Neil Ashby
Slides from Ashby's tutorial talk,
"Relativity in the Global Positioning System," are given here in six web pages. In the narrative below you can
click on any subject to go to its
page. Click here for a full
PDF version.
**pg. 1** Ashby showed
how the
Global Positioning System is an excellent context
for teaching relativity. The fundamental
principles of relativity are central to a working GPS; an
understanding of reference frames, including
accelerating frames and the principle of
equivalence, is necessary to understand GPS; dealing with the
relativity of simultaneity is a central
problem of operating a GPS.
**pg. 2** Ashby
shows ways to present to students these basic ideas:
**pg. 3 **Combining
the equivalence principle
with these ideas Ashby shows that a clock will have its frequency
shifted in a gravitational field (gravitational
redshift), and, therefore, two identical clocks at different heights
will run at different rates. How big will be the difference in
rates of a clock on Earth and a clock in orbit? Big enough to
cause a navigational error of 13
km in a day if it is not corrected for.
**pg. 4 **Currently,
clock frequencies can be stable
to parts in 10^{14}
over a week. In one day an error of 10^{-14}
would correspond to an uncertainty of
.26 m in position.
With such precise timing GPS
can use the constancy of *c*
and the accurately known positions
of orbiting satellites to determine the distance **|r-r**_{j}|
of the GPS receiver from the j^{th}
satellite. Four independent measurements
suffice to locate the receiver
on the surface of Earth to within a meter.
**pg. 5** An
orbiting clock shows the effects of both the
time dilation
predicted by special relativity,
and the gravitational redshift predicted by general relativity. These
tend to cancel. Time
dilation in the GPS is negative, arising because the orbiting the
satellite is moving relative to the receiver on the ground. The effect
of gravitational red shift is
positive because the satellite sits in a weaker gravitational field than
the receiver.
**pg. 6 **On
Earth's geoid the two effects
cancel. A clock at the pole runs faster than at the equator because the
velocity of the surface is less at the pole. But because Earth is
oblate, a clock at the pole is closer to Earth's center than at the
equator and so sits in a stronger gravity field and runs more slowly by
just the amount to offset time dilation.
For a clock in an orbiting
satellite the two effects only partially cancel.
A change in orbit changes the rate
at which the satellite clock runs. A change of 20 km changes the
frequency by 1.88 parts in 10^{13}. Whenever a satellite's
orbit is changed, the clock rate is corrected accordingly.
GPS illustrates many
principles of GR and special relativity without new mathematics.
To explore further, however, you will need to
introduce the metric.
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