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

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.