millicosm.substack.com/p/in-partial-pugnacious-defense-of?r=dwyb4&utm_campaign=p...
1 correction found
are only defined anywhere if they are defined everywhere
This is too strong. At least two of the examples named in the sentence—Maxwell’s equations and Einstein’s equations in general relativity—are standard local differential-equation systems with local initial/boundary value formulations.
Full reasoning
The statement says these theories “are only defined anywhere if they are defined everywhere.” That is not correct for the examples named.
- A peer-reviewed arXiv paper on Maxwell theory is literally titled “Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions” and proves existence/uniqueness of local solutions for Maxwell equations on domains with boundary conditions. That is the opposite of a theory that can only be defined globally.
- A classic AMS publication on general relativity studies “the Einstein equations of evolution” as partial differential equations and proves that differentiability of the Cauchy data is maintained for short time. Again, that is a local initial-value formulation, not something that only exists if specified everywhere at once.
- More generally, gauge theory is routinely formulated using locally trivial principal fibre bundles, which is standard language for a theory built from local patches plus compatibility conditions.
So while gauge theories can have important global/topological structure, it is false to say they are only defined anywhere if they are defined everywhere.
3 sources
- Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions
In this article we develop the local wellposedness theory for quasilinear Maxwell equations in H^m for all m >= 3 on domains with perfectly conducting boundary conditions... We prove the existence and uniqueness of local solutions...
- General relativity partial differential equations and dynamical systems
In this paper we study two aspects of the Einstein equations of evolution for an empty spacetime. In the first part (§§1-3) we give a simple direct proof that the differentiability of the Cauchy data is maintained for short time.
- Gauge transformations on locally trivial quantum principal fibre bundles
...the structure group of a locally trivial principal fibre bundle (QPFB)... gauge transformations are defined ... related to the local trivializations.