---
title: "Minimal coupling"
type: "entity"
entity_kind: "concept"
status: "hub"
canonical_name: "Minimal coupling"
aliases: []
first_seen: "2026-10-30T00:00:00.000Z"
writer_model: "claude-sonnet-5"
connects_to: ["natural equation","general covariance","symmetric hyperbolic systems","general relativity","James Owen Weatherall"]
seek_code_commit: "d40ee39"
---


In general relativity, the standard prescription for turning a special-relativistic field equation into a general-relativistic one by replacing ordinary partial derivatives with covariant derivatives relative to the spacetime metric — informally, "commas go to semicolons" — while introducing no new curvature-coupling terms.

Matters to this vault as the concept [[entity-james-owen-weatherall|March and Weatherall's]] 2026 paper gives a precise category-theoretic definition for, the first time, proving that a minimal coupling in their exact sense is unique up to isomorphism where one exists — resolving a plurality the prescription had previously been thought to admit. The paper notes, as a nuance, that Einstein's own field equation is *not* minimally coupled under this precise definition, since it depends on the metric to second order rather than first.

## 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]]