The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature

Abstract

We prove a synthetic Bonnet-Myers rigidity theorem for globally hyperbolic Lorentzian length spaces with global curvature bounded below by K < 0 and an open distance realizer of length L = π √ |K| . In the course of the proof, we show that the space necessarily is a warped product with warping function cos : (− π 2 , π 2 ) → R + .

Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds

“…We also introduced a diversity of metric tools, and thereby furthering the development of the theory considerably. Moreover, these tools and methods have further applications in Lorentzian geometry and general relativity: The most striking example is the recently established splitting theorem for Lorentzian length spaces with non‐negative curvature in [9]. The splitting theorem can be understood as a rigidity statement of the classical Hawking–Penrose singularity theorem [28] and as such is of fundamental importance in Einstein's theory of gravity.…”

On curvature bounds in Lorentzian length spaces

“…We intend to form an implication circle, so the converse implication is proven later. In that proof, a technical detail requires us to assume the triangle inequality of angles as displayed in (8), which was achieved in [9, Theorem 4.5(ii)] using geodesic prolongation, cf. [9,Definition 4.2].…”

“…[5,11,12]) has had on the field of Riemannian geometry, after its introduction in [24] the theory has quickly branched out from Lorentzian Alexandrov geometry (e.g. [4,8,9]) into a variety of fields, in particular into optimal transport and metric measure geometry (e.g. [10,15,27]), causality theory (e.g.…”

Gluing of Lorentzian length spaces and the causal ladder

“…The same holds for causal pasts and futures. 5 This follows by [6, remark 3.1.7]: points outside of A form their own equivalence class, so these 'trivial' equivalences can be immediately cut out via the transitivity of the relations. 6 By ∪ x 1 ≪a 1 I + 2 (a 2 ) we mean the union of I + 2 (a 2 ) over all a 1 ∈ I + 1 (x 1 ) ∩ A 1 .…”

“…In addition to a monotonicity formulation of curvature bounds, the authors also introduce a formulation relating the size of angles and their comparison angles, similar to the classical angle condition in metric geometry, cf [9, definition 4.1.5]. (vii) [5] formulates and proves a splitting theorem for (globally hyperbolic) Lorentzian length spaces.…”