Comparison To The Continuum
Given a causal set we may ask whether it can be embedded into a Lorentzian manifold. An embedding would be a map taking elements of the causal set into points in the manifold such that the order relation of the causal set matches the causal ordering of the manifold. A further criterion is needed however before the embedding is suitable. If, on average, the number of causal set elements mapped into a region of the manifold is proportional to the volume of the region then the embedding is said to be faithful. In this case we can consider the causal set to be 'manifold-like'
A central conjecture to the causal set programme is that the same causal set cannot be faithfully embedded into two spacetimes that are not similar on large scales. This is called the hauptvermutung, meaning 'fundamental conjecture'. It is difficult to define this conjecture precisely because it is difficult to decide when two spacetimes are 'similar on large scales'.
Modelling spacetime as a causal set would require us to restrict attention to those causal sets that are 'manifold-like'. Given a causal set this is a difficult property to determine.
Read more about this topic: Causal Sets
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