Constructing an interval of Minkowski space from a causal set

A criticism sometimes made of the causal set quantum gravity program is that there is no practical scheme for identifying manifoldlike causal sets and finding embeddings of them into manifolds. A computational method for constructing an approximate embedding of a small manifoldlike causal set into Minkowski space (or any spacetime that is approximately flat at short scales) is given, and tested in the 2 dimensional case. This method can also be used to determine how manifoldlike a causal set is, and conversely to define scales of manifoldlikeness. In any quantum gravity program in which spacetime is replaced by a discrete structure, it is important to relate continuum approximations to the underlying structures, in order to relate known physics to the new theory (see e.g. [1]). In the causal set program [2], the discrete/continuum correspondence relies on embeddings of the fundamental causal set (or causet – for definitions see [2, 3, 4]) into Lorentzian manifolds, and the concept of sprinkling. A sprinkling is a random selection of points from the manifold made according to a Poisson process – this means that the number of points sprinkled into any region depends only upon the volume V of that region, according to