Geometric methods in Lorentzian settings illuminate shock formation in fluids

Demetrios Christodoulou, Emeritus Professor of Mathematics and Physics at ETH Zurich and one of the most influential figures in modern mathematical relativity, has published a far-reaching perspective article in the journal General Relativity and Gravitation that connects two of his most celebrated achievements: the geometric analysis of spacetime developed in his work on black hole formation, and the notoriously difficult problem of how shocks emerge and evolve in compressible fluids. The article, published on 24 August 2026 as a review corresponding to a lecture delivered in Granada on 9 May 2025, distills decades of technical innovation into a single narrative about how ideas originally forged to understand the structure of curved four-dimensional spacetime can be transplanted into the study of ordinary matter in motion.

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.