Spacetime geometry from causal structure and a measurement

The causal structure of spacetime defines a partial order on the events of spacetime. In an earlier paper, using techniques from domain theory, we showed that for globally hyperbolic spacetimes one could reconstruct the topology from the causal structure. However, the causal structure determines the metric only up to a local rescaling (a conformal transformation); in a four dimensional spacetime, the metric tensor has ten components, and thus effectively only nine are determined by the causal structure. After establishing the relationship between measurement in domain theory, the concept of global time function and the Lorentz distance, we are able to domain theoretically recover the final tenth component of the metric tensor, thereby obtaining causal reconstruction of not only the topology of spacetime, but also its geometry.