A minimally-coupled naturally symmetric hyperbolic equation's causal cones must coincide exactly with the metric's light cones

philosophy-of-physicsgeneral-relativitysymmetric-hyperbolic-systemsnatural-bundlesminimal-couplingcausal-structurearxiv

Weatherall and Eleanor March show that for a first-order, quasilinear, minimally-coupled natural equation that is "naturally symmetric hyperbolic" — one that admits a natural hyperbolization in their technical sense — the equation's causal structure is forced: every null covector must be characteristic, and the causal cones of any maximal natural hyperbolization must be exactly the future- or past-directed timelike covectors relative to the background metric. The equation's own causal cones, in other words, can be neither wider nor narrower than the metric's own light cones (Theorem 8, Lemma 6, and Corollary 1 jointly establish this; the paper's abstract states the conclusion directly):

"the causal cones associated with a (first order, quasilinear, minimally coupled) naturally symmetric hyperbolic natural equation must always coincide with the metric lightcones"

The result sits inside the classical theory of symmetric hyperbolic systems, originated by Kurt Otto Friedrichs in 1954 to guarantee well-posed initial-value problems for first-order PDE systems, where a disturbance's propagation speed is bounded by the system's characteristic cones. What this paper adds is specific to the natural-equation / minimal-coupling setting: naturality and minimal coupling together leave no room for a theory's own causal cones to diverge from gravity's.