[Paper Review] The Large Numbers Hypothesis: Outline of a self-similar quantum-cosmological Model

H. Genreith|ArXiv.org|Sep 2, 1999

Dark Matter and Cosmic Phenomena18 references3 citations

TL;DR

This paper proposes a self-similar fractal model of the universe that unifies Einstein's early conjecture on gravity's role in elementary particles with Dirac's Large Numbers Hypothesis, linking cosmological and quantum scales through scale-invariant geometry. The model derives consistent scaling relations between fundamental constants and cosmic parameters, offering a unified framework where the universe exhibits fractal-like self-similarity across length and time scales.

ABSTRACT

In 1919 A. Einstein suspected first that gravitational fields could play an essential role in the structure of elementary particles. In 1937, P.A.M. Dirac found a miraculous link between the properties of the visible Universe and elementary particles. Both conjectures stayed alive through the following decades but still no final theory could be derived to this issues. The herein suggested fractal model of the Universe gives a consistent explanation to Dirac's Large Numbers Hypothesis and combines the conjectures of Einstein and Dirac.

Motivation & Objective

• To resolve the long-standing puzzle of the numerical coincidences between fundamental constants and cosmological parameters, known as Dirac's Large Numbers Hypothesis.
• To reconcile Einstein's early speculation that gravitational fields might structure elementary particles with Dirac's observation of large dimensionless numbers in physics.
• To develop a consistent theoretical framework where quantum-scale and cosmological-scale phenomena are linked through self-similar scaling laws.
• To provide a geometric and dynamical model that explains the observed proportionality between the size of the universe and the age of elementary particles.
• To explore whether the universe's structure exhibits fractal-like properties across different scales, preserving physical laws under scale transformations.

Proposed method

• Proposes a self-similar spacetime model where physical laws remain invariant under scale transformations, inspired by fractal geometry.
• Applies scale-invariant field equations to relate the gravitational constant, elementary particle masses, and cosmological parameters.
• Uses dimensionless ratios of fundamental constants (e.g., ratio of electromagnetic to gravitational forces) to derive scaling relations between particle and cosmic scales.
• Introduces a recursive geometric structure where the universe's large-scale properties mirror those of elementary particles through scaling symmetry.
• Employs a formalism based on conformal invariance and dimensionless unification to maintain consistency across quantum and cosmological regimes.
• Derives scaling laws that connect the age of the universe to the Compton wavelength of elementary particles, suggesting a deep link between quantum and cosmic scales.

Experimental results

Research questions

• RQ1 Can the observed large dimensionless numbers in physics—such as the ratio of the electromagnetic to gravitational force—be explained by a self-similar structure of spacetime?
• RQ2 Is there a consistent theoretical framework that unifies Einstein’s idea of gravity shaping elementary particles with Dirac’s Large Numbers Hypothesis?
• RQ3 Do the physical constants and parameters of the universe exhibit scale-invariant relationships across quantum and cosmological scales?
• RQ4 Can a fractal-like, self-similar model of spacetime reproduce known physical constants and cosmological observations without introducing arbitrary parameters?
• RQ5 What are the implications of scale invariance for the unification of quantum mechanics and general relativity in this model?

Key findings

• The model establishes a consistent scaling relation between the age of the universe and the Compton wavelength of elementary particles, supporting Dirac’s Large Numbers Hypothesis.
• The ratio of the gravitational to electromagnetic force constants is shown to scale with cosmic time, aligning with Dirac’s original observation of large dimensionless numbers.
• The self-similar structure implies that the same geometric and physical principles govern both quantum-scale particles and large-scale cosmic structures.
• The model predicts that the gravitational constant G is not constant but varies inversely with cosmic time, consistent with Dirac’s varying-G hypothesis.
• The framework provides a geometric explanation for the observed proportionality between the number of particles in the universe and its age, without requiring fine-tuning.
• The model remains consistent under conformal transformations, suggesting a deeper symmetry underlying the unification of quantum and cosmological physics.