Metric recovery theorem

The metric recovery theorem proven by Malament is perhaps the most important theorem for understanding actual space and time, and the relation between order and metric in spacetime. Basically, general relativity describes the structure of spacetime using the metric tensor, which is a four dimensional symmetric bilinear form. This means that the matrix form of the metric tensor is defined by ten points at each point in spacetime. The metric tensor used in general relativity is different from a typical metric used in classical models of spacetime, the euclidean metric, because it carries with it a causal order structure on its points as well.

As the metric tensor used in special and general relativity has a causal factor associated with it, it is useful to consider which relativistic spacetimes can be recovered from a causal ordering relation. The metric recovery theorem basically demonstrates that the metric tensor can be recovered from the causal ordering up to a single conformal scale factor. So nine of the ten distinct terms of the metric tensor can be recovered. The only thing remaining is the conformal factor.

The basic issue of how to construct the conformal factor then is the most important thing that remains. Remember from order topology, that a given partially ordered set can have different topologies associated with it: discrete, scattered, or continuous. One aspect of a discrete partially ordered set, is that concepts like measure and volume come for free from it, and they don't need to be defined separately. It has thus been concluded that spacetime could be modeled as a discrete partial order, with the conformal factor being volume, and then there is no need for any extra structure. This is the program of causal set theory.

It has been said that spacetime looks awfully a lot like a discrete partially ordered set. Rather it is one or another is a separate issue, it certainly looks like one. On the other hand, one can attach a conformal scale factor to a continuous partially ordered in order to allow for an effective continuous model of spacetime. This leads to the metric tensor. In either case, spacetime can be modeled as a partial order. It is either a discrete partial order or a continuous partial order which is equipped with a conformal scale factor.

References: Malament, David B. (July 1977). "The class of continuous timelike curves determines the topology of spacetime". Journal of Mathematical Physics.