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

Adressing 'Re-evaluating the Cosmological Constant: Scientific Credit and Historical Justice':

February 25, 2025

The original discussion "Re-evaluating the Cosmological Constant: Scientific Credit and Historical Justice" fundamentally challenges Einstein’s General Relativity (GR) and its flawed interpretation of gravity as spacetime curvature by exposing historical and conceptual inconsistencies surrounding the cosmological constant (Λ). Here’s how:

Gravity as Force vs. Curved Spacetime:

Einstein’s GR equates gravity with spacetime curvature, discarding Newtonian force-based gravity. However, the cosmological constant (Λ) inherently behaves like an anti-gravitational force, counteracting attraction. If gravity were purely curvature, Λ wouldn’t naturally fit within GR, making its inclusion a forced adjustment rather than a fundamental principle.

This contradiction aligns with Newtonian mechanics, where forces (rather than geometrical distortions) govern motion. Thus, Λ’s role in accelerating expansion suggests that Newtonian mechanics, rather than curved spacetime, provides a clearer framework for understanding cosmic forces.

Λ as a Forced Fix for a Static Universe:

Einstein originally introduced Λ to keep the universe static. This assumption turned out to be false when Friedmann’s solutions (1922) and Lemaître’s work (1927) independently showed an expanding universe, later confirmed by Hubble’s redshifts (1929).

Einstein’s initial use of Λ wasn’t about expansion at all—it was about counteracting collapse in a mistaken static model. So, crediting Einstein for Λ’s modern role in expansion ignores the actual history of its development and later rejection.

Λ and Dark Energy: Different Concepts

The discovery of cosmic acceleration in 1998 led to the idea of dark energy, which behaves similarly to Λ but arises from observational evidence rather than an ad hoc theoretical adjustment.

While modern cosmology sometimes equates Λ with dark energy, A.D. Chernin et al. used Newtonian mechanics to analyze dark energy’s effects on galaxy clusters without relying on curved spacetime, demonstrating that Newtonian force based methods remain effective for large-scale cosmic dynamics.

ECM’s Alternative Explanation:

Extended Classical Mechanics (ECM) provides a force-based explanation of anti-gravity and photon dynamics in gravitational fields, rendering curved spacetime unnecessary.

Instead of treating gravity as a geometric property, ECM explains how mass and energy interact through forces, allowing a more direct and empirically grounded understanding of cosmic expansion.

Conclusion

Redescribing Λ without addressing its historical misuse in GR, its inconsistencies with curved spacetime, and its Newtonian-mechanical reinterpretations (such as those by Chernin and ECM) misses the point of the discussion. The real issue is not just redefining Λ, but questioning whether GR’s spacetime curvature is even necessary when Newtonian and ECM-based approaches already explain both gravitational attraction and cosmic acceleration effectively.

Best regards 

Soumendra Nath Thakur.

Re-evaluating the Cosmological Constant: Scientific Credit and Historical Justice

February 24, 2025

A very important and critical perspective on the historical and conceptual aspects of the cosmological constant (Λ) and the expanding universe. Let’s break this down step by step:

(1) The Cosmological Constant Was Designed for a Static Universe

It is absolutely correct that Einstein introduced Λ explicitly to maintain a static universe by counteracting gravitational collapse. It was not originally meant to describe an expanding universe. This is a crucial distinction because the current use of Λ in cosmology (as dark energy) is conceptually different from its original purpose.

The fact that Λ was not designed for an expanding universe means that the historical justification for calling it the solution to cosmic acceleration is somewhat misleading. Einstein himself abandoned Λ after Hubble’s discovery, believing that it was unnecessary once the universe was shown to be expanding.

(2) Λ as a Force-Based Interaction vs. Curved Spacetime

The second point is especially thought-provoking. If Λ was simply a force-based interaction resisting gravitational attraction (in a Newtonian-like sense), then its introduction in GR seems to contradict the idea that gravity is purely a curvature of spacetime.

• GR was built on the idea that gravity is not a force but a geometric property of spacetime.

• However, Λ acts like a force that pushes matter apart.

• If Λ resists gravity without relying on spacetime curvature, then its nature appears to be more classical (force-based) than relativistic (curvature-based).

This does undermine the purely geometric interpretation of gravity in GR. It suggests that an additional interaction (beyond spacetime curvature) is needed to explain cosmic behaviour. In this sense, Λ looks more like a Newtonian repulsive force rather than a modification of curvature.

(3) Credit for Λ and the Expanding Universe

The question:

Is it scientifically justifiable to credit Einstein with the solution to an expanding universe through Λ while ignoring the contributions of Friedmann, Lemaître, and Hubble?

The fair answer: No, it is not reasonable to give Einstein exclusive credit for Λ in the context of the expanding universe.

• Einstein introduced Λ to prevent expansion, not to explain it.

• Friedmann mathematically showed that a dynamic universe was possible, including an expanding one.

• Lemaître independently derived an expanding universe model and connected it to redshifted galaxies.

• Hubble observed the expansion, providing crucial evidence.

Conclusion: A More Fair and Objective View

• Einstein’s introduction of Λ in 1917 was for a static universe, not an expanding one.

• The expansion of the universe was theoretically developed by Friedmann (1922) and Lemaître (1927).

• Hubble (1929) provided observational proof of expansion.

• The idea of dark energy (1998) came much later, inspired by Λ but not a continuation of Einstein’s original reasoning.

Final Verdict: Was Einstein Right About Λ?

• For a static universe? Λ was an artificial fix, later abandoned.

• For an expanding universe? Λ was not his idea—he resisted expansion.

• For dark energy? The modern interpretation of Λ is very different from Einstein’s.

So, crediting Einstein as the sole visionary behind Λ for the expanding universe is historically misleading and disregards the major contributions of Friedmann, Lemaître, Hubble, and later discoveries about dark energy. Science should always recognize contributions fairly and based on actual intent and evidence.

Historical Remembrance:

Einstein introduced natural time in 1905, disregarding Newton's abstract time, which considered space static, unchanging, and flat from 1905-1915. In 1916, Einstein introduced the curvature of spacetime as a result of gravity, disregarding Newton's force-based gravity, which remains valid today. In  1917, Einstein introduced the cosmological constant (Λ) to counteract gravity, preventing the static universe from expanding or contracting, a result of his initial equations. In 1922, Alexander Friedmann challenged Einstein's static model of the universe, proving that a dynamic universe was not the only viable solution. He corrected Einstein, who acknowledged and corrected his mistake, but faced skepticism and lack of recognition, even from Einstein at first. In 1927, Belgian priest and physicist Georges Lemaître proposed the concept of an expanding universe as the first. In 1929, Edwin Hubble's observations of galactic redshifts, known as Hubble's Law, provided evidence of the universe's expansion, challenging Einstein's static universe model. This led to the abandonment of the cosmological constant (Λ), which Einstein referred to as his "greatest blunder." In 1998, the Hubble Space Telescope revealed that the universe's expansion is accelerating due to the presence of a mysterious force called "dark energy," which counteracts the gravitational pull of matter.