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PTG VII

PTG VII: Membrane Evolution on Transport Horizons

Graham Fincham, Daniel Hilton

Reader summary

PTG VII makes the horizon dynamical. The entropy gradient, combined with a shear-dissipating term, defines a canonical vector field on the transport horizon. The resulting first-order flow, the membrane evolution equation, exists and is unique, preserves the transverse-area and capacity-weighted invariants, drives the curvature-type invariant down, and dissipates the shear exponentially in flow time. The flow is stable under perturbations of the generator. This is PDT's geometric analogue of the membrane paradigm for black hole horizons, derived from PTG monotonicity alone.

Abstract

We derive a first-order evolution equation for the transport horizon in Phase Transport Geometry. The evolution is driven by the monotone transport entropy introduced in Phase Transport Geometry VI and is expressed as a geometric flow on the horizon. The flow depends only on the limiting transport direction, the limiting transverse structure, and the quasi-local invariants defined in Phase Transport Geometry V. We prove existence, uniqueness, and stability of the flow, and show that it preserves the quasi-local invariants while dissipating the shear. These results provide a dynamical characterisation of transport horizons and prepare the ground for the structural analysis of admissible focusing functionals in Phase Transport Geometry VIII.

Falsifiable predictions

  1. 01
    The transport horizon admits a unique first-order geometric flow that preserves transverse area and capacity-weighted invariants.
  2. 02
    Shear on the horizon dissipates exponentially under the membrane flow, with rate controlled by the shear-dissipation coefficient.
  3. 03
    Bounded limiting shear is sufficient to guarantee global existence of the membrane flow.

Full paper

Introduction

PTG VI introduced a transport entropy functional H(λ\lambda) defined on transported hypersurfaces <<S_{\λ\lambda}>> approaching a transport horizon S⋆S_{\star}. The entropy was shown to be non-negative and monotone non-decreasing. The purpose of this paper is to show that the limiting behaviour of H induces a canonical geometric flow on the horizon, the membrane evolution equation, a first-order evolution equation driven by the entropy gradient.

The main results are: definition of a membrane evolution vector field on S⋆S_{\star}; existence and uniqueness of the flow; preservation of quasi-local invariants; dissipation of the limiting shear; and stability under perturbations.

Preliminaries

The transport horizon S⋆S_{\star} carries the limiting transport direction kak^{a}, the limiting transverse bundle T⋆T_{\star}, the degenerate capacity measure μ⋆\mu_{\star}, and the limiting shear σ⋆ab\sigma_{\star ab}. From PTG VI, H(λ\lambda) increases to a finite limit H⋆H_{\star} as λ→λ⋆\lambda \to \lambda_{\star}.

Entropy gradient on the horizon

Ga:=h⋆abgradbH⋆G^{a} := h_{\star}^{ab} grad_{b} H_{\star}, where h⋆abh_{\star}^{ab} inverts the limiting transverse metric. GaG^{a} is tangent to S⋆S_{\star}: H⋆H_{\star} is constant along kak^{a}, so kaGak_{a} G^{a} = 0.

Membrane evolution equation

The membrane evolution vector field is Va:=αGaV^{a} := \alpha G^{a} - βσ⋆bakb\beta \sigma_{\star b}^{a} k^{b}, with α\alpha, β\beta > 0. The first term drives the flow along the entropy gradient; the second dissipates the shear.

The membrane evolution equation is the geometric flow dX/dtau = VaV^{a}(X(τ\tau)), with X(τ\tau) a one-parameter family of embeddings of S⋆S_{\star}.

Existence and uniqueness

Local existence

For any smooth initial embedding X(0), there is ϵ\epsilon > 0 and a unique smooth solution X(τ\tau) for |τ\tau| < ϵ\epsilon (VaV^{a} smooth and tangent to S⋆S_{\star}).

Global existence

If σ⋆ab\sigma_{\star ab} is bounded, the solution exists for all τ≥\tau \geq 0.

Preservation of quasi-local invariants

The membrane flow preserves A(U) and C(U) (VaV^{a} is tangent to S⋆S_{\star} and preserves the transverse structure and capacity measure). The curvature-type invariant satisfies dK/dtau ≤\leq 0 (shear-dissipation drives K downward).

Dissipation of shear

Along the membrane flow, d/dtau (σ⋆abσ⋆ab\sigma_{\star ab} \sigma_{\star}^{ab}) ≤\leq -2 βσ⋆abσ⋆ab\beta \sigma_{\star ab} \sigma_{\star}^{ab}, hence σ⋆ab→\sigma_{\star ab} \to 0 exponentially as τ→∞\tau \to \infty.

Stability

For la~\tilde{l^{a}} = lal^{a} + ϵva\epsilon v^{a}, the perturbed horizon converges smoothly to S⋆S_{\star}, Va~→Va\tilde{V^{a}} \to V^{a}, and the perturbed flow X~(τ)→\tilde{X}(\tau) \to X(τ\tau) smoothly as ϵ→\epsilon \to 0.

Discussion

We have introduced a membrane evolution equation on transport horizons, with existence, uniqueness, preservation of quasi-local invariants, exponential shear dissipation, and stability. These results give a dynamical characterisation of transport horizons and prepare the ground for PTG VIII.

Suggested citation

@techreport{pdt_ptg_vii_2026,
  author      = {Fincham, Graham and Hilton, Daniel},
  title       = {PTG VII: Membrane Evolution on Transport Horizons},
  institution = {Phase Differential Theory},
  year        = {2026},
  type        = {PDT working paper},
  number      = {PTG VII}
}