[Paper Review] Solutions of Conformal Gravity with Dynamical Mass Generation in the Solar System

Joshua Wood, William Moreau|ArXiv.org|Feb 13, 2001

Cosmology and Gravitation Theories4 citations

TL;DR

This paper proposes that conformal gravity with dynamical mass generation—via a scalar Higgs field—can explain the anomalous constant radial acceleration observed in Pioneer 10/11, Galileo, and Ulysses spacecraft, while preserving standard solar system tests. Numerical solutions of the full field equations show that a slowly varying Higgs field gradient produces a nearly constant inward acceleration at 40 AU, consistent with data, without violating Schwarzschild-like constraints on light bending and perihelion precession.

ABSTRACT

The field equations of Mannheim's theory of conformal gravity with dynamic mass generation are solved numerically in the interior and exterior regions of a model spherically symmetric sun with matched boundary conditions at the surface. The model consists of a generic fermion field inside the sun, and a scalar Higgs field in both the interior and exterior regions. From the conformal geodesic equations it is shown how an asymptotic gradient in the Higgs field causes an anomalous radial acceleration in qualitative agreement with that observed on the Pioneer 10/11, Galileo, and Ulysses spacecraft. At the same time the standard solar system tests of general relativity are preserved within the limits of observation.

Motivation & Objective

• To investigate whether conformal gravity with dynamical mass generation can account for the anomalous radial acceleration in deep-space spacecraft without violating standard solar system tests.
• To resolve the incompatibility between general relativity and dynamical mass generation, which leads to a traceless energy-momentum tensor.
• To numerically solve the coupled nonlinear fourth-order field equations for a spherically symmetric Sun model with matched interior and exterior boundary conditions.
• To determine whether the scalar Higgs field, required for mass generation, can produce a nearly constant radial acceleration consistent with Pioneer mission data.
• To explain why planets do not exhibit the same anomalous acceleration as spacecraft, despite the scalar field's influence.

Proposed method

• Formulate a conformally invariant action including a self-interacting scalar Higgs field and a fermion field with Yukawa coupling, ensuring dynamical mass generation.
• Derive the Weyl field equations from the total action, with the energy-momentum tensor sourced by fermions and scalar fields.
• Implement numerical solutions for the interior (Sun) and exterior (interplanetary space) regions using matched boundary conditions at the solar limb.
• Apply Newton-Raphson iterations to refine the exterior solution, ensuring agreement with the Schwarzschild metric to within 1 part in 10^15.
• Use the observed Pioneer acceleration to constrain the scalar field gradient at 40 AU, setting boundary conditions for numerical integration.
• Analyze the scalar field profile and metric behavior across the solar system to verify consistency with observational limits on light bending and perihelion precession.

Experimental results

Research questions

• RQ1 Can conformal gravity with dynamical mass generation reproduce the observed constant radial acceleration of ~8.5×10⁻¹⁰ m/s² in the outer solar system, as seen in Pioneer 10/11 and Galileo?
• RQ2 Does the inclusion of a scalar Higgs field for dynamical mass generation allow the theory to remain consistent with standard solar system tests like light deflection and perihelion precession?
• RQ3 Why do planets not exhibit the same anomalous acceleration as spacecraft, despite the presence of the scalar field?
• RQ4 What boundary conditions on the scalar field gradient are required to match the Pioneer anomaly data at 40 AU?
• RQ5 How does the scalar field profile evolve across the Sun’s interior and exterior, and does it yield a metric that approximates Schwarzschild in the weak-field limit?

Key findings

• The numerical solution shows that a scalar Higgs field with a very small gradient (~10⁻²⁴ m⁻² at 40 AU) produces a nearly constant radial acceleration toward the Sun, matching the observed Pioneer anomaly magnitude.
• The metric field matches the Schwarzschild solution to within one part in 10¹⁵ at the solar limb, confirming compatibility with standard solar system tests.
• The scalar field increases smoothly through the Sun’s interior and assumes a very shallow, asymptotically decreasing gradient in the exterior region, consistent with long-range behavior.
• The theory requires a specific value of α ~ 3.5×10¹⁵ and a small coupling λ ≤ 10⁻⁴⁵ to ensure field stability at large radii.
• The scalar field gradient is significantly reduced within planetary bodies due to superposition effects, explaining why planets do not exhibit the same anomalous acceleration as spacecraft.
• The full system of field equations was solved numerically for the first time in this work, confirming consistency between interior and exterior solutions with matched boundary conditions.