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