Reader summary
PTG II derives PDT's version of the Raychaudhuri equation, the workhorse identity that controls how bundles of trajectories spread or focus. We do it without a metric. With only a torsion-free connection, the deformation tensor, and a non-negative focusing functional, we obtain an inequality of the same shape as the classical one. For twist-free congruences with negative initial expansion, focusing happens in finite parameter: the expansion blows up to minus infinity in a bounded interval. This finite-parameter focusing is the analytical engine behind every incompleteness theorem in the rest of the series.
Abstract
We derive a Raychaudhuri-type evolution equation for the expansion of transport congruences in the metric-free framework of Phase Transport Geometry. The derivation requires only a torsion-free affine connection, the deformation tensor of a transport generator, and a non-negative focusing functional. No metric, curvature tensor, or dynamical field equations are assumed. Under a twist-free condition, the resulting inequality yields finite-parameter focusing for initially converging congruences. These results form the analytical foundation for the incompleteness theorems developed in Phase Transport Geometry III.
Falsifiable predictions
- 01Twist-free transport congruences with negative initial expansion focus in finite affine parameter, bounded by -3/theta_0.
- 02The Phase-Raychaudhuri inequality holds with no metric input; only the transport cone, the connection, and a non-negative focusing functional are required.
- 03Twist alone cannot prevent focusing once the shear and focusing functional are accounted for.
Full paper
Introduction
Building on the kinematic structures introduced in PTG I, we derive an evolution equation for the expansion of a transport congruence using only affine differentiation and the transport cone structure. The resulting inequality is analogous in form to the classical Raychaudhuri equation but is obtained without any metric or curvature assumptions.
The main results are: an evolution identity for the deformation tensor; a Raychaudhuri-type inequality for the expansion; and a finite-parameter focusing theorem for twist-free congruences under a non-defocusing condition.
Preliminaries
We recall the minimal data from PTG I: transport generators l^a in L_p, the deformation tensor B_ab = grad_b l_a, the projected deformation tensor tilde B_ab on the transverse bundle, and the optical scalars (expansion theta, shear sigma_ab, twist omega_ab). Transport differentiation along l^a is d/dlambda = l^a grad_a.
The focusing functional
A focusing functional F(l^a) is smooth on the interior of each cone L_p, homogeneous of degree two, non-negative, and absorbs curvature-type terms in the evolution identity. Its existence is assumed here and justified structurally in PTG VIII.
Evolution of the deformation tensor
Affine commutator identity
For any vector field l^a, the standard torsion-free identity gives grad_a (l^b grad_b l_c) = l^b grad_b (grad_a l_c) + (grad_a l^b)(grad_b l_c) - R^d_cab l^a l^b.
Evolution of B_ab
Applying the commutator and contracting yields dB_ab/dlambda = grad_b(l^c grad_c l_a) - B_ac B^c_b + R_cabd l^c l^d.
The Phase-Raychaudhuri equation
Projecting the evolution identity onto the transverse bundle and contracting with h^ab gives the main analytical result.
Phase-Raychaudhuri equation
The expansion satisfies dtheta/dlambda = -(1/3) theta^2 - sigma_ab sigma^ab - omega_ab omega^ab - F(l^a).
Non-defocusing inequality
If F(l^a) >= 0, then dtheta/dlambda <= -(1/3) theta^2 - sigma_ab sigma^ab - omega_ab omega^ab.
Finite-parameter focusing
A transport generator is twist-free if omega_ab = 0.
Focusing inequality
If l^a is twist-free and F(l^a) >= 0, then dtheta/dlambda <= -(1/3) theta^2.
Finite-parameter focusing theorem
If l^a is twist-free with F(l^a) >= 0 and initial expansion theta_0 < 0, then there exists 0 < lambda_* <= -3/theta_0 such that theta(lambda) -> -infinity as lambda -> lambda_*. Proof by comparison with the explicit solution of dpsi/dlambda = -(1/3) psi^2.
Discussion
We have derived a Raychaudhuri-type inequality and established finite-parameter focusing for twist-free transport congruences. These results require only affine differentiation and the transport cone structure, and form the analytical foundation for the incompleteness theorems in PTG III.
BibTeX
@techreport{pdt_ptg_ii_2026,
author = {Fincham, Graham and Hilton, Daniel},
title = {PTG II: The Phase-Raychaudhuri Equation and Finite-Parameter Focusing},
institution = {Phase Differential Theory},
year = {2026},
type = {PDT working paper},
number = {PTG II}
}