Induced Spatial Geometry from Causal Structure

Astrid Eichhorn, Sumati Surya, Fleur Versteegen

Research output: Contribution to journalJournal articleResearchpeer-review


Motivated by the Hawking-King-McCarthy-Malament (HKMM) theorem and the associated reconstruction of spacetime geometry from its causal structure (M,≺) and local volume element ϵ, we define a one-parameter family of spatial distance functions on a Cauchy hypersurface ς using only (M,≺) and ϵ. The parameter corresponds to a 'mesoscale' cut-off which, when appropriately chosen, provides a distance function which approximates the induced spatial distance function to leading order. This admits a straightforward generalisation to the discrete analogue of a Cauchy hypersurface in a causal set. For causal sets which are approximated by continuum spacetimes, this distance function approaches the continuum induced distance when the mesoscale is much smaller than the scale of the extrinsic curvature of the hypersurface, but much larger than the discreteness scale. We verify these expectations by performing extensive numerical simulations of causal sets which are approximated by simple spacetime regions in 2 and 3 spacetime dimensions.

Original languageEnglish
JournalClassical and Quantum Gravity
Issue number10
Publication statusPublished - 15. Apr 2019


  • causal sets
  • causal structure
  • quantum gravity


Dive into the research topics of 'Induced Spatial Geometry from Causal Structure'. Together they form a unique fingerprint.

Cite this