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PTG V

PTG V: Quasi-Local Invariants on Transport Horizons

Graham Fincham, Daniel Hilton

Reader summary

PTG V equips the transport horizon with three quasi-local invariants: a transverse area, a capacity-weighted invariant, and a curvature-type invariant built from the limiting shear. Each is well-defined on compact subsets of the horizon, independent of how you approach it, invariant under admissible rescalings of the transport generator, and stable under perturbations. These quasi-local quantities are the geometric handles PDT uses to define entropy in PTG VI and to drive membrane dynamics in PTG VII.

Abstract

We introduce three quasi-local invariants associated with transport horizons in Phase Transport Geometry. These invariants are defined using the limiting transverse structure, the limiting transport direction, and the degenerate capacity measure established in Phase Transport Geometry IV. We prove that the invariants are well-defined, independent of the choice of transported hypersurface, stable under perturbations of the transport generator, and invariant under admissible rescalings. These quasi-local quantities form the basis for the entropy and monotonicity structures developed in Phase Transport Geometry VI.

Falsifiable predictions

  1. 01
    The transverse area, capacity-weighted, and curvature-type invariants on a transport horizon depend only on the horizon, not on the family of hypersurfaces used to approach it.
  2. 02
    All three invariants are unchanged under positive rescalings of the transport generator.
  3. 03
    Quasi-local invariants vary smoothly under admissible perturbations of the generator, giving stable horizon-level observables.

Full paper

Introduction

PTG IV established that incomplete transport congruences induce a canonical terminal hypersurface, the transport horizon, equipped with a limiting transport direction kak^{a}, a limiting transverse bundle T⋆T_{\star}, and a degenerate capacity measure μ⋆\mu_{\star}. The purpose of this paper is to define and analyse quasi-local invariants associated with this boundary structure.

We introduce three invariants: a transverse area invariant, a capacity-weighted invariant, and a curvature-type invariant derived from the limiting deformation tensor. We prove that each invariant is well-defined on S⋆S_{\star}, independent of the choice of transported hypersurface approaching the horizon, invariant under admissible rescalings of the transport generator, and stable under perturbations.

Preliminaries

Let S⋆S_{\star} be the transport horizon of a twist-free transport generator lal^{a} satisfying F(lal^{a}) ≥\geq 0 and θ\theta < 0 on some initial hypersurface of finite transport capacity. The limiting deformation tensor is B⋆ab~:=lim⁡Bab~(λ)\tilde{B_{\star ab}} := \lim \tilde{B_{ab}}(\lambda) as λ→λ⋆\lambda \to \lambda_{\star}.

Quasi-local invariant I: transverse area

For compact U in S⋆S_{\star}, A(U) :=∫:= \int over U of h⋆abdμ⋆h_{\star ab} d\mu_{\star}. This is well-defined and finite (limiting transverse metric exists and is smooth, μ⋆\mu_{\star} is finite on compact sets) and independent of approach (smooth convergence of <<S_{\λ\lambda}>> and habh_{ab}).

Quasi-local invariant II: capacity-weighted invariant

C(U) :=∫:= \int over U of ρ(ka)dμ⋆\rho(k^{a}) d\mu_{\star}. It is well-defined and non-negative (ρ\rho non-negative and homogeneous of degree one; μ⋆\mu_{\star} non-negative). Under rescaling ka→αkak^{a} \to \alpha k^{a} with α\alpha > 0, C is invariant: homogeneity of ρ\rho cancels the rescaling.

Quasi-local invariant III: curvature-type invariant

K(U) :=∫:= \int over U of σ⋆abσ⋆abdμ⋆\sigma_{\star ab} \sigma_{\star}^{ab} d\mu_{\star}, where σ⋆ab\sigma_{\star ab} is the limiting shear. Well-defined and finite (limiting shear smooth; measure finite), and independent of approach (smooth convergence of σab\sigma_{ab}).

Stability of the invariants

Let la~\tilde{l^{a}} = lal^{a} + ϵva\epsilon v^{a}. The perturbed horizon S⋆~\tilde{S_{\star}} converges smoothly to S⋆S_{\star}; the limiting structures (kak^{a}, T⋆T_{\star}, μ⋆\mu_{\star}) vary smoothly with ϵ\epsilon. Hence AepsA_{eps}(U) →\to A(U), CepsC_{eps}(U) →\to C(U), KepsK_{eps}(U) →\to K(U) as ϵ→\epsilon \to 0.

Discussion

The three quasi-local invariants are well-defined, independent of approach, rescaling-invariant, and stable. They form the basis for the entropy and monotonicity structures in PTG VI.

Suggested citation

@techreport{pdt_ptg_v_2026,
  author      = {Fincham, Graham and Hilton, Daniel},
  title       = {PTG V: Quasi-Local Invariants on Transport Horizons},
  institution = {Phase Differential Theory},
  year        = {2026},
  type        = {PDT working paper},
  number      = {PTG V}
}