Manifold-Topology from K-Causal Order

Abstract

To a significant extent, the metrical and topological properties of spacetime can be described purely order-theoretically. The K + relation has proven to be useful for this purpose, and one could wonder whether it could serve as the primary causal order from which everything else would follow. In that direction, we prove, by defining a suitable order-theoretic boundary of K + (p), that in a K-causal spacetime, the manifold-topology can be recovered from K + . We also state a conjecture on how the chronological relation I + could be defined directly in terms of K + .

Steps towards Lorentzian quantum gravity with causal sets

“…Therefore one could indeed expect that the causal relations together with the number-to-volume correspondence could be sufficient to recover a sufficiently coarse-grained geometry from a causal set. Explicit examples that information on continuum geometry can be reconstructed explicitly include the spacetime dimension [11][12][13][14], timelike and spacelike distances [15], the spatial topology [16][17][18], the scalar d'Alembertian [19][20][21] and the scalar curvature in arbitrary dimensions [22,23].…”