arXiv subjects

Jona Röhrig

Publications and source records attributed to Jona Röhrig.

5 recordsLinked to original sources

arXiv2026

Ollivier-Ricci curvature for causal sets

We introduce a novel notion of Ollivier-Ricci curvature for causal sets using Lorentzian optimal transport. The construction is motivated by a new Lorentzian asymptotic formula of independent interest, which recovers timelike Ricci curvature, up to higher-order terms, from the transport distance between probability measures on nearby causal diamonds. Passing to the discrete setting, this leads to a mesoscopic notion of Ricci curvature defined along maximal chains and built from probability measures on causal diamonds. We study several variants, including idle and Lin-Lu-Yau type curvatures, prove local-to-global propagation results and timelike Bonnet-Myers theorems, and compute the curvature for a range of explicit causal sets. We design high-density Poisson sprinkling numerical experiments recovering the expected constant-curvature signatures of Minkowski, de Sitter, and anti-de Sitter space. These results provide evidence that the construction captures timelike Ricci curvature from order-theoretic data.

math.DG↗

arXiv2026

Space of Timelike Directions and Curvature Bounds

We investigate the consequences of timelike sectional curvature bounds in Lorentzian length spaces for the existence and structure of the space of directions at a point. It is established that, under upper timelike sectional curvature bounds, the space of directions exists and is itself a metric space with curvature bounded above by $-1$. Furthermore, the metric cone over the space of directions, which canonically models the tangent space at a given point, is shown to constitute a Lorentzian length space with timelike sectional curvature bounded above by $0$. To do this, we introduce the notion of $ε$-$μ$ timelike sectional curvature bounds, which are compatible with pre-existing synthetic curvature conditions. These results extend the comparison-geometric framework to the Lorentzian setting, providing a synthetic characterization of geodesics, tangent cones, and curvature under causal constraints.

math.MG↗

arXiv2026

A Splitting Theorem for non-positively curved Lorentzian spaces

We prove a splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature. Additionally, we extend the first variation formula to spaces with any timelike curvature bound, either from above or below, and different from 0.

math.DG↗

arXiv2025

Non Hilbertian (Lorentzian) Length Spaces

In this note, the idea of finite dimensional $L^p$ spaces is transferred to Lorentzian length spaces to provide an example that is locally nowhere Minkowskian. Looking at the sectional curvature bounds of this example leads to the more general statement that normed spaces in which the norm does not come from an inner product, have no sectional curvature bounds. This statement holds in the Riemannian and Lorentzian cases. In addition, the Lorentzian $L^p$ space can be used as an example in the context of Lorentzian Gromov-Hausdorff convergence, to show that unbounded sectional curvature or geodesic regularity is in general not preserved in the GH limit, and as an example of a sequence of uniform bounded length spaces which are not GH pre-compact.

math.DG↗

arXiv2022

Yang-Mills solutions on Minkowski space via non-compact coset spaces

We find a two-parameter family of solutions of the Yang-Mills equations for gauge group SO(1,3) on Minkowski space by foliating different parts of it with non-compact coset spaces with SO(1,3) isometry. The interior of the lightcone is foliated with hyperbolic space $H^3\cong \text{SO}(1,3)/\text{SO}(3)$, while the exterior of the lightcone employs de Sitter space dS$_3\cong \text{SO}(1,3)/\text{SO}(1,2)$. The lightcone itself is parametrized by SO(1,3)/ISO(2) in a nilpotent fashion. Equivariant reduction of the SO(1,3) Yang-Mills system on the first two coset spaces yields a mechanical system with inverted double-well potential and the foliation parameter serving as an evolution parameter. Its known analytic solutions are periodic or runaway except for the kink. On the lightcone, only the vacuum solution remains. The constructed Yang-Mills field strength is singular across the lightcone and of infinite action due to the noncompact cosets. Its energy-momentum tensor takes a very simple form, with energy density of opposite signs inside and outside the lightcone.

hep-th↗