A natural hyperbolization tensor must be divergence-free relative to the Levi-Civita connection, as a consequence of naturality alone

philosophy-of-physicsgeneral-relativitysymmetric-hyperbolic-systemsnatural-bundlesconservation-lawsarxiv

Weatherall and Eleanor March prove (Theorem 11) that the divergence of a natural hyperbolization tensor — taken relative to the Levi-Civita derivative operator determined by the background metric — is forced to vanish, not because of any dynamical equation of motion but because the only first-order natural operator from the metric bundle to the relevant tensor bundle is the zero operator (citing Kolář–Michor–Slovák's classification of natural operators):

"a certain rank-2 tensor, associates with a natural hyperbolization, must always be divergence free relative to the associated Levi-Civita derivative operator"

(The word "associates" appears exactly so in the paper's own abstract; reproduced verbatim here, not a transcription error.) Full covariant conservation — the stronger statement a reader might expect — holds only when a further term involving the derivative of the field itself vanishes (Corollary 2). The result pairs with the paper's dominant-energy-condition result: both are conservation-type properties that fall out of naturality for a natural, minimally-coupled symmetric hyperbolic system, rather than being imposed as physical postulates.