Massive Nordström scalar (density) gravities from universal coupling

TL;DR

The paper derives uncountably infinitely many massive scalar gravities by universal coupling to the trace of the total stress-energy tensor, generalizing Nordström’s massless scalar gravity. All are new except the Freund-Nambu theory, and the massive variants are strictly special relativistic, invariant only under the 10-parameter Poincaré group rather than the 15-parameter conformal group.

Key findings

• Universal coupling yields a distinct massive scalar gravity for every real density weight \(w\); the case \(w=1/2\) is the previously known Freund-Nambu theory, while all other cases are new.
• The mass term breaks conformal symmetry: massive variants are invariant only under the 10-parameter Poincaré group, whereas Nordström’s massless theory is invariant under the 15-parameter conformal group.
• The theories have a smooth massless limit, producing underdetermination between massless and sufficiently small-mass massive scalar gravities.
• Stability analysis finds the only critical point is the expected vacuum \(\sqrt{-g}=\sqrt{-\eta}\) (giving \(\gamma=0\) for all \(w\)), and it is the ground state for all \(w\); some theories repel infinitely from the singularity \(\sqrt{-g}=0\), while others do not.

Methods

• Theoretical derivation using Lagrangian densities, scalar density variables, and the Rosenfeld-type metrical definition of stress-energy; universal coupling adds the trace of the total stress-energy tensor as source.
• A bimetric change of variables \(g=\eta+k\gamma\) is used, with \(k^2=64\pi G w^2\) and \(k=8\sqrt{\pi G}w\); special cases \(w=1\) and \(w=0\) are handled separately, including by l’Hôpital’s rule.
• No empirical data or sample is analyzed; the work is a formal field-theoretic construction, including spinor coupling via density-weighted Ogievetsky-Polubarinov spinors of weight \(3/8\) in four spacetime dimensions.

Limitations

• The paper states that no claim of exhaustiveness is made: “There might be still other massive scalar gravities worthy of discovery.”
• It states that checking stability for all \(w\) perturbatively would be substantial and that treating all infinitely many theories perturbatively is not viable; using the neutral \(w=0\) field gives infinite series with no polynomial case, making analysis difficult.
• Evident from the text: the paper performs no observational fitting or empirical test of the theories, so the claimed empirical viability of sufficiently small masses is not demonstrated here.