The brachistochrone problem in a stationary space‐time

In a stationary space‐time the brachistochrone problem can be formulated in two different ways, viz., to find the path of shortest travel time with prescribed specific energy from one space‐point to another (i) measured in terms of proper time or (ii) measured in terms of the time coordinate distinguished by the stationarity assumption. It is shown that in the static case both brachistochrone problems can be reduced to geodesic problems of appropriate Riemannian three‐metrics, in close analogy to the brachistochrone problem in a Newtonian potential. In the stationary but nonstatic case, however, this is true only for the proper time brachistochrones, whereas the coordinate time brachistochrones are influenced by a sort of Coriolis force. These results are illustrated by calculating the brachistochrones in Rindler, Schwarzschild, and Gödel space‐times.