Lorentzian Geometry Outside of General Relativity: an Application to Airline Boarding

Two-dimensional Lorentzian geometry has recently found application in some models of nonrelativistic systems, most profitably for the process of boarding an aeroplane. The duration of the boarding process is assumed to be a result of passengers standing in the isle and blocking each others’ way. According to this model, the expected boarding time is equal to the length of the longest timelike geodesic on a certain Lorentzian manifold. Although the asymptotic approach used is valid only for large numbers of passengers, it is useful in comparing the effectiveness of various airline policies such as asking passengers with seats in the back half of the plane to board first. To explain the ideas involved, we first treat a related but simpler question, stemming from the idea of the longest increasing subsequence of a permutation. A description of the boarding model is followed by its translation into Lorentzian geometry and a sample calculation. Finally, we discuss the validity and practical implications of the model. Many of the theorems we will quote have probabilistic statements, which we will not attempt to prove because the methods involved are not relevant to general relativity. Instead we will often provide a heuristic argument.