[Paper Review] Massive conformal gravity

F. F. Faria|arXiv (Cornell University)|Dec 19, 2013

Geophysics and Gravity MeasurementsEarth and Planetary Sciences2 references3 citations

TL;DR

This paper proposes a massive conformal gravity theory invariant under conformal transformations, combining the Weyl action and a conformally coupled scalar field to generate a massive graviton. The theory yields a modified Newtonian potential with a Yukawa-like term, successfully reproducing galactic rotation curves with a graviton mass of ~10⁻²⁶ eV/c² and a range λ ≈ 20–30 kpc, offering a potential solution to dark matter without introducing new particles.

ABSTRACT

In this article we construct a massive theory of gravity that is invariant under conformal transformations. The massive action of the theory depend on the metric tensor and a scalar field, which are considered as the only field variables. We find the vacuum field equations of the theory and the solution of its Newtonian limit.

Motivation & Objective

• To construct a massive gravity theory that is invariant under conformal transformations, addressing limitations of standard massive gravity like the vDVZ discontinuity and BD ghost.
• To unify conformal invariance with massive gravity by coupling the Weyl action to a scalar field with conformal symmetry.
• To derive the vacuum field equations and analyze the weak-field and Newtonian limits to test cosmological viability.
• To explore whether the modified potential can explain galactic rotation curves without dark matter.
• To lay the foundation for a renormalizable, unitary quantum theory of massive conformal gravity.

Proposed method

• Construct a conformally invariant gravitational action using the Weyl tensor squared and a conformally coupled scalar field, with a mass term proportional to λ⁻².
• Derive field equations by varying the action with respect to the metric and scalar field, yielding the Bach tensor and scalar wave equation.
• Apply weak-field approximations to linearize the equations, assuming hμν and σ are small perturbations around Minkowski spacetime.
• Impose the unitary gauge φ = φ₀ = constant to simplify the geodesic and field equations, recovering general relativistic behavior in the limit.
• Solve the linearized field equations in the Newtonian limit to derive a modified Poisson-like equation with a massive graviton contribution.
• Use spherical symmetry and boundary conditions to solve the resulting differential equation, yielding a potential with Newtonian and Yukawa terms.

Experimental results

Research questions

• RQ1 Can a massive gravity theory be constructed that is invariant under conformal transformations, avoiding the vDVZ discontinuity and BD ghost?
• RQ2 Does the inclusion of a conformally coupled scalar field allow for a consistent massive graviton with physical mass generation?
• RQ3 What is the Newtonian limit of the theory, and does it yield a potential that can explain galactic rotation curves without dark matter?
• RQ4 How do the parameters γ and λ in the modified potential affect the gravitational potential at different scales?
• RQ5 Can the theory be consistently coupled to matter and quantized in a unitary and renormalizable way?

Key findings

• The theory yields a modified gravitational potential φ(r) = -GM/[r(1+γ)](1 + γ e^(-r/λ)), combining Newtonian and Yukawa-like terms.
• For γ = -0.92 and λ ≈ 20–30 kpc, the potential accurately reproduces observed galactic rotation curves.
• The graviton mass is estimated at m ~ 10⁻²⁶ eV/c², consistent with cosmological observations.
• The scalar field equation reduces to □φ - (1/6)Rφ = 0 in vacuum, preserving conformal invariance.
• The vacuum field equations in the unitary gauge yield Wμν - λ⁻²Gμν = 0 and R = 0, leading to a wave equation with massive mode.
• The solution exhibits a repulsive Yukawa term at large distances (r > λ), which may explain the formation of galaxy clusters.