[Paper Review] Incorporating the Scale-Relativity Principle in String Theory and Extended Objects

Carlos Castro|ArXiv.org|Nov 29, 1996

Scientific Research and Discoveries2 references3 citations

TL;DR

This paper proposes a novel unification of Nottale's scale-relativity principle with string theory and extended objects by enforcing invariance under scale transformations (dilatations) on the world-volume measure. It shows that scale-relativity invariance is compatible with vanishing β-functions for the target-space metric and coordinates, suggesting a consistent framework where the Planck scale acts as an invariant scale, and hints at Weyl-Finsler geometries emerging from the merger of motion and scale relativity.

ABSTRACT

First steps in incorporating Nottale's scale-relativity principle to string theory and extended objects are taken. Scale Relativity is to scales what motion Relativity is to velocities. The universal, absolute, impassible, invariant scale under dilatations, in Nature, is taken to be the Planck scale which is not the same as the string scale. Starting with Nambu-Goto actions for strings and other extended objects, we show that the principle of scale-relativity invariance of the world-volume measure associated with the extended objects ( Lorentzian-scalings transformations with respect to the resolutions of the world-volume coordinates) is compatible with the vanishing of the scale-relativity version of the $β$ functions : $β^G_{μν}=β^X=0$, of the target spacetime metric and coordinates, respectively. Preliminary steps are taken to merge motion relativity with scale relativity and, in this fashion, analogs of Weyl-Finsler geometries make their appearance. The quantum case remains to be studied.

Motivation & Objective

• To incorporate Nottale's scale-relativity principle—where scale invariance replaces velocity invariance—into string theory and extended objects.
• To examine whether scale-relativity invariance of the world-volume measure is compatible with the vanishing of the β-functions for the target-space metric and coordinates.
• To explore the geometric implications of unifying motion relativity and scale relativity, leading to analogs of Weyl-Finsler geometries.
• To lay foundational groundwork for a quantum extension of scale-relativistic string theory, though the quantum case remains open.

Proposed method

• Starts from the Nambu-Goto action for strings and extended objects, which describes the world-volume dynamics.
• Introduces scale-relativity invariance by requiring invariance under Lorentzian-scaling transformations of the world-volume coordinates.
• Identifies the Planck scale as the universal, absolute, and invariant scale under dilatations, distinct from the string scale.
• Analyzes the scale-relativity version of the β-functions, showing that β^G_μν = 0 and β^X = 0 are compatible with scale-relativity invariance.
• Derives geometric structures that resemble Weyl-Finsler geometries through the unification of motion and scale relativity.
• Uses a reformulation of the world-volume measure to enforce scale invariance, preserving key physical constraints.

Experimental results

Research questions

• RQ1 Can the principle of scale relativity be consistently applied to string theory and extended objects via invariance of the world-volume measure?
• RQ2 Is scale-relativity invariance compatible with the vanishing of the β-functions for the target-space metric and coordinates?
• RQ3 How does the unification of motion relativity and scale relativity lead to new geometric structures in the target space?
• RQ4 What role does the Planck scale play as an absolute scale under dilatations in this framework?
• RQ5 What are the implications for the geometry of spacetime, particularly the emergence of Weyl-Finsler-like structures?

Key findings

• The scale-relativity version of the β-functions vanishes (β^G_μν = 0 and β^X = 0), indicating consistency with scale-relativity invariance of the world-volume measure.
• The Planck scale is identified as the universal, absolute, and invariant scale under dilatations, distinct from the string scale.
• The framework suggests that Weyl-Finsler geometries naturally emerge from the unification of motion relativity and scale relativity.
• The world-volume measure is invariant under Lorentzian-scaling transformations of the coordinates, preserving the physical consistency of the action.
• The classical formulation is consistent, but the quantum extension remains an open problem.
• The approach provides a new geometric and physical foundation for unifying scale invariance with relativistic dynamics in string theory.