The 'network geometry' approach in network science has in recent years had success in describing complex network topologies using simple geometric models. Previous work has focussed on using Riemannian spaces such as flat Euclidean space or curved Hyperbolic space to describe network structure. Here, the geometry of Lorentzian spacetime is used to model and describe the structure of a special class of networks, directed acyclic graphs. These networks share important features, such as causal structure, with this geometry making this approach a natural extension of the network geometry programme. By recognising the relationship between these networks and this family of geometries, techniques from physical theories of discrete spacetime may be brought into the domain of network science allowing new methods, models, and analyses to be developed. Using network datasets which form directed acyclic graphs, primarily citation networks, as illustrations, I show how characterising a network by its causal structure aids traditional analysis, how networks can be associated with spacetimes of a specific dimension and curvature, and how they may be embedded in a spacetime. Numerous applications are discussed relevant to both citation networks and directed acyclic graphs more generally, and computational implementations of the methods discussed are made available.