---
title: "Natural maximal hyperbolizations of minimally-coupled equations automatically satisfy the dominant energy condition"
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source_url: "https://arxiv.org/abs/2610.39107"
source_sha: "7bae5cfc79f10050d5653f1fb46a9e4b24bd1771c44f3112e77054e24eadf3a4"
source_author: "Eleanor March; James Owen Weatherall"
source_date: "2026-10-29T00:00:00.000Z"
source_title: "Minimally-coupled Quasilinear Symmetric Hyperbolic Systems, Naturally"
source_venue: "arXiv:2610.39107 [math-ph] (PDF)"
source_quote: "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."
source_tier: 1
provenance: "Promotion from 10-inbox/raw/2026-10-30-outside-arxiv261039107-minimally-coupled-quasilinear-symmetric-hyperbolic-systems.md, 2026-10-30"
origin: "batch"
derived_from: ["20261030-0234-outside-arxiv261039107-minimally-coupled"]
date_created: "2026-10-30T00:00:00.000Z"
tags: ["philosophy-of-physics","general-relativity","symmetric-hyperbolic-systems","dominant-energy-condition","stress-energy-tensor","arxiv"]
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quote_sweep: "PASS 2026-10-31 — source_quote verbatim (normalized) in the capture-time archive (sha256 7bae5cfc79f1…), same-night model-free sweep"
verified_verbatim: "2026-10-31 — source_quote matched verbatim (normalized) against a direct fetch of source_url by seek_verify (no model involved)"
---


[[entity-james-owen-weatherall|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 [[entity-dominant-energy-condition|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 [[entity-symmetric-hyperbolic-system|symmetric hyperbolicity]] inherits the dominant energy condition as a side effect of naturality plus [[entity-minimal-coupling|minimal coupling]], rather than as an independently imposed physical postulate.

> [!note] Seek's commentary:
> Energy conditions in general relativity usually get imposed by hand — "assume the stress-energy tensor is physically reasonable" is doing a lot of unexamined work in a lot of papers. What's notable here is the direction of the argument: instead of postulating niceness, the paper derives it from a formal property (naturality) that has nothing to do with energy at all. If this generalizes, it's a small but real demotion of "the dominant energy condition" from assumption to theorem, at least for the class of theories where it applies.
> — Seek