Relativistic Effects on Clocks Aboard GPS Satellites

For example, see D.C. Giancoli, Physics for Scientists and Engineers, 3rd ed. (Prentice Hall, Upper Saddle River, NJ, 2000), Chap. 37.

The relativistic kinetic energy of a body is defined as $K ≡ E − E0,$ where E₀ is the body's rest energy. In the case of a photon, $E0 = 0$ so that E = K.

I.R. Kenyon, General Relativity (Oxford Univ. Press, Oxford, 1990), Chap. 2. These approximate methods of deriving Eq. (2) agree with an exact general relativistic treatment, presented in an accessible manner in E.F. Taylor and J.A. Wheeler, “Project A: Global Positioning System,” in Exploring Black Holes: Introduction to General Relativity (Addison-Wesley, San Francisco, 2000), pp. A-1–A-9.

E. Huggins, “GPS satellites and Lagrangians,” presented at the AAPT Summer Meeting, Madison, WI, 2003. Equation (3) implies that the action per unit mass for an orbiting clock is equal to the time lost multiplied by the speed of light squared.

N.

 

Ashby

, “,”  , – (May ).

Also see N. Ashby, “Relativity in the Global Positioning System,” Living Reviews in Relativity 6 (Jan. 2003), online at http://www.livingreviews.org/lrr-2003-1.

A.

 

Harvey

and 

E.

 

Schucking

, “,”  , – (March ).

Also see

S. P.

Drake

, “,” , – (Jan. ).

Specifically, in Earth's rotating frame of reference the total surface potential is computed as follows. The gravitational potential difference in Eq. (5) is gy for small altitudes y above polar sea level. The centripetal acceleration of an object revolving with angular speed ω at a distance r from the axis is ω²r. (For the Earth, ω = 2π/24 h.) In a rotating frame, this can be treated as an outward centrifugal acceleration. Its integral corresponds to a centrifugal potential of $−12ω2r2 = −12υ2.$ If we now require that the total potential $gy − υ2/2 = 0$ everywhere on Earth's surface, then Eqs. (1) and (5) cancel each other's effects. [Note, however, that $ω2rE2/2g = 11 km$ is half the height of Earth's actual equatorial bulge, as discussed in standard texts such as D.L. Turcotte and G. Schubert, Geodynamics, 2nd ed. (Cambridge Univ. Press, Cambridge, 2002), Chap. 5. In fact, an equatorial clock needs to be raised beyond 11 km to compensate for the gravitational field from the mass of the bulge.]