Conformal Transformations and Nordström's Scalar Theory of Gravity

TL;DR

Nordström’s scalar theory of gravity is observationally ruled out, but it serves as a useful toy model for non-minimal coupling and conformal transformations. The authors show that a conformal transformation recasts Nordström’s theory in a Jordan-frame metric form and can extend manifolds through curvature singularities at the classical level.

Key findings

• Nordström’s theories are ruled out by Solar System observations: light bending gives PPN parameter γ = −1, whereas observation agrees with GR’s γ = 1; perihelion shift gives β = 1/2 for F(Φ)=Φ instead of GR’s β = 1; geodetic precession is minus one third the GR prediction.
• The conformal transformation ḡμν = (1+F)² ℓμν turns Nordström’s theory into a metric “Jordan representation”; for F(Φ)=Φ the equations reduce to R = 24πG/c⁴ T_m, Dν T^μν_m = 0, and Cμνρσ = 0.
• In cosmology with F=Φ and w > 1/3 or w < −2/3, a(t) ≡ (1+Φ) ∝ t^{2/(4+3w)} and density ∝ t^{−6(1+w)/(4+3w)}; T=0 is a Big-Bang with infinite density and infinite redshift.
• For a static spherically symmetric field, Φ = −M/r and r = M is an infinite-redshift surface; the Jordan metric has a curvature singularity with Rμν Rμν = 4M²(M²−4Mr+6r²)/(r−M)⁸, but geodesics can cross it and the manifold can be extended via the conformal transformation.

Methods

• Theoretical analysis of Nordström’s scalar theory in both its original “Einstein frame” formulation and its conformally transformed “Jordan frame” metric formulation.
• Derives field equations, particle equations of motion, redshift formulas, cosmological solutions, and static spherically symmetric “black hole” solutions analytically.
• Uses gedanken experiments and PPN parameters to compare Nordström predictions with GR and observations; no observational dataset or statistical analysis is used.

Limitations

• The paper states that Nordström’s theory is observationally ruled out and that the extension through the horizon singularity is only at the classical level; it also notes that particles beyond r = M being treated as antiparticles “probably leads to quantum instabilities.”
• The text makes evident that the analysis is purely theoretical, with no new observational data, and that the antiparticle interpretation is imposed “somewhat arbitrarily.”