page_title: Causal sets
section_title: Geodesics
hierarchical_section_title: Causal sets / Geometry / Geodesics
context_page_description: The causal sets program is an approach to quantum gravity. Its founding principles are that spacetime is fundamentally discrete and that spacetime events are related by a partial order. This partial order has the physical meaning of the causality relations between spacetime events.
  The program is based on a theorem by David Malament that states that if there is a bijective map between two past and future distinguishing space times that preserves their causal structure then the map is a conformal isomorphism. The conformal factor that is left undetermined is related to the volume of regions in the spacetime. This volume factor can be recovered by specifying a volume element for each space time point. The volume of a space time region could then be found by counting the number of points in that region.
  Causal sets was initiated by Rafael Sorkin who continues to be the main proponent of the program. He has coined the slogan "Order + Number = Geometry" to characterize the above argument. The program provides a theory in which space time is fundamentally discrete while retaining local Lorentz invariance.
context_section_description: A link in a causal set is a pair of elements  such that  but with no  such that .
  A chain is a sequence of elements  such that  for . The length of a chain is .
  If every  in the chain form a link, then the chain is called a path.
  We can use this to define the notion of a geodesic between two causal set elements, provided they are order comparable, that is, causally connected (physically, this means they are time-like).  A geodesic between two elements  is a chain consisting only of links such that
   and 
  The length of the chain, , is maximal over all chains from  to .
  In general there can be more than one geodesic between two comparable elements.
  Myrheim first suggested that the length of such a geodesic should be directly proportional to the proper time along a timelike geodesic joining the two spacetime points. Tests of this conjecture have been made using causal sets generated from sprinklings into flat spacetimes. The proportionality has been shown to hold and is conjectured to hold for sprinklings in curved spacetimes too.
caption: A plot of geodesics between two points in a 180-point causal set made by sprinkling into 1+1 dimensions
exif: {}