PTG I: PTG I: A Metric-Free Transport Framework for Scalar Fields

Reader summary

PTG I lays the foundation. Strip a manifold down to the bare minimum: a smooth scalar field (the phase), a cone of allowed transport directions at every point, and a torsion-free connection so you can differentiate. No metric, no curvature tensor, no Einstein equations. From this skeleton we define transport generators, transport curves, a deformation tensor, a transverse bundle with its own inner product, and the optical scalars (expansion, shear, twist). Everything that follows in the PTG arc, including focusing, incompleteness, horizons and entropy, is built on these objects.

Abstract

We introduce Phase Transport Geometry, a metric-free geometric framework constructed from a smooth scalar field, an admissible transport cone bundle, and a torsion-free affine connection. From these minimal ingredients we define transport generators, transport curves, the deformation tensor, a transverse bundle equipped with a positive-definite inner product, and the associated optical scalars. All constructions are purely geometric and require no metric, curvature tensor, or dynamical field equations. This paper establishes the foundational kinematic structures that underpin the subsequent development of transport focusing, incompleteness, boundary geometry, quasi-local invariants, entropy, and membrane dynamics.

Falsifiable predictions

01
Transport kinematics on a manifold can be fully described using only a scalar field, an admissible cone bundle, and a torsion-free affine connection.
02
The projected deformation tensor admits a unique decomposition into expansion, shear, and twist without invoking any metric.
03
All higher PTG structures (focusing, horizons, entropy, membrane flow) are determined by this minimal kinematic data.

Full paper

Introduction

This paper develops the foundational geometric structures of Phase Transport Geometry. The framework is constructed from minimal data: a smooth manifold, a scalar field, a cone of admissible transport directions, and a torsion-free affine connection. No metric, curvature tensor, or physical interpretation is assumed. The purpose of this paper is to define the basic kinematic objects associated with transport: transport generators, transport curves, the deformation tensor, the transverse bundle, and the optical scalars.

The results of this paper are purely local and structural. No focusing inequalities, completeness criteria, boundary structures, or entropy-type quantities are introduced here. These appear in subsequent papers of the series.

Minimal geometric data

Underlying manifold

Let M be a smooth, connected, Hausdorff, second-countable 4-manifold. No metric structure is assumed. All constructions are invariant under diffeomorphisms of M.

Scalar field and phase orientation

A smooth scalar field

\Phi

: M

\to

R is fixed. A vector

l^{a}

T_{pM}

is phase-oriented if

l^{a} grad_{a} \Phi \geq

Transport cone bundle

At each point p in M, let

L_{p}

be a nonempty, closed, convex cone satisfying: positive scaling, convexity, and phase orientation (

l^{a} grad_{a} \Phi \geq

0 for all

l^{a}

L_{p}

). The assignment p

\to L_{p}

is upper semicontinuous.

Transport generators and curves

A smooth vector field

l^{a}

is a transport generator if

l^{a}

(p) in

L_{p}

for all p. A C1 curve

\gamma

: I

\to

M is a transport curve if its tangent lies in L at every parameter.

Affine connection and transport differentiation

The manifold carries a torsion-free affine connection used solely for covariant differentiation. Differentiation along a transport generator is d/dlambda

:= l^{a} grad_{a}

Deformation tensor and transverse geometry

The deformation tensor of a transport generator

l^{a}

B_{ab} := grad_{b} l_{a}

. The transverse bundle at p is

T_{p perp} :=

{

X^{a}

X^{a} l_{a}

= 0 }, equipped with a fixed positive-definite inner product

h_{ab}

. The projected deformation tensor is the double projection of

B_{ab}

onto the transverse bundle.

Optical scalars and decomposition

The expansion is

\theta := h^{ab} B_{ab}

. The shear

\sigma_{ab}

is the trace-free symmetric part. The twist

\omega_{ab}

is the antisymmetric part.

Optical decomposition theorem

The projected deformation tensor decomposes as (1/3)

\theta h_{ab}

\sigma_{ab}

\omega_{ab}

. The projection induces an orthogonal decomposition of

T_{perp}

tensor

T_{perp}

into trace, symmetric trace-free, and antisymmetric components.

Discussion

This paper establishes the foundational kinematic structures of Phase Transport Geometry. Beginning with only a scalar field, a transport cone bundle, and a torsion-free affine connection, we have defined transport generators, transport curves, the deformation tensor, the transverse bundle, and the optical scalars. These structures form the basis for the analytical, global, and boundary results developed in subsequent papers of the series.

Suggested citation

@techreport{pdt_ptg_i_2026,
  author      = {Fincham, Graham and Hilton, Daniel},
  title       = {PTG I: A Metric-Free Transport Framework for Scalar Fields},
  institution = {Phase Differential Theory},
  year        = {2026},
  type        = {PDT working paper},
  number      = {PTG I}
}