---
title: "Natural equation"
type: "entity"
entity_kind: "concept"
status: "hub"
canonical_name: "Natural equation"
aliases: ["natural bundle"]
first_seen: "2026-10-30T00:00:00.000Z"
writer_model: "claude-sonnet-5"
connects_to: ["general covariance","minimal coupling","symmetric hyperbolic systems","Kretschmann objection","Robert Geroch"]
seek_code_commit: "d40ee39"
---


A field equation built on a natural bundle — one that transforms functorially under diffeomorphisms of the underlying spacetime manifold — which is the technical, category-theoretic content behind what physicists informally call "general covariance."

Matters to this vault as the framework [[entity-james-owen-weatherall|Weatherall]] and Eleanor March's 2026 paper works within throughout: its central project is showing how close natural equations can come to having a well-posed initial-value problem (never exactly, per a companion paper, but "almost" for physically important cases like Einstein's equation), building on [[entity-robert-geroch|Geroch's]] technical apparatus and addressing, in spirit, the [[entity-erich-kretschmann|Kretschmann objection]] that covariance alone is not a substantive constraint.

## References
- [[claim-march-weatherall-minimal-coupling-unique-up-to-isomorphism]]
- [[claim-march-weatherall-causal-cones-coincide-with-metric-lightcones]]
- [[claim-march-weatherall-natural-maximal-hyperbolizations-satisfy-dominant-energy-condition]]
- [[claim-march-weatherall-hyperbolization-tensor-divergence-free-by-naturality]]