The central subject of the new perspective is the shock development problem: the question of what happens to a compressible fluid after a shock wave has begun to form. Shocks are among the most familiar singularities in nature, responsible for the crack of a sonic boom and the abrupt pressure fronts that propagate through air after an explosion. Mathematically, however, they pose a profound challenge. The equations of inviscid compressible fluid dynamics, the Euler equations, describe smooth flow up to a moment when the gradient of certain quantities blows up. Before that moment, the solution is classical and differentiable; after it, discontinuities appear and the equations in their classical form cease to make sense. Understanding how to continue the solution past this breakdown point, in a way that is both physically meaningful and mathematically rigorous, had remained open for more than a century and a half.

Christodoulou’s most recent monograph, The Shock Development Problem, published by the EMS Publishing House in 2019, finally resolved this question. The new article reviews the mathematical methods of that monograph, and in doing so it reveals an unexpected intellectual lineage. The machinery used to track the formation and propagation of shocks was not invented in isolation; it is, in essential respects, an adaptation of the geometric framework that Christodoulou had earlier built together with Sergiu Klainerman in their landmark 1993 work, The Global Nonlinear Stability of the Minkowski Space, published in the Princeton Mathematical Series. That result demonstrated, using purely geometric methods, that Minkowski spacetime, the flat spacetime of special relativity, is stable against small nonlinear perturbations, thereby establishing the nonlinear stability of the simplest solution of Einstein’s field equations.

The bridge between these two domains lies in what mathematicians call Lorentzian geometry, the geometry of spacetimes in which the metric has a signature that distinguishes timelike directions from spacelike ones. In general relativity, the fundamental objects of study are Lorentzian manifolds, and the natural way to analyze them is through foliations, families of hypersurfaces slicing the manifold, and through the behavior of null surfaces, the analogues of light fronts. Christodoulou realized that even when one studies a compressible fluid in flat spacetime, a problem that superficially has nothing to do with gravity, the same Lorentzian geometric structures are the correct tools. The characteristic surfaces along which information propagates in a fluid, and along which shocks in particular form, are naturally viewed as hypersurfaces in a Lorentzian metric constructed from the fluid equations themselves. This reframing allows the powerful estimates and construction techniques of geometric analysis to be applied to fluid mechanics.

This is not the first time Christodoulou has shown such a connection. In 2007 he published The Formation of Shocks in 3-Dimensional Fluids, also in the EMS Monographs in Mathematics, in which he proved that shocks are an inevitable feature of smooth, arbitrarily small initial data for compressible Euler flow in three dimensions. The result was striking because it showed that the nonlinearity of the fluid equations is so persistent that no smooth initial condition, however small and however carefully chosen, can avoid developing infinite gradients in finite time. Two years later, in 2009, he applied analogous methods to the relativistic setting in The Formation of Black Holes in General Relativity, proving that trapped surfaces, the hallmark boundaries of black holes, can form in the evolution of spacetimes from smooth data that is nowhere trapped initially. The structural parallels between these two problems, shock formation in fluids and trapped surface formation in gravitation, are no accident; both concern the focusing of characteristic cones and the breakdown of smooth evolution.

The new perspective article addresses the step beyond formation: what happens past the point where the shock first appears. This is the shock development problem proper. In the 2007 monograph, Christodoulou characterized the geometry of the boundary of the region of smooth flow at the exact moment of shock formation, showing that this boundary possesses a singular structure, often described in terms of a crease set, where the smooth solution meets the incipient discontinuity. The question then becomes how to define and construct the solution for later times, when the shock surface separates regions of fluid in different states. The answer required constructing the shock front itself as a free boundary, tracking its evolution with estimates of a precision rarely attempted in fluid mechanics, and verifying that the resulting weak solution satisfies the physical conservation laws in the correct distributional sense.

Technically, the framework demands an intricate interplay between hyperbolic estimates and differential geometry. One must control the derivatives of the fluid variables up to very high order on the smooth side of the shock, establish that the shock surface remains regular in an appropriate sense, and match the two across the discontinuity. The Lorentzian viewpoint is indispensable here because the shock surface and the characteristic hypersurfaces emanating from it form a geometric system whose curvature properties dictate where and how the solution can be continued. Quantities analogous to the curvature components studied in general relativity play the role of error terms that must be estimated; the deformation tensors of the foliations play the role of connection coefficients. In this way, a problem of classical fluid dynamics is recast as a problem in the geometric analysis of a Lorentzian manifold, and the hard-won lessons of decades spent on the Einstein equations become directly transferable.

The significance of this unification extends beyond fluid mechanics itself. The methods developed for the shock development problem have already stimulated further research in the mathematical community. Among the works building on these foundations is a 2022 preprint by Leon Abbrescia and Jared Speck investigating the emergence of the singular boundary from the crease in three-dimensional compressible Euler flow, which takes up precisely the geometric structures that Christodoulou’s monograph isolated. Christodoulou himself also provided a shorter account of the shock development program in a 2022 article in the Journal of Mathematical Physics, offering a compact summary of the ideas now expanded in the present review. The appearance of this new perspective in General Relativity and Gravitation, a journal at the heart of the relativity community, underscores the message that the boundaries between gravitational physics and fluid dynamics are far more porous than traditional disciplinary divisions suggest.

There is also a broader conceptual payoff. The fact that the same geometric techniques resolve questions in both general relativity and classical fluid mechanics points toward a unified theory of singularities in hyperbolic systems, the class of equations, including the wave equation, the Einstein equations, and the Euler equations, whose solutions propagate at finite speed and whose nonlinearities tend to concentrate energy into ever thinner regions. Whether the singularity is a black hole horizon forming in a collapsing spacetime or a pressure discontinuity racing through air, the mathematical pathology is structurally related: the characteristic cones focus, gradients blow up, and the smooth description of the system terminates. Geometric analysis in the Lorentzian setting provides a language in which this commonality can be made precise and, more importantly, exploited.

For Christodoulou, whose career has been marked by results of extraordinary technical depth, from his early work on black hole dynamics and the memory effect to his proofs of global stability and singularity formation, the new article serves as both a retrospective and a roadmap. It presents the shock development problem not as an isolated curiosity of fluid mechanics but as a chapter in a larger story about the geometry of propagation, one that began with the stability of spacetime itself and continues wherever waves and fronts evolve in a Lorentzian world. Readers of the journal, whether they come to it from relativity or from the theory of partial differential equations, will find in it a demonstration of how the deepest tools of one field can illuminate the oldest open problems of another.

To appreciate the significance of the work, it helps to understand what Lorentzian distance measures. In general relativity, spacetime is modeled as a four-dimensional manifold equipped with a Lorentzian metric, a geometric object that defines causal relationships between events. The Lorentzian distance between two causally related points is the length of the longest timelike curve—the worldline of a permissible observer—connecting them. Unlike ordinary Riemannian distance, which is smooth and well-behaved, Lorentzian distance is only continuous in general, and it vanishes whenever the two points cannot be causally connected. This inherent roughness makes Lorentzian geometry a delicate subject, and many of its deepest theorems rely on careful control of how these distance functions behave.

The classical Lorentzian splitting theorems, which trace back to work in the late 1980s by researchers including Richard Newman, J.-H. Eschenburg, and Galloway himself, are a case in point. These theorems assert, roughly, that if a spacetime satisfies the strong energy condition—the statement that gravity, as encoded in the Einstein equations, tends to focus matter—and contains a complete timelike line, meaning an inextendible geodesic that maximizes the time separation between every pair of points along it, then the spacetime splits as a product. In physical terms, the universe must decompose into a static spatial part crossed with ordinary time, much like the flat spacetimes of special relativity. Splitting theorems are rigidity statements: they show that under physically reasonable hypotheses, the universe cannot have an exotic global shape.

A crucial ingredient in all known proofs of these splitting theorems is the local equi-Lipschitz continuity of families of Lorentzian distance functions—often constructed as Busemann functions—associated with a complete timelike line. In essence, equi-Lipschitzness means that an entire family of functions shares a common bound on how fast they can change: there is a single Lipschitz constant that controls them all in a neighborhood of the line. This uniform control allows mathematicians to extract smoothly converging subsequences, take limits of Busemann functions, and ultimately produce the preferred time coordinate whose existence forces the splitting. Without it, the limiting machinery at the heart of the proofs would collapse.

Galloway, McCann, and Ohanyan asked a natural next question: does the same uniform regularity hold when the distance functions are taken not to a single point, but to the level sets of a Cauchy temporal function? A Cauchy temporal function is a smooth time function that increases along every future-directed timelike curve and whose level sets are Cauchy surfaces—spatial slices that every inextendible timelike curve crosses exactly once. Such functions are the gold standard for imposing a clean global notion of time on a spacetime, and they exist in all globally hyperbolic spacetimes, the class of spacetimes most physicists consider reasonable models of our universe. One might therefore expect the distances to these well-behaved foliations to inherit the same regularity as distances to points.

The authors show that this expectation is false. In general, families of Lorentzian distances to and from the level sets of a Cauchy temporal function fail to be locally equi-Lipschitz. The failure is not an artifact of pathological metrics or exotic causal structures; it is an intrinsic feature of how Lorentzian geometry treats spatial slices. Whereas a complete timelike line provides a rigid scaffold—its maximizing property propagates uniform control through the neighborhood—the foliation by Cauchy surfaces offers no such mechanism. The distance to a surface can concentrate its variation in ways that no single Lipschitz constant can tame, even locally. This negative result matters because several proposed approaches to splitting conjectures for Cauchy surfaces have implicitly assumed or hoped for exactly this kind of regularity, and the new theorem rules out the naive strategy of extending the classical point-based arguments wholesale to foliations.

The stakes become clear when the authors connect their findings to Bartnik’s splitting conjecture. In 1988, Robert Bartnik conjectured that a cosmological spacetime—one that admits a compact Cauchy surface—satisfying the strong energy condition should admit a splitting of a related kind, with deep ties to the existence of constant mean curvature surfaces. Decades of partial progress have been recorded; the authors note in their notes that specific classes of Cauchy surfaces have been treated in earlier works, yielding splitting results subject to additional conditions. But a general proof remains elusive. Galloway, McCann, and Ohanyan formulate new conjectures based on the existence of Cauchy temporal functions in cosmological spacetimes and in timelike geodesically complete spacetimes, conjecturing that the Lorentz distances to the level sets of such functions are equi-Lipschitz precisely in the circumstances that matter. Strikingly, they prove that these conjectures are equivalent to Bartnik’s splitting conjecture.

This equivalence is the conceptual heart of the paper. It transforms an analytic question about the regularity of distance functions into a statement about the global causal structure of the cosmos, and vice versa. On the one hand, if Bartnik’s conjecture holds, then the equi-Lipschitz property must follow in the conjectured settings, providing a new handle on spacetime rigidity. On the other hand, anyone seeking to prove Bartnik’s conjecture can now target the equi-Lipschitz property directly, using tools from analysis, partial differential equations, and geometric measure theory that were developed for related regularity problems. The authors’ negative theorem serves as a warning sign along the way: the property does not come for free, so any successful proof must identify precisely which additional structure restores uniform control.

The work also sits within a broader modern program to extend Lorentzian geometry beyond smooth manifolds. Recent research, including the authors’ own collaborations with Mathias Braun, Nicola Gigli, Clemens Sämann, and others, has developed splitting theorems and causal calculus on nonsmooth and metric-measure spacetimes, inspired in part by optimal transport and by the synthetic treatment of curvature. In such settings, where the metric may be only continuous or the spacetime may carry a measure-theoretic weight, questions of Lipschitz regularity become even more delicate. The new negative result calibrates expectations across this program, indicating that techniques tied to distance functions along lines—which do enjoy local equi-Lipschitzness—cannot be transplanted blindly to settings built around temporal functions and their foliations.

For physicists, the paper is a reminder that the geometry underlying cosmological models is governed by rigid mathematical constraints that remain only partially mapped. Splitting theorems constrain the possible global shapes of universes obeying the energy conditions that dominate classical cosmology, and Bartnik’s conjecture sits at the frontier of this understanding. By demonstrating exactly where the standard toolkit fails—and by proving that a specific regularity conjecture is equivalent to that frontier problem—Galloway, McCann, and Ohanyan have redrawn the map of what is known and what must be proven. The full paper, including the precise counterexamples and the proofs of the equivalences, is available in General Relativity and Gravitation.

In ordinary general relativity, light travels along null geodesics: curves whose tangent vectors lie on the boundary of the light cones determined by a Lorentzian metric. These paths represent the possible histories of photons, although the theory itself treats them geometrically rather than as individual particles. Penrose recognized that the totality of light rays can sometimes form a smooth manifold in its own right. Instead of studying each light ray inside spacetime, one studies the space whose points are entire unparametrized rays. That space naturally carries a contact structure, a geometric condition that describes how hyperplanes twist and fail to fit together into ordinary surfaces. Hedicke’s work replaces the metric-dependent language of Lorentzian geometry with the broader setting of strongly convex cone structures, where the allowed future-directed causal directions are specified directly by cones in each tangent space.

A cone structure can be imagined as assigning a future-pointing cone to every point of a manifold. Vectors inside the cone represent timelike or causal directions, while vectors on its smooth boundary represent null directions. Strong convexity imposes a crucial form of regularity: the cone is convex, its boundary is smooth, and its curvature behaves positively in directions transverse to the rays generated by scaling. This condition ensures that the boundary resembles a well-behaved light cone even when no quadratic metric is available. The framework therefore includes Lorentzian spacetimes but also more flexible Lorentz-Finsler geometries, in which the “speed of light” may depend asymmetrically on direction. Cone geodesics are defined without selecting a particular metric: they are curves that remain locally on the boundary of the causal relation, meaning that nearby points along the curve are connected by null, or horismotic, propagation.

The central geometric object is constructed from the dual cone in the cotangent bundle. Every cone of allowed tangent vectors determines a dual cone consisting of covectors that evaluate non-negatively on all causal directions. The boundary of this dual cone carries the canonical Liouville one-form from cotangent geometry. When the null or cone-geodesic flow is factored out, this form produces a natural contact structure on the space of cone geodesics. In practical terms, the construction compresses the full dynamics of null propagation into a manifold whose dimension is typically two less than that of the spacetime’s cotangent bundle. The resulting contact structure is not an arbitrary decoration: it records which infinitesimal variations of a cone geodesic remain geometrically compatible with the lightlike condition. Its coorientation is inherited from the Liouville form, although the paper emphasizes that a specific contact form depends on a choice of section, while the underlying cooriented contact structure is canonical.

Global hyperbolicity makes this abstract space especially manageable. If a cone spacetime has a Cauchy hypersurface, every inextendible causal curve intersects that hypersurface exactly once. Consequently, every cone geodesic can be represented by the point where it crosses a chosen spatial slice together with its direction there. Hedicke proves that the space of cone geodesics is then contactomorphic to the spherical cotangent bundle of the Cauchy hypersurface, denoted (ST^{*}\Sigma). This bundle consists of nonzero covectors at points of (\Sigma), considered up to positive scaling, and carries the standard contact structure obtained by restricting the Liouville form. The identification is explicit: a covector along a null geodesic is restricted to the tangent space of the Cauchy slice. A warning about orientation is important here—the natural coorientation inherited from the cone construction is reversed under this contactomorphism, a sign convention that becomes essential when interpreting causal motion as positive motion.

Once all Cauchy slices are identified with a fixed manifold (\Sigma), the changing light-ray geometry becomes a time-dependent family of contact transformations. Hedicke shows that a globally hyperbolic, strongly convex cone structure on (\mathbb{R}\times\Sigma) generates a path ((\varphi_t)) of contactomorphisms of (ST^{*}\Sigma). The path is positive: its velocity points consistently through the chosen cooriented contact hyperplanes. The meaning is physical and geometric at once. If (v) labels an initial cone geodesic, its spatial position at time (t) is obtained by projecting (\varphi_t(v)) to (\Sigma), producing a curve of the form (t\mapsto (t,\pi(\varphi_t(v)))). Thus the entire family of null trajectories is encoded by a positive contact evolution. In this picture, causality in spacetime is transformed into an order-like property in the contactomorphism group.

The reverse construction is even more striking. Starting with any positive path of contactomorphisms on (ST^{*}\Sigma), the paper defines a cone structure on (\mathbb{R}\times\Sigma). The infinitesimal generator (X_t^f) of the contact path determines a time-dependent contact form (\alpha_t^f), normalized so that (\alpha_t^f(X_t^f)=1). At each point (p\in\Sigma), the contact form on the cotangent fibre determines a star-shaped set (K_t^f(p)). The allowed spacetime directions are then obtained from the polar body of this set. In formula form, the cone is generated by vectors (\partialt+w) satisfying (\max{v\in K_t^f(p)}v(w)\leq 1). Equivalently, the associated function is (G_f(w_0,w)=w0-\max{v\in K_t^f(p)}v(w)). Convex duality guarantees that these are proper closed cones, even when their boundaries are not smooth or strictly convex.

This reverse correspondence also reveals why generalized Lorentz-Finsler geometry is needed. The maximum over a cotangent fibre is typically only Lipschitz or nonsmooth when the underlying star-shaped body has corners or flat portions. The resulting cone structure may therefore fail to be strongly convex, yet it remains sufficiently regular to define a locally Lipschitz Lorentz-Finsler space. The function (G_f) is positively homogeneous and concave, properties that replace the smooth quadratic behavior of a classical Lorentzian metric. Hedicke proves that the cone family varies locally Lipschitz-continuously and that its causal geometry can still be studied. When the normalized contact forms arise from genuine Finsler metrics, the construction simplifies to the familiar expression (G_f=dt-H_t), where (H_t) is a time-dependent Finsler norm on (\Sigma), and the cone is the usual future region described by (dt\geq H_t).

The study establishes a complete agreement in the strongly convex case. If a cone structure produces a positive path (\varphi_t), reconstructing a cone structure from that path returns the original cones and the original Lorentz-Finsler function. Conversely, if a positive contactomorphism path is generated by a family of Finsler metrics, the induced cone structure is globally hyperbolic and strongly convex, and the contact path recovered from its cone geodesics is the one with which researchers began. The infinitesimal generator has an especially elegant interpretation: at each fixed time it is the Reeb vector field of the contact form determined by the instantaneous Finsler metric. In familiar Riemannian examples, this is the cogeodesic flow, whose projections are ordinary geodesics. The theorem therefore links light propagation, Reeb dynamics and Finsler geodesic flow within one unified mechanism.

The broader implications reach beyond a technical correspondence. In the space of cone geodesics, the set of all rays passing through a spacetime point forms a Legendrian sphere called its sky. Timelike motion of the point produces a positive Legendrian isotopy, while causal motion produces a non-negative one. This translates the distinction between timelike and lightlike travel into the language of contact topology, where positivity can be studied using linking, orderability and dynamical invariants. The work also points toward a major open question: whether every positive path of contactomorphisms, even one producing nonsmooth cone boundaries, necessarily defines a globally hyperbolic cone structure. Hedicke conjectures that it does. If confirmed, the result would establish a remarkably broad dictionary in which complete positive contact dynamics automatically encode a globally well-behaved causal spacetime, offering a new route for studying relativity through the topology of transformations rather than through metrics alone.

Subject of Research: Cone structures, Lorentz-Finsler geometry, light rays, contact geometry, and positive paths of contactomorphisms

Article Title: On the Space of Cone Geodesics and Positive Paths of Contactomorphisms

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03554-x

Keywords: Cone structures; Lorentz-Finsler geometry; light rays; contact geometry; cone geodesics; contactomorphisms; global hyperbolicity; spherical cotangent bundles