Math 2.0

Relativity and gravitational theory Definition

Causal sets

Builds on Partial orderLocally finite posetCausal structurePoisson point process

Definition

A causal set is a set CC with a binary relation ⪯\preceq such that:

  1. ⪯\preceq is a partial order: reflexive, antisymmetric and transitive;
  2. ⪯\preceq is locally finite: for all x,z∈Cx, z \in C the interval {y∈C:x⪯y⪯z}\set{y \in C : x \preceq y \preceq z} is finite.

So a causal set is a locally finite poset. Let (M,g)(M, g) be a Lorentzian manifold and let ϕ ⁣:C→M\phi \colon C \to M be an injective map. The map ϕ\phi is a faithful embedding if two conditions hold. First, x⪯yx \preceq y exactly when ϕ(x)\phi(x) is in the causal past of ϕ(y)\phi(y). Second, the image looks like a Poisson sprinkling of density ρ\rho: each region of volume VV contains about ρV\rho V points.

Remarks

Example: take a Poisson sprinkling of density ρ\rho into a bounded region of Minkowski space, with the order "lies in the causal past of". This is a causal set, since every interval lies in a compact causal diamond, which almost surely contains finitely many points. Non-example: the set of all points of Minkowski space with the causal order is a partial order but not locally finite, since every nontrivial interval is uncountable. The use of Poisson sprinkling makes the discreteness compatible with Lorentz invariance in a statistical sense. The central conjecture of the programme (the Hauptvermutung) is that a causal set faithfully embeds into two spacetimes only if they are approximately isometric at scales large compared to ρ−1/4\rho^{-1/4}.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Causal structure, concept, no evidence levelCausal structureconceptLocally finite poset, concept, no evidence levelLocally finite posetconceptPartial order, concept, no evidence levelPartial orderconceptPoisson point process, concept, no evidence levelPoisson point processconceptCausal sets, definition, no evidence levelCausal setsdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Causal structure concept
  2. Locally finite poset concept
  3. Partial order concept
  4. Poisson point process concept
  5. Causal sets definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Causal_sets
About this page
TagH8JC4PE6Z5
URIm2://H8JC4PE6Z5
Kinddefinition
WritingW0 stub
MSC83C45, 06A06
DefinesCausal sets
UsesPartial order, Locally finite poset, Causal structure, Poisson point process
Changed3 November 2026

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