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PTG VI

PTG VI: Transport Entropy and Monotonicity

Graham Fincham, Daniel Hilton

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

  1. 01
    Transport entropy is monotone non-decreasing along any admissible transport congruence, in direct analogue to the area theorem for black hole horizons.
  2. 02
    The horizon-limit entropy H⋆H_{\star} is finite and stable under admissible perturbations of the transport generator.
  3. 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_{\λ\lambda}>> be the transported hypersurfaces generated by a twist-free transport generator lal^{a} with F(lal^{a}) ≥\geq 0 and θ\theta < 0 on S. Let A(U), C(U), K(U) denote the quasi-local invariants on compact U in S⋆S_{\star}.

Transport entropy

Definition

H(λ\lambda) :=∫:= \int over <<S_{\λ\lambda}>> of ( αhabhab\alpha h_{ab} h^{ab} + βρ(la)\beta \rho(l^{a}) + <<\gamma \sigma_{ab} \σ\sigma^{ab}>> ) dmu, where α\alpha, β\beta, γ\gamma > 0 are fixed constants whose admissible ranges are characterised in PTG VIII.

Well-definedness and non-negativity

H(λ\lambda) is finite for all λ\lambda < λ⋆\lambda_{\star} (smooth, non-negative integrand; finite capacity), and H(λ\lambda) ≥\geq 0 (all terms non-negative).

Evolution of the entropy

Differentiating under the ∫\int and using the evolution of the hypersurface measure gives dH/dlambda as an ∫\int over <<S_{\λ\lambda}>> of d/dlambda terms plus θ\theta times the integrand Ξ\Xi. Since θ\theta < 0 and Ξ≥\Xi \geq 0 for λ\lambda < λ⋆\lambda_{\star}, the expansion contributes a non-negative term to dH/dlambda. The shear term satisfies d/dlambda(<<\sigma_{ab} \σ\sigma^{ab}>>) ≥\geq -(2/3) <<\theta \sigma_{ab} \σ\sigma^{ab}>>.

Monotonicity

Entropy monotonicity

For all λ\lambda < λ⋆\lambda_{\star}, dH/dlambda ≥\geq 0. Combine the evolution identity with the focusing inequality and the non-negativity of the integrand. H(λ\lambda) is therefore non-decreasing.

Limiting behaviour

The limit H⋆:=lim⁡H_{\star} := \lim H(λ\lambda) as λ→λ⋆\lambda \to \lambda_{\star} exists (monotonicity) and is finite (quasi-local invariants are finite).

Stability

Under la~\tilde{l^{a}} = lal^{a} + ϵva\epsilon v^{a}, the perturbed entropy H~(λ)→\tilde{H}(\lambda) \to H(λ\lambda), and H⋆~→H⋆\tilde{H_{\star}} \to H_{\star}, as ϵ→\epsilon \to 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.

Suggested citation

@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}
}