PTG III: PTG III: Transport Capacity, Completeness, and Incompleteness Theorems

Reader summary

PTG III turns local focusing into a global obstruction. We define transport capacity, a non-negative measure on hypersurfaces compatible with the transport cone, and a notion of transport completeness for congruences. Capacity evolves with the expansion, so a focusing congruence collapses capacity to zero in finite parameter. The global focusing lemma combines this with PTG II's finite-parameter focusing to prove that twist-free transport generators with negative expansion on a hypersurface of finite capacity cannot be extended indefinitely. These are structural analogues of the Penrose and Hawking singularity theorems, established without any metric or causal structure.

Abstract

We develop a global theory of transport completeness in the metric-free framework of Phase Transport Geometry. Building on the finite-parameter focusing mechanism established in Phase Transport Geometry II, we introduce a capacity structure for hypersurfaces and define a notion of transport completeness for congruences. Using these constructions, we prove incompleteness theorems for twist-free transport generators under a non-defocusing condition. The results are purely geometric and require no metric, curvature tensor, or causal structure.

Falsifiable predictions

01
Any hypersurface of finite transport capacity with strictly negative expansion lies at the start of an incomplete transport congruence.
02
Localised regions of negative expansion suffice to force incompleteness; compactness of the initial hypersurface is not required.
03
Transport incompleteness is a purely geometric phenomenon, independent of any metric, causal structure, or curvature tensor.

Full paper

Introduction

This paper extends the analytical results of PTG II into a global framework. The finite-parameter focusing theorem implies that twist-free transport congruences with negative initial expansion cannot be extended indefinitely. To convert this local divergence into a global obstruction, we introduce a notion of transport capacity for hypersurfaces and define transport completeness for congruences.

The main results are: a definition of transport capacity compatible with the transport cone structure; a global focusing lemma combining finite-parameter focusing with capacity collapse; and incompleteness theorems for twist-free transport generators under a non-defocusing condition. These are structural analogues of classical incompleteness theorems but require no metric or curvature assumptions.

Preliminaries

Transport generators are smooth vector fields

l^{a}

with

l^{a}

(p) in

L_{p}

. The expansion

\theta

of the congruence generated by

l^{a}

satisfies finite-parameter focusing (PTG II): for twist-free

l^{a}

with F(

l^{a}

)

\geq

0 and

\theta

(0) < 0, there is finite

\lambda_{\star}

with

\theta \to

\infty

\lambda \to \lambda_{\star}

Transport capacity

Hypersurfaces

A smooth embedded hypersurface S is transversely admissible if

L_{p}

contains at least one vector transverse to S at each p in S.

Capacity density and measure

Let

\rho(l^{a})

be a non-negative, degree-one homogeneous scalar density on admissible transport generators. The capacity measure on S is <<d\mu_{S \

\rho

} := \rho(l^{a}) d\mu_{S}>>, where

d\mu_{S}

is the induced hypersurface measure.

Transport capacity

J(S, l)

:= \int

over S of

\rho(l^{a}) d\mu_{S}

. A hypersurface has finite transport capacity if this

\int

is finite.

Transport completeness

A transport generator

l^{a}

is future transport complete if every

\int

curve starting on S is defined for all

\lambda \geq

Evolution of capacity

Capacity evolution

dJ/dlambda =

\int

over <<S_{\

\lambda

}>> of

\theta \rho(l^{a})

dmu. Differentiate under the

\int

and use the definition of expansion.

Monotonicity

\theta

< 0 on <<S_{\

\lambda

}>>, then dJ/dlambda < 0.

Capacity collapse

\theta(\lambda) \to

\infty

\lambda \to \lambda_{\star}

, then J(<<S_{\

\lambda

}>>, l)

\to

Global focusing lemma

Let S have finite transport capacity, and let

l^{a}

be twist-free with F(

l^{a}

)

\geq

0 and

\theta

< 0 on S. Then the congruence is not future transport complete. Finite-parameter focusing forces capacity collapse; a transported hypersurface with zero capacity cannot exist as a smooth admissible hypersurface, so the congruence cannot be extended.

Incompleteness theorems

Compact initial hypersurfaces

If S is compact with finite transport capacity and

\theta

< 0 on S, then the congruence is not future transport complete.

Non-compact hypersurfaces

Compactness is unnecessary: finite capacity and

\theta

< 0 on S suffice.

Localised negative expansion

\theta

< 0 on a subset U of S of positive capacity, then the congruence is incomplete.

Discussion

We have introduced transport capacity and transport completeness and proved incompleteness theorems for twist-free transport generators. These results are structural analogues of classical incompleteness theorems but arise without any metric or curvature assumptions, and form the global geometric foundation for the boundary structures in PTG IV.

Suggested citation

@techreport{pdt_ptg_iii_2026,
  author      = {Fincham, Graham and Hilton, Daniel},
  title       = {PTG III: Transport Capacity, Completeness, and Incompleteness Theorems},
  institution = {Phase Differential Theory},
  year        = {2026},
  type        = {PDT working paper},
  number      = {PTG III}
}