Natural maximal hyperbolizations of minimally-coupled equations automatically satisfy the dominant energy condition

philosophy-of-physicsgeneral-relativitysymmetric-hyperbolic-systemsdominant-energy-conditionstress-energy-tensorarxiv

Weatherall and Eleanor March prove (Theorem 10) that any maximal natural hyperbolization of a minimally-coupled, naturally symmetric hyperbolic equation — a rank-2 tensor field built naturally from the spacetime metric — must satisfy the dominant energy condition, or its sign-reversed counterpart:

"we show that natural maximal hyperbolizations satisfy the dominant energy and conservation conditions. This provides a link between natural hyperbolizations and nice properties of stress–energy tensors in general relativity."

The authors note that the Hilbert stress-energy tensor is, in many physical cases, exactly such a hyperbolization. The upshot: any matter theory whose stress-energy tensor secures symmetric hyperbolicity inherits the dominant energy condition as a side effect of naturality plus minimal coupling, rather than as an independently imposed physical postulate.