Properties of Geodesics: Resolving an Apparent Conflict of Global Positioning System Evidence with General Relativity
Keywords: Global Positioning System, GPS, General Relativity, Special Relativity, Geodesic, Orbit, Timekeeping, Proper Time
A gedanken experiment is described that exposes an apparent conflict between the treatment of proper timekeeping on geodesics according to general relativity theory, as customarily understood, and empirical evidence such as that of the Global Positioning System. The paradox is resolved by noting that there may be many geodesics between two spacetime events, only one of which represents a global maximum of proper time. The cardinality of such nonuniqueness (which may be that of the continuum) at first seems to violate the property that a geodesic between two events always incurs a (local) extremum of proper time. However, to first order (hence to observationally significant order), all free-fall orbits that have the same period have the same proper time, so no first variations of the orbits within our solution set change the proper time?a consistency check on the geodesic (extremum) interpretation of such orbits.