PTG IV: PTG IV: Transport Horizons and Boundary Geometry

Reader summary

PTG IV asks what the failure of completeness leaves behind. Each incomplete congruence ends on a canonical terminal hypersurface, the transport horizon, that carries a limiting transport direction, a limiting transverse bundle, and a degenerate capacity measure. We prove the horizon exists, is smooth, has a unique limiting direction independent of approach, and is structurally stable under perturbations of the generator. This is the boundary object PDT uses in place of a classical event horizon, and it is the support on which the entropy and membrane flow of later papers live.

Abstract

We develop the boundary geometry associated with incomplete transport congruences in Phase Transport Geometry. Building on the incompleteness theorems of Phase Transport Geometry III, we show that the failure of transport completeness induces a canonical terminal hypersurface equipped with a limiting transport direction, a limiting transverse structure, and a degenerate capacity measure. This hypersurface, called a transport horizon, arises as the boundary of transported hypersurfaces approaching the focusing parameter. We establish the existence, smoothness, and structural properties of transport horizons and prove their stability under perturbations of the transport generator.

Falsifiable predictions

01
Every incomplete twist-free transport congruence terminates on a unique smooth transport horizon with a well-defined limiting direction.
02
The capacity measure on a transport horizon is identically zero, distinguishing it from any ordinary smooth hypersurface.
03
Transport horizons are structurally stable: small admissible perturbations of the generator deform the horizon smoothly.

Full paper

Introduction

PTG III established that twist-free transport generators with negative initial expansion cannot be extended indefinitely when emanating from hypersurfaces of finite transport capacity. The purpose of this paper is to analyse the geometric boundary induced by this incompleteness.

We show that the transported hypersurfaces converge to a smooth terminal hypersurface with vanishing capacity and a canonical limiting transport direction. This hypersurface, called a transport horizon, carries a natural boundary geometry derived from the limiting behaviour of the congruence.

Preliminaries

Let S be a transversely admissible hypersurface of finite transport capacity, and

l^{a}

a twist-free generator with F(

l^{a}

)

\geq

0 and

\theta

< 0 on S. The transported hypersurface at parameter

\lambda

is <<S_{\

\lambda

} :=>> {

\gamma_{p}(\lambda)

: p in S }. From PTG III there is finite

\lambda_{\star}

with

\theta \to

\infty

and J(<<S_{\

\lambda

}>>, l)

\to

0 as

\lambda \to \lambda_{\star}

Formation of the transport horizon

Candidate limiting hypersurface

S_{\star}

is the closure of the union of <<S_{\

\lambda

}>> for

\lambda

\lambda_{\star}

, minus that union. It is non-empty (nested family with finite limit parameter), and smooth (smooth limits of embedded hypersurfaces).

Limiting direction

For any sequence

\lambda_{n} \to \lambda_{\star}

, the limit

k^{a} := \lim \hat{l^{a}}(\lambda_{n})

exists, is non-zero, and is independent of the approach sequence. Normalisation gives boundedness; smoothness of

l^{a}

and the cone structure give convergence; uniqueness follows from smooth convergence of <<S_{\

\lambda

}>>.

Definition of the transport horizon

The transport horizon is

S_{\star}

equipped with: the limiting direction field

k^{a}

, the limiting transverse bundle

T_{\star}

, and the degenerate capacity measure

\mu_{\star}

Boundary geometry

Intrinsic geometry

S_{\star}

inherits a smooth intrinsic geometry from the transported hypersurfaces.

Transverse geometry

The transverse bundles converge smoothly to

T_{\star}

S_{\star}

, since they are defined by orthogonality to

l^{a}

which converges to

k^{a}

Degenerate capacity measure

The limiting capacity measure

\mu_{\star}

is identically zero (capacity collapse).

Stability under perturbations

Let

\tilde{l^{a}}

l^{a}

\epsilon v^{a}

be a perturbed admissible generator. The perturbed focusing parameter satisfies |tilde

\lambda_{\star}

\lambda_{\star}

| = O(

\epsilon

). The perturbed capacity collapses. The perturbed horizons

\tilde{S_{\star}}

converge smoothly to

S_{\star}

, and the limiting directions

\tilde{k^{a}} \to k^{a}

, as

\epsilon \to

Structural stability

The boundary data (

TS_{\star}

T_{\star}

k^{a}

) are stable under admissible perturbations.

Discussion

Incomplete transport congruences induce a canonical terminal hypersurface equipped with a limiting direction, a limiting transverse structure, and a degenerate capacity measure. These transport horizons are the boundary objects for the quasi-local invariants and monotonicity structures in PTG V and VI.

Suggested citation

@techreport{pdt_ptg_iv_2026,
  author      = {Fincham, Graham and Hilton, Daniel},
  title       = {PTG IV: Transport Horizons and Boundary Geometry},
  institution = {Phase Differential Theory},
  year        = {2026},
  type        = {PDT working paper},
  number      = {PTG IV}
}