This book had its origin in a one-year course that I taught at Yale
throughout the decade of the 1970’s. The course was for non-science majors
who were interested in learning about the major branches of Physics. In the
first semester, emphasis was placed on Newtonian and Einsteinian Relativity.
The recent popularity of the Einstein exhibit at the American Museum of
Natural History in New York City, prompted me to look again at my fading
lecture notes. I found that they contained material that might be of interest to
today’s readers. I have therefore reproduced them with some additions, mostly
of a graphical nature. I recall the books that were most influential in my
approach to the subject at that time; they were Max Born’s The Special Theory
of Relativity, Robert Adair’s Concepts of Physics, and Casper and Noer’s
Revolutions in Physics. These three books were written with the non-scientist
in mind, and they showed what could be achieved in this important area of
teaching and learning; I am greatly indebted to these authors.
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1. INTRODUCTION
This brief book is for the inquisitive reader who wishes to gain an understanding of the
immortal work of Einstein, the greatest scientist since Newton. The concepts that form the
basis of Einstein’s Theory of Special Relativity are discussed at a level suitable for Seniors in
High School. Special Relativity deals with measurements of space, time and motion in inertial
frames of reference (see chapter 4). An introduction to Einstein’s Theory of General Relativity,
a theory of space, time, and motion in the presence of gravit y, is given at a popular level. A
more formal account of Special Relativity, that requires a higher level of understanding of
Mathematics, is given in an Appendix.
Historians in the future will, no doubt, choose a phrase that best characterizes the 20th-
century. Several possible phrases, such as “the Atomic Age”, “the Space Age” and “the
Information Age”, come to mind. I believe that a strong case will be made for the phrase “the
Age of Einstein”; no other person in the 20th-century advanced our understanding of the
physical universe in such a dramatic way. He introduced many original concepts, each one of
a profound nature. His discovery of the universal equivalence of energy and mass has had, and
continues to have, far-reaching consequences not only in Science and Technology but also in
fields as diverse as World Politics, Economics, and Philosophy.
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The topics covered include:
a) understanding the physical universe;
b) describing everyday motion;
relative motion,
Newton’s Principle of Relativity,
problems with light,
c) Einstein’s Theory of Special Relativity;
simultaneity and synchronizing clocks,
length contraction and time dilation,
examples of Einstein’s world,
d) Newtonian and Einsteinian mass;
e) equivalence of energy and mass, E = mc2;
f) Principle of Equivalence;
g) Einsteinian gravity;
gravity and the bending of light,
gravity and the flow of time, and
red shifts, blue shifts, and black holes.
2. UNDERSTANDING THE PHYSICAL UNIVERSE
We would be justified in thinking that any attempts to derive a small set of fundamental
laws of Nature from a limited sample of all possible processes in the physical universe, would
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lead to a large set of unrelated facts. Remarkably, however, very few fundamental laws of
Nature have been found to be necessary to account for all observations of basic physical
phenomena. These phenomena range in scale from the motions of minute subatomic systems
to the motions of the galaxies. The methods used, over the past five hundred years, to find the
set of fundamental laws of Nature are clearly important; a random approach to the problem
would have been of no use whatsoever. In the first place, it is necessary for the scientist to have
a conviction that Nature can be understood in terms of a small set of fundamental laws, and
that these laws should provide a quantitative account of all basic physical processes. It is
axiomatic that the laws hold throughout the universe. In this respect, the methods of Physics
belong to Philosophy. (In earlier times, Physics was referred to by the appropriate title,
“Natural Philosophy”).
2.1 Reality and Pure Thought
In one of his writings entitled “On the Method of Theoretical Physics”, Einstein stated:
“If, then, experience is the alpha and the omega of all our knowledge of reality, what then is the
function of pure reason in science?” He continued, “Newton, the first creator of a
comprehensive, workable system of theoretical physics, still believed that the basic concepts
and laws of his system could be derived from experience.” Einstein then wrote “But the
tremendous practical success of his (Newton’s) doctrines may well have prevented him, and
the physicists of the eighteenth and nineteenth centuries, from recognizing the fictitious
character of the foundations of his system”. It was Einstein’s view that “. the concepts and
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fundamental principles which underlie a theoretical system of physics are free inventions of the
human intellect, which cannot be justified either by the nature of that intellect or in any other
fashion a priori.” He continued, “If, then, it is true that the axiomatic basis of theoretical
physics cannot be extracted from experience but must be freely invented, can we ever hope to
find the right way? .. Can we hope to be guided safely by experience at all when there exist
theories (such as Classical (Newtonian) Mechanics) which to a large extent do justice to
experience, without getting to the root of the matter? I answer without hesitation that there is, in
my opinion, a right way, and that we are capable of finding it.” Einstein then stated
“Experience remains, of course, the sole criterion of the physical utility of a mathematical
construction. But the creative principle resides in Mathematics. . . I hold it true that pure
thought can grasp reality, as the ancients dreamed.”
3. DESCRIBING EVERYDAY MOTION
3.1 Motion in a straight line (the absence of forces)
The simplest motion is that of a point, P, moving in a straight line. Let the line be
labeled the “x-axis”, and let the position of P be measured from a fixed point on the line, the
origin, O. Let the motion begin (time t = 0) when P is at the origin (x = 0). At an arbitrary time,
t, P is at the distance x:
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P [x, t]
-x O position, x +x
at time t
If successive positions of P are plotted, together with their corresponding times, we can
generate what is called the “world line” of P.
Let us observe a racing car moving at high speed along the straight part of a race track
(the x−axis), and let us signal the instant that it passes our position, x = 0, by lowering a flag:
An observer, standing at a measured distance D, from x = 0, starts his clock at the instant, t = 0,
when he sees the flag lowered, and stops his clock at the instant t = T, as the car passes by.
We can obtain the average speed of the car, v, during the interval T, in the standard way; it is
v = D/T (in units of velocity).
If, for example, D = 1 mile, and T = 20 seconds (1/180 hour), then
v = 1 (mile)/(1/180) (hour) = 180 miles per hour.
This is such a standard procedure that we have no doubt concerning the validity of the result.
3.2 The relativity of everyday events
Events, the description of when and where happenings occur, are part of the physical
world; they involve finite extensions in both time and space. From the point of view of a
theory of motion, it is useful to consider “point-like” events that have vanishingly small
extensions in time and space. They then can be represented as “points” in a space-time
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geometry. We shall label events by giving the time and space coordinates: event E → E[t, x],
or in three space dimensions, E[t, x, y, z], where x, y, z are the Cartesian components of the
position of the event. There is nothing special about a Cartesian coordinate system, it is a
mathematical construct; any suitable coordinate mesh with a metrical property (measured
distances defined in terms of coordinates) can be used to describe the spatial locations of
events. A familiar non-Cartesian system is the spherical polar coordinate system of the lines of
latitude and longitude on the surface of the earth. The time t can be given by any device that is
capable of producing a stable, repetitive motion such as a pendulum, or a quartz-controlled
crystal oscillator or, for high precision, an atomic clock.
Suppose we have an observer, O, at rest at the origin of an x−axis, in the F-frame. O
has assistants with measuring rods and clocks to record events occurring on the x-axis:
F
-x O +x
We introduce a second observer, O′, at rest at the origin of his frame of reference, F′. O′ has
his assistants with their measuring rods (to measure distances, x′) and clocks (to measure times,
t′) to record events on the x′-axis. (The F′-clocks are identical in construction and performance
to the clocks in the F-frame). Let O′ coincide with O at a common origin O = O′ (x = x′ = 0), at
the synchronized time zero t = t′ = 0. At t = t′= 0, we have
F′
-x′ O′ +x′
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Suppose that the observer O′, and his assistants with rods and clocks, move to the right with
constant speed V along the common x, x′−axis. At some later time t, the two sets of observers,
represented by O and O′, record a common event that they write as E[t, x] and E′[t′, x′],
respectively. The relative positions of the two observers at time t is:
F E[t. x]
-x O x +x
D F′⇒ E′[t′, x′]
-x′ O′ x′ +x′
where D = Vt is the distance that O′ moves at constant speed V, in the time t.
We therefore write the relationship between the two measurements by the plausible equations
(based on everyday experience):
t′ = t (everyday identical clocks tick at the same rate)
and
x′ = x − D = x − Vt.
These are the basic equations of relative motion according to the concepts first put forward by
Galileo and Newton. They are fully consistent with measurements made in our real world (the
world of experience). They are not, however, internally consistent. In the equation that relates
the measurement of distance x′ in the F′-frame to the measurements in the F-frame, we see that
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the space part, x′, in the F′-frame, is related to the space part, x, and the time part, t, in the F-
frame: space-time in one frame is not related to space-time in the other frame! Furthermore,
the time equation makes no mention of space in either frame. We see that there is a
fundamental lack of symmetry in the equations of relative motion, based on everyday
experience. The question of the “symmetry of space-time” will lead us to Einstein’s
philosophy of the “free invention of the intellect”.
3.3 Relative velocities
We have seen that the position of an event, E[t, x], measured by an observer O, is
related to the position of the same event, E′[t′, x′], measured by an observer O′, moving with
constant speed V along the common x, x′−axis of the two frames, by the equation
x′ = x − Vt .
The speed v of a point P[t, x], moving along the x−axis, is given by the ratio of the finite
distance the point moves, Δx, in a given finite time interval, Δt:
v = Δx/Δt .
We can obtain the speeds v, and v′ of the same moving point, as measured in the two frames,
by calculating v = Δx/Δt and v′ = Δx′/Δt′, as follows:
Δx′/Δt′ = v′ = Δx/Δt − VΔt/Δt (where we have used Δt′ = Δt because t = t′ in everyday
experience).
We therefore find
v′ = v − V,
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the speeds differ by the relative speed of the two frames. This is consistent with experience: if
a car moves along a straight road at a constant speed of v = 60 mph, relative to a stationary
observer O, and an observer O′ follows in a car at a constant speed of V = 40 mph relative to
O, then the speed of the first car relative to the occupant of the second car is v′ = 20 mph.
3.4 The Newtonian Principle of Relativity
The Newtonian Principle of Relativity asserts that, in the inertial frames F, F′, the
following two situations
x = x′ = 0 at t = t′ = 0
F
−x O +x
D = Vt ⇒ +V F′ (moves to right
at speed V)
O′ +x′
D′ = Vt′ F′
O′ +x′
−V ⇐ F (moves to left at speed V)
O +x
cannot be distinguished by experiments that involve mechanical systems (classical systems
that obey Newton’s Laws of Motion).
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The speed V has been written in bold face to remind us that here we are dealing with a
vector quantity that has both magnitude (the speed in mph) and a sense of direction: +V in the
+x-direction and −V in the −x-direction.
3.5 Problems with light
We are accustomed to the notion that waves propagate through a medium, required to
support the waves. For example, sound waves propagate as pressure variations in air, and water
waves propagate as coupled displacements of the water molecules, perpendicular to the
direction of the wave motion. In the 19th-century, Maxwell discovered that light waves are
electromagnetic phenomena. This great work was based on theoretical arguments, motivated
by the experimental results of Faraday and Henry. One of the most pressing questions facing
scientists of the day was:
“what is waving when a beam of light propagates through empty space?”
It was proposed that the universe is filled with a medium called the aether with the property of
supporting light waves, and having no other physical attributes. (For example, it would have
no effect on the motion of celestial bodies). In the latter part of the 19th-century, Michelson
and Morley carried out a famous experiment at the Case Institute in Cleveland that showed
there is no experimental evidence for the aether. Light travels through the void, and that is that.
Implicit in their work was the counter-intuitive notion that the speed of light does not depend
on the speed of the source of the light.
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The Aether Theory was popular for many years. Non-traditional theories were
proposed to account for the null-result of the Michelson-Morley experiment. Fitzgerald
(Trinity College, Dublin) proposed that the Michelson-Morley result could be explained, and
the Aether Hypothesis retained, if the lengths of components in their apparatus were “velocity-
dependent” − lengths contract in the direction of motion, and lengths remain unchanged when
perpendicular to the direction of motion. He obtained the result
L = [1/
0
√(1 − (v/c)2)] L ≡ γL
⇑ ⇑
(length of rod at rest) (length of rod moving at speed v)
Here, c is the constant speed of light (2.99 … × 108 meters/second).
All experiments are consistent with the statement that the ratio v/c is always less than 1, and
therefore γ is always greater than 1. This means that the measured length of the rod L , in its
0
rest frame, is always greater than its measured length when moving.
At the end of the 19th-century, Larmor introduced yet another radical idea: a moving
clock is observed to tick more slowly than an identical clock at rest. Furthermore, the
relationship between the clock rates in the moving and rest frames is given by the same factor,
γ, introduced previously by Fitzgerald. Specifically,
Δt = γΔt
0
⇑ ⇑
(an interval on a moving clock) (an interval on a clock at rest)
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Since the velocity-dependent term γ is greater than 1, the intervals of time Δt (moving), are
greater than the intervals Δt (at rest).
0
Fitzgerald, Larmor, and other physicists at that time considered length contraction and
time dilation to be “real” effects, associated with minute physical changes in the structure of
rods and clocks when in motion. It was left to the young Einstein, working as a junior Patent
Officer in Bern, and thinking about space, time, and motion in his spare time, to introduce a
new Theory of Relativity, uninfluenced by the current ideas. Although, in later life, Einstein
acknowledged that he was aware of the Michelson-Morley result, and of the earlier work on
length contraction and time dilation, he neither used, nor referred to, the earlier results in his first
paper on Relativity.
4. EINSTEIN’S THEORY OF SPECIAL RELATIVITY
In 1905, Einstein published three great papers in unrelated areas of Physics. In this
chapter, we shall discuss his new ideas concerning the relative motion of beams of light, and of
objects that move at speeds close to that of light. His independent investigations were based
upon just two postulates:
1. The generalized Principle of Relativity: no mechanical or optical experiments can
be carried out that distinguish one inertial frame of reference from another. (This is a
development of Newton’s Principle of Relativity that is limited to mechanical experiments,
involving speeds much less than the speed of light; it applies to experiments in the everyday
world).
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Inertial frames of reference are non-rotating, and move in straight lines at constant
speed. They are non-accelerating.
2. The speed of light in a vacuum is a constant of Nature, and is independent of the
velocity of the source of the light.
Einstein was not concerned with questions having to do with the Aether; for him, a true theory
of the physical properties of the universe could not rest upon the mysterious qualities of such an
unobservable. As we shall see, Einstein was concerned with the precise meaning of
measurements of lengths and time intervals. In his later years, he recalled an interesting
thought that he had while in school. It had to do with the meaning of time. Our lives are
dominated by “psychological time”; for example, time seems to go by more quickly as we
grow older. For the young Einstein, time in the physical world was simply the reading on a
clock. He therefore imagined the following: if the schoolroom clock is reading 3PM, and I
rush away from the clock at the speed of light, then the information (that travels at the speed of
light) showing successive ticks on the clock, and therefore the passage of time, will never reach
me, and therefore, in my frame of reference, it is forever 3PM − time stands still. He therefore
concluded that the measurement of time must depend, in some way, on the relative motion of
the clock and the observer; he was, by any standards, a precocious lad.
If we apply the Galilean-Newtonian expression for the relative velocities v, v′,
measured in the inertial frames F, F′, (moving with relative speed V), to the measurements of
flashes of light, v = c, the speed of light in F, and v′ = c′, the speed of light in F′, we expect
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c′ = c − V (corresponding to v′ = v − V for everyday objects).
Modern experiments in Atomic, Nuclear, and Particle Physics are consistent with the fact that
c′ = c, no matter what the value of the relative speed V happens to be.
At the end of the 19th-century, a key question that required an answer was therefore: -
why does the Gali