Space-time constructivism vs. modal provincialism: Or, how special relativistic theories needn't show Minkowski chronogeometry

Abstract

Already in 1835 Lobachevski entertained the possibility of multiple (one might say “rival”) geometries of the same type playing a role. This idea of rival geometries has reappeared from time to time (including Poincaré and several 20th century authors) but had yet to become a key idea in space-time philosophy prior to Brown's Physical Relativity. Such ideas are emphasized towards the end of Brown's book, which I suggest as the interpretive key. A crucial difference between Brown's constructivist approach to space-time theory and orthodox “space-time realism” pertains to modal scope. Constructivism takes a broad modal scope in applying (at least) to all local classical field theories—modal cosmopolitanism, one might say, including theories with multiple geometries. By contrast the orthodox view is modally provincial in assuming that there exists a unique geometry, as the familiar theories (Newtonian gravity, Special Relativity, Nordström's gravity, and Einstein's General Relativity) have. These theories serve as the “canon” for the orthodox view. Their historical roles also suggest a Whiggish story of inevitable progress. Physics literature after c. 1920 is relevant to orthodoxy primarily as commentary on the canon, which closed in the 1910s. The orthodox view explains the spatio-temporal behavior of matter in terms of the manifestation of the real geometry of space-time, an explanation works fairly well within the canon. The orthodox view, Whiggish history, and the canon have a symbiotic relationship.

If one happens to philosophize about a theory outside the canon, space-time realism sheds little light on the spatio-temporal behavior of matter. Worse, it gives the wrong answer when applied to an example arguably within the canon, a sector of Special Relativity, namely, massive scalar gravity with universal coupling. Which is the true geometry—the flat metric from the Poincaré symmetry group, the conformally flat metric exhibited by material rods and clocks, or both—or is the question faulty? How does space-time realism explain the fact that all matter fields see the same curved geometry, when so many ways to mix and match exist? Constructivist attention to dynamical details is vindicated; geometrical shortcuts can disappoint. The more exhaustive exploration of relativistic field theories in particle physics, especially massive theories, is a largely untapped resource for space-time philosophy.