Space-time Philosophy Reconstructed via Massive Nordstr\"om Scalar Gravities? Laws vs. Geometry, Conventionality, and Underdetermination

TL;DR

This paper explores massive scalar gravity theories, their geometric implications, and their philosophical significance, challenging traditional views on spacetime structure and conventionality in light of particle physics and historical context.

Contribution

It reconstructs space-time philosophy through massive Nordström scalar gravities, highlighting their implications for geometry, conventionality, and the underdetermination problem in gravitational theories.

Findings

Massive scalar gravity is Poincaré invariant but not conformally invariant.

Multiple theories exhibit the same bimetric geometry, unrelated to field equations.

The massless limit reveals underdetermination between massless and massive theories.

Abstract

Klein-Gordon gravity, 1920s-30s particle physics, and 1890s Neumann-Seeliger modified gravity suggest a "graviton mass term" *algebraic* in the potential. Unlike Nordstr\"om's "massless" theory, massive scalar gravity is invariant under the Poincar\'e group but not the 15-parameter conformal group. It thus exhibits the whole Minkowski space-time structure, indirectly for volumes. Massive scalar gravity is plausible as a field theory, but violates Einstein's principles of general covariance, general relativity, equivalence, and Mach. Geometry is a poor guide: matter sees a conformally flat metric due to universal coupling, but gravity sees the rest of the flat metric (on long distances) in the mass term. What is the `true' geometry, in line with Poincar\'e's modal conventionality argument? Infinitely many theories exhibit this bimetric `geometry,' all with the total stress-energy's trace as source; geometry does not explain the field equations. The irrelevance of the Ehlers-Pirani-Schild construction to conventionalism is evident given multi-geometry theories. As Seeliger envisaged, the smooth massless limit yields underdetermination between massless and massive scalar gravities---an unconceived alternative. One version easily could have been developed before GR; it would have motivated thinking of Einstein's equations along the lines of his newly reappreciated "physical strategy" and suggested a rivalry from massive spin 2 for GR (massless spin 2, Pauli-Fierz 1939). The Putnam-Gr\"unbaum debate on conventionality is revisited given a broad modal scope. Massive scalar gravity licenses a historically plausible rational reconstruction of much of space-time philosophy in light of particle physics. An appendix reconsiders the Malament-Weatherall-Manchak conformal restriction of conventionality and constructs the `universal force' in the null cones.