The Scalar–Flux Law: A Unifying Principle for Constraints, Fluxes, and Causal Evolution

This paper isolates a simple structure that recurs across very different physical theories. A scalar field is fixed at each instant by an elliptic equation; its total “charge” can change only by boundary flux; and the truly propagating content lives in a transverse sector that evolves causally. We illustrate this Scalar–Flux Law (SFL), map it across multiple domains, and show how to turn it into fast, stable numerical solvers.

The general idea: any system that processes information locally in spacetime must implicitly define a "present" to separate state from dynamics.  Every local information processor must define a present to function, not because the universe has a special "now" but because that's how local computation works.

Efficient computation requires:

• State (encoded in scalar constraints, solved elliptically on a spatial slice)
• Dynamics (flux evolution, propagated hyperbolically)
• Boundaries (where information enters/leaves the local region)

What we discovered

• One template, many theories. We show that EM (Coulomb gauge), lapse-first GR, Madelung quantum hydrodynamics, incompressible flow, optimal-transport/thermodynamics (JKO), Yang–Mills (in a gauge), and black-hole mechanics all share the same three-part pattern:

  1. an elliptic constraint for a scalar φ,

  2. a continuity law for its density q, and

  3. transverse, wave-like degrees of freedom.

• Minimal SFL template (precise).
  Elliptic constraint: ℒ[φ] = q
  Continuity: ∂ₜq + ∇·F = 0
  Global flux law: d/dt ∫Ω q dV = −∮∂Ω F·n dS
  The scalar is “instantaneously” determined by (q, boundary data); only boundary flux can change the total charge. Waves live in the transverse sector.

• Gravity connection. In lapse-first GR the scalar is the time potential Φ, yielding a concrete flux law (∂ₜΦ = −4πG r T_tr). This provided the seed example that led us to formulate SFL as a general organizing principle.

• From principle to practice. We give a worked EM example (Coulomb gauge) and an algorithmic blueprint: solve one global elliptic problem per step, then advance transverse fields with a hyperbolic update. This split is numerically attractive (projection methods, multigrid/FFT Poisson solves, clean CFL control) and easy to parallelize.

Why it matters

• Clarity. SFL explains why “constraint-plus-waves” formulations feel the same across fields: scalars enforce structure; transverse parts carry signals.

• Computation. The elliptic–hyperbolic split turns constraints into solvable projections and keeps dynamics stable. We outline complexity and implementation choices (FFT vs. multigrid; domain decomposition; boundary handling).

• Theory. A covariant phase-space account ties SFL to Noether charges: the “constraints” C_χ that make dJ_χ = −C_χ appear are exactly where the elliptic pieces come from.

Clarity gained

Reformulating into a scalar constraint + transverse evolution turns several “mysteries” into accounting: causality (only transverse waves transport energy/information), work (radiative work is JT⋅ET; scalar terms are storage), near–far separation (reactive vs. propagating energy), and in GR the “problem of time” (the lapse Φ is a constraint variable; only TT modes propagate). The result is a minimal solver (one Poisson + one wave) and a clean, cross-domain story for how fields store vs. transmit energy.

What’s in the paper

• A concise SFL “minimal template” and conditions for validity (regularity, boundary data, well-posedness of ℒ).

• A slim cross-theory quick map (with expanded tables in the appendix).

• A worked Coulomb-gauge EM example (complete decomposition and two-step solver).

• Limitations/failure modes (foliation/gauge dependence, relativistic mixing, strong coupling/singular geometry, non-Abelian complications).

• A covariant Noether/symplectic clarification showing how constraints generate the elliptic sector.

Provenance

The lapse-first GR refactor and the GR scalar–flux law that motivated SFL were first introduced in:
Snyder, A. Gravity as Temporal Geometry: a quantizable reformulation of GR (2024). DOI: 10.5281/zenodo.16878018.

Another full refactoring using this method applied to ElectroMagnetism: https://doi.org/10.5281/zenodo.16968711

Yang-Mills Theory in Coulomb Gauge: https://doi.org/10.5281/zenodo.17057632