Relativity in the Global Positioning System

Abstract

The Global Positioning System (GPS) uses accurate, stable atomic clocks in satellites and on the ground to provide world-wide position and time determination. These clocks have gravitational and motional frequency shifts which are so large that, without carefully accounting for numerous relativistic effects, the system would not work. This paper discusses the conceptual basis, founded on special and general relativity, for navigation using GPS. Relativistic principles and effects which must be considered include the constancy of the speed of light, the equivalence principle, the Sagnac effect, time dilation, gravitational frequency shifts, and relativity of synchronization. Experimental tests of relativity obtained with a GPS receiver aboard the TOPEX/POSEIDON satellite will be discussed. Recently frequency jumps arising from satellite orbit adjustments have been identified as relativistic effects. These will be explained and some interesting applications of GPS will be discussed.

Interpretation of Mössbauer experiment in a rotating system: A new proof for general relativity

“…Following [9] let us consider a transformation from an inertial frame, in which the space-time is Minkowskian, to a rotating frame of reference. Using cylindrical coordinates, the line element in the starting inertial frame is [9] …”

“…The transformation to a frame of reference {t ′ , r ′ , φ ′ z ′ } rotating at the uniform angular rate ω with respect to the starting inertial frame is given by [9] …”

“…To compute this second contribution we use eq. (10) of [9] which represents the proper time increment dτ on the moving clock having radial coordinate r ′ for values v ≪ c…”

“…Indeed, if one simply considers the Schwarzschild gravitational redshift but neglects the effect of the Earth's rotation, GPS would not work [15]! In fact, following Chapters 3 and 4 of [9] in order to address the problem of clock synchronization within the GPS one starts from an approximate solution of Einstein's field equations in isotropic coordinates in a locally inertial, non-rotating, freely falling coordinate system with origin at the earth's center of mass, i.e. eq.…”

“…where M E is the earth's mass, J 2 the earth's quadrupole moment coefficient, a 1 the earth's equatorial radius and P 2 the Legendre polynomial of degree 2 [9]. Usually, one retains only terms of first order in the small quantity V c 2 in eq.…”