Kinematic Time Shift Calculation If the kinematic time dilation expression is expanded in a binomial expansion, then for small velocities it becomes This expression can be used to compute the time dilation in the Hafele-Keating experiment in which an atomic clock was taken aboard an aircraft and compared to a ground-based closk. The problem encountered with measuring the difference between a surface clock and one on an aircraft is that neither location is really an inertial frame. If we take the center of the earth as an approximation to an inertial frame, then we can compute the difference between a surface clock and the aircraft clock. Taking a "proper time" at the earth's center as if the master clock were there, the time measured by a clock on the surface would be larger and that for the airborne clock would be approximately since to the level of the approximations used, the height of the aircraft does not significantly change the radius R. The difference in the times compared to our hypothetical master clock would then be Now this relationship is just the reverse of the actual experiment, since we have assumed that the clock is at the center of the earth, whereas the actual clocks are in the frames which are moving with respect to the center. The time difference expression should be valid, but in comparing the aircraft clock to the surface clock, we should find that it has fallen behind, so we can model that time difference by Application Note that the "earth center" time has been replaced by the surface time in this expression. This is a valid approximation in this case since the time difference is many orders of magnitude smaller than the time itself, and this allows us to model the difference between two measurable times.