[Paper Review] The Conformal Universe I: Physical and Mathematical Basis of Conformal General Relativity

Renato Nobili|arXiv (Cornell University)|Jan 11, 2012

Cosmology and Gravitation Theories40 references3 citations

TL;DR

This paper proposes Conformal General Relativity (CGR), a conformally invariant extension of Einstein's General Relativity that introduces a ghost scalar field σ(x) and a physical massless scalar field φ(x), both with non-zero vacuum expectation values. The interaction between these fields enables a geometric-to-matter energy transfer that drives an inflationary big bang, with CGR reducing to GR as the Higgs boson mass stabilizes, resolving issues like the cosmological constant problem and initial singularity.

ABSTRACT

This is the first of three papers on Conformal General Relativity (CGR), which differs from Einstein's General Relativity (GR) in that it requires action--integral invariance under local scale transformations in addition to general coordinate transformations. The theory is here introduced in the semiclassical approximation as a preliminary approach to a quantum theoretical implementation. The idea of a conformal--invariant extension of GR was introduced by Weyl in 1919. For several decades it had little impact, as CGR implies that all fields are massless. Today this does not appear to be an unsurmountable difficulty since nonzero mass parameters may result from the spontaneous breakdown of conformal symmetry. The theory leads to very interesting results and predictions: 1) the spontaneous breakdown of conformal symmetry is only possible in a 4D--spacetime with small negative curvature; 2) CGR requires the introduction of a ghost scalar field $σ(x)$ invested with geometric meaning and a physical scalar field $φ(x)$ of zero mass, both of which have nonzero vacuum expectation values; 3) in order to preserve $S$--matrix unitarity, $σ(x)$ and $φ(x)$ must interact in such a way that the total energy density is bounded from below; 4) this interaction makes $φ(x)$ behave like a Higgs field of varying mass, which is capable of promoting a huge energy transfer from geometry to matter identifiable as the big bang; 5) in the course of time, the Higgs boson mass becomes a constant and CGR converges to GR.

Motivation & Objective

• To address critical shortcomings of General Relativity, including the initial singularity, the cosmological constant problem, and the need for unnatural Higgs masses in inflationary models.
• To formulate a conformal-invariant extension of GR that preserves unitarity and allows for a semiclassical quantum implementation.
• To explain the origin of the big bang as a geometric-to-matter energy transfer driven by spontaneous conformal symmetry breaking.
• To reconcile the observed small cosmological constant with quantum vacuum energy by requiring exact cancellation of zero-point contributions.
• To demonstrate that CGR reduces to standard GR in the late-time limit as the Higgs-like field acquires a constant mass.

Proposed method

• Formulates a conformal-invariant action integral under local Weyl transformations, requiring invariance under both general coordinate and scale transformations.
• Introduces a ghost scalar field σ(x) with geometric meaning and a physical massless scalar field φ(x), both acquiring non-zero vacuum expectation values.
• Derives the conformal transformation laws for the metric, connection, curvature, and energy-momentum tensors under Weyl rescaling gμν → e^{2α}gμν.
• Constructs the conformal Einstein tensor Ĝμν and shows its invariance under Weyl transformations when combined with the conformal curvature tensor C².
• Imposes unitarity on the S-matrix by requiring the total energy density to be bounded from below through interaction between σ and φ fields.
• Demonstrates that the φ field behaves as a Higgs-like field with varying mass, enabling a large-scale energy transfer from geometry to matter during early-time evolution.

Experimental results

Research questions

• RQ1 Can a conformal-invariant extension of General Relativity resolve the initial singularity and the cosmological constant problem?
• RQ2 How can a conformal symmetry breaking mechanism generate a finite, non-zero mass for the Higgs-like field while preserving unitarity?
• RQ3 What role does the ghost scalar field σ(x) play in enabling a geometric-to-matter energy transfer that mimics the big bang?
• RQ4 Why is 4D spacetime with small negative curvature necessary for spontaneous conformal symmetry breaking in CGR?
• RQ5 How does the interaction between σ and φ fields ensure bounded energy density and S-matrix unitarity?

Key findings

• Spontaneous conformal symmetry breaking is only possible in a 4D spacetime with small negative curvature, which stabilizes the vacuum structure.
• The ghost scalar field σ(x) and the physical scalar field φ(x) both acquire non-zero vacuum expectation values, with σ having geometric meaning and φ acting as a Higgs-like field with varying mass.
• The interaction between σ and φ ensures the total energy density is bounded from below, preserving S-matrix unitarity in the quantum regime.
• The φ field's varying mass enables a massive transfer of energy from spacetime geometry to matter, identified as the big bang, with the energy scale matching inflationary requirements.
• As the Higgs-like mass of φ stabilizes to a constant value, CGR asymptotically converges to standard General Relativity, resolving the initial singularity and explaining the observed universe's large-scale homogeneity.
• The conformal curvature term C² in the action is invariant only in 4D, and its inclusion via Stelle's mechanism ensures renormalizability of quantum gravity with gravitational ghosts as regulators.