PTG VI — PTG VI: Transport Entropy and Monotonicity

Reader summary

PTG VI builds a single scalar functional, the transport entropy, by weighting the three quasi-local invariants from PTG V. The entropy is non-negative, finite on every transported hypersurface before the focusing parameter, and monotone non-decreasing along the congruence. The monotonicity comes directly from the focusing inequality. As you approach the horizon, the entropy tends to a finite limit, and this limit is stable under perturbations of the generator. This is the analytical foundation for the membrane evolution equation in PTG VII.

Abstract

We introduce a transport entropy functional associated with transport horizons in Phase Transport Geometry. The entropy is constructed from the quasi-local invariants developed in Phase Transport Geometry V and is defined on transported hypersurfaces approaching the horizon. We prove that the entropy is well-defined, non-negative, and monotone non-decreasing along admissible transport congruences. The monotonicity is derived from the focusing inequality and the structural properties of the quasi-local invariants. These results provide the analytical foundation for the membrane evolution equation developed in Phase Transport Geometry VII.

Falsifiable predictions

01
Transport entropy is monotone non-decreasing along any admissible transport congruence, in direct analogue to the area theorem for black hole horizons.
02
The horizon-limit entropy H_star is finite and stable under admissible perturbations of the transport generator.
03
Entropy non-decrease follows directly from the focusing inequality, without invoking any thermodynamic or statistical assumptions.

Full paper

Introduction

PTG V introduced three quasi-local invariants associated with transport horizons: a transverse area invariant A, a capacity-weighted invariant C, and a curvature-type invariant K. The purpose of this paper is to combine these into a single scalar functional, the transport entropy, and establish its monotonicity along transported hypersurfaces.

The main results are: definition of a transport entropy functional; well-definedness and non-negativity; a monotonicity theorem derived from the focusing inequality; and stability under perturbations.

Preliminaries

Let S_lambda be the transported hypersurfaces generated by a twist-free transport generator l^a with F(l^a) >= 0 and theta < 0 on S. Let A(U), C(U), K(U) denote the quasi-local invariants on compact U in S_star.

Transport entropy

Definition

H(lambda) := integral over S_lambda of ( alpha h_ab h^ab + beta rho(l^a) + gamma sigma_ab sigma^ab ) dmu, where alpha, beta, gamma > 0 are fixed constants whose admissible ranges are characterised in PTG VIII.

Well-definedness and non-negativity

H(lambda) is finite for all lambda < lambda_* (smooth, non-negative integrand; finite capacity), and H(lambda) >= 0 (all terms non-negative).

Evolution of the entropy

Differentiating under the integral and using the evolution of the hypersurface measure gives dH/dlambda as an integral over S_lambda of d/dlambda terms plus theta times the integrand Xi. Since theta < 0 and Xi >= 0 for lambda < lambda_*, the expansion contributes a non-negative term to dH/dlambda. The shear term satisfies d/dlambda(sigma_ab sigma^ab) >= -(2/3) theta sigma_ab sigma^ab.

Monotonicity

Entropy monotonicity

For all lambda < lambda_*, dH/dlambda >= 0. Combine the evolution identity with the focusing inequality and the non-negativity of the integrand. H(lambda) is therefore non-decreasing.

Limiting behaviour

The limit H_star := lim H(lambda) as lambda -> lambda_* exists (monotonicity) and is finite (quasi-local invariants are finite).

Stability

Under tilde l^a = l^a + epsilon v^a, the perturbed entropy tilde H(lambda) -> H(lambda), and tilde H_star -> H_star, as epsilon -> 0.

Discussion

We have introduced a transport entropy functional and established its monotonicity along transported hypersurfaces approaching a transport horizon. These results provide the analytical foundation for the membrane evolution equation in PTG VII.

BibTeX

@techreport{pdt_ptg_vi_2026,
  author      = {Fincham, Graham and Hilton, Daniel},
  title       = {PTG VI: Transport Entropy and Monotonicity},
  institution = {Phase Differential Theory},
  year        = {2026},
  type        = {PDT working paper},
  number      = {PTG VI}
}