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PTG VI

PTG VI: Transport Entropy and Monotonicity

Graham Fincham, Daniel Hilton

Original research seriesPublic manuscript availableJournal-review status not confirmed.

Earlier research paper

Earlier research paper

This paper remains available as part of the development of Phase Differential Theory. The R00–R13 foundational suite provides the current formulation of the theory. Read this paper in its original model and publication context.

Current foundationsResearch history

Current foundational treatment

Specific claim correctedPrincipal topic: Gravitational dynamics.

A transport entropy built on PTG V. Gravitational dynamics are treated in R10; verification practice in R13.

Specific qualification: Entropy sign argument requires correction.

A topic connection does not mean that every claim in this paper is retained or that a current paper reproduces its proof.

Context for this earlier argument

Entropy sign argument requires correction

The displayed argument for the sign of the entropy change requires correction. It should not be cited as an established monotonicity theorem.

Current context

A present-day website summary. It is not part of the original manuscript.

PTG VI builds a single scalar functional, the transport entropy, by weighting the three quasi-local invariants from PTG V. The paper argues that this entropy is monotone along the congruence; that sign argument requires correction and is not relied on as an established theorem.

Original abstract

Preserved from the original manuscript.

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.

Claims stated in the original paper

  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

Graham Fincham, Daniel Hilton. “PTG VI: Transport Entropy and Monotonicity.” Phase Differential Theory research programme. https://www.phasedifferentialtheory.com/papers/ptg-vi

@misc{pdt_ptg_vi,
  title = {PTG VI: Transport Entropy and Monotonicity},
  author = {Graham Fincham and Daniel Hilton},
  url = {https://www.phasedifferentialtheory.com/papers/ptg-vi}
}

No DOI recorded. No repository record verified. Journal-review status not confirmed.