Lorentz Distances to Cauchy Surface Foliations Can Fail Local Equi-Lipschitzness

Mathematicians have uncovered a subtle but fundamental failure in one of the technical pillars of Lorentzian geometry, with consequences for long-standing conjectures about the structure of the universe. In a new paper published in General Relativity and Gravitation, Gregory J. Galloway of the University of Miami, Robert J. McCann of the University of Toronto, and Argam Ohanyan of the University of Toronto demonstrate that a regularity property known as local equi-Lipschitzness—which is guaranteed for families of Lorentzian distance functions to and from individual points—breaks down in general when one instead considers distances to entire families of Cauchy surfaces. The result, published on 9 September 2026 as Volume 58, article number 105 of the journal, reshapes the technical landscape surrounding Bartnik’s cosmological splitting conjecture, one of the most important open problems linking spacetime geometry to cosmology.

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.