TOOLS AND MODELS FOR TEACHING
EINSTEINIAN PHYSICS
This page contains a set of inexpensive resources to aid the teaching of Einsteinian physics. First
we describe the need for special equipment for visualizing the concepts of Einsteinian physics.
Then, we provide brief descriptions of each teaching aid. The talk will include video clips to
illustrate all of the teaching aids discussed here.
Significance
Einstein developed his general theory of relativity which describes space, time and gravity in a
beautiful unified framework. The theory predicts many subtle and strange phenomena which
today are commonly observed. Quantum mechanics which arose from Einstein and Planck’s
prediction of the photon underpins the electronics industry. It is becoming widely recognized that
the school physics curriculum should introduce special and general relativity and quantum
mechanics from an early age because, these theories provide our best understanding of the nature
space-time, gravity, matter and radiation. To introduce such new concepts requires new
resources for teachers. This paper describes some teaching aids and resources that provide
effective tools for teaching the new concepts of modern physics.
Design and Procedure
1) Tools for visualizing for general relativity: To teach general relativity it is necessary to have
resources to help visualization of the central concepts the curvature of space, spacetime diagrams
and the time warps. The following five paragraphs describe simple teaching aids, resources and
activities on this topic. The following five paragraphs describe tools for visualizing quantum
mechanics.
a) Lycra curved space classroom exhibit: Einsteinian general theory of relativity can be
expressed as follows:
Matter tells space how to curve; Space tells matter how to move.
According to general relativity, gravity is an effect of space-time curvature that arises from the
effect of masses. An ideal material for simulating the curvature of space by matter is Lycra
sheet- an elastic textile material used for sports clothing. We use a 1.5m x1.5m sheet of lycra
stretched across 3m x 3m wooden frame. In the absence of matter we observe the flat surface of
sheet. Now, when we place a single golf ball on the lycra sheet it produces almost imperceptible
dent. However, a second ball placed on the sheet rolls toward the first due to the curvature of
sheet created by the first ball. Using up to hundred balls we gradually increase the curvature. We
see apparent attraction between the balls on the lycra sheet. Whole classes participate in this
activity. Students can study orbital motions on the lycra sheet and even observe three body
interactions. The model can be interpreted as demonstrating the Einsteinian curvature of the
space-time but it also allows the Newtonian concept of the gravitational force of attraction to be
conceptualized. However, it is important to emphasis that this is a model, and that the true
curvature of space is a curvature of both space and time in four dimensions. The figures showing
this exhibit are given in figure 1(a) and 1(b):
Fig. 1(a) Golf ball placed on lycra sheet
produced dent in the lycra sheet
Figure 1(b) Showing as golf ball increases,
curvature is also increasing
b) Geometry on curved space: When space is curved as observed in the lycra sheet model
above, geometry is altered. This activity allows students to investigate geometry on curved 2D
space. The key point is that normal Euclidean geometry fails in curved space. First it is essential
for students to understand the meaning of a straight line: a) as the shortest distance between two
points and b) as a path that in practice we can only define by light. When we survey straight lines
(like fences or roads) we use light. We ask students to investigate geometry, on curved 2
dimensional outer surface of a wok. Straight lines can be defined by tightly stretched string, and
also by surveying “fences” created by attaching magnet poles (from magnetic construction toys)
to form triangles. By measuring angles with protractor students observe that the sum of angles of
a triangle increases above 180 degrees as the triangle perimeter increases. Students graph sum of
angles versus perimeter and note that only in the small triangle limit does the Euclidean result
hold. Videos about NASA’s Gravity Probe B and images of gravitational lenses are used to show
how these concepts extrapolate into 3D space.
c) Space-time diagrams, the Macdonald’s billboard and warped time: Once students have
been introduced to the idea that space is curved and studied geometry of curved space, it is
essential for students to become familiar with space-time through the 2D space-time diagram.
Students easily grasp the concept of reducing 4 dimensional space-time into a 2 dimensional
version by say-plotting distance against time. They drawing their own journey to school in
space-time and learn to recognize that speed are defined by the gradient. We then successively
introduce two key concepts. First through discussion, we lead students to recognize that we
measure space and time in completely different and arbitrary units such as meters (related to a
platinum bar in Paris) and seconds (probably originally related to the time for one human
heartbeat). How can you plot a meaningful graph of space-time when the units are arbitrary and
disconnected? Macdonald’s billboards that proclaim “Macdonald’s 3 minutes ahead” provide the
clue. Speed links space and time so you can use time to measure distance if you can agree on a
speed. Through discussion students discover that the speed of light is the one universal
connection. The second key concept is that, gravity arises from the warping of the time
dimension. We have used both a power point animation and stretchable lycra graph to illustrate a
space-time diagram with warped time. We use a lycra sheet stretched on a frame which can be
distorted into a rhombus. After the activity described below students come to recognize that a
tiny warped time is sufficient to make free fall trajectories, the shortest path in space-time, as
suggested by Einstein.
d) Universality of free fall: According to Einstein’s equivalence principle, it is impossible to
distinguish between a uniform acceleration and a uniform gravitational field. This is the
universality of free fall discovered by Galileo in the context of dropping objects from the
Leaning Tower of Pisa. The remarkable independence of free fall on the composition of the
objects convinces us that gravity is not a force like say magnetism, which does depend on the
composition of the bodies. General relativity postulates that gravity is the natural tendency of an
object to move in a “straight line” or shortest path in space-time. At the time of Galileo most
people believed Aristotle’s statement that things fall with constant speed proportional to their
mass. We ask students to this statement by dropping two objects with large mass difference as
full and empty drink bottles. Students did this experiment on the day of NYSF with water
balloons as shown in figure given below 2(a). We discuss the confounding variable of wind
resistance and show the NASA YouTube video of astronaut David Scott performing a hammer
and feather free fall experiment on the surface of the Moon (NASA, 1971). The main aim of
these activities is to encourage students to recognize that free fall is free float: natural motions in
space. Gravity is the force you have you apply to prevent free fall. In the context of space-time
diagrams, the free fall trajectory is the shortest path in space-time if time depends on height by
an amount g/c2 per meter, a small effect that is easily measured by modern clocks.
Figure 2(a) Dropping water balloons from
by leaning tower
Fig.2(b) Gravitational lensing effect as observed
the Hubble space telescope in Abell 1689.
e) Gravitational lens images: The curvature of space seen in the lycra sheet is frequently
observed in astronomical images as a result of light passing through curved space. Due to this
effect, stars appear to be in false position when their light passes close to the Sun. The first
observation of light deflection was observed by Arthur Eddington and his collaborators during a
total solar eclipse in 1919 (Eddington et al., 1920). Precision astronomical images today show
multiple examples of curved space such as the figure 2(b) above. Google images show a large
gallery of gravitational lens images. We ask the students to search and investigate gravitational
lensing images available on the internet. The base of wine glass simulates a gravitational lens.
We ask students to photograph each other through such lenses to observe amusing distortions.
2) Tools for visualizing quantum physics: The key aspect of quantum mechanics is the light
comes in photons but it has wave like behavior. The teaching aids and activities below are
designed first to emphasis the particle like aspect of light discovered by Einstein. For this we use
foam bullets fired from popular children’s toys called Nerf guns. Having emphasized the key
properties that arise from photons as particles we go on to emphasize the wave like behavior.
Finally, we point out that when interference is taking place there often may be only one photon
present at a time in the apparatus. This leads to the recognition that interference involves the
quantum weirdness that no one can understand, as highlighted by Richard Feynman, Hawking
and many others (Feynman, 1964).
a) Nerf gun photography: We ask students to “photograph” each other using nerf gun foam
bullets that are used to represent photons. One student stands against a glossy wall or whiteboard
while, others “illuminate” the subject with bullets as shown in figure 3(a). Bullets stick only to
the glossy surface, and create a silhouette “photograph” of the student. In discussion we
emphasized that photons have properties analogous to the bullets, including momentum and
discuss how this phenomenon is used by spacecraft using solar sails.
Fig. 3(a) A girl is standing against a glossy
Surface to create silhouette photograph
Fig. 3(b) Ping pong balls ejected from saucers
with Nerf gun bullets
b) Photoelectric effect: The ejection of electrons from the surface of a metal, when light of
suitable frequency falls on it, is known as the photoelectric effect. This concept was given by
Einstein and won the Noble prize in 1921. To explain this phenomenon Einstein predicted that
light comes in quanta called photons. Nerf gun photons again allow the process to be visualized.
We use saucers to represent atoms that confine electrons in the form of ping pong balls as shown
in figure 3(b). We create an array of saucers and ping pong balls. A Nerf gun “photon” source is
used to “illuminate” the saucers. Some bullets transfer kinetic energy to the balls which fly out in
close analogy to the photoelectric effect. We then discuss applications of photoelectric effect
such as digital cameras, and solar cells.
c) Nerf gun photography of a suspended balloon: According to the Heisenberg uncertainty
principle, we cannot measure the position and momentum of an object simultaneously with
arbitrary precision. Measurement involves intrinsic uncertainty. The uncertainty principle is
easily demonstrated with Nerf guns-by applying Nerf gun photography to a low mass object such
as balloon. When students try to photograph a balloon suspended by a string, the momentum of
the bullets displaces the position of the balloon causing intrinsic fuzziness in the “image” of the
balloon. This is a vivid realization of the uncertainty principle.
Having emphasized the particle like property of light, we go on to emphasize wave like property
of light using green laser pointers and appropriate safety measures.
d) Laser diffraction measurements of a human hair: We begin by showing aerial photographs
of ocean waves diffracting around an island. The images show the waves from both sides of the
island creating a pattern of constructive and destructive interference on the shoreline behind the
island. Then we use a laser pointer as our source of waves, a human hair as an island and a
screen as the shoreline. The similarity of the interference patterns demonstrates that light is
indeed a wave. Students observe and measure the pattern of light and dark bands on the screen as
shown in fig. 5(a). We then apply this to measurement of the students own hair diameters by
determining the position of the first dark fringe. Hair diameter is given by laser wavelength x
distance from the hair to the screen/ dark fringe distance from central maximum. In this way
students use the wavelike properties of light to compare each other’s hair in an enjoyable and
engaging activity.
Fig. 5(a) Diffraction pattern of human hair
with laser light
Fig.5 (b) Interference pattern of soap
bubble with laser light
e) Soap film interference: Laser pointers allow interference to be observed in light reflected
from soap films. The interference arises from the phase difference of the reflected light from the
front and rear faces of the soap film. This activity is particular interesting because the size of the
interference “apparatus” is the thickness of soap bubble film only a matter of few microns. It is
easy to show that under most circumstance there is never more than one photon present inside
the film at any one time. Yet clear high contrast interference is easily obtained as shown in figure
5 (b). This experiment proves that the interference observed is not the interference of photons but
of some other entity which physicists call the wave function, but cannot really explain. This
interference when photons arrive individually is one of the hallmarks of quantum weirdness.