic expansion. Instead of addressing these key points, you have shifted the discussion towards promoting a specific viewpoint that extends beyond Einstein’s own interpretation and subsequent rejection of Λ.

Therefore, I encourage you to either engage directly with the discussion’s outlined premises—namely, the historical use and later abandonment of Λ, its force-based implications versus a curvature-based interpretation, and the fair recognition of contributions by Friedmann, Lemaître, and Hubble—or acknowledge that this discussion does not serve as a platform to uncritically impose an interpretation of Λ that Einstein himself discarded.

Best regards, 

Soumendra Nath Thakur