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Contents
02
About Perimeter Institute
03
Introduction
04
Curriculum Links & Suggested Ways
to Use this Resource
05
06
07
08
09
10
Student Activities
Worksheet 01: GPS Activities
Worksheet 02: G
PS Simulation
Worksheet 03: E
nergy in Satellites
Worksheet 04: Relativity
Frames of Reference and General Relativity: Demonstrations
Worksheet 05: General Relativity
11
12
13
15
17
20
Supplementary Information
What is the Global Positioning System?
How Does the GPS Work?
Special Relativity and the GPS
General Relativity and the GPS
Worksheet Solutions
Who Are the People in the Video? Credits
About Perimeter Institute
Perimeter Institute
Canada’s Perimeter Institute for Theoretical Physics is an
independent, non-profit, scientific research and educational
outreach organization where international scientists gather
to push the limits of our understanding of physical laws and
explore new ideas about the very essence of space, time,
matter, and information. The award-winning research centre
provides a multi-disciplinary environment to foster research
in the areas of Cosmology, Particle Physics, Quantum
Foundations, Quantum Gravity, Quantum Information,
Superstring Theory, and related areas. The Institute,
located in Waterloo, Ontario, also provides a wide array of
educational outreach activities for students, teachers, and
members of the general public in order to share the joy of
scientific research, discovery, and innovation. Additional
information can be found online at www.perimeterinstitute.ca
Perimeter Inspirations
This series of in-class educational resources is designed to
help teachers inspire their students by sharing the mystery
and power of science. Perimeter Inspirations is the product
of extensive collaboration between experienced teachers,
Perimeter Institute's outreach staff and international
researchers. Each module has been designed with both
junior and senior high school youth in mind and has been
throughly tested in classrooms.
About Your Host
Dr. Damian Pope is Senior Manager of Scientific
Outreach at Perimeter Institute for Theoretical
Physics, Waterloo, Ontario, Canada. He holds
a PhD in theoretical physics from the University
of Queensland, Australia in the area of quantum
physics. He also has extensive experience in
explaining the wonders of physics to people of
all ages and from all walks of life.
02
Introduction
The Global Positioning
System (GPS)
is a modern technology that has changed the face
of navigation forever. Based on a network of over 30
satellites, it broadcasts signals that anyone with a GPS
receiver can use to pinpoint their location on Earth
to within a few metres.
The GPS has a vast range of applications. Every day
it is used by a diverse group of people including drivers,
pilots, surveyors, construction workers, farmers, couriers,
hikers, and many others. Economically, it plays a critical
role in industries that are worth billions of dollars annually.
A key feature of the GPS is the fact that the signals it
broadcasts include incredibly precise timing information.
To facilitate this, each GPS satellite houses an atomic
clock capable of measuring time to within a fraction of a
nanosecond. The GPS is so precise it must take into account
the effects of Einstein’s theory of relativity- both special
relativity and general relativity.
Special relativity tells us that ‘moving clocks run slow’.
The faster an object moves relative to us, the slower we
see its time passing. General relativity says that gravity
slows down time. The closer an object is to a large mass,
the slower time passes. Together, these two effects mean
that clocks inside GPS satellites run faster than clocks in
GPS receivers on Earth. If not corrected, this would lead to
timing errors that would result in GPS measurements rapidly
accumulating errors. These errors would build up at the rate
of 11 km per day. So, far from being purely abstract and
detached from reality, Einstein’s theory of relativity is critical
to the operation of an extremely practical technology.
This resource uses the GPS as a vehicle to introduce
students to Einstein’s theory of relativity and includes a fiveminute in-class video and an accompanying teacher’s guide.
It also touches on a number of other topics including satellite
motion, geometry, energy, and precision. We trust that you
find it a useful addition to your science and physics courses.
03
Curriculum
Links
Suggested Ways
to Use this Resource
Intermediate Science (Grades 9 and 10)
Topic
Connection
to the GPS
Relevant
Materials
Process of
science &
innovation
Science involves the discovery
of new ideas that are used to
create new technologies.
Worksheet 1
Science and
society
The GPS is used by millions of
people every day in thousands
of applications.
Worksheet 1
Science and
Numbers
The ability to comprehend and
manipulate numbers is useful for
understanding how the GPS works.
Worksheet 1
Space
GPS satellites orbit in space
20 200 km above Earth.
Worksheet 1
Senior Physics
Topic
04
Connection
to the GPS
Relevant
Materials
Uniform
circular
motion
GPS satellites move
in circular orbits.
Worksheet 3:
Question 1
Universal
gravitation
GPS satellites are held in orbit
by the gravitational force between
them and Earth.
Worksheet 3:
Questions 1, 2
Momentum
and energy
The orbit of a GPS satellite
dictates the values of its
kinetic and gravitational
potential energy.
Worksheet 3:
Questions 1, 2
Special
relativity
The GPS is so precise that it
must take into account the
effects of time dilation.
Worksheet 3:
Question 4
Worksheet 4
General
relativity
The GPS is so precise that it
must take into account the
effects of general relativity.
Worksheet 3:
Question 4
Worksheet 4:
Question 3
Worksheet 5
Measurement
The GPS uses a large number
of significant figures in its
calculations of distance to
achieve its high degree of accuracy.
Worksheet 3:
Question 2
Worksheet 4:
Question 1
Frames of
reference
The effect of gravity on time
can be predicted by considering
accelerated frames.
Worksheet 5:
Questions 1-6
Kinematics
The GPS uses the equation
d = vt to locate your position.
Worksheet 2
This flexible resource includes a classroom
video and five student worksheets.
The worksheets are provided in editable
electronic form so that you can modify
them as you wish.
Intermediate Science (Grades 9 and 10)
Outline for a Single Period
Video (5 minutes)
Worksheet 1 (15-60 minutes)
Senior Physics
Outline for a Single Period: Satellite Motion
Worksheet 2: GPS Simulation (45-60 minutes)
Video (5 minutes)
Homework: Worksheet 3: Energy in Satellites
Outline for a Single Period: Special Relativity
Worksheet 2: GPS Simulation (45-60 minutes)
Video (5 minutes)
Homework: Worksheet 4: Relativity
Student Activities
Worksheet 1: GPS activities — Series of activities
suitable for intermediate students
Worksheet 2: GPS simulation — Hands-on activity
that models how the GPS works.
Worksheet 3: Energy in Satellites — Problems
related to satellite motion and energy
Worksheet 4: Relativity — Problems on special
and general relativity
Worksheet 5: General Relativity — Demonstrations
and concept questions
Student Activities
Worksheet 1:
GPS Activities
Below are four activities that you can do with an intermediate science class to explore the ideas presented in the video.
Each activity is independent of the others and takes 10 to 15 minutes. You can choose to do any number of them.
1. Discussion Questions
Discuss the following questions with a partner and write
down your answers.
What is the GPS?
Who uses the GPS?
How does the GPS work?
2. Researching Applications of the GPS
After watching the video, form small groups and make a list
of five different uses of the GPS. Next, create a three-column
chart as shown below. Write a brief description of each GPS
use from your list. Then describe a few of the advantages
GPS offers over traditional methods. For example:
use
Description
Advantages
Farming spraying
pesticide
on crop
fields
Farmers use the
GPS to more
accurately spray
chemicals on their
crops.
The GPS allows farmers
to be much more
efficient and accurate
because: i) chemicals
are sprayed exactly
where they are needed.
ii) no area is sprayed
twice. iii) the amount of
chemicals deposited on
the crops is accurate as
both location and speed
of travel are known
precisely.
3. Number Crunching
The video mentions a lot of numbers. The following questions
are designed to help give meaning to these numbers.
a) GPS satellites orbit at a height of 20 200 km above
Earth’s surface. The radius of Earth is 6370 km.
Break into small groups and draw large scale diagrams
of the GPS orbits and Earth.
b) GPS satellites move at a speed of 14 000 km/h.
How does this compare to a car? A jet? Two times faster?
Twenty times? Two hundred times? Two thousand times?
c) The speed of light is 300 000 000 m/s. How long does it
take for a radio signal, traveling at the speed of light,
to get from a GPS satellite to Earth’s surface?
d) How long does it take light to travel the length
of a 30 cm ruler?
4. Exploring how the GPS locates objects
Background
The Global Positioning System is a navigational tool that
locates your position by sending distance information from
GPS satellites to GPS receivers.
Overview
You will be given some information about how far you are
from three cities and your goal is to find your location.
Materials
Compasses, maps of Canada
Instructions FOR STUDENTS
1.You are located 975 km from Vancouver.
Using a compass, draw a circle on the map that is centred
on Vancouver and has a radius of 975 km.
2.You are also located 2716 km from Toronto.
Draw a circle on the map that is centred on Toronto
and has a radius of 2716 km.
3.You are also located 3028 km from Montreal.
Draw a circle on the map that is centred on Montreal
and has a radius of 3028 km.
4.The three circles should intersect at a single point.
Mark their intersection with a cross. This is the city at
which you are located. Write down the name of this city.
Discussion
1.If you only knew how far you were from Vancouver,
at which points on the map could you be located?
Label these points on the map.
2. If you only knew how far you were from Vancouver
and Toronto, at which points on the map could you
be located? Label these points on the map.
3. The GPS uses information from four satellites.
What extra information does using this many
satellites provide?
05
Student Activities
Worksheet 2:
GPS Simulation
Background
The Global Positioning System (GPS) consists of more than 30
satellites orbiting around the Earth at an altitude of 20 200 km.
The satellites send out signals in the microwave band that
contain information about where the satellite is and what time
the transmission is sent. GPS receivers use this information
to determine how far a receiver is from the satellite. The
receiver can only determine its location if it can obtain signals
from four different satellites at the same time.
Overview
In this activity, you will work in groups of three to model how
the GPS works. Your first task is to generate a GPS message
to describe where a receiver is located. Your second task is
to use messages from another group to locate their receiver.
(Note: Only three messages are needed because we are
working in two dimensions not three dimensions).
Materials
• calculators
• rulers
• compasses
• 1 map for each group
(preferably at least 1:10 000 000 scale)
1. Work in a group. Choose a major city on the map for your
GPS receiver. Write the location on a slip of paper and give
it to your teacher.
2. Each student in the group represents a GPS satellite.
Choose a different location on the map that your satellite
will be passing over at the time the message is sent to the
GPS receiver (City X in figure below).
3. Calculate the distance dsignal that your signal must travel
to reach the receiver using Pythagoras’ Theorem. Two of
the sides of the triangle in the figure below are the altitude
of the satellite (20 200 km) and the distance dground from
the receiver to the point directly below the satellite (City X).
Measure dground remembering to take the scale of the map
into account.
Satellite
dsignal
GPS Receiver
06
20 200 km
dground
City X
4. Now you have found the signal distance, determine the
transit time ∆t for the signal (to three significant digits),
using the speed of light c = 3.00 x 108 m/s and
5. Compose a message as follows: “Satellite 1 is 20 200 km
above city X (the location on the map) and this signal has
taken ∆t to reach you”.
6. Once your group has produced messages for all three
satellites, leave them with the map and go to another
group’s map. Use the information you find to locate
the other group’s receiver.
(i) Use the time given to calculate the distance each
signal travelled.
(ii) Use Pythagoras’ Theorem to determine the distance
from the point on the ground directly below the satellite
(i.e. City X) to the receiver’s location.
(iii) Draw a circle with the appropriate radius, beneath
each satellite location.
(iv) The receiver is located where the three circles intersect.
Discussion
1. Why can the transit time never be less than 0.067 s?
2. List some possible errors in the GPS calculations that
have not been taken into account.
3. A standard GPS receiver is accurate to within about 10 m.
How many significant digits does the GPS need to use in
the value of dsignal to achieve this accuracy? What value for
c must be used in the calculations to obtain this level of
precision?
Student Activities
Worksheet 3:
Energy in Satellites
Useful equations:
v=
c = 2.997 924 58 x 108 m/s
mearth= 5.97 x 1024 kg
d
G = 6.67 x 10-11 N·m2/(kg2)
rearth = 6.37 x106 m
QUESTION 1
Each GPS satellite orbits at a distance of 2.66 x 107 m from
Earth’s centre at a speed of 3.874 x 103 m/s. Each satellite
has a mass of 2.0 x 103 kg.
a) What is the kinetic energy of a GPS satellite?
b) What is the gravitational potential energy of
a GPS satellite?
c) What is the total energy of a GPS satellite?
QUESTION 2
Each GPS satellite is launched from Earth’s surface
by rocket. How much work must the rocket do on the
satellite so that it reaches the height of its orbit?
QUESTION 3
The GPS calculates the distance d from a GPS satellite to a
receiver by multiplying the speed c of a GPS signal by the
time ∆t the signal takes to travel from satellite to receiver:
c) Use your answers to a) and b) to explain why the GPS
needs to use atomic clocks accurate to at least 10-9 s,
instead of regular quartz clocks accurate to only 10-6 s.
QUESTION 4
Einstein's theories of special relativity and general relativity
have opposing effects on time in the GPS. Einstein’s theory
of special relativity states that the clocks inside GPS satellites
run slower than a stationary clock on Earth by 8.3 x 10-11 s per
second. This is due to the speed of the satellites. Einstein’s
theory of general relativity says the satellite clocks also run
faster than those on Earth by 5.2 x 10-10 s per second because
Earth’s gravity is weaker at the satellites’ altitude.
a) How much slower does a GPS clock run each day due to
special relativity? How much faster does it run each day
due to general relativity?
b) GPS satellites emit signals that travel at the speed of
light c. Any timing error in the GPS translates into
a distance error equal to:
d = c ∆t
Let ∆t = 0.068 503 387 s
a) Calculate d using i) all of the digits in ∆t and c and ii) rounding off ∆t and c to three significant figures.
b) What is the difference between your answers to
parts i) and ii)?
Distance Error = c ∆terr
Calculate the daily distance error from both special
and general relativity.
c) Calculate the difference between these two distances to
find the overall distance error per day from relativity.
07
Student Activities
Worksheet 4:
Relativity
Useful equations:
v = d/t G = 6.67 x 10-11 N·m2/(kg2)
c = 2.997 924 58 x 108 m/s
QUESTION 1
GPS satellites send time signals to GPS receivers.
A receiver gets a signal that reads t1 = 9:00:27.723 119 038
(i.e. 9 am and 27.723 119 038 seconds). The signal is received
at t2 = 9:00:27.790 249 045 according to the receiver.
a) How long did it take the signal to travel from
the satellite to the receiver?
b) How far is the receiver from the satellite?
QUESTION 2
In special relativity, the relationship between the time
elapsed for a GPS satellite clock and a clock on Earth is
where the term v2/(2c2) is the rate at which a GPS clock runs
slowly from the perspective of someone on Earth.
a) Calculate v2/(2c2) for a GPS satellite.
b) How slowly does someone on Earth see a GPS satellite
running over the course of a day?
QUESTION 3
In addition to the timing error in the GPS from special
relativity, GPS satellite clocks run faster by 5.2 x 10–10 s
per second due to effects from general relativity.
a) How large a timing error does this correspond to over the
course of a day?
b) How large is the daily distance error from the combined
effects of special and general relativity?
where ∆t is the time elapsed according to the satellite clock,
∆t’ is the time elapsed on Earth and v is the satellite’s speed
relative to Earth.
GPS satellites move at v = 3.874 x 103 m/s. When v is much
less than c, the relationship is well approximated by
08
QUESTION 4
A friend says “The theory of relativity is just that, a theory.
It’s not real, unlike Newton’s laws of motion which are laws.”
Is your friend correct? Explain why or why not using the
information you have learned about the effects of relativity
on the GPS.
Student Activities
Frames of Reference
and General Relativity
Demonstration 1: Water in Freefall
Demonstration 2: Swinging Tray
This demonstration illustrates one half of the equivalence
principle — the fact that the experience of being in freefall in
a constant gravitational field is like being in zero gravity.
This demonstration illustrates the other half of the equivalence
principle — the fact that acceleration can mimic many features
of gravity.
Take a plastic bottle of any size. Drill a ½ cm hole in the cap.
Using a thumbtack, make a second hole on the side of the
bottle near the top. (This is to ensure there is enough air
pressure inside the bottle to let the water come out freely.)
Take a cafeteria tray and tape four 1.5 m lengths of strong
string underneath as shown in Figure 2. Bring the eight ends
around and tie them in a knot. Grasp the knot. Place a cup
of water on the tray and start moving the tray back and forth,
steadily increasing the amplitude until you swing it all the
way around vertically and then horizontally! The cup of water
stays on the tray.
Fill the bottle up to the second hole. When you hold the
bottle upside down it should leak noticeably. Show the leaky
bottle to your students and have them thoroughly discuss
Question 1 on Worksheet 5. Then drop the bottle. Next,
have students discuss Question 2. It tends to cause
more disagreement.
After your students have shared their thoughts, demonstrate
what happens by repeatedly tossing the bottle straight up
for about a metre and then catching it. Draw your students’
attention to the water on the way up, at the top and on the
way down. The water only pours out when the bottle is in
your hands. It does not leak out while the bottle is in motion
because both the water and the bottle are accelerating
downwards at exactly the same rate due to gravity (9.8 m/s2).
Have students discuss the explanation for this phenomenon
via Question 3. It looks as if there is an outward force holding
the glass onto the tray – but there is no such force. The
explanation in an inertial frame of reference is that the normal
force acting on the glass is pushing inwards and keeping the
glass accelerating in a circle. However, from the perspective
of the glass, the acceleration feels like a gravitational force
pulling it towards the bottom of the tray.
String
String wrapping
underneath
Pinhole
Hole in cap
Tape
underneath tray
Figure 2 String attached to tray
Figure 1 Water bottle
Figure 3 Glass upside down on tray
09
Student Activities
Worksheet 5:
General Relativity
1. A plastic water bottle has a small hole in the cap. It is
turned upside down and dropped with the hole uncovered.
What happens to the water as the bottle falls?
a) It pours out at the same rate as when the
bottle is stationary. b) It pours out more slowly than when the bottle is stationary.
c) It pours out faster than when the bottle is stationary.
d) It stays in the bottle.
2. A plastic water bottle has a small hole in the cap and is
turned upside down. It is thrown upwards with the hole
uncovered. What happens to the water while the bottle is
in the air?
a)
b)
c)
d)
It pours out on both the way up and down.
It stays in the bottle.
It pours out only on the way up.
It pours out only on the way down.
3. A cup of water is on a tray. The tray is swung rapidly in a
horizontal circle. The water stays in the cup and the cup
stays on the tray because there is a large acceleration
a) inwards which resembles a large gravitational
field outwards.
b) outwards which resembles a large gravitational
field outwards.
c) inwards which resembles a large gravitational
field inwards.
d) outwards which resembles a large gravitational
field inwards.
5. Figure 2 shows the positions of the same rocket when it
is accelerating upwards. How often does Alice receive the
pulses now?
a) every 100 ns
b) more frequently than every 100 ns
c) less frequently than every 100 ns
6. The rocket is stationary in a constant gravitational field.
How often does Alice receive the pulses in this situation?
a) every 100 ns
b) more frequently than every 100 ns
c) less frequently than every 100 ns
7. General relativity states that the ratio of the times elapsed
for Alice and Bob is tA/ tB = 1 – (g∆h)/c2, where g = 9.8 m/s2,
∆h is the rocket’s height and c is the speed of light.
In 1960, an experiment was done to test this relationship.
Pulses of light were sent from the top floor of a building
to the basement, a distance of 20 m. How large is the
relativistic effect (g∆h)/c2 in this case?
a)
b)
c)
c)
2 x 10–9
2 x 10 –12
2 x 10–15
2 x 10 –18
5
4
3
2
1
6
5
4
3
2
4. Figure 1 shows the positions of a rocket every 100 ns,
going from left to right. The rocket is moving up at a
constant velocity. Bob is at the rear and sends pulses of
light towards Alice in the nose of the rocket every 100 ns.
How often does Alice receive the pulses?
1
Figure 1 Rocket moving with constant velocity
5
a) every 100 ns
b) more frequently than every 100 ns
c) less frequently than every 100 ns
4
3
2
1
5
4
3
2
1
Figure 2 Rocket moving with constant acceleration.
010
Student Activities
What is the
Global Positioning System?
The Global Positioning System (GPS) is a network of
more than 30 satellites operated by the US Air Force that
broadcasts signals accessible to anyone with a GPS receiver.
The satellites are orbiting Earth at an altitude of 20 200 km
and a speed of 3874 m/s. The satellites follow one of six
orbital planes inclined to the equator by 55o and separated
around the equator by 60 o. This pattern ensures that at least
four satellites are visible from any point on Earth’s surface at
any one time.
The master control station for the GPS is at the Schriever
Air Force Base in Colorado Springs, USA. This station,
along with several other bases around the world, tracks the
satellites and provides regular updates to the information in
the radio signals broadcast by the satellites.
GPS receivers are essentially sophisticated radios tuned to
1575.42 MHz, which decipher and compare the signals from
several satellites to determine where the receiver is located.
The original purpose for the GPS was to provide a navigational
aid for the US military. As the system developed it became
evident that it would also be very useful for civilian purposes.
In 1983, the accidental shooting down of a commercial
passenger plane, which had strayed from its scheduled
path by mistake into restricted air space over the Soviet
Union, prompted then US President Ronald Reagan to make
GPS available to everyone. Since the system became fully
functional in 1995, thousands of applications have
been developed.
GPS is not just used for navigation. It has become a
standard tool for surveyors, builders, and farmers. When
combined with a transmitter it is a powerful tracking device
used by hospitals, police, and wildlife biologists. GPS signals
can also be used to generate very precise timestamps, which
are used by cell-phone networks, financial institutions, and
computer companies to establish highly accurate transaction
times. The overall economic impact of the GPS runs into
billions of dollars annually.
011
Supplementary Information
How Does
the GPS Work?
Figure 4
Each GPS satellite broadcasts a continuous signal containing
information about where the satellite is (ephemeris), what
time it is (a timestamp), and the general health of the system
(almanac). The receiver uses this signal to estimate how far
away it is from the transmitting satellite by multiplying the
signal’s travel time by the speed at which the signals travel
― the speed of light. The receiver is located on the surface
of an imaginary sphere centred on the satellite (see Figure 4).
The receiver repeats this process for three more satellites
to produce four overlapping spheres. Using a geometrical
method called trilateration the receiver determines its
location as the unique point where the surfaces of the four
imaginary spheres intersect (see Figure 4). Three spheres are
needed to determine the location, and the fourth signal is
needed to establish the precise time used in the calculations.
With four signals the receiver is able to determine its position
within a few metres.
To achieve this degree of accuracy, the GPS must use
very precise information. The speed of light used in the
calculations is c = 299 792 458 m/s and the timestamps
must be accurate to within 20 to 30 ns. To generate
these timestamps, each GPS satellite contains several
atomic clocks that generate very precise, highly stable
measurements of time. Atomic clocks use the oscillation
of electrons rather than a pendulum or a piece of quartz to
012
keep track of time. The standard clock uses cesium-133,
which has one valence electron that can be excited so that
it undergoes a transition with a very specific energy and
frequency. This transition is used to produce a resonant
vibration at 9 192 631 770 Hz which is extremely sensitive to
variations in frequency and thus produces a highly accurate
measurement of time. This resonant frequency is now used
as the global basis for the definition of a second.
The incredible precision required by the GPS makes it very
sensitive to error. The scientists and engineers who operate
the system have to take into account many sources of error.
The atomic clocks have to be adjusted to compensate
for relativity (both special and general relativity). The
slight variations in the satellite orbits caused by the tiny
gravitational pulls of the Sun and Moon have to be corrected
regularly. The satellite signals travel through the various
layers of the atmosphere, each with a slightly different index
of refraction. The signals also bounce off objects such
as mountains and buildings and can follow various paths
before reaching the receiver. There are also rounding errors
introduced by the receiver’s processor during calculations.
All of these errors must be taken into account to ensure that
the GPS can locate objects to within a few metres, even
though the satellites are orbiting over 20 000 km away.
Supplementary Information
Special Relativity
and the GPS
Introduction to Special Relativity
Special relativity is a theory of how motion affects
measurements of length and time. Developed by Einstein
in 1905, it is based on two postulates:
1. The speed of light in a vacuum is c = 2.997 924 58 x 108 m/s
in all inertial reference frames. All observers in these frames
measure light as travelling at this speed independent of
their own speed relative to the light’s source.
2. The laws of physics are the same in all inertial
reference frames.
Everything in special relativity can be derived from these
two statements.
Time Dilation
One of the most significant consequences of the postulates
is that “moving clocks run slow”. Stated more accurately,
special relativity says an observer in an inertial reference
frame sees a clock that is moving relative to them as
running slow. They observe that less time elapses
according to such a clock than one in their own frame. This
phenomenon is called time dilation and is governed by
the following equation:
(1)
where ∆t is the amount of time elapsed for the observer, ∆t
is the amount of time elapsed for the moving clock, v is the
clock’s speed relative to the observer, and c is the speed
of light.
Equation 1 shows that the observer sees the clock as slowed
by an amount that depends on the clock’s relative speed v.
Since the factor containing v appears frequently in special
relativity, Equation 1 is written as
where
(3)
The GPS satellites move at 3.874 km/s relative to Earth, a
speed that is 0.0013% of the speed of light. Calculating γ
from Equation 3, we get γ =1.000 000 000 0 83. This means
that for every second we observe as passing for a GPS
satellite clock, we observe 1.000 000 000 0 83 s passing on
Earth’s surface. This represents a slowing down of time by
8.3 x 10–11 s per second. Although this deviation is extremely
small, it has a critical impact on the operation of the GPS.
Note that because the value of γ for GPS satellites is so
close to one, most classroom calculators will return the
answer ‘1’. See the answer to Worksheet 4, Question 2 on
page 18 for information on how to get around this issue.
Because GPS satellite clocks run slow by 8.3 x 10–11 s per
second in our reference frame, they gradually fall behind the
clocks in the GPS receivers. Over the course of a day, the
amount they fall behind is (8.3 x 10–11 s)(24)(60)(60) = 7.2 μs
If this difference was not corrected for, GPS satellite clocks
would become unsynchronized from GPS receiver clocks.
This would mean that the receivers would measure the travel
time ∆t for signals inaccurately. In turn, this would result in
errors in GPS distance measurements equal to
d = c ∆t = (2.997 295 x 108 m/s) (7.2 x 10–6 s) = 2.1 km
So, the effect of time dilation would cause an error of more
than 2 km per day, something that would render the
GPS useless.
(2)
013
Supplementary Information
Frequently Asked Questions
Q - At different positions in its orbit, a GPS satellite will
have differing speeds relative to different GPS receivers.
Given this, do we need to adjust the speed used in the
equation for time dilation to account for this variation?
A - In principle, we do need to use a different value for v
in Equation 1 depending on the precise speed of a given
satellite relative to a particular receiver. However, the speed
of the satellites (3874 m/s) is much larger than the speed
of a GPS receiver as it moves with Earth’s rotation (465
m/s at the equator). Differences in the values of the relative
speed between a satellite and a receiver result in variations
in the amount of time dilation of just 1% at most and so are
insignificant for the current accuracy of the GPS.
Q - GPS satellites are in orbit and so are accelerating.
They are not in inertial reference frames. Similarly, GPS
receivers are accelerating due to Earth’s rotation and so
are also not in inertial frames. Given this, how can we
use special relativity, which primarily deals with inertial
frames, to calculate the amount of time dilation?
A - The reason we can use this theory is that the acceleration
of GPS receivers (0.034 m/s2) is so small that we can ignore
it. Over the course of one second, the acceleration changes
each receiver’s speed by just 0.034 m/s. For a receiver at
the equator, this is just 0.007% of its speed due to Earth’s
rotation. So, the effect the acceleration has on the amount
of time dilation is at most only about 0.007% of the total
value per day of 7 μs. This corresponds to just 0.0005 μs,
a negligible effect.
Imagine a second object with the same velocity but which
is not accelerating (see Figure 5). This object is in an inertial
frame and so, using Equation 1, we can calculate that we
see its clock running slow by 8.3 x 10-11 s per second. The
GPS satellite shares the same instantaneous motion and so
we will also see its clock running slow by the same amount.
In the next instant, the satellite clock shares the same motion
as a third object moving at 3874 km/s in a slightly different
inertial frame. So, its clock runs slow by the same amount as
in the previous instant.
Continuing this process over the satellite’s entire orbit, we
find that the satellite’s clock runs slow by 8.3 x 10-11 s per
second throughout its orbit. We can use special relativity at
each instant of the satellite’s motion and then add up all of
the amounts of time dilation to calculate the total amount.
Even though the satellite is accelerating, by comparing
it to other objects in inertial frames moving at the same
instantaneous speeds, we can use special relativity to
determine how slowly its clock runs.
Object in
inertial frame
v = 3.874 km/s
v = 3.874 km/s
a=0
a
Earth
Approximating GPS receivers as being in inertial frames, a
GPS satellite moves at a speed of 3874 m/s relative to this
frame. At each moment in time, it has an instantaneous
velocity of 3874 m/s along its orbit.
Figure 5
014
GPS
Satellite
Supplementary Information
General Relativity
and the GPS
Why does gravity slow down time?
One of the key ideas in the video is that gravity slows down
time. That is, clocks in a stronger gravititional field run slower
than clocks in weaker gravitational field. For the GPS, this
means that the clocks on Earth run slower than those in the
satellites by 45 μs per day. At first glance, gravity and time
seem to be two unrelated phenomena. Why would gravity
affect time? Einstein discovered this connection by considering
frames of reference.
In 1905, Einstein published his theory of special relativity which
deals with frames of reference moving at constant velocity. It
took him another ten years to extend this theory so it could
handle accelerating frames and gravity—the theory of general
relativity. His first breakthrough was the equivalence principle,
which he described as “the happiest thought of my life.” This
principle says that the laws of physics in a reference frame in
freefall are equivalent to those in a frame with no gravity.
This can be seen in Figure 6. In Figure 6a the frame is far
from Earth or any other large mass. In Figure 6b the frame is
in freefall. In both frames of reference Alice feels weightless
and her ball appears to her not to fall down. The equivalence
principle applies to all situations where a frame is freely falling
under gravity. An object in orbit is also in freefall and this is
why the astronauts on the International Space Station feel
weightless even though the gravitational field on the station is
more than 9 N/kg.
Einstein next considered what it would be like in a frame
accelerating in the absence of a gravitational force. In
Figure 7a, Alice is once again way out in space far from
Earth or any other large mass, but this time she is in a frame
accelerating at 9.8 m/s2 relative to her original frame in
Figure 6a. It feels like being on Earth’s surface (Figure 7b).
The ball appears to fall down and she feels a normal force
pushing up from the floor.
The equivalence principle tells us that any phenomenon that
occurs in a frame with constant acceleration also happens in
a frame with a constant gravitational field. Einstein used this to
predict that gravity slows time.
Consider two people, Alice and Bob, in a rocket. Bob is at the
rear of the rocket and Alice is at the front. Bob sends Alice
signals every 100 ns, which is about how long it takes light
to travel 30 m. If the rocket moves with a constant velocity
(see Figure 8) Alice receives each pulse at constant intervals
Earth
Figure 6a Alice and the ball are
weightless because there are no
large masses nearby.
Figure 6b Alice feels weightless and
the ball appears to her not to fall
Earth
Figure 7a Alice feels heavy and the
ball appears to fall down because
the frame is accelerating up.
Figure 7b Alice feels heavy and
the ball appears to fall down
because of the Earth’s gravity
and so receives the pulses 100 ns apart. However, if the rocket
is accelerating upwards (see Figure 9) the flight time of each
pulse is longer because Alice is moving away from successive
pulses at a greater speed.
Because Alice receives the pulses less frequently when the
rocket accelerates, the equivalence principle says she will
also receive them less frequently when the rocket is parked
in a gravitational field. Once again, Bob is sending the pulses
every 100 ns and Alice receives them separated by more than
that. Alice infers that Bob’s time is running slower than hers.
Conversely, if Alice sends pulses 100 ns apart to Bob, he will
receive them more frequently and he will say that her time is
running fast. This result is very different from the time dilation of
special relativity where if Alice and Bob are moving relative to
each either, they both say that the other’s time is passing more
slowly. This is one place where general relativity is easier to
comprehend than special relativity.
015
Supplementary Information
5
4
3
Substituting the values rE = 6.4 x 106 m and rsat = 2.64 x 107 m
into the equation, we get
2
1
2
6
5
4
3
2
1
Figure 8 Rocket moving with constant velocity
5
4
3
2
1
5
4
3
2
1
Figure 9 Rocket moving with constant acceleration.
This slowing down of time is known as gravitational time
dilation and the ratio of the times in a gravitational field is
given by
where ∆tR is the time elapsed for the receiver, ∆tS is the
time elapsed at the source, ∆Φ is the gravitational potential
difference between the source and the receiver, and c is the
speed of light.
For Alice and Bob sitting in a rocket on the Earth, the
gravitational potential difference, ∆Φ, can be approximated by
gΔh, where g is 9.8 N/kg and Δh is their vertical separation.
Thus the ratio of times for Alice (tA) and Bob (tB) will be given by
For the GPS, the ratio of the satellite time tsat and Earth-based
reciver time tE is given by
where rE and rsat are, respectively, the radius of Earth’s surface
and the satellite's orbit, G = 6.67 x 10-11 Nm2/(kg2) is the
universal gravitational constant, and M = 5.97 x 1024 kg is
Earth’s mass.
016
So, due to gravitational time dilation, for every second that
passes for a clock inside a GPS receiver on Earth, 1.000 000
000 52 s passes for a clock inside a GPS satellite. That is, the
satellite clocks run faster by 5.2 x 10-10 seconds each second.
Multiplying this result by the number of seconds per day
(24 x 3600 = 86 400), we arrive at the result in the video that
gravitational time dilation makes GPS satellite clocks run fast
by (86 400) (5.2 x 10-10) = 45 x 10-6 s per day.
Another Way to Understand Why
Gravity Slows Down Time
There is another very different way to derive gravitational time
dilation. It involves the conservation of energy, special relativity,
and quantum mechanics. Consider one of the photons sent up
by Bob.
As the photon rises, there is an increase in gravitational
potential energy. In order to conserve total energy, the photon
must lose energy. Light always travels at speed c, so it can’t
lose energy by slowing down. A photon’s energy is given by
E = hf (where h is Planck’s constant, 6.626 x 10–34 J·s and f
is the photon's frequency) and the only way the photon can
lose energy is by decreasing its frequency. This decrease
in frequency to conserve energy is exactly the same as the
change derived from the equivalence principle.
Earth
Figure 10 The only way the photon can lose energy is by decreasing its frequency.
More about General Relativity
It is important to realize that the equivalence principle is only
one part of the story behind general relativity. For the full story
including black holes, gravitational lenses, and gravitational
waves you also need the second key idea — that mass causes
space and time (spacetime) to curve. However, the equivalence
principle alone is responsible for around 99% of the 45 μs per
day time difference between the receiver and satellite clocks in
the GPS. This is because spacetime is only very slightly curved
near the Earth.
Supplementary Information
Worksheet
Solutions
Worksheet 1: GPS Activities
3. Number Crunching
a) This is an opportunity to examine ratios. The satellite's
height is roughly three times Earth's radius, so the
satellites should be four times as far away from Earth’s
centre as the surface is.
b) Cars typically move at speeds from 40 to 100 km/h.
If you take an average of 70 km/h, then GPS satellites
move 200 times faster. Commercial jets move at
around 700 km/h; the satellites are 20 times faster.
c) GPS satellites orbit 20 200 km above Earth’s surface.
Using the equation t = d/v we get
t = (20 200 000 m)/(300 000 000 m/s) = 0.067 s
d) This is an opportunity to use powers of ten. A ruler is
30 cm = 0.30 m long. The time needed is 0.30/300 000
000 = 1 billionth of a second = 1 nanosecond.
4. Exploring How The GPS Locates Your Position
The DVD contains an electronic copy of a map of Canada
you can print and use for this activity.
The city students are located in is Calgary. It is 975 km
from Vancouver, 2716 km from Toronto and 3028 km
from Montreal.
Discussion
1. Anywhere on the edge of the circle centred on
Vancouver with a radius of 975 km.
2. Either of the two points where the circles centred on
Vancouver and Toronto intersect.
3. Altitude; this activity is two-dimensional and involved
determining your location on Earth's surface. Actual
GPS navigation is three-dimensional and the GPS
determines your altitude as well.
Worksheet 2: GPS Simulation
This activity works best when students use a large map of a
sizeable country, a continent or the entire world. However, if
none of these maps are available, you can also use world map
on 8.5" x 11" paper.
To find the distance from Calgary to Thunder Bay:
To find the distance the signal travels:
To find the time taken for the signal to reach the receiver:
Discussion Questions
1. The satellites are in orbit 20 200 km above Earth's surface
so the shortest possible distance for the signal to travel is
20 200 km which would take 0.0675 s.
2. There are many errors that GPS must account for
to produce accurate results, including atmospheric
refraction, reflections off building and mountains, orbital
shifts, clock errors, and rounding errors. In this activity
we have ignored the curvature of the Earth and did not
account for the satellite position with much precision.
3. GPS satellites are in orbit 20 200 km above Earth's surface
which means the signal must travel at least
2 × 107 m. To determine the location to within 10 m, the
GPS must use at least seven significant digits in the
calculations (2.000001 × 107 m). The value for c used
by the GPS in its calculations is 299 792 458 m/s.
Worksheet 3: Energy in Satellites 874
1.a)
The kinetic energy of a GPS satellite is 1.5 × 1010 J.
b)
Sample calculations
Assume the receiver is located at Calgary, which is 4.5
cm from Thunder Bay on a 1:52 000 000 scale map. The
message will be: “Satellite 1 is 20 200 km above Thunder
Bay and this signal has taken 0.0679 s to reach you”.
66
3.0
017
Supplementary Information
The gravitational potential energy of a GPS satellite is
3.0 × 1010 J.
c)
ET = EK + EG
= 1.5 × 1010 J – 3.0 × 1010 J
Because of this, the GPS needs more accurate clocks
and uses atomic clocks onboard each GPS satellite.
4. a) 7.2 μs slow due to special relativity and 45 μs
fast due to general relativity
b) special relativity: 2.1 km; general relativity: 13 km
c) 11 km
ET = –1.5 × 1010 J
Worksheet 4: Relativity
The total energy of a GPS satellite is –1.5 × 1010 J.
2. To get to a GPS satellite to the height of its orbit, the
rocket needs to increase the satellite's gravitational
potential energy
Let rS be the radius of Earth’s surface and rS be the orbital
radius of a GPS satellite. The difference in gravitational
potential energy for a GPS satellite in orbit and on Earth’s
surface is
1. a) 27.790 249 045 s – 27.723 119 038 s = 0.067 130 007 s
The signal took 0.067 130 007 s
b) d = c (t2 – t1)
= (2.997 924 58 × 108 m/s) (0.067 130 007 s)
= 20 125 069.8 m
The receiver is 20 125 069.8 m from the satellite.
34
2. a)
We can derive the equation
37
66
as follows:
The binomial theorem implies that
ΔEG = 9.5 × 1010 J
Applying this to the equation
So, 9.5 x 1010 J of work needs to be done on the satellite.
3. a) i) d = c ∆t
= (2.997 924 580 × 108 m/s) (0.068 503 387 s)
=20 536 798.8 m
ii) d = c ∆t
= (3.00 × 108 m/s) (0.0685 s)
= 2.055 × 107 m
b) The difference between answers to i) and ii) is
13 201.2 m or 13.2012 km
c) If the GPS used quartz clocks, measurements of ∆t
would only be accurate to the sixth significant figure
in d, which corresponds to hundreds of metres.
So, the GPS would only be accurate to about 100 m.
This level of accuracy is not great enough for many
GPS applications (such as navigation in cars).
yields
as v2/c2 << 1
018
Supplementary Information
b) There are (3600) (24) = 86 400 seconds per day.
(8.34 × 10-11) (86 400 s) = 7.2 ×10-6 s So, the timing
error from special relativity each day is 7.2 ×10-6 s
3. a) (5.2 × 10-10) (86 400 s) = 4.5 × 10-5 s So, the timing error
from general relativity each day is 4.5 × 10-5 s
b) daily overall timing error = 45 μs - 7.2 μs
= 38 μs
distance error = 11 km
4. A good answer to this question would include the point
that the GPS makes necessary corrections for the effects
of relativity and that it would not work without doing so.
Worksheet 5: General Relativity
1. d) The water stays in the bottle.
2. b) The water stays in the bottle. Most students correctly
predict that the water stays in on the way down, but
they think it will come out on the way up. They are
confusing the concepts of velocity and acceleration.
While in freefall (going up or going down) the bottle is
accelerating at 9.8 m/s2 down along with the water. This
is a demonstration of the first half of the equivalence
principle — the fact that physics in freefall is like being
in zero gravity — and it shows why astronauts in orbit
(another form of freefall) feel weightless.
3. a) There is a large acceleration inwards which resembles
a large gravitational field outwards. Students are often
confused about the direction of the acceleration. Most
of them have experienced centrifugal forces, and these
fictional forces feel more real to them than centripetal
forces. This is a demonstration of the second half of
the equivalence principle ― the fact that acceleration
in one direction resembles a gravitational field in the
opposite direction. It also shows how artificial gravity
can be created in space stations.
019
4. a) Alice receives the pulses every 100 ns. They are slightly
delayed because the rocket is moving. However, they
are all delayed by the same amount so the intervals
between them will be identical.
5. c) Alice receives the pulses less frequently than every
100 ns. Draw the students’ attention to differences
between Figures 1 and 2. Alice is moving away
faster from the pulses in Figure 2 and this produces
a Doppler shift. The greater the acceleration of
the rocket, the greater the change in frequency. To
increase student engagement, you may wish to have
your students give their answers by clapping their
hands. Clapping once a second can represent a), really
rapid clapping is b) and really slow clapping is c).
6. c) Alice receives the pulses less frequently than every
100 ns. The equivalence principle says this situation is
like the accelerating rocket. Bob sends signals every
100 ns according to his time, but Alice receives them
at greater intervals and concludes that Bob’s time is
running slow. If Alice sent signals to Bob, he would
conclude that her time is running fast. Notice that this
is different from how time dilation in special relativity
works where each person concludes the other person’s
time is running slow.
7. c) The relativistic effect is the part that differs from one,
g∆h/c2 = (9.8 m/s2) (20 m)/(9 x 1016 m2/s2) ≈ 2 x 10-15.
The experiment mentioned in the question was done
by physicists Pound and Rebka at Harvard University.
It was an incredibly difficult experiment to perform
but their results were within 1% of the predictions of
general relativity. Given the miniscule size of the result
2 × 10-15, it is tempting to think this effect has no
practical consequences. However, the GPS shows
the effect is indeed important.
Supplementary Information
Who Are the People
in the Video?
LATHAM BOYLE
Faculty Member, Perimeter Institute
Boyle is a cosmologist who specializes in
studying gravitational waves and the early
universe. He undertook his graduate studies
at Princeton University.
CLIFF BURGESS
Professor, McMaster University
Associate Faculty Member, Perimeter Institute,
Burgess is a physicist with a broad range of
interests including string theory, cosmology
and particle physics. He also has a passionate
interest in outreach.
JORGE ESCOBEDO
Graduate Student, Perimeter Institute
Escobedo is doing research in string theory
under Professor Rob Myers. One of his main
areas of interest is gauge/gravity dualities.
GHAZAL GESHNIZJANI
Postdoctoral Researcher, Perimeter Institute
Geshnizjani works in the field of cosmology
and focuses on researching the early universe.
She obtained her PhD from Brown University in 2005.
Credits
AUTHOR TEAM
Damian Pope
Perimeter Institute for Theoretical Physics
Dave Fish
Sir John A Macdonald Secondary School
Waterloo, Ontario
Roberta Tevlin
Danforth Collegiate and Technical Institute
Toronto, Ontario
Tim Langford
Newtonbrook Secondary School
Toronto, Ontario
SCIENCE ADVISORS
Professor Cliff Burgess
McMaster University
Perimeter Institute for Theoretical Physics
Bruno Hartmann
Perimeter Institute for Theoretical Physics
GRAPHIC DESIGNERS
Andrea Sweet
Senior Communications Specialist
Perimeter Institute for Theoretical Physics
Jeff Watkins
EDITORS
Julia Hubble
Susan Fish
3D IMAGES
Steve Kelly
SPECIAL THANKS TO
James Ball
John F. Ross Collegiate Vocational Institute
Guelph, Ontario
Darlene Fitzner
Ernest Manning High School
Calgary, Alberta
SEAN GRYB
Perimeter Institute for Theoretical Physics
Doris LaChance
Ecole Beausjour
Plamodon, Alberta
EDUCATIONAL PRODUCER
Damian Pope
Perimeter Institute for Theoretical Physics
Lisa Lim-Cole
Uxbridge Secondary School
Uxbridge, Ontario
EXECUTIVE PRODUCERS
Greg Dick
General Manager of Outreach
Perimeter Institute for Theoretical Physics
Dennis Mercier
Turner Fenton Secondary School
Brampton, Ontario
John Matlock
Director of External Relations and Communications
Perimeter Institute for Theoretical Physics
Sean Bradley
Redhand Productions
020
Christine Nichols
Castle View High School
Castle Rock, Colorado, USA
Duncan Smith
Bishop Grandin Senior High School
Calgary, Alberta
Richard Taylor
Merivale High School
Ottawa, Ontario
John Wright
Wanganui Collegiate School
Wanganui, New Zealand