Volume 21: Pages 57-64, 2008
Properties of geodesics: Resolving an apparent conflict of Global Positioning System evidence with general relativity
M. H. Brill 1,2,3,4, R. R. Hatch 1,2,3,4, T. E. Phipps Jr. 1,2,3,4, T. C. Van Flandern 1,2,3,4
1Datacolor, 5 Princess Road, Lawrenceville, New Jersey 08648, USA
2NavCom Technology, Inc., 105 West Torrence Blvd., Suite 108, Redondo Beach, California 90277, USA
3908 South Busey Avenue, Urbana, Illinois 61801, USA
4994 Woolsey Court, Sequim, Washington 98382–5058, USA
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.
Keywords: Global Positioning System, GPS, General Relativity, Special Relativity, Geodesic, Orbit, Timekeeping, Proper Time
Received: October 17, 2007; Accepted: April 6, 2008; Published Online: December 15, 2008
