Gravity, Orbits, and Special Relativity
Objectives:
• The student will compare and contrast the Newtonian view of gravity and
the current view as described by Einstein’s Theory of Special Relativity
• The student will create models to help visualize 4-dimensional space-time,
and the effect of massive objects (such as the Sun) on planetary and other
orbits
• The student will relate how theories may be revised in the light of new
discoveries and observations
Suggested Grade Level:
Ninth - Twelfth
Subject Areas:
Physics
Astronomy
Timeline:
One to two 50 minute periods
Background:
Teacher - This lesson is intended to be presented after students have studied
motion (including circular motion and centripetal acceleration,) Newton’s Laws of
Motion and Newton’s Law of Universal Gravitation.
The current theory of gravity follows from Einstein’s Theory of Special Relativity.
Isaac Newton’s concept of gravity was that it is a force of attraction between two
objects which have mass, acting over a distance. The strength of this force may
be calculated using the equation F=Gm1m2/d2 (Where F=force of gravity, G=the
Universal Gravitational Constant 6.67x10-11 N m2/kg2, m1= mass of object 1 in
kilograms, m2=mass of object 2 in kilograms, d= distance separating the objects
in meters.)
Newton’s great insight was that gravity is a universal force; the same force that
causes objects to fall to the ground applies to all objects in the universe – and
that gravity is responsible for holding the moon in orbit about the earth (or the
planets around the sun) in much the same way as centripetal acceleration keeps
an object moving in a circle.
Newton was unable to explain exactly how gravity works, however – but since his
Law of Gravitation was able to mathematically predict the observed motion of
celestial objects successfully, it was widely accepted and became the foundation
of “Newtonian Physics.”
In the 20th century, Einstein’s Theory of Special Relativity proposed a new
explanation of gravity. Einstein proposed that instead of being separate, that
space and time were linked inseparably in a 4-dimensional manner, forming
“space-time,” and that gravity is a consequence of the effect of an object’s mass
on the curvature of space-time. The activities in this lesson will help students to
visualize how gravity may be explained according to Einstein’s theory.
Students – Know the background of the above elements.
Materials:
ping-pong balls
string
tape
metal shot (such as lead fishing sinkers, or buckshot)
ball-bearings
tennis balls
golf balls
blue Jell-O (clear gelatin and blue food coloring may be substituted)
rectangular or large square glass pans
construction paper
ruler
scissors
clear plastic cube-shaped Ziplock or Glad storage containers
flashlights
notebooks or journals.
Lesson:
1. Vocabulary
Centripetal Acceleration - The acceleration toward the center, that holds a
satellite in elliptical orbit.
Circular Motion - In physics, circular motion is rotation along a circle: a
circular path or a circular orbit. The rotation around a fixed axis of a threedimensional body involves circular motion of its parts.
Space-time - World line of the orbit of the Earth depicted in two spatial
dimensions X and Y (the plane of the Earth orbit) and a time dimension,
usually put as the vertical axis. Note that the orbit of the Earth is an ellipse
in space, but its world line is a helix in space-time.
Tesseract - In geometry, the tesseract is the 4-dimensional analog of the
cube. That is, the tesseract is to the cube as the cube is to the square.
More formally, the tesseract can be described as a regular convex 4polytope with eight cubical cells.
2. Pre-activity:
Materials for this portion: Jell-O, glass pans, beakers for boiling water,
Bunsen burners, hot plates or microwave oven, stirring spoon.
In advance, students working in groups of 3-4 will prepare a rectangular
(or square) mold of Jell-O following the directions listed on the package.
Clear gelatin may be substituted, using 2-3 drops of blue food coloring. (If
there is no access to a refrigerator at school, students may prepare this at
home and bring to school.)
3. Activity 1:
Materials for this portion: tape, ping-pong balls, string, scissors
Working in groups, students will tape a ping-pong ball to a length of string
approximately 2 ft. in length. Have the students whirl the ball in a
horizontal circle; inform the students that the ball models the Earth as it
orbits the Sun.
Have the students answer the following question in their journals: “What
would happen to the motion of the Earth if the Sun suddenly exploded,
and the Sun ceased to exist “?
After making their prediction, have the groups model such an event by
whirling the ping-pong ball as before, and suddenly letting go. (Students
should predict that the Earth’s motion would move off in a straight line
tangent to the circle.)
Pose the following thought question, and have students discuss in groups;
upon reaching a conclusion, have students answer in their journal.
“Einstein’s Theory of General Relativity states that nothing can travel
faster than the speed of light. It takes about 8 minutes for light from the
Sun to reach the Earth. If the Sun exploded, how could the Earth instantly
start moving off in a straight line – if nothing can travel faster than light
speed – before news of the event (in the form of light) reaches the Earth?”
• It is highly unlikely that students will be able to provide an explanation;
accept all answers. (The true answer, according to theory, is that the
Earth would continue to orbit for about 8 minutes, until moving off in a
straight line. Why?)
4. Activity 2:
Materials for this portion: construction paper, tape, scissors, strings,
Ziplock or Glad storage containers, flashlights
Part 1: Explain to students that Einstein resolved this dilemma by
proposing that space and time are linked in a 4-dimensional “fabric” called
“space-time.” To help students accept this abstract notion, have them
answer the following question in their journals after discussing it in groups.
“Suppose you wish to meet with a friend from out-of-town, who is
unfamiliar with your town, tomorrow morning? What would you have to
specify in the way of directions, in order for your friend to meet you at the
coffee shop on the second floor of the town mall?”
Discuss the answers to this question as a class. (Students would need to
specify 3 spatial coordinates – an address; or length, width, height above
the ground – and in addition, the time of the meeting.)
Part 2: While it is impossible to visualize 4-dimensional space, it is
possible to illustrate a 3-dimensional projection of 4-D cube – or
“hypercube,” just as it is possible to draw 3-D cube on a 2-dimensional
plane, such as a chalk board. This projection of 4-D space-time is known
as a “tesseract” (Charles Howard Hinton, British Mathematician 18531907.) The tesseract is to a cube, as a cube is to a square (see
Addendum.)
Have student groups construct a cube using the following pattern
(Addendum.) Turn off the lights, and have a student shine a flashlight on
one of the cubes held before the chalk-board. Then draw a representation
of a cube on the board. Students should have no problem recognizing the
illustration or “projection” as a cube.
Working in groups, have students construct a tesseract. Tape a section of
string (all of equal length) to the corners of their construction paper cube.
Tape the bottom four strings to the bottom inside corners of the Ziplock
storage container. Then tape the top gour strings to the top corners of the
container, and attach the lid.
The smaller cube should be centered within the larger storage container,
and the strings cut to the appropriate length so there is no excess. (See
Addendum)
Inform students that the tesseract is the 3-D representation or “projection”
in 3-D space of a 4-dimensional object (a hypercube.).
Have students answer in their journals: “Do you think there might exist
more than four dimensions? What do you think higher dimensions might
be like if you could visit them?”
5. Activity 3: Gravity
Materials for this portion: ping-pong balls, metal shot, scissors, tape,
tennis balls, golf balls, ball-bearings, marbles, and Jell-O mold (prepared
in advance.)
Have student groups fill a ping-pong ball or tennis ball with some metal
shot so that it is very heavy by cutting a small incision, inserting the metal,
and taping over the hole.
(A large ball bearing or golf ball may be a suitable substitute, but needs to
be very heavy.)
Have students place the ball in the center of their Jell-O mold. Inform the
students that the mold represents space-time, and the ball is the Sun.
Have students note the “warp” or curvature of the Jell-O around the Sun,
similar to the dent a bowling ball would make on a stretched sheet or
trampoline.
Each student in the group will make several attempts, using marbles (or
ball bearings) to roll a marble across the Jell-O so that it passes near the
Sun, and hits a target on the opposite side. Have students note how
variables such as speed, closeness to the Sun, and starting position affect
the outcome.
Explain to students that the path or orbit of an object, according to
Einstein, is a result of it following the curvature of space-time.
Objects (such as planets) have sufficient speed if they are in a circular or
elliptical orbit to continue moving around the “gravity well” of the Sun
without falling in.
Have students answer in their journals: “Why do satellites orbiting the
Earth eventually get closer and closer until they enter the atmosphere and
burn up? Do you think the Earth’s orbit (or other planets’) gets smaller and
smaller over time? What would a space ship need to do in order to escape
from a planet’s gravity well? Scientists say there is no such thing as zerogravity; relate this idea to the curvature of space-time.”
Some experiments have suggested that Einstein’s theory is valid. Further
investigations are currently being conducted in several countries to serve
as a final test of the theory of Special Relativity.
Extensions:
1. Have students watch part 1 of the video “The Elegant Universe” (available
for download at www.pbs.org under the television program “NOVA.”)
2. Have students read the book “Flatland: A Romance of Dimensions” by
Edwin A. Abbott (copy New York: Penguin Books, 1998) and conduct a
whole-class discussion on the book.
3. Have students write a report on Black Holes
4. Have students visit the website for the LIGO (Laser Interferometer Gravitywave Observatory) and show the video (real-audio download) “Einstein’s
Messengers.”
5. Provide students groups with a GPS unit and a set of GPS coordinates;
have them rendezvous at a given location at a specified time to find a set
of clues for a treasure-hunt.
Evaluation/Assessment:
• The student did compare and contrast the Newtonian view of gravity and
the current view as described by Einstein’s Theory of Special Relativity
• The student did create models to help visualize 4-dimensional space-time,
and the effect of massive objects (such as the Sun) on planetary and other
orbits
• The student did relate how theories may be revised in the light of new
discoveries and observations
• Answers to journal entries and discussion questions
• Informal observation of group activities
• Essay responses to critical thinking questions included on formal
assessments on Newton’s Laws and Gravity
Resources:
Greene, B. (Brian) 1963 – “The Elegant Universe: superstrings, hidden
dimensions, and the quest for the ultimate theory/Brian Greene. New York:
Vintage Books, c2003.
PBS: NOVA
www.pbs.org/wbgh/nova/elegant
(copy of the book) Abbott, Edwin A. “Flatland: A Romance of Many Dimensions”
(New York: Penguin Books, 1998 (original version 1884)
Web site Resources:
http://en.wikipedia.org/wiki/Tesseract
http://csep10.phys.utk.edu/astr161/lect/history/newtongrav.html
http://ligo.caltech.edu
http://wwwligo-la.caltech.edu
http://www.exploratorium.edu/
Addendum:
Cube pattern for student construction:
A Tesseract or Hypercube
A hypercube “unfolded” into 3-D space
Animation of the unfolding of a hypercube (tesseract) and a cube
http://en.wikipedia.org/wiki/Image:Hcube_fold.gif