PTG V: Quasi-Local Invariants on Transport Horizons

Daniel Hilton

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 foundations Research history

Current foundational treatment

Specific claim correctedPrincipal topic: Gravitational dynamics.

Quasi-local invariants on horizons relate to gravitational dynamics in R10.

Specific qualification: Limiting horizon requires further conditions.

R10: Gravity and conditional Einstein dynamics

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

Limiting horizon requires further conditions

The invariants rely on the limiting horizon of PTG IV, whose convergence and regularity requirements are not stated. Ordinary integration against an identically zero measure cannot yield nonzero invariants.

Current context

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

PTG V equips the transport horizon with three quasi-local invariants: a transverse area, a capacity-weighted invariant, and a curvature-type invariant built from the limiting shear. Each is well-defined on compact subsets of the horizon, independent of how you approach it, invariant under admissible rescalings of the transport generator, and stable under perturbations. These quasi-local quantities are the geometric handles PDT uses to define entropy in PTG VI and to drive membrane dynamics in PTG VII.

Original abstract

Preserved from the original manuscript.

We introduce three quasi-local invariants associated with transport horizons in Phase Transport Geometry. These invariants are defined using the limiting transverse structure, the limiting transport direction, and the degenerate capacity measure established in Phase Transport Geometry IV. We prove that the invariants are well-defined, independent of the choice of transported hypersurface, stable under perturbations of the transport generator, and invariant under admissible rescalings. These quasi-local quantities form the basis for the entropy and monotonicity structures developed in Phase Transport Geometry VI.

Claims stated in the original paper

01
The transverse area, capacity-weighted, and curvature-type invariants on a transport horizon depend only on the horizon, not on the family of hypersurfaces used to approach it.
02
All three invariants are unchanged under positive rescalings of the transport generator.
03
Quasi-local invariants vary smoothly under admissible perturbations of the generator, giving stable horizon-level observables.

Full paper

Introduction

PTG IV established that incomplete transport congruences induce a canonical terminal hypersurface, the transport horizon, equipped with a limiting transport direction

k^{a}

, a limiting transverse bundle

T_{\star}

, and a degenerate capacity measure

\mu_{\star}

. The purpose of this paper is to define and analyse quasi-local invariants associated with this boundary structure.

We introduce three invariants: a transverse area invariant, a capacity-weighted invariant, and a curvature-type invariant derived from the limiting deformation tensor. We prove that each invariant is well-defined on

S_{\star}

, independent of the choice of transported hypersurface approaching the horizon, invariant under admissible rescalings of the transport generator, and stable under perturbations.

Preliminaries

Let

S_{\star}

be the transport horizon of a twist-free transport generator

l^{a}

satisfying F(

l^{a}

)

\geq

0 and

\theta

< 0 on some initial hypersurface of finite transport capacity. The limiting deformation tensor is

\tilde{B_{\star ab}} := \lim \tilde{B_{ab}}(\lambda)

as

\lambda \to \lambda_{\star}

.

Quasi-local invariant I: transverse area

For compact U in

S_{\star}

, A(U)

:= \int

over U of

h_{\star ab} d\mu_{\star}

. This is well-defined and finite (limiting transverse metric exists and is smooth,

\mu_{\star}

is finite on compact sets) and independent of approach (smooth convergence of <<S_{\

\lambda

}>> and

h_{ab}

).

Quasi-local invariant II: capacity-weighted invariant

C(U)

:= \int

over U of

\rho(k^{a}) d\mu_{\star}

. It is well-defined and non-negative (

\rho

non-negative and homogeneous of degree one;

\mu_{\star}

non-negative). Under rescaling

k^{a} \to \alpha k^{a}

with

\alpha

> 0, C is invariant: homogeneity of

\rho

cancels the rescaling.

Quasi-local invariant III: curvature-type invariant

K(U)

:= \int

over U of

\sigma_{\star ab} \sigma_{\star}^{ab} d\mu_{\star}

, where

\sigma_{\star ab}

is the limiting shear. Well-defined and finite (limiting shear smooth; measure finite), and independent of approach (smooth convergence of

\sigma_{ab}

).

Stability of the invariants

Let

\tilde{l^{a}}

=

l^{a}

+

\epsilon v^{a}

. The perturbed horizon

\tilde{S_{\star}}

converges smoothly to

S_{\star}

; the limiting structures (

k^{a}

,

T_{\star}

,

\mu_{\star}

) vary smoothly with

\epsilon

. Hence

A_{eps}

(U)

\to

A(U),

C_{eps}

(U)

\to

C(U),

K_{eps}

(U)

\to

K(U) as

\epsilon \to

0.

Discussion

The three quasi-local invariants are well-defined, independent of approach, rescaling-invariant, and stable. They form the basis for the entropy and monotonicity structures in PTG VI.

Suggested citation

Graham Fincham, Daniel Hilton. “PTG V: Quasi-Local Invariants on Transport Horizons.” Phase Differential Theory research programme. https://www.phasedifferentialtheory.com/papers/ptg-v

@misc{pdt_ptg_v,
  title = {PTG V: Quasi-Local Invariants on Transport Horizons},
  author = {Graham Fincham and Daniel Hilton},
  url = {https://www.phasedifferentialtheory.com/papers/ptg-v}
}

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