[Paper Review] Scalar Gravitation and Extra Dimensions

F. Ravndal|ArXiv.org|May 6, 2004

Relativity and Gravitational Theory1 references16 citations

TL;DR

This paper revisits Gunnar Nordström's early relativistic scalar theory of gravitation and its unification of gravity and electromagnetism via extra dimensions, presenting it in a modern framework. It demonstrates how compactified extra dimensions naturally give rise to a scalar radion field that mimics Nordström's scalar gravity, with the radion coupling to matter and modifying fundamental constants—offering a dynamical mechanism consistent with modern cosmological models like quintessence and Brans-Dicke theory.

ABSTRACT

Gunnar Nordstrom constructed the first relativistic theory of gravitation formulated in terms of interactions with a scalar field. It was an important precursor to Einstein's general theory of relativity a couple of years later. He was also the first to introduce an extra dimension to our spacetime so that gravitation would be just an aspect of electromagnetic interactions in five dimensions. His scalar theory and generalizations thereof are here presented in a bit more modern setting. Extra dimensions are of great interest in physics today and can give scalar gravitational interactions with similar properties as in Nordstrom's original theory.

Motivation & Objective

• To re-express Nordström's 1913 relativistic scalar theory of gravitation in a modern field-theoretic framework.
• To explore how extra compactified dimensions can generate scalar gravitational interactions analogous to Nordström's theory.
• To connect the radion field, which encodes the size of compactified dimensions, to scalar gravity and Brans-Dicke-like dynamics.
• To investigate the cosmological and phenomenological implications of such scalar fields, particularly their time-varying coupling to matter.
• To assess the viability of these models in light of solar system tests and dark energy observations.

Proposed method

• Formulates a five-dimensional action with a flat metric and a compactified extra dimension to derive a four-dimensional effective theory.
• Applies a conformal transformation to map the Jordan frame (with non-minimal coupling) to the Einstein frame, yielding a canonical scalar field with a potential.
• Derives the radion field equation in the Einstein frame, showing it takes the form of a wave equation with a source term from the trace of the matter stress-energy tensor.
• Relates the Brans-Dicke parameter α to the field’s coupling strength, with α² = 1/(4ω + 6), linking to observational constraints.
• Analyzes quantum corrections to the radion potential, showing it acquires a more realistic form than a pure exponential.
• Demonstrates that the radion couples to matter via a conformal factor A²(ϕ), leading to time-varying fundamental constants.

Experimental results

Research questions

• RQ1 How can a scalar field in four dimensions emerge from a higher-dimensional theory with compactified extra dimensions?
• RQ2 To what extent does the radion field in a compactified extra dimension reproduce the dynamics of Nordström's scalar gravity?
• RQ3 What are the cosmological implications of a radion field with a potential derived from higher-dimensional gravity?
• RQ4 How do quantum corrections affect the radion potential and its consistency with solar system tests?
• RQ5 Can the radion field explain time-varying fundamental constants, such as the fine-structure constant?

Key findings

• The radion field in a compactified five-dimensional spacetime naturally gives rise to a scalar gravitational interaction that closely resembles Nordström’s original theory.
• After a conformal transformation, the radion field acquires a canonical kinetic term and a potential that, when the original potential is constant, becomes an exponential, matching quintessence models.
• The Brans-Dicke parameter α is related to the field’s coupling strength, with α² = 1/(4ω + 6), and quantum corrections lead to a more realistic potential than a pure exponential.
• The radion couples to matter via a conformal factor A²(ϕ), implying that fundamental constants, such as the fine-structure constant, may vary over cosmic time.
• The equation of motion for the radion field in the Einstein frame is of the same form as Nordström’s scalar field equation when the potential is zero, confirming the dynamical equivalence.
• The model remains consistent with solar system gravity tests due to quantum corrections that drive the Brans-Dicke parameter to large values today.