[Paper Review] Special Relativity and Theory of Gravity via Maximum Symmetry and Localization -- In Honor of the 80th Birthday of Professor Qikeng Lu

Han-Ying Guo|ArXiv.org|Jul 26, 2007

Cosmology and Gravitation Theories7 references3 citations

TL;DR

This paper proposes a generalized framework of special relativity and gravity based on maximum symmetry and localization, unifying Minkowski, de Sitter (dS), and anti-de Sitter (AdS) spacetimes as equivalent under a principle of relativity on maximally symmetric spaces. It introduces dS-special relativity with invariant constants c and R, leading to a dS-gravity model with a dimensionless coupling g ≈ 10⁻⁶¹ that may explain dark matter effects beyond general relativity, suggesting dS physics as a foundation for large-scale cosmology.

ABSTRACT

Like Euclid, Riemann and Lobachevsky geometries on an almost equal footing, based on the principle of relativity of maximum symmetry proposed by Lu and the postulate on invariant universal constants, dS/AdS SR can be set up on an almost equal footing with Einstein's SR. For dS-case, there is a coin-like relation: A law of inertia in Beltrami atlas with Beltrami time simultaneity for the PoR on one side. The proper-time simultaneity and a RW-like dS-space with entropy and an accelerated expanding S^3 fitting the cosmological principle on another. If our universe is asymptotic to the RW-like dS-space, it should be slightly closed with an entropy bound. Contrarily, via its asymptotic behavior, it can fix on Beltrami frames without `an argument in a circle' and acts as the origin of inertia. There is a triality of conformal extensions of three kinds of SR and their null physics on the projective boundary of a 5-d AdS-space. Thus there is a dS-spacetime on the boundary of a vacuum of supergravity. Gravity should be based on the localized PoR of full maximum symmetry with a gauge-like dynamics. Thus, this may lead to theory of gravity of corresponding local maximum symmetry. A simple model of dS-gravity characterized by a dimensionless constant shows the features. Our universe may already indicate that the dS SR and the dS-gravity be the foundation of large scale physics.

Motivation & Objective

• To generalize Einstein’s special relativity to de Sitter and anti-de Sitter spacetimes by extending the principle of relativity to all maximally symmetric spaces.
• To resolve the cosmological constant problem by treating Λ as a fundamental constant like c, G, and ℏ, rather than a quantum vacuum energy.
• To establish a consistent theory of gravity based on localization of maximum symmetry, analogous to gauge theories.
• To explore the conformal triality of null physics on Minkowski, dS, and AdS spacetimes via projective boundary structures.
• To propose dS-gravity as a candidate foundation for large-scale physics, with observable effects potentially explaining dark matter.

Proposed method

• Formulates dS-special relativity on the dS-hyperboloid HR+ ⊂ M1,4 using Beltrami coordinates and Beltrami time simultaneity to define inertial motion.
• Introduces two simultaneity conventions: Beltrami simultaneity for the principle of relativity and proper-time simultaneity for a Robertson-Walker-like dS-space with accelerated S³ expansion.
• Applies Klein’s Erlangen program to unify kinematics via maximal symmetry groups, with Minkowski space as the R → ∞ limit.
• Uses conformal extensions of Minkowski, dS, and AdS spacetimes on a null cone modulo projective equivalence, [N] ≅ ∂P(AdS⁵), to establish a triality of null physics.
• Derives a gauge-like dynamics for dS-gravity through localization of dS-invariance, characterized by a dimensionless coupling g ≈ (ΛGℏ/3c³)¹ᐟ² ≈ 10⁻⁶¹.
• Analyzes the geometry of umbilical Riemann-Cartan manifolds to describe local dS-invariance and identify deviations from general relativity.

Experimental results

Research questions

• RQ1 How can the principle of relativity be generalized to spacetimes of constant curvature, such as de Sitter and anti-de Sitter, beyond Minkowski space?
• RQ2 What is the role of the cosmological constant Λ as a fundamental constant in a generalized special relativity framework?
• RQ3 How does the dS-gravity model with local dS-invariance differ from general relativity, and what observable gravitational effects might arise?
• RQ4 Can the conformal triality of null physics on Minkowski, dS, and AdS spacetimes be unified via a common projective boundary structure?
• RQ5 Does the asymptotic behavior of our universe toward a dS-space with R ≈ (3/Λ)¹ᐟ² support dS-special relativity as a foundation for large-scale physics?

Key findings

• dS-special relativity is formulated on dS/AdS spacetime with radius R as a natural extension of Einstein’s special relativity, with Minkowski space recovered as the R → ∞ limit.
• The universe may be slightly closed in O(Λ) with R ≈ (3/Λ)¹ᐟ², and the entropy bound is S ≈ 3πc³kB/(ΛGℏ), consistent with cosmological observations.
• Two simultaneity conventions—Beltrami and proper-time—support the principle of relativity and the cosmological principle, respectively, on dS-spacetime.
• A triality of conformal extensions exists for Minkowski, dS, and AdS spacetimes, with null physics unified on the projective boundary [N] ≅ ∂P(AdS⁵), linking to AdS/CFT correspondence.
• dS-gravity is proposed as a gauge-like theory with a dimensionless coupling g ≈ 10⁻⁶¹, exhibiting features on umbilical manifolds of local dS-invariance.
• Gravitational effects in the dS-gravity model that exceed general relativity may account for dark matter, suggesting a new mechanism for large-scale structure.