Massive Nordstr\"om Scalar (Density) Gravities from Universal Coupling

TL;DR

This paper explores massive scalar gravities derived from universal coupling, highlighting their unique properties, geometric interpretations, and the continuum between massless and massive theories, with implications for particle physics and gravity modeling.

Contribution

It introduces a broad class of massive scalar gravities with a bimetric structure, extending Nordström's theory, and analyzes their geometric and physical properties, including the massless limit.

Findings

Many new massive scalar gravity theories with bimetric geometry.

Theories exhibit a smooth transition to massless scalar gravity.

Compatibility with fermions using density-weighted spinors.

Abstract

Both particle physics and the 1890s Seeliger-Neumann modification of Newtonian gravity suggest considering a "mass term" for gravity, yielding a finite range due to an exponentially decaying Yukawa potential. Unlike Nordstr\"{o}m's "massless" theory, massive scalar gravities are strictly Special Relativistic, being invariant under the Poincar\'{e} group but not the conformal group. Geometry is a poor guide to understanding massive scalar gravities: matter sees a conformally flat metric, but gravity also sees the rest of the flat metric, barely, in the mass term. Infinitely many theories exhibit this bimetric 'geometry,' all with the total stress-energy's trace as source. All are new except the Freund-Nambu theory. The smooth massless limit indicates underdetermination of theories by data between massless and massive scalar gravities. The ease of accommodating electrons, protons and other fermions using density-weighted Ogievetsky-Polubarinov spinors in scalar gravity is noted.