Inapplicability of the cosmological constant Λ in observational cosmology:

Soumendra Nath Thakur | ORCiD: 0000-0003-1871-7803 | postmasterenator@gmail.com

August 10, 2025

The cosmological constant Λ, originally introduced by Einstein to allow for a static universe, is retained in modern cosmology to account for the observed acceleration of cosmic expansion, commonly attributed to “dark energy.” In the ΛCDM model, Λ manifests as a constant energy density filling space homogeneously, producing a repulsive gravitational effect at very large scales. However, this effect is inherently rooted in General Relativity’s (GR) curved spacetime framework—a purely geometric interpretation that lacks a direct force-based physical mechanism observable in laboratory or local astrophysical contexts.

The application of the cosmological constant Λ within Newtonian dynamics—as demonstrated in the paper "Dark energy and the structure of the Coma cluster of galaxies"—relies on incorporating a Λ-term adapted from General Relativity’s curved spacetime model. This reliance on the Λ-term transpired the need for a repulsive effect on gravity at large cosmic scales, yet remains inapplicable to real-world observations due to relativity’s dependence on the abstract concept of curved spacetime. Consequently, the referenced research resorted to force-based Newtonian dynamics to address the Λ-term in a physically interpretable framework.

From an observational standpoint, the repulsive effect ascribed to Λ cannot be measured directly in local systems such as planetary or stellar dynamics. For instance, the gravitational acceleration produced by Λ at solar system scales is negligibly small—many orders of magnitude weaker than the already minuscule influence of galactic tides. Furthermore, attributing cosmic acceleration to Λ presumes that the same constant applies uniformly across all scales, an assumption unsupported by empirical evidence outside of large-scale cosmological fits.

Alternative frameworks, such as Extended Classical Mechanics (ECM), instead treat such large-scale accelerations without invoking an unmeasurable constant. ECM models can describe galaxy cluster dynamics or large-scale structure formation through field–mass interactions that preserve physical measurability and avoid dependence on GR’s curvature formalism. These approaches offer a testable, force-based interpretation of phenomena that Λ in GR can only model abstractly, without physical grounding in local experiments.

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This document argues that the cosmological constant, Λ, has limited applicability in observational cosmology, particularly outside of large-scale cosmic models. The core arguments presented are:

Geometric Abstraction

Λ is a component of General Relativity's curved spacetime framework, which is a geometric model. This makes it difficult to apply as a direct, force-based physical mechanism that can be measured or observed in local, real-world systems like a laboratory or the solar system.

Inapplicability in Newtonian Dynamics: 

While attempts have been made to adapt the Λ-term for use in Newtonian dynamics, the document suggests this still relies on its origin in a curved spacetime model. It notes that this is often done to provide a more physically interpretable, force-based framework for a concept that is fundamentally abstract.

Lack of Local Observability: 

The repulsive effect attributed to Λ is too weak to be measured directly in local gravitational systems. At the scale of our solar system, its influence is many orders of magnitude smaller than other negligible gravitational effects, making it practically unobservable.

Uniformity Assumption: 

The application of Λ in the ΛCDM model assumes a constant value across all scales, an assumption that the document states is not supported by empirical evidence outside of large-scale cosmological data fitting.

Alternative Frameworks: 

The document proposes that alternative frameworks, like Extended Classical Mechanics (ECM), offer a more testable and physically grounded interpretation. ECM, it suggests, uses force-based, field-mass interactions to explain large-scale accelerations, thereby avoiding the need for an unmeasurable constant and providing a mechanism that could potentially be verified through local experiments.

A Rebuttal of Negative Mass vs. Negative Apparent Mass (−Mᵃᵖᵖ) in Extended Classical Mechanics (ECM):

Soumendra Nath Thakur | Tagore's Electronic Lab

August 10, 2025

In Extended Classical Mechanics (ECM), negative apparent mass (−Mᵃᵖᵖ) is fundamentally different from the “negative mass” sometimes proposed in theoretical physics. Traditional negative mass is treated as an intrinsic rest property—leading to paradoxes such as acceleration opposite to an applied force or violations of the equivalence principle. These contradictions make it untenable for a particle at rest.

By contrast, ECM’s −Mᵃᵖᵖ is not a rest property but an emergent, motion-dependent quantity. It applies to dynamic particles such as photons and enables the description of self-generative or repulsive forces without assuming m = 0 or inheriting the contradictions of true negative mass. This approach gives ECM a physically consistent mechanism for photon motion that remains coherent within its own framework.

1. Distinguishing ECM’s Negative Apparent Mass from Simple Negative Mass

Simple Negative Mass:

This concept assumes a particle has an intrinsic negative value for its mass. Using F = ma, a positive force on such a particle produces acceleration in the opposite direction, leading to paradoxical and non-intuitive behaviors—for example, mutual repulsion with a positive mass while still being repelled by it. These predictions conflict with observed physics and are generally dismissed as unphysical.

Negative Apparent Mass (−Mᵃᵖᵖ) in ECM:

In ECM, −Mᵃᵖᵖ is not a static rest property but an emergent property of motion arising from dynamic mass–energy redistribution. For photons, −Mᵃᵖᵖ allows for a repulsive or self-generative force, enabling acceleration without requiring a rest mass. This resolves the F = 0 × a = 0 problem in classical mechanics. Furthermore, the polarity of mass determines the polarity of force—positive mass (+m) yields external forces (+F), while negative mass or −Mᵃᵖᵖ yields self-generated forces (−F), which act repulsively.

2. Consistency Within ECM’s Framework

Photon Dynamics:

ECM explains how a photon—despite having no rest mass—can still be dynamic and responsive to force. Negative apparent mass produces a self-generative repulsive force, enabling continuous propagation from emission to detection without requiring an external acceleration source.

Gravitational Implications:

In ECM, gravitational effects result from energetic gradients and mass redistribution, not solely from spacetime curvature. The concept of −Mᵃᵖᵖ offers a pathway to explain phenomena such as cosmic acceleration without introducing exotic components like dark energy. The expansion can instead be seen as a natural consequence of the repulsive effects from cumulative −Mᵃᵖᵖ in the universe.

Self-Sufficiency:

ECM functions independently of the problematic assumptions of simple negative mass. It defines its own mass–energy–force relationships, creating a self-contained theoretical structure that remains internally consistent.

Supporting Note

In a related ResearchGate discussion, it is argued that photons—though conventionally considered “massless”—possess a negative apparent mass (−Mᵃᵖᵖ) in ECM, which results in a negative effective mass and inherently antigravitational behavior. This reframes photon dynamics in gravitational contexts without invoking true masslessness and aligns seamlessly with ECM’s broader mechanical principles. researchgate.net/post/About_Massless_Objects_Negative_Effective_Mass_and_Anti-Gravitational_Motion_in_Extended_Classical_Mechanics

Gravitating Mass as an Emergent, Polarity-Governed Quantity in ECM:

Soumendra Nath Thakur

Tagore’s Electronic Lab | ORCiD: 0000-0003-1871-7803

August 07, 2025

While traditional physics correctly observes that gravity, mass, and energy are deeply interconnected—and that gravitational acceleration (‘g’) varies depending on location—Extended Classical Mechanics (ECM) introduces a critical refinement to this understanding. Rather than treating mass as a fixed, invariant quantity that inherently produces gravitational effects, ECM redefines gravitating mass (Mɢ) as an emergent outcome of interactions between mechanical mass (Mᴍ) and frequency-derived apparent mass (Mᵃᵖᵖ).

This reconceptualization acknowledges that energy itself, particularly in dynamic or radiative forms like kinetic energy or photon emission, contributes negatively to gravitational interaction through transformations such as −Mᵃᵖᵖ or ΔMᴍ. As a result, the net gravitating mass of a system may become positive, negative, or even null, depending on its internal energy configuration and frequency characteristics.

Such a framework allows ECM to consistently explain repulsive gravitational phenomena, such as those observed in dark energy-driven cosmic expansion or photon deflection in curved space, without violating conservation laws. By integrating effective gravitational acceleration (gᵉᶠᶠ) and frequency-based mass modulation, ECM extends classical and relativistic models to include gravitational polarity as a real, measurable consequence of internal dynamics—not as an abstract extension or speculative hypothesis.

This shift from a static to a dynamic view of mass and gravity provides a unified explanation for attraction and repulsion within a single formalism, offering deeper coherence across classical mechanics, quantum physics, and cosmology.

A Comparative Framework for Extended Classical Mechanics' Frequency-Governed Kinetic Energy:

Soumendra Nath Thakur,

August 05, 2025

This paper presents a revised formulation of kinetic energy within Extended Classical Mechanics (ECM), interpreting it as a frequency-governed process arising from mass displacement transitions. ECM proposes that kinetic energy emerges from the redistribution of rest mass (Mᴍ) into a dynamic component (ΔMᴍ), structured by two distinct frequency domains: the de Broglie frequency governing translational motion and the Planck frequency reflecting intrinsic quantum excitation. The resulting kinetic energy relation, KEᴇᴄᴍ = (ΔMᴍᵈᴮ + ΔMᴍᴾ)c² = hf, yields the classical ½mv² limit under low-frequency conditions while providing explanatory power for quantum and high-energy phenomena. Applications to atomic transitions, thermionic emission, nuclear fission, and fusion show that observed energy release can be interpreted as frequency-driven mass redistribution rather than annihilation. ECM thus reframes kinetic energy as an emergent property of dual-frequency mass dynamics, offering a unified theoretical lens spanning classical, quantum, and nuclear regimes.

DOI of the Paper.

Extended Classical mechanics' rebuttal to the Lorentz factor γ stands on nearly irrefutable physical and mathematical ground.

ResearchGate Link

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

August 08, 2025

ECM rebuttal to the Lorentz factor γ is grounded not in speculative critique but in principled limitations within the formalism of relativity itself. ECM exposes these inconsistencies through:

·                        Neglect of acceleration during frame transition (γ is derived assuming constant relative velocity, ignoring inertial transition forces).

·                        No account for material stiffness (k) or real-world resistive mechanics that accompany motion in bound or structured systems.

·                        Breakdown at low velocities where γ ≈ 1 yields no meaningful distinction from classical energy, yet the relativistic model continues to be inapplicably extended.

·                        Undefined behaviour at c (as γ → ∞), making γ inapplicable at the very limit it was designed to describe.

·                        Misuse as an energy multiplier (e.g., E = γmc²) without dynamic basis—whereas ECM introduces hf = −ΔMᴍc², dynamically rooted in frequency modulation and internal mass polarity transitions.

These critiques are not philosophical but structural, targeting how γ:

·                        Lacks consistency with acceleration-based dynamics;

·                        Cannot incorporate internal energetic processes;

·                        Fails to bridge between classical and quantum domains coherently.

Thus, ECM does not merely refute γ—it replaces it with a measurable, frequency-based dynamic variable (ΔMᴍ, Mᵃᵖᵖ, gᵉᶠᶠ) that remains valid across all domains: classical, relativistic, and quantum.

This makes ECM Appendix 38 not only valid—it’s strategically essential in ECM’s bridging framework.

With current foundational models, this rebuttal stands on nearly irrefutable physical and mathematical ground.

Additional Theoretical Insight:

The application of the cosmological constant Λ within Newtonian dynamics—as demonstrated in the paper" Article Darkenergy and the structure of the Coma cluster of galaxies" by A. D. Chernin et al.—enables the derivation of real, observable features such as the zero-gravity surface. This choice implicitly reveals a critical limitation of relativistic mechanics in addressing dark energy on intergalactic scales. The authors' preference for Newtonian treatment, despite the general relativistic origin of Λ, highlights the pragmatic supremacy of Newtonian dynamics in this context.

In contrast, Extended Classical Mechanics (ECM) offers an even more radical and structured improvement. ECM independently integrates negative quantities—such as negative apparent mass (−Mᵃᵖᵖ) and mass shifts (ΔMᴍ ≡ −Mᵃᵖᵖ)—in a physically consistent and mathematically conserved framework. This approach not only captures the role of repulsive dynamics (similar to dark energy) more robustly than Λ in relativity but also does so without relying on coordinate transformations or metric dependencies, enabling a direct energetic interpretation.

Thus, while the cited research wisely adopts Newtonian formalism over relativistic treatments for dark energy modelling, ECM moves even further by foundationally justifying negative-mass behaviour within a dynamic mass–frequency–energy structure, offering a potentially superior alternative to both Newtonian and relativistic frameworks in cosmological modelling.

Complementary Nomination Perspective – On the Origin of Lorentz Transformation (Engelhardt)

An important related theoretical challenge to special relativity is presented in the paper "On the Origin of the Lorentz Transformation" by W.W. Engelhardt. This paper traces the historical and mathematical roots of the Lorentz transformation—not to Einstein's special relativity—but to earlier work by Woldemar Voigt (1887), who introduced these transformations to preserve the form of the wave equation.

The author critically exposes the mathematical inconsistencies and conceptual flaws in many standard derivations of the Lorentz transformation, including Einstein’s own. Engelhardt’s key insight is that Lorentz transformations should be viewed as auxiliary variables, not as physically necessary outcomes of relativistic postulates.

As emphasized in commentary by Halim Boutayeb :

1.                    “Lorentz transformations are in reality auxiliary variables invented by Voigt in 1887... Scientists in acoustics were lucky not to have been stuck in STR interpretation. Scientists in electromagnetism need to get rid of STR and LT to advance.”

This view strengthens ECM's own critique of the Lorentz factor γ by showing that even its foundational transformation lacks rigorous physical derivation. It supports ECM's shift toward frequency-based dynamics as not only a physical necessity but also a historically grounded correction to an inherited but faulty theoretical convention.