24-07-2012, 04:29 PM
Global Positioning System
Global Positioning.pdf (Size: 36.41 KB / Downloads: 105)
Operation of the GPS
Do you think that general relativity concerns only events far from common
experience? Think again! Your life may be saved by a hand-held
receiver that “listens” to overhead satellites, the system telling you where
you are at any place on Earth. In this project you will show that this system
would be useless without corrections provided by general relativity.
The Global Positioning System (GPS) includes 24 satellites, in circular
orbits around Earth with orbital period of 12 hours, distributed in six
orbital planes equally spaced in angle. Each satellite carries an operating
atomic clock (along with several backup clocks) and emits timed signals
that include a code telling its location. By analyzing signals from at least
four of these satellites, a receiver on the surface of Earth with a built-in
microprocessor can display the location of the receiver (latitude, longitude,
and altitude). Consumer receivers are the approximate size of a
hand-held calculator, cost a few hundred dollars, and provide a position
accurate to 100 meters or so. Military versions decode the signal to provide
position readings that are more accurate—the exact accuracy a
military secret. GPS satellites are gradually revolutionizing driving, flying,
hiking, exploring, rescuing, and map making.
Stationary Clocks
Earth rotates and is not perfectly spherical, so, strictly speaking, the
Schwarzschild metric does not describe spacetime above Earth’s surface.
But Earth rotates slowly and the Schwarzschild metric is a good approximation
for purposes of analyzing the Global Positioning System.
Clock Velocities
Now we need to take into account the velocities of Earth and satellite
clocks to apply the more complete equation [3] to our GPS analysis. What
are the values of the clock velocities vEarth and vsatellite in this equation,
and who measures these velocities? For the present we find the simplest
measure of these velocities, using speeds calculated from Euclidean geometry
and Newtonian mechanics. Newton uses a fictional “universal” time
t, so Newtonian results will have to be checked later in a more careful
analysis.
Justifying the Approximations
We calculated the speed of a satellite in circular orbit and the speed of the
clock on Earth’s surface using Euclidean geometry and Newtonian
mechanics with its “universal time.” Now, the numerator in each expression
for speed, namely rdφ, is the same for Euclidean geometry as for
Schwarzschild geometry because of the way we defined r in Schwarzschild
spacetime. However, the time dt in the denominator of the speed is
not the same for Newton as for Schwarzschild. In particular, the derivation
of equation [3] assumes that the speeds in that equation are to be
calculated using changes in far-away time dt. Think of a spherical shell
constructed at the radius of the satellite orbit and another “shell” that is
the surface of Earth.
Summary
A junior traveler, making her first trip by train from the United States into
Mexico, sees the town of Zacatecas outside her window and reassures herself
by the marginal note in the guidebook that this ancient silver-mining
town is 1848 kilometers from San Diego, California, and 1506 kilometers
from New Orleans, Louisiana. On a surface, two distances thus suffice to
fix location. But in space it is three. Find those three distances, to each of
three nearest satellites of the Global Positioning System, by finding the
time taken by light or radio pulse to come from each satellite to us. Simple
enough! Or simple as soon as we correct, as we must and as we have, for
the clock rates at each end of the signal path. (1) General relativity predicts
that both the relative altitudes and the relative speeds of satellite and
Earth clocks affect their relative rates.
Global Positioning.pdf (Size: 36.41 KB / Downloads: 105)
Operation of the GPS
Do you think that general relativity concerns only events far from common
experience? Think again! Your life may be saved by a hand-held
receiver that “listens” to overhead satellites, the system telling you where
you are at any place on Earth. In this project you will show that this system
would be useless without corrections provided by general relativity.
The Global Positioning System (GPS) includes 24 satellites, in circular
orbits around Earth with orbital period of 12 hours, distributed in six
orbital planes equally spaced in angle. Each satellite carries an operating
atomic clock (along with several backup clocks) and emits timed signals
that include a code telling its location. By analyzing signals from at least
four of these satellites, a receiver on the surface of Earth with a built-in
microprocessor can display the location of the receiver (latitude, longitude,
and altitude). Consumer receivers are the approximate size of a
hand-held calculator, cost a few hundred dollars, and provide a position
accurate to 100 meters or so. Military versions decode the signal to provide
position readings that are more accurate—the exact accuracy a
military secret. GPS satellites are gradually revolutionizing driving, flying,
hiking, exploring, rescuing, and map making.
Stationary Clocks
Earth rotates and is not perfectly spherical, so, strictly speaking, the
Schwarzschild metric does not describe spacetime above Earth’s surface.
But Earth rotates slowly and the Schwarzschild metric is a good approximation
for purposes of analyzing the Global Positioning System.
Clock Velocities
Now we need to take into account the velocities of Earth and satellite
clocks to apply the more complete equation [3] to our GPS analysis. What
are the values of the clock velocities vEarth and vsatellite in this equation,
and who measures these velocities? For the present we find the simplest
measure of these velocities, using speeds calculated from Euclidean geometry
and Newtonian mechanics. Newton uses a fictional “universal” time
t, so Newtonian results will have to be checked later in a more careful
analysis.
Justifying the Approximations
We calculated the speed of a satellite in circular orbit and the speed of the
clock on Earth’s surface using Euclidean geometry and Newtonian
mechanics with its “universal time.” Now, the numerator in each expression
for speed, namely rdφ, is the same for Euclidean geometry as for
Schwarzschild geometry because of the way we defined r in Schwarzschild
spacetime. However, the time dt in the denominator of the speed is
not the same for Newton as for Schwarzschild. In particular, the derivation
of equation [3] assumes that the speeds in that equation are to be
calculated using changes in far-away time dt. Think of a spherical shell
constructed at the radius of the satellite orbit and another “shell” that is
the surface of Earth.
Summary
A junior traveler, making her first trip by train from the United States into
Mexico, sees the town of Zacatecas outside her window and reassures herself
by the marginal note in the guidebook that this ancient silver-mining
town is 1848 kilometers from San Diego, California, and 1506 kilometers
from New Orleans, Louisiana. On a surface, two distances thus suffice to
fix location. But in space it is three. Find those three distances, to each of
three nearest satellites of the Global Positioning System, by finding the
time taken by light or radio pulse to come from each satellite to us. Simple
enough! Or simple as soon as we correct, as we must and as we have, for
the clock rates at each end of the signal path. (1) General relativity predicts
that both the relative altitudes and the relative speeds of satellite and
Earth clocks affect their relative rates.