PTG V — PTG V: Quasi-Local Invariants on Transport Horizons

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

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.
02
All three invariants are unchanged under positive rescalings of the transport generator.
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 k^a, a limiting transverse bundle T_star, and a degenerate capacity measure 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_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_star be the transport horizon of a twist-free transport generator l^a satisfying F(l^a) >= 0 and theta < 0 on some initial hypersurface of finite transport capacity. The limiting deformation tensor is tilde B_star_ab := lim tilde B_ab(lambda) as lambda -> lambda_*.

Quasi-local invariant I: transverse area

For compact U in S_star, A(U) := integral over U of h_star_ab dmu_star. This is well-defined and finite (limiting transverse metric exists and is smooth, mu_star is finite on compact sets) and independent of approach (smooth convergence of S_lambda and h_ab).

Quasi-local invariant II: capacity-weighted invariant

C(U) := integral over U of rho(k^a) dmu_star. It is well-defined and non-negative (rho non-negative and homogeneous of degree one; mu_star non-negative). Under rescaling k^a -> alpha k^a with alpha > 0, C is invariant: homogeneity of rho cancels the rescaling.

Quasi-local invariant III: curvature-type invariant

K(U) := integral over U of sigma_star_ab sigma_star^ab dmu_star, where sigma_star_ab is the limiting shear. Well-defined and finite (limiting shear smooth; measure finite), and independent of approach (smooth convergence of sigma_ab).

Stability of the invariants

Let tilde l^a = l^a + epsilon v^a. The perturbed horizon tilde S_star converges smoothly to S_star; the limiting structures (k^a, T_star, mu_star) vary smoothly with epsilon. Hence A_eps(U) -> A(U), C_eps(U) -> C(U), K_eps(U) -> K(U) as epsilon -> 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.

BibTeX

@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}
}