On the Uniqueness of Embeddings of Causal Sets

Abstract

We introduce the notion of a well-conditioned embedding of a causal set into a Lorentzian manifold and prove that if a causal set admits well-conditioned embeddings into two manifolds, then their interiors are related by an $\varepsilon$-approximate isometry. To justify the definition, we show that in the high-density limit a Poisson sprinkling almost surely yields a causal set possessing a well-conditioned embedding. The error $\varepsilon$ is given explicitly and tends to zero in the high-density limit.