Can geometrizing pressure unify vortex identification criteria in fluid flows?

Pressure becomes curvature, not an external force

The paper asks whether the long-noticed analogy between pressure in fluids and gravity in general relativity can be turned into an explicit geometric framework, and whether that framework can explain why so many different vortex identification criteria exist. Its answer is a theoretical construction: model non-relativistic flow on a Newton–Cartan geometry, then apply a conformal transformation to the spatial metric so that the pressure field itself becomes the source of an effective spatial curvature.

The starting point is a formal parallel the paper states directly: the pressure Poisson equation and the gravitational Poisson equation share the same structure, with the velocity gradient and mass density acting as source terms. The authors argue this parallel is more than surface resemblance. Just as Einstein recast gravity from a force into spacetime curvature, they recast the pressure gradient from an external force into a geometric property of an effective space, so that a fluid particle under pressure alone follows a geodesic.

A conformal metric turns pressure into geometry

The construction begins with a Newton–Cartan geometry, the natural setting for non-relativistic gravity and fluids, built from a clock one-form that keeps time absolute and a degenerate spatial metric whose constant-time slices are flat Euclidean space. Because incompressible flow lives in absolute time and flat space, there is no obvious stage for curvature; the paper creates one by postulating an effective spatial metric that depends on the pressure field through a conformal factor.

The scalar potential is defined as pressure divided by density times a characteristic velocity squared, with the sound speed set to unity in natural units so the dimensionless potential reduces to pressure over density. The physical reading the authors give is that high-pressure regions correspond to metric dilation, where space is effectively expanded, and low-pressure regions to metric contraction, where space is effectively compressed. This is a definitional choice within the model, not a measured deformation of physical space.

Curvature is dominated by pressure Hessian and Laplacian

From the effective metric the paper derives a spatial connection, and pressure gradients enter the connection coefficients directly. Computing the spatial Ricci tensor for a three-dimensional conformally flat metric yields an expression whose second-order terms are the pressure Hessian and the pressure Laplacian, so the curvature is controlled by how pressure varies and how that variation changes across space, not by pressure magnitude alone.

The authors then define an effective spatial tensor in the format of the Einstein tensor. In the weak-field limit, where the dimensionless pressure potential is much smaller than one, as in incompressible flows, this tensor reduces to a form still dominated by the pressure Hessian and Laplacian, and its trace becomes twice the flat-space Laplacian of the potential. The paper separates two effects that readers could easily conflate: a metric scaling effect that sets the effective distance per unit coordinate length, and a curvature effect that makes geodesics converge or diverge. Both act together on fluid trajectories.

Geodesics recover the Euler equation in the weak-field limit

The central claim is that the pressure gradient can be absorbed into spatial geometry so that it no longer appears as an external force. Setting the four-acceleration of a fluid particle to zero makes its trajectory a geodesic in the Newton–Cartan spacetime. Projecting to spatial components, substituting the conformal metric and the pressure potential, and using incompressibility, the weak-field limit reduces to the classical incompressible Euler equation.

That recovery is the self-consistency check for the construction: the geometric picture must reproduce standard fluid mechanics where standard fluid mechanics is known to hold. The paper also shows that the pressure Poisson equation follows from the geodesic equation in the same limit, so it is not an independent dynamical input but a consequence of geometric consistency. The authors describe pressure as an instantaneous geometric constraint rather than an external force, and they distinguish their approach from Arnold's geometrization of the configuration space: here it is physical space that is geometrized.

Vortex criteria become projections of one curvature

The paper's answer to the vortex-identification question rests on a stated gap: most widely used criteria are computed from the velocity gradient tensor alone, so dynamic information such as pressure and its derivatives is neither included nor implicitly embedded. Criteria such as the second invariant, the swirl strength, and the lambda-2 criterion define scalar fields that reflect local vortical intensity, while other schemes extract vortex orientation from the real eigenvector of the velocity gradient tensor.

Within the geometric framework, these prevailing Eulerian and Lagrangian schemes are interpreted as different geometric invariants or projections of the effective curvature. That is the sense in which the criteria are unified: they are not shown to give identical numbers or identical vortex boundaries, but to be different ways of reading the same underlying geometric object. The paper presents this as a reinterpretation that adds dynamic content to criteria built on kinematics, not as a numerical benchmark showing that the criteria agree in a specific flow.

In two dimensions the curvature tensor vanishes identically

The paper examines the degenerate two-dimensional limit separately, because dimensional reduction changes what the geometry can express. In that limit the effective spatial tensor vanishes identically, and the global integral of the total curvature vanishes as well. This is a topological constraint on the construction, not a statement that two-dimensional flow lacks structure.

The authors use that constraint to offer a new reading of the dual-cascade scenario in two-dimensional turbulence. In their interpretation, the inverse energy cascade arises from geometric condensation of positive-curvature structures toward larger scales, while the forward enstrophy cascade results from geometric dispersion of negative-curvature structures toward smaller scales. This is presented as an interpretation of the cascade picture within the framework; the paper does not report a simulation or experiment that measures these curvature structures in a turbulent flow.

The unification is geometric, not numerical

The framework is developed for inviscid flow in flat Euclidean space, and the authors state that their subsequent discussion focuses mainly on the weak-field limit, meaning incompressible flows where the dimensionless pressure potential is small. The Euler-equation recovery, the pressure Poisson derivation, and the reduced form of the effective tensor all carry that weak-field condition. The two-dimensional results are a separate degenerate limit with its own vanishing-tensor constraint.

The paper is a theoretical construction: the evidence is the derivation itself, the recovery of known equations, and the reinterpretation of existing criteria, not a comparison against measured vortex fields. Its most decision-relevant claim for readers is therefore a priority about how to read vortex criteria: treat Eulerian and Lagrangian schemes as different geometric invariants or projections of pressure-induced effective curvature rather than as rival definitions, while remembering that the unification is established at the level of geometric interpretation and, for the cascade reading, at the level of a proposed interpretation of two-dimensional turbulence.

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