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Chapter 4 - Einstein’s Relativity
Light: The Gateway to Relativity
An in-depth study will show that light is a very strange entity that shows no respect for
our common sense understanding of the universe. Of course, it is not nature’s job to fit
the existing theories; rather, it is the theoretical physicist’s job to match theories to
nature. One of Einstein’s famous statements is “reality is the real business of science.”
When the theory does not match nature, the theory, in this case the NewtonianTheory,
must be wrong. Consequently, the seemingly unrealistic properties of light became the
gateway to the next two revolutions of science, the development of relativity and the
development of quantum mechanics. In the next chapter we will look at the properties of
light that led to quantum mechanics and in this chapter we will look at the properties of
light that led to the development to the Special and General Theories of Relativity.
The speed of light
We will start our review of the quest for an understanding of light with the French
physicist Foucault (foo-koh’), who lived from 1819-1868, and was one of the first to
develop accurate measurements of the speed of light. He did this using a toothed wheel
as others had done, and then developed a method of his own using rotating mirrors along
with the toothed wheel. He found the speed of light to be 3 x 108 meters/second.
The nature of light - what is it?
The next major development in the understanding of light came in the 1860s: James
Clark Maxwell (a Scottish physicist) developed a theory of light that was not based on
the Newtonian theory. The Newtonian theory was based on forces, while Maxwell’s
theory was based on a concept of fields. Maxwell’s work on light could be thought of as
a pillar of science in of itself. What follows in the next five paragraphs is the
development of Maxwell’s theory. Einstein wrote that the field concept was “the most
profound and fruitful that physics has experienced since Newton.”
A number of scientists were studying magnetic and electric fields, which at this time
were mysterious and unexplained phenomena. It was shown that if one shook a magnet
above iron filings, it would take a short time for iron filings to move. This proved that it
took time for at least one field, a magnetic field, to move across space. This was
surprising because Newton’s theory had the gravitational field working instantaneously.
Perhaps it took time for all fields to move.
Danish physicist Hans Oersted was the first to fine the surprising connection between
magnetism and electricity. Michael Faraday then exhaustively explored the relationship
through thousands of experiments. What was this relationship? It was found that if
magnets moved past a loop of wire (a completed circuit) an electric current was
produced. Furthermore, electricity moving through a wire created a magnetic field
around it. A moving magnetic field creates an electric field and a moving electric field
creates a magnetic field. Faraday was demonstrating this during a lecture and a lady in
the audience asked him what the value of this was. Faraday’s famous response was
“Madam, of what value is a newborn child?” Faraday’s curt response shows his passion
for his research; however, the question the lady asked was actually a very good question
and is the type of question that needs to be asked in science. The questions is basically,
why are we studying this and does the reason justify the time and expense being allocated
to the research? In applied science the time and expense can be more directly compared
to the expected outcome. Pure research is a different. The driving force behind it is
typically our human desire to discover the secrets of nature. There is no way to predict
how future discoveries will benefit humankind. Debates as to if we should spend billions
of dollars to build bigger and bigger particle accelerators are good modern examples of
this question being asked. It turns out that the research on magnetism and electricity was
time and money well spent as the discovery of their connection later led to the
development of both the generator and electric motor.
Maxwell studied Faraday’s experimental results and, using his superior mathematical
skills, was able to find the elusive mathematic relationship between magnetism and
electricity. He then hypothesized that, since electricity creates magnetism and magnetism
creates electricity, that an electric field spontaneously creates a magnetic field, which
will, in tern, create an electric field. He claimed that the electric and magnetic fields
moved out through space by creating each other in leapfrog fashion, with one creating the
other indefinitely. This process of leapfrog wave creation would continue through space
until the double wave system would strike something that could absorb its energy. This
double wave moving through space could be thought of as a composite wave composed
of both an electronic field and a magnetic field. As the wave moves forward it pulses
with an electronic component, then a magnetic component, back to an electronic
component and so on and on. The wave would move out like ripples from a pebble
dropped into a pond, only the composite electro-magnetic wave would be in three
dimensions instead of two.
Maxwell then used the equations he developed from Faraday’s work to calculate the
speed of this composite electro-magnetic wave. Surprisingly, his calculations told him
that this composite wave that he had been imagining traveled at the same speed as light.
In a bold move he then further hypothesized that this electromagnetic wave that he had
theoretically constructed from the study of magnetism and electricity was light.
Imagine the tremendous thrill he must have felt. His study of electricity and magnetism
had led him to the discovery of the true nature of light. He was the first person to
understand the core of what light actually is. It is a compound electromagnetic wave
created, and propagated, through space by electric and magnetic fields creating each other
in leapfrog fashion. Different colors and different types of light were simply different
frequencies of these electromagnetic waves.
Maxwell’s hypothesis was supported by the experiments of Heinrich Hertz. Hertz
created a low-frequency electromagnetic radiation (radio waves). It was found that the
only difference between these waves and light was the frequency. We now know that
electromagnetic radiation is a class of “light” energy that includes visible light and ranges
in frequency from low frequency radio waves to very high frequency gamma rays.
Faraday was able to show a connection between magnetic and electric fields; however it
was Maxwell who was able to mathematically unify these two forces, and in doing so
gave us a much deeper insight as to what light actually was.
The mathematics of Maxwell’s theory has become a cornerstone of science. The math
consists of eight partial differential equations which together define electricity and
magnetism. These equations (known as Maxwell’s equations) are part of the curriculum
that must be mastered by graduate students who study electricity and light. These
equations are so well known and important that there are T-shirts made with the saying “In the beginning, God said (then the 8 equations are written) and then there was light.”
Maxwell’s equations are given below. Again we see, mathematics is the language of the
universe. These equations are beyond the math skills of most of us. Those who have not
had sufficient calculus to use them should use these equations as motivation to continue
studies in mathemataics.
Summary of Maxwell’s equations
Name
Partial Differential
form
Integral form
Gauss’ Law
Gauss' law for magnetism
(absence of magnetic
monopoles):
Faraday’s law of
induction
Ampere’s law +
Maxwell’s extension
Where:
ρ is the free electric charge density (SI unit: coulomb per cubic meter), not
including dipole charges bound in a material.
is the magnetic flux density (SI unit: tesla, volt × second per square meter),
also called the magnetic induction..
is the electric displacement field (SI unit: coulomb per square meter).
is the area of the Gaussian surface.
is the electric field (SI unit: volt per meter).
is the magnetic field strength (SI unit: ampere per meter).
is the current density (SI unit: ampere per square meter).
is the divergence operator (SI unit: 1 per meter).
is the curl operator (SI unit: 1 per meter).
Note that although SI units are given here for the various symbols, Maxwell's equations
will hold unchanged in many different unit systems (and with only minor modifications
in all others). The most commonly used systems of units are SI units, which are used for
engineering, electronics and most practical physics experiments. Planck units (also
known as "natural units"), are used in theoretical physics, quantum physics and
cosmology. An older system of units, the cgs system, is sometimes also used.
The development of Maxwell’s equations gave scientists new confidence. Einstein saw
the field concept as progress towards a better understanding of the mysteries of nature.
Other scientists were overconfident. Some believed that, between Newton’s work and
Maxwell’s equations, nature was fully understood. When Max Planck asked his advisor
about becoming a physicist he was told to switch fields because physics was basically
finished. His advisor believed that all the major discoveries that could be made in
physics had already been made. Fortunately Max Planck did not take his advisor’s
advice and went on to become the founder of quantum theory.
Hints that there were problems with Newtonian Physics
Despite the overconfidence of many of the older physicists, the late eighteen hundreds
and early nineteen hundreds were an exciting time in science. Marie Curie discovered
the mysterious new element radium that glowed in the dark. The light energy given off
by radium was endless. One kilogram of radium gives off enough energy to boil away a
kilogram of water every half hour “forever.” Where was this energy coming from? How
did radium produce this light and heat? Newton had established the law of conservation
of energy. This law states that energy cannot be created or destroyed; it can only be
changed from one form to another. Was this never-ending supply of energy seemingly
produced from nothing a violation to Newton’s law of conservation of energy?
There were also several problems concerning aether, which was the material assumed to
fill the void of space. Since it was known that light was an extremely fast wave,
scientists could use the known properties of waves to determine some of the properties
that aether must have. The fact that light was a wave helped substantiate the existence of
aether because all waves needed a medium to travel through. The speed of a wave
through its medium is dependent on the elasticity of the medium. Elasticity is the ability
of an object to return to its original shape once it has been deformed. On a molecular
level it is the ability of molecules to return to their original position once disturbed. A
solid has more elasticity than a liquid or a gas so sound, for example, will travel fastest
through a solid. Since light travels so fast through space, the elasticity of aether must be
greater than the most elastic materials we are familiar with (such as very hard spring
steel). Since light is a transverse wave, the aether must be a solid. A transverse wave
vibrates perpendicular to the direction of motion, like a water wave moving across the
water. The water moves only up and down while the wave moves forward across the
water. The other type of wave is a longitudinal wave such as sound. A longitudinal
wave move back in forth in a compression and stretching movement that is in the same
direction that the wave travels. If we consider a wave moving through a medium that
takes up all space, transverse waves need a solid to travel through. Gases and liquids can
support only longitudinal waves such as sound. The fact that light was known to be a
transverse wave created more problems for understanding aether. Solids allow both
longitudinal and transverse vibrations, thus earth quakes move with both vibrations
simultaneously. What is it about aether that prevents any back and forth vibration but yet
allowed perpendicular vibrations to propagate with such extreme speed?
Although aether was reasoned to be a transparent, hard, elastic solid, it offered absolutely
no resistance to the transfer of matter through it. Planets could orbit around the sun
forever without the aether slowing them down or affecting their orbit in any way. The
new understanding of light was forcing aether to take on more and more bizarre
properties. Historically, as additional facts about the universe are discovered, an old,
incorrect theory is patched and modified to accept the new information. With patch upon
patch, the old theory that was at one time streamlined and beautiful becomes awkward
and ugly. This is a sign that the theory is nearing the end of its useful life. Eventually a
new, neat, clean theory will replace it. The Newtonian view of the universe was in its
first stages of becoming awkward. It was only a matter of time before it would be
replaced.
The feeling of scientific revolution was starting to grow among some of the younger and
more radical thinkers. When Einstein worked at the patent office he needed to find a way
to stay in touch with the more progressive ideas and new scientific discoveries. He and
some of his former classmates met regularly and became an informal study group which
they jokingly called the “Olympian Academy.” They met in Zurich’s coffee houses and
beer halls and discussed philosophy and science. They studied and discussed the
controversial work of Ernst Mach, a Viennese physicist and philosopher who challenged
many of the accepted scientific ideas of the time, especially those that were beyond our
senses. His controversial book was called “The Science of Mechanics.” In this book he
challenged the idea of atoms which were beyond the realm of direct observation and
measurement. Mach also criticized the ideas of aether and absolute motion.
Mach believed that absolute space and absolute time could not be measured. He claimed
that there was no way to prove that time and space (distance) was the same every where
and for all conditions. Relative motion could be measured, but absolute motion could not
because it depended on a stationary aether as the reference frame. Aether could not be
measured or even substantiated; in fact, as noted earlier there was mounting evidence
against the existence of aether. Since absolute space and time could not be proven, we
could not assume them. Furthermore, since Newton assumed absolute space and time
when he developed is laws of motion and his theory gravity, Newton’s work was without
a provable foundation and therefore must be thrown out.
Most physicists believed Mach’s objections were more philosophical rather than real.
Certainly, the great success of Newtonian Physics was proof enough that the original
assumptions about time and space had to be correct. The constant speed of light;
however, was different. This turned out to be a huge problem for the Newtonian view of
the universe. At the age of 16 Einstein conducted a thought experiment. He asked
himself what would light look like if we could run along side it at the same speed. The
answer to this question was that it would be a stationary “frozen wave.” He knew there
was no such thing in nature and that a frozen wave was impossible. While attending the
Polytechnic he substantiated this for himself. When Einstein learned Maxwell’s
equations and worked with them he found that there was no solution to the equations in
which light was frozen in time. He was also the first, or one of the first, to correctly
interpret the speed of light given by Maxwell’s equations. Maxwell’s equations gave the
speed of light as 3 x 108 meters/second; however, there was no reference frame. It was
assumed by most that this meant with respect to aether. The problem was that aether
could not be detected or measured. Einstein had the correct interpretation in that the
equation meant that the speed of light was 3 x 108 meters/second in every reference
frame. Regardless of the direction and speed you were moving, you would always
measure light to be 3 x 108 meters/second. No matter how fast you moved you could
never catch up to a light beam because it always sped away from you at the same speed.
Michelson and Morley
Michelson and Morley’s “failed” experiments on aether and the speed of light proved
what Einstein had already learned from Maxwell’s equations. Michelson (1852-1931)
added his own improvements to Foucault’s speed of light experiments and ran his own
experiments. The basic set-up is shown below. The beam splitter is a half-silvered
mirror. The mirrors are placed at right angles to each other and at equal distance from the
beamsplitter, which is oriented at an angle of 45° relative to the two mirrors.
Michelson’s goal was to determine the earth’s speed and direction through aether using
the speed of light as a reference. His experiment utilized the concept that aether was the
one constant from which the “absolute” velocity of all objects in space could be
determined.
The following analogy may help you understand the concept behind their experiment.
Imagine that we are trying to determine the velocity of a boat on a lake by firing nonexploding torpedoes at it from various directions. The torpedoes all travel at 50 miles per
hour over the surface of the water. The speed of the torpedoes, as measured by the
occupants of the boat, is determined by the speed of the torpedo in the water and by the
speed and direction of the boat. The torpedoes that approach the boat from the front will
seem much faster than the torpedoes that approach the boat from the rear. In fact, if the
boat is moving faster than the torpedoes, the torpedoes will never be able to catch up to
the boat from the rear. The occupants of the boat analyze the torpedo speed data of
torpedoes coming from several different directions. They find that the fastest torpedo
approach the boat at 60 miles per hour and that the slowest torpedo approach the boat
from the opposite direction at a speed of 40 miles per hour. They deduce from their data
that the torpedoes must be traveling at 50 miles per hour and that the boat must be
traveling 10 miles per hour in the direction from which the 60 mile per hour torpedoes
came.
In comparing Michelson’s experiments to the above analogy, the earth is the boat, space
(aether) is the water, and light is the torpedo. Light traveling at different directions
should be clocked at different velocities because of the motion of the earth. The results
of these experiments should not only give the speed of the surface of the earth as it
rotates through the aether, it should also give the best possible value for the speed of
light. Using mirrors, a single light beam was split into two identical beams which were
sent out ninety degrees to each other. The apparatus was rotated and tested at several
different orientations with the intent to maximize the difference in relative speed between
the two beams. The light beam sent in the direction of the earth’s motion ought to travel
faster than light sent at right angles to the earth’s motion. When the split light beams
were brought back together they could recombine in phase or out of phase. If light meets
in phase the positive lobe of one light wave adds to the positive lobe of another light
wave (like the crest on one water wave combining with the crest of a second water wave
to make a wave twice as large). If light returning on different paths met out of phase,
there would be interference fringes and if the light beams would meet in phase there
would be no interferences fringes. As the apparatus was rotated there was no change in
the image on the detector. This meant that the beams could not be changing speed as
they changed direction. This was seen as an impossible result. In comparing to the boat
analogy it would be like the torpedoes striking the boat at the same speed regardless of
the speed of the boat.
Michelson believed his first experiments in 1881 failed because he could detect no
difference in the speed of the two light beams. In 1881 he tried the experiment again
with Morley who was noted for experimental detail and precision. They ran the
experiment again with additional modifications that would insure absolute precision. The
apparatus was mounted on a bed of liquid mercury so it could be rotated freely. This
time their apparatus was so delicate that it easily picked up the vibrations caused by
passing horse carriages. This attempt turned out to be the most famous “failed”
experiment in history. Light seemed to traveled the same speed in any direction,
regardless of the speed of the detector. How could this be explained? It smacked
common sense and traditional physics in the face.
The common sense, traditional approach was first worked out by Galileo and then
incorporated by Newton as one of the basic assumptions of his work. The following is an
example. A woman is walking three miles per hour on a train that is moving 10 miles per
hour in the same direction she is walking. How fast is she moving? Of course it depends
on the frame of reference. From the frame of reference of someone standing on the
ground, she is moving 13 miles per hour. From the frame of reference of someone on the
train, she is traveling 3 miles per hour. Each frame of reference is valid. The math to
switch from one frame of reference to the other is called the Galilean transformation.
Galileo’s principle of relativity holds that the laws of mechanics are invariant under
Galilean transformations (i.e. both observers can use the same laws to make the same
kind of measurements and both of their accounts can be shown to be equivalent accounts
of the same situation.)
The problem is that this common sense treatment of relative velocity does not work for
light. According to the Galilean/Newtonian/common sense view of velocity, if we were
moving away from light at ¼ the speed of light, the light should only strike us at ¾ the
normal speed of light. Similarly, if we were traveling towards the light at ¼ the speed of
light, the light should strike us at 1¼ times the normal speed of light. The MichelsonMorley experiment proved that nature does not work this way. Light strikes us at the
same speed regardless of our motion relative to the light. This means that the
Galileo/Newton/common sense view of velocity must be wrong. The question now
facing scientists was whether the Newtonian concept of the universe be salvaged, or
would the fact that light travels the same speed for every observer overturn more than
200 years of physics along with our comfortable common sense notions of relative
motion?
Dutch theorist Hendrik Lorentz, along with Irish physicist George Fitzgerald, was able to
salvage the Newtonian view; however, it was at a high price. To do so they had to
suggest that an object actually changed shape as it accelerated, that it would become
shorter along the direction it moved. Lorentz demonstrated that just the right amount of
shrinkage would result in both the observer (who would not shrink) and the person
moving (who would actually shrink along with their measuring stick) measuring the same
speed for a light beam.
As wild as this was, it was accepted because it preserved the concepts of absolute time
and space while still explaining the experimental evidence. This idea, and the math
associated with it, became known as the Lorentz-Fitzgerald contraction. The math
worked well and is still used to relate moving frames of reference; however, was this
really why different observers traveling at different speeds measure the same speed for
light coming from any direction? Is this a true explanation of nature?
Einstein accepted the fact that light did indeed travel at the same speed for different
observers; however, he thought that the Lorentz explanation was a bit contrived and not a
true reflection of the way nature behaved. He realized that this bizarre property of light
was incompatible with Newtonian concepts. He searched for an alternative explanation.
Einstein wrestled with this problem and became stymied. He discussed his ideas with his
friend Besso, who was a coworker at the patent office. This helped Einstein clarify
questions and problems; however, the answer still eluded him. Emotionally, he was at a
very low point. Then one afternoon he noticed a clock on a building as he went home
from work. He again asked himself the question “what would I see if I were moving at
the speed of light?” He imagined that as he moved away from the clock at the speed of
light, the clock would slow down because it would take more time for the image of each
consecutive tick of the clock to reach him. At the speed of light he envisioned the clock
stopping. As he would later recall, at that point “A storm broke loose in my mind.” He
felt that he had finally tapped into “God’s thoughts.” Time was different for observers
traveling at different velocities. He later expressed this concept in the statement, “Our
concepts and laws of space and time can only claim validity insofar as they stand in a
clear relation to our experiences.”
He now had the concept. His next task was to explore it further and develop the
supporting mathematics. Can two observers agree as to the time of an event? Can there
be an absolute frame of reference for time? Apparently there can’t be, according to
epiphany Einstein had while on his way home from work. To analyze this further, he
developed the following thought experiment:
A passenger is traveling on a flat car located in the center of a moving train. At the
instant when the passenger is along side someone on the ground, two bolts of lightning
strike. One strikes the engine and one strikes the caboose. The person on the ground
sees the lightning strikes as simultaneous. Will the person on the train agree that the
strikes are simultaneous? The answer is no, light from the lightning bolt that strikes the
engine will reach her first. Since she is traveling towards that direction, the light will
have less distance to travel than the light coming from the caboose. The two strikes will
occur at different times for the passenger. The concept of simultaneity is an illusion that
is dependent on the frames of references for the observers. Einstein is famous for saying
that the past, present and future are an illusion - although they are a very persistent
illusion. Einstein realized that the study of physics required a new way of thinking about
space and time.
Using reasoning similar to that of the example with lighting bolts, he showed that
observers moving at different velocities with respect to each other will also come up with
different measurements of distance. The faster the observer, the shorter the distance
measured.
Once he had quantified relative differences in time and distance (space), he was able to
demonstrate why all observers obtained the same value for the speed of light, no matter
how fast they were moving with respect to each other or the light source. Since speed is
simply distance/time, all observers will come up with the same value if both the distance
and time change proportionally for each observer. The ratio of distance over time will
always be 3 X 108 meters/sec because, as someone speeds up, both the time and distance
change by the same percentage, thus maintaining the same ratio.
His final equations turned out to be exactly the same as the Lorentz’s equations; however
the reasons behind the equations were totally different. Instead of the object in motion
shrinking, as Lorentz proposed, space and time would both change as an object increased
its velocity. The Lorentz equations are still used; however, instead of being called the
Lorentz contraction they are now called the Lorentz transformation.
Einstein completed his special theory of relativity in 1905 and it was published in
Annelen der Physik under the title “on the electrodynamics of moving bodies.” The
simplified equations for relativistic length (distance), time, and mass are given below.
Any units will work in these equations, as long as the same units are chosen for the
velocities of the object and the speed of light.
Where:
l is the length of the body while in motion
lo is the length while the object is at rest
t is the time elapsed for the body in motion
to is the time elapsed if the body was at rest
m is the mass of the body while in motion
mo is the rest mass (proper mass)
v is the velocity of the body
c is the velocity of light
You may be asking yourself “Since velocity is relative, how do we know which observer
is moving and which observer is stationary?” From the vantage point of each observer, it
is the other that is moving; therefore, the length, time and mass for the observer will be lo,
to, and mo. The relative length, time, and mass will be what each observer will see and
measure for the other. If Sam and Sue pass each other in identical rocket ships at a
relative speed of 80% of the speed of light, each will measure their own mass, length, and
tick of a clock to be what it would be as if they were stationary. To ask which one is
really moving in an invalid question because all motion is relative. All that matters is
that the difference in their velocities is 80% of the speed of light. If Sam and Sue both
held up meter sticks to the window as the rockets passed each other, each would note that
the other meter stick would look shorter than their own. Even though the other’s meter
stick would still read one meter, it would measure 0.6 meters long to the other observer.
In fact, each one would see the other as being flattened or squished in the direction of
motion. They could argue about which of them has the correct measurement; however, if
they understand relativity they would know that they are each right with respect to their
own frame of reference.
Similarly, if they devised an experiment to determine the mass of the other’s rocket, they
would find that the rockets were not identical as they once thought. The mass that they
would each measure would result in a mass that is 1.67 times greater than their own. If
they held their watches up as they passed, each one would perceive the other watch to be
slower than their own. While the other watch ticked off 10 seconds, their own watch
would tick off 16.7 seconds. If one or both turned their rocket and adjusted their speed so
they were going the same speed and the same direction a second, a meter, and a kilogram
would again be the same for both because they would share the same frame of reference.
This all works out nicely for mass and distance; however, time can be more difficult to
grasp. As Sam and Sue passed each other, Sam would see Sue aging slower and Sue
would see Sam aging slower. Again both are right from their own perspective. Well,
you ask, which one will live longer? If both are 40 when they pass and both live to be 60
(according to their own reference frames), Sue (and the occupants of her craft) would
perceive that Sue died first and they would be correct. On the other hand, Sam (and the
occupants of his craft) would perceive that he died first, and they would be right. If the
occupants of each craft had powerful telescopes so they could watch what happens within
the other craft as they continued to speed away from each other, they would all verify that
they are right (remember that it takes time for light to travel). If your mind is starting to
hurt right now, you are starting to grasp relativity; or at least realizing that our every day
common sense perception of reality is not valid.
This might be a good place to take a break. Relax a few minutes and come back for
another example. I promise this one will not be so bad; in fact it is fun in that it
demonstrates that one type of time travel is possible.
For this example, let’s have Sue and Sam as twins on planet earth. On Sue’s 40th
birthday she takes off in a space craft and travels at 80% of the speed of light for ten
years (from her frame of reference, 5 years out and 5 years back) and then returns to her
home town of Spring Valley. Now, as Sue is leaving planet earth, she looks back through
her telescope. She sees her brother aging slower than she, and as her brother looks up at
Sue through his telescope he sees that Sue is aging slower than he is. The key to this
problem is that Sue turns around and eventually enters Sam’s reference frame. As Sue
turns around and returns, her reference frame, while always correct from her perspective,
changes due to the acceleration and deceleration involved and comes into line with Sam’s
frame of reference. The simplified version of this calculation is to use Sam’s frame of
reference to calculate Sue’s age. During this trip Sam watched Sue age 10 years while he
aged 16.7 years. Sue will be 50 when she returns and Sam will be 56.7 years old. She
will actually be younger than Sam.
Atomic clocks placed in jets and flown around the earth have demonstrated that this type
of “time travel” is possible. Atomic clocks were needed because the jets do not travel
fast when compared to the speed of light; therefore, the slowing down of time for the
clock and airborne occupants of the jet was barely measurable. If the occupants were
able to approach the speed of light time would still seem to go on normally for them;
however, when they returned to earth they would find that it was at a near stand-still as
compared to the time that passed on earth.
Note that the above example includes acceleration. Changing direction is one type of
acceleration and since Sue changed direction she underwent acceleration. The
acceleration aspect of this problem places it out of the realm of special relativity and into
general relativity theory. The purpose of giving the example here is simply to help you
understand the nature of time with respect to speed.
For the final example, imagine that we are in a rocket traveling at 90% the speed of light.
From the rocket we then fire a bullet in the same direction that the rocket is traveling.
Let’s have the bullet also travel at 90% the speed of light with respect to the rocket. The
Galilean (and our every day common sense answer) would be to add these two velocities
together to obtain the expected velocity of the bullet of 1.8 times the speed of light. Of
course, we understand that nothing can exceed the speed of light so we realize that this
must be incorrect. Since meter sticks are shortening and time is slowing down the sum of
the velocities as determined by an observer on the ground will actually be about 99% the
speed of light. Our ever day experience of simply adding velocities is not correct, at least
for objects moving very fast. The way nature “really” behaves is given by the following
formula where u is the velocity of the rocket, v is the velocity of the bullet and c is the
speed of light. Note that for ever day speeds the denominator simplifies to
approximately one and the Galilean method of simply adding the velocities reappears.
Velocity (from ground) = (u + v)/(1 + uv/c2)
Even though the bullet is traveling at 90% the speed of light from the perspective of
someone on the rocket, it is only traveling 9% faster than the rocket from the reference
frame of someone on the ground. What is the trade off here? What happened to the rest
of the velocity of the bullet? The trade off is that the person on the ground observes the
bullet as shorter, “aging slower,” and more massive. As Einstein would later show, some
of the energy of the bullet (the velocity that is “lost”) shows up as an increase in mass.
To help visualize the relationship between time and speed, think of our journey through
space-time as an arrow of given length. The length of this arrow represents how rapidly
we are moving through space and time. Mathematically the length of the arrow is equal
to how fast we are moving through space added to how fast we are moving through time.
The length of the arrow never changes; however the time component and the speed in
space component can each change at the expense of the other. In order for the length of
the arrow to remain the same, if the speed in space component goes up, the speed in time
component must go down. If something is moving close to the speed of light, the speed
through time component is nearly the entire length of the arrow and the movement
through time component is near zero. If something is stationary, its speed through space
component is zero so all of its movement is through time. Something is moving through
time at maximum speed if it is stationary. The slowest something can move through time
would be if it is moving at the speed of light. Since all motion is relative to the reference
frame of the observer, the observer is always stationary and it is the observer’s
surroundings that seem to be moving. Time is stopped only for those things that are
moving at the speed of light as compared to us and time is stopped for them from our
frame of reference only, it continues on at maximum pace from their reference frame
because from their reference frame they are stationary. It is when they change direction
and speed to enter our frame of reference will they agree that they have been aging much
slower than us. Einstein later shows that objects that have mass cannot quite make it to
the speed of light.
To summarize, the length of the arrow is the speed of light. The combined motion of an
object’s speed through space and the object’s speed through time is always equal to the
speed of light. As an object moves faster through space its passage through time slows
down. In taking this concept to its limit, space turns into time and time turns into space
as a function of velocity.
E=mc2
In 1905, three months after submitting his paper on special relativity, Einstein submitted
for publication a three page paper that developed the equivalence between mass and
energy. The paper was titled “Does the Inertia of a Body Depend Upon Its EnergyContent?” and was published in Annalen der Physik in 1905. Einstein developed this
relationship using a relativistic analysis of kinetic energy and mass. His original form of
the equations was Ko - Ki = ½(L/c2) v2 where Ko - Ki is the change in kinetic energy, L is
an amount of radiant energy given off, c is the speed of light and v is the velocity of the
object. After he derives this equation he goes on to state:
“From this equation it directly follows that:
If a body gives off the energy L in the form of radiation, its mass diminishes by L/c2.
The fact that the energy withdrawn from the body becomes energy of radiation evidently
makes no difference, so that we are led to the more general conclusion that: the mass of a
body represents its energy-content; if the energy changes by L, the mass changes in the
same sense by L/c2.” Here, as throughout the rest of his work, Einstein is able to draw a
much broader meaning from a much more narrow equation. If we form an equation from
his last statement given above, we have ∆m = ∆E/c2 where ∆ is the Greek letter delta
which means change. He also states that the mass of a body represents its energy-content
so we can drop the delta sign and rearrange the formula to its more familiar form E =
mc2. In using this equation, energy will have units of Joules, mass will have units of
kilograms and the speed of light is 3 X 108 m/sec. These units work together because 1
Joule = 1 kg m2/sec2.
He ends the paper by stating that perhaps his theory could be tested using high energycontent compounds such as radium salts. Nuclear reactions, such as the decay of radium,
are perhaps the most obvious examples where we can see Einstein’s mass-energy
equivalence in action; however, the most profound implication of his equation is that all
energy transformations result in a change in mass. Mass is, in fact, a measure of the total
energy an object contains.
Einstein’s special theory of relativity had overturned 200 years of physics. There is no
longer an absolute frame of reference for time or motion. Time and distance are
dependent on velocity. As something approaches the speed of light, time slows to a near
standstill and distance shrinks to nearly nothing. The universe has a speed limit which is
the speed of light. The speed of light is always measured to be the same regardless of the
position or velocity of the observer because of the simultaneous changes in time and
distance. Mass and energy are interchangeable; in fact, the total energy of an object is
incorporated in its mass. This includes every type of energy including nuclear, chemical,
thermal, as well as changes in kinetic energy. Give this, the total amount of energy
contained in an object can simply be found by measuring its mass. As an object speeds
up or changes in energy in any way its mass also changes. Conceptually this idea is
sound; however, from a practical perspective, we often cannot measure the change in
mass involved. Since a very small change in mass represents a tremendous amount of
energy, we do not have balances precise enough to measure the small change of mass that
accompanies most types of energy changes. Although often difficult to measure, these
changes are real and have been demonstrated for particles moving at very high speeds
and for nuclear reactions.
Einstein’s discoveries required an entirely new way of thinking about the universe. This
is difficult because it violates our common sense. Einstein once said that common sense
is that prejudice learned before the age of eighteen. Our common sense is based on our
everyday experiences from childhood. The relativistic effects are not measurable unless
something is moving at very high velocities. Since these relationships are not part of our
experience we have a difficult time accepting them.
As difficult as Einstein’s ideas were to comprehend and accept, his theories were great
news for physics. The problem of the speed of light had been solved. Marie Curie’s
“unlimited” energy in radium could now be explained in terms of a transformation of
mass to energy. Furthermore, Newton’s work was not lost; his equations emerged out of
Einstein’s for speeds and situations within our everyday experience. The good news was
that relativity yielded a richer, more accurate, and much deeper understanding of the
universe.
1905 was an extraordinary year for Einstein. During this time:

He explained the Photoelectric Effect which helped initiate quantum theory.

He correctly explained Brownian motion, which was the first hard physical
evidence for the existence of atoms and molecules.

He put forth his special theory of relativity which toppled the framework of
classical Newtonian physics.
Recall from the last unit that the only other year to even come close to this was in 1666
when the 27-year-old Newton:

Created his universal law of gravitation.

Invented integral and differential calculus.

Developed his theory of white light being composed of all colors of light.
Development of General Relativity
In 1905, almost as soon as Einstein worked out the special theory of relativity, he set his
sights on bigger game: general relativity. There was a glaring hole in Einstein’s special
theory of relativity- his theory only dealt with objects that were not being accelerated;
that is, not changing direction or changing speed. Most objects in the universe are
orbiting, which is a change in direction (i.e. a type of acceleration). Objects are
constantly speeding up or slowing down. The special theory was not general enough to
deal with most objects in the universe.
In 1907 Einstein had been asked to write an article summarizing the state of relativity.
This gave him the opportunity to step back and analyzing relativity in the context of all of
physics. He realized that the accepted theory of gravity did not fit well within the
relativistic interpretation of the universe. This presented Einstein with a real challenge.
Gravity in action dealt with what the special theory left out, velocities that are not
constant, including the acceleration produced by the pull of gravity. Gravity also
highlighted a clash between Newton’s gravity and relativity. The clash dealt with time.
Time plays no role in Newton’s theory of gravity. Gravity is always there, to be felt
immediately whenever two objects appear. In Newton’s universe, gravity acts
instantaneously across space. Einstein’s universe had a speed limit. Nothing can travel
faster that the speed of light, not even the force of gravity.
So Einstein set out to develop a theory of gravity of his own. He outlined the beginnings
of the work in the 1907 review paper on the applications of his special theory of relativity
to various branches of physics. He was sitting at his desk at the patent office mulling
over the nagging question of gravity. He looked out the window and spotted a worker on
the roof, and this inspired a thought experiment. What if the man should fall, what would
he feel as he was falling? He would feel no forces at all acting on him; he would not feel
his own weight. The feeling (or rather the lack of feeling of gravitational pull) of
someone in free fall was identical to the feeling someone would have while floating in
outer space. He later changed his thought experiment to someone in an elevator that was
in freefall because of a broken cable. Since the person could not see their surroundings,
there would be no way for them to tell if they were in free fall or floating in outer space.
They could move around the elevator as if they were weightless.
Einstein had stated that it was deeply dissatisfying to him that, although the relation
between inertia and energy is so beautifully derived (in the equation E = mc2), there was
as yet no relation between inertia and weight. Inertia (mv) is a function of mass and
weight is a measure of force (the pull of gravity on an object). Newton’s equation for
gravity does relate mass with the force of gravity. Einstein’s original form of E = mc2
was in terms of kinetic energy and mass. Force does not show up in his equation.
Einstein’s derivation of E = mc2 showed that as the kinetic energy of an object increased
the objects mass also increased by a specific amount. It is in the broadest interpretation
of the equation E = mc2 that we find that the mass of an object is actually a measure of
the total amount of energy an object contains, and that mass and energy are
interchangeable. His simple equation gives us a deep insight into nature. He wanted to
find a similar insight concerning gravity. Einstein needed a new way of looking at
gravity and he decided to use the observation that freefall in gravity was “equivalent” to
weightlessness in space as the starting point to develop a new theory of gravity. His hope
was that this insight would be as fruitful as his insight of the speed of light being
constant. Recall that he used the speed of light insight as the starting point for his Special
Theory of Relativity. Einstein coined the phrase “the equivalence principle” to relate
equivalent experiences such as free fall and floating in outer space.
Einstein recognized another equivalence, that of acceleration and gravity. Another
thought experiment demonstrates this. If we place a person in a rocket being accelerated
in outer space they are going to be pulled toward the direction opposite to the direction of
motion. Think of pull you feel when a car is accelerating. A person in a rocket that is
under constant acceleration feels the same pull that they would feel from gravity while on
earth. One way to create “gravity” in space is to spin or otherwise accelerate a space
station. Acceleration creates the same effect as gravity. If you experience two G’s of
force while on a roller coaster, the acceleration is creating a force twice the pull of
Earth’s gravity. Gravity is equivalent to acceleration. Any experiments conducted in
each situation would yield the same result. All physical laws remain the same in the
reference frame of someone under constant acceleration and someone who is standing on
a planet experiencing the pull of gravity. This is the equivalence principle at work.
Weight is simply the perception of a change in the inertial motion (velocity) of an object,
regardless of what causes the change. The change could be caused by acceleration or a
gravitational field on earth. Einstein felt that, as he moved forward with his theory, he
needed to describe gravity as a property of mass in motion.
Einstein has already realized that gravity bent light. Since energy and mass are
interchangeable, light should be affected by gravity according to the mass equivalent of
its energy. Einstein explored this concept with the use of his equivalence principle and
another thought experiment. The thought experiment consists of a rocket ship with a
flashlight fastened outside a window and with the light beam being directed from the
window the opposite wall, perpendicular to the direction of motion. If the rocket ship is
stationary or moving at constant velocity, the light beam will go directly from one side to
the other in a straight line; however, what will happen to the light if the rocket is
continually being accelerated? This is easiest to visualize if we have the rocket starting
from a stationary position and accelerating. Before takeoff the beam goes directly across
the cabin. As the rocket takes off it will move upward as the light travels across the
cabin. Since it takes a small amount of time for the light to travel from one side of the
rocket to the other and the rocket will have moved upward in that time, the light will
strike the opposite wall slightly lower that it did when the rocket was at rest. To someone
in the rocket, the light beam would appear bent. The conclusion is that an accelerating
reference frame bends light. Now, using Einstein’s equivalence principle, we know that
an accelerating frame of reference is equivalent to a frame of reference that is in a
gravitational field. We can conclude from this that light will also be bent by a
gravitational field. The greater the acceleration, or the greater the “gravitational pull,”
the more the light will be bent.
The next step is to find the relativistic connection between gravity and time. Einstein’s
original thought experiment utilized the red shift of light due to gravity. His experiment
and conclusions were valid but difficult to follow and understand. The following thought
experiment presented by Einstein in Berlin is easier to understand. We will use the same
rocket we used in the last thought experiments only now we fit it with two clocks. One
clock is at the top of the rocket and the other clock is at the base of the rocket. The clock
at the top of the rocket sends out a pulse of light every one second on that clock. While
the rocket is stationary both clocks are synchronized and ticking away the one second
intervals in unison. As the rocket accelerates, the rear clock will accelerate towards the
light coming from the rocket at the nose of the rocket. Since the light from the front now
has to travel less distance to reach the rear clock, it will arrive at the rear clock slightly
sooner than it would have if it had to travel the full distance (if the rear clock hadn’t
moved forward to meet it). This will result in the light pulses arriving at the rear clock in
less that one second intervals. The rear clock will be ticking slower than the observed
ticking of the forward clock. Note that the light beams given off of the front clock do not
have to be flashes of light, they can simply be the light beams reflected from the front
clock giving us the image of the front clock. The conclusion is that the rear clock is
running slower (or that the front clock is running faster). Since it is the rear clock that is
being accelerated towards the light rays that have already left the surface of the front
clock, we conclude that acceleration slows time down. Even though this experiment
makes it seem as if the slowing down of the rear clock is the result of some type of
optical illusion having to do with acceleration and light, Einstein had the genius to realize
that the rear clock was actually physically running slower and this was no mere illusion.
Time slows down for objects being accelerated. The next insight was that, since
according to the equivalence principle, acceleration and gravity are equivalent, gravity
also slows time down. The greater the acceleration, or the greater the gravity, the slower
time will move.
The concept of spacetime as a unified whole is a consequence of special relativity;
however it was first explicitly proposed by someone other than Einstein. In 1908 one of
Einstein’s old math professors from the polytechnic (Hermann Minkowski) rewrote
Einstein’s special theory of relativity in a more graceful and rigorous mathematical
treatment. Einstein and his old professor did not get along. Einstein had cut classes to
focus on areas of study that he thought were more exciting and important. His old
professor had called him a lazy dog and was surprised at Einstein’s success in creating
his special theory of relativity. When Einstein first saw the rewrite, he commented that
now that the mathematicians had gotten hold of his theory, even he no longer understood
it. Einstein was paying the price for skipping out on his math class.
Minkowski saw something deeper in Einstein’s theory, and his new math treatment of the
theory brought out a much more elegant and profound relationship between time and
space. Space and time form a four dimensional unity. In 1908 Minkowski wrote:
“The views of space and time which I wish to lay before you have sprung from the soil of
experimental physics, and therein lies their strength. They are radical. Henceforth space
by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind
of union of the two will preserve an independent reality.”
Few people are able to see the beauty and significance of this insight and it even took
Einstein two years to come to grips with it. For most people this is simply the three
dimensions of space added to time to make four dimensions. In relativity, what happens
in space (i.e. objects in motion) affects time. Space and time cannot be separated.
Minowiki linked time and space into a four dimensional fabric using the power of
symmetry. Other examples of this type of unification are energy/matter and
electricity/magnetism. Einstein’s theory basically said that different observers will
disagree about events and then went on to explain how and why they disagree.
Minkowski created a 4 dimensional coordinate system in which time had equal footing to
the three space coordinates. Minkowski reworked the Pythagorean Theorem into four
dimensions and showed how to calculate the shortest time-space distance between two
events. This shortest time-space interval was called the absolute interval. He was able to
create a fabric of space/time and show the relationship between events within that fabric
in such a way that all observers could agree on that relationship. He coined the term
“world-line” to refer to the space-time track that an individual person or particle makes in
the four dimensional time-space fabric as we move and interact during our existence.
As relativity became better understood there was a growing realization that space-time
might be curved. The following thought experiment can be used to demonstrate this.
Imagine a disk (like an old vinyl record on a turntable). This turntable can turn
exceedingly fast, so fast that the outside of the record will be traveling close to the speed
of light. We know that as objects move faster they become shorter. If we take the
circumference of the outside of the record we see that it is traveling much faster than a
circumference near the center of the record. They both have the same revolutions per
minute; however, the outside circumference is traveling a greater distance for one
revolution. Since objects shorten in the direction of motion as a function of velocity, the
outside circumference must be shortened by a greater percentage. This distorts the shape
of the record into a dome - thus the flat two dimensional disk is transformed into curved
space. While the record is stationary, it is flat; however, the faster it spins the more of
dome shape it must take.
The spinning of the earth must create the same phenomena. The poles must be shortened
proportionally less than the equator because of their difference in speed. Although it is
very difficult to imagine the curvature of space in either three or four dimensions it
became clear that space must be curved.
It became apparent that the curvature of space due to relativity was going to do away with
another long held assumption about reality. It had been assumed since ancient times that
the normal geometry relationships (the ones you studied in geometry) were valid
everywhere. One of the books that had a great impact on Einstein as a youth was a math
book on Euclidean Geometry. The logic and beauty of geometry impressed him. It had
always been assumed that space was linear and that these laws all held within our
universe. This assumption turned out to be false. If space can be curved, as relativity
indicates, the shortest distance between two points is not necessary a straight line, the
three angles of a triangle do not always add up to 180 degrees, and all of the other
geometrical relationships we assumed to be valid were no longer always valid. This also
meant that x,y,z coordinate system we use in math does not necessary hold for our
universe. Einstein realized that he needed a mathematics that applied to curved space.
For Einstein to continue with his theory he has to pick up where Minkowski left off,
apply it to curved space-time and then determine the degree of curvature using the new
physical principles he was developing. The problem was that Einstein did not have the
math skills to do this. This required tensor calculus (the mathematics of curved surfaces).
It is ironic that this was once considered the most useless branch of mathematics and it
was now required to define the fabric of the universe. Einstein did pretty much what he
did while he was attending the polytechnic: he asked his former polytechnic classmate
(Marcel Grossmann) for help. While at the polytechnic Einstein would use Grossmann’s
notes to help get him through the classes, such as math, that he was skipping. Now it was
Grossmann who helped guide Einstein through the difficult geometry of curved space.
This geometry had been developed in the 1850s by a German mathematician named
George Friedrich Riemann. Riemann generalized geometry to the point where he
considered geometry in any number of dimensions and situations in which measurements
changed from point to point in space but in such a way that one could transform one set
of measurement into another according to a fixed rule.
Einstein and Grossmann reworked Minkowski’s four dimensional, linear space-time. In
doing so they created a four dimensional positively curved space-time in which the
curvature of space-time was determined by the amount of mass and energy contained
within it. Within this new theory, gravity is simply the curvature of space-time as
determined by the amount of mass and energy. Large masses, like a planet or a star,
deform the space-time around them. When a comet or floating space ship is deflected by
the earth’s gravitational field, it is not being “pulled” by the earth, but rather is merely
following the shortest path available to it as determined by the increased curvature of
space-time near massive objects. The occupants of the spacecraft feel no acceleration
acting on them when they are in freefall, because there are no forces acting on them; they
are merely following the natural curvature of space-time. It is when they fight to leave
this natural path with their booster rockets that they feel G-forces acting on them. If you
feel the G-force, you are being accelerated. We feel a G-force of one because the ground
we are standing on is preventing us from following the natural curvature of space-time
we should be taking to the center of the earth. The acceleration is us simply standing on
the surface of the earth. It is exactly backwards of what we are used to thinking. The
force field of gravity does not pull and thereby accelerate objects; rather, it distorts space-
time itself and objects follow this curvature of space unless they are acted upon by an
unbalanced force, such as the surface of the earth pushing against them.
Finally in 1913 Einstein published what he believed was the final form of the new, more
general relativistic theory. It contained specific predictions, including the amount of
deflection of light around the sun - the number he hoped to confirm with Freundlich’s
eclipse expedition.
Einstein felt that they had worked out the gravitation problem to his satisfaction; however
a flaw in the theory eventually emerged concerning the equivalence principle. According
to Einstein’s original equivalence principle, observers in gravity or observers in an
accelerating rocket should tell no difference. His theory was now prediction that there is
a difference. In gravity, especially strong fields like a black hole, subjects get physically
closer. The closer the objects get to each other, the greater the “attraction” and this, in
turn, increases the rate of acceleration. Two people in different the different reference
frames could tell which one they were in by measuring the acceleration. If the rate of
acceleration was increasing they would be in a gravitational field. His theory
contradicted its own starting point; obviously this was a big problem.
Another problem was that Einstein’s equations could not explain the slight irregularity in
Mercury’s orbit. As Mercury orbits the sun, each path shifts slightly to the previous one.
Newton’s equations could not explain this; however, if Einstein’s equations were correct
they should be able to predict the actual orbit of Mercury. A third problem was that, in
its present form, Einstein’s theory also had trouble dealing with the gravitational situation
around a large rotating star. Einstein’s theory still needed work.
Einstein reworked the math of his theory using a more rigorous treatment and the
problems all evaporated. The equivalence principle held for all observers and the wobble
in Mercury’s orbit was explained. Also, Newton’s formulas fell out of Einstein’s
equations for situation involving relatively low gravity, such as planet earth.
The year Einstein completed and presented the final form of this general theory of
relativity was 1915. In 1919 there was additional confirmation of his General Theory of
Relativity. Two teams of astronomers positioned themselves to photograph the
“apparent” position of stars during a solar eclipse. As the starlight passed close to the sun
on its journey to the earth it would be bent by the sun’s gravitational field. The correct
amount of bending would confirm Einstein’s theory. The correct result made him even
more of a celebrity. He wrote concerning his theory, “I am now completely satisfied and
no longer doubt the correctness of the whole system.”
In all, it took him more than ten years from the completion of special relativity, eight
years from his first formal statement of the problem, to reach his final answer. He
stubbornly worked on, overcoming extreme difficulties one at a time, because solving
this problem was the next step in obtaining a better understanding of the universe. The
fact that the Special Theory of Relativity did not include gravity or acceleration made in
obvious that is was a very incomplete theory. Einstein wrote “The simpler our picture of
the external world and the more facts it embraces, the stronger it reflects in our minds the
harmony of the universe.” Einstein believed that a theory had to be beautiful and formally
graceful if it was to capture a glimpse of nature in action.
Another insight of Einstein’s was that the energy within a gravitational field itself added
to the gravity, because energy and mass are equivalent. Gravity, to a very small extent,
feeds itself. Newton’s theory did not recognize this, the final form of Einstein’s theory
did. This effect is very small so it is only noticeable under very strong gravity situations.
This effect is slightly noticeable in the orbit of Mercury and is not noticeable in the orbits
of the other planets. This is because Mercury is the planet closest to the sun and therefore
experiences the highest gravity. This additional small factor of gravity “feeding itself” is
mathematically nonlinear which makes the mathematics of the relationship much more
difficult to work with. This was one of the factors that had created additional problems
for Einstein.
In 1917 Einstein’s general theory of relativity was vindicated by the results obtained by
photographing the apparent position of distant stars during a solar eclipse. The
photographs would indicated how much the light is bent by a very strong gravitational
field as it passes next to the sun on its way to earth. From a theoretical perspective there
were three possible outcomes. The first would be that light would not be bent at all
because it has not mass. This outcome would support Newton’s strictest interpretation of
gravity. The second possibility would be that the light would be bent as per its equivalent
mass (using E = mc2) being inserting into Newton’s equation. This outcome would
support both E = mc2 and Newton’s work. The third possibility was that the light would
be bent even more as predicted by inserting the equivalent mass of light into Einstein’s
general theory of relativity equation. The results supported the general theory of
relativity and Einstein became a superstar overnight.
Einstein’s work was truly revolutionary, although Einstein never used that term to
describe his theories. He did not seek any glorification for his work; however he became
a celebrity none the less. In fact, the public seemed to be craving a superstar, and
Einstein was it. Einstein continually appeared in the newspapers and newsreels. The
following is a limerick style poem written during that time period.
Relativity Poem
There once was a young lady named Bright,
Who could travel much faster than light.
She set out one day,
In a relative way,
And came back the previous night.
His equation in its simplest form is called the Einstein equation. It reads Guv = 8πTuv
The above equation is actually far more complex than the above symbols make it appear.
Einstein’s equation is really several tensor calculus equations where both sides of the
simplified equation represent 4 X 4 matrices. The right side of the formula represents
space-time while the left side represents matter-energy. The equation as a whole can
represent the entire universe (matter energy and space time). The physicist John Wheeler
describes it this way: the matter and energy side tell the space time (the universe) what
shape to be, while the space-time tells matter-energy (all that the universe contains) how
to move. Put the two sides of the equation together and the result is a universal theory, an
account of the shape of the cosmos, its evolution, and even, potentially, its ultimate fate.
The equation is used by first making a few assumptions to define the system the equation
is being applied to. For example, using the equation to describe a star would require
different mathematical input assumptions than if the equation was being used to describe
the universe as a whole. The equation is then solved and each particular solution can be
considered another prediction made by the relativity theory. Some of these initial
predictions were so weird that they were not taken seriously by most physicists. For
example, in 1915 a brilliant mathematician named Karl Schwarzschild solved the
relativity equation for an idealized star that was squeezed smaller and smaller to create an
object of greater and greater density. He found that at a certain point the curvature of
space time becomes infinite; that is, space time itself would curve in on itself. The
distance from the center of the object where space time curvature becomes infinite is a
boundary beyond which there is no escape. If anything, including light, got that close to
the object it would be “pulled in” by space-time with no hope of escape. Even Einstein
did not believe this represented a real object in nature, although he recognized the
validity of the math and presented the paper to the Academy in January of 1916 on behalf
of Karl Schwarzschild. Schwarzschild could not present because this was during WWI
and Schwarzschild was at the German front using his math skills to solve ballistics
problems for German artillery batteries. Schwarzschild was completing his relativity
work during any “down time” he had at the front. Unfortunately, Schwarzschild died in
May of 1916 at the early age of 42 from a skin disease he contracted while working on
the front lines.
You may have already recognized that what Schwarzschild was predicting with
Einstein’s equation. Schwarzschild’s math indicated the possibility of black holes. In
1938 J. Robert Oppenheimer and George Volkoff showed that black holes can form when
stars collapse. Black holes have since been located in space and are now well accepted.
That region of a black hole beyond which nothing can escape is called the event horizon.
The relativity equation can bring us to limit of infinite space time curvature (which is the
event horizon); however, at that point the equation breaks down and cannot penetrate into
the black hole. When space time curvature reaches infinity we have what physicist call a
singularity which is a point were the physics equations break down. Beyond this point
the laws of physics break down and because of this nothing can be said about the interior
of a black hole.
One of Einstein’s solutions to the relativity equation predicted the existence of “gravity
waves.” These waves would move through a gravitational field in the same way that
electromagnetic waves move through an electromagnetic field. They are also created in
much the same way as an electromagnetic wave and would also travel at the speed of
light. Electromagnetic waves form when electrically charged masses accelerate relative
to each other. According to relativity, gravity waves should be created by two
gravitational masses accelerating relative to each other (like two stars colliding). There is
still no hard evidence for the existence of gravity waves; however, there is indirect
evidence and the general consensus is that they exist.
In 1916 Einstein started to question the structure of the universe as a whole. When he
was done in February of 1917, he had founded a new branch of physics which is modern
cosmology. He first argued that Newton’s universe would eventually result in a near zero
density of matter throughout the universe. This was because as stars collide and ricochet
into infinity the overall density should decrease. He then developed his own model of the
universe by starting with the following two assumptions.
1.
The universe contains an average density of matter which “is everywhere the
same and different from zero.”
2.
The universe is static (unchanging) with no change in its structure over time.
He expressed the above assumption mathematically, inserted it into his relativity
equation, and then solved the equation. The universe that emerged from the solution had
not limits, but was finite. It existed as a kind of four dimensional ball which has a finite
volume but around which one could travel forever without finding an end.
The above description is the basic structure of the universe hat we hold today; however,
there was a problem with this model. Einstein realized that a universe of this type would
collapse because of the total gravity of the system. We will revisit Einstein’s universe
and discuss how he dealt with the collapsing universe problem in the chapter on
cosmology. However, for you to understand modern cosmology you first need to learn
the basics of the other branch of physics that Einstein’s early work created, which is
quantum theory. Quantum theory is the topic we will cover in the next chapter.
In summary, the strengths of Einstein’s relativity theories are:

They work for all situations, even those with objects moving near the speed of
light, solving the shortcomings of Newton’s theory.

They explained the bending of light due to gravity.

They explain what causes gravity, space-time being warped by mass. Freefall is
“floating in space,” following the natural curves of space time.
The problem with relativity is that it is not a complete theory in that all forces are not
included. Einstein’s theory included the electromagnetic force and gravitational force;
however, the other two fundamental forces were discovered after Einstein’s theory.
These are two forces that act within the atom and are called the strong force and the weak
force. Since his theory does not include these forces it cannot be considered a complete
theory. There is also a need for the branch of physics called quantum mechanics to
describe the very small objects in the universe. Furthermore, the uncertainty principle of
quantum mechanics seems to disagree with the “spirit of relativity and Newtonian
physics”. Relativity and Newtonian physics were based on cause and effect, in which the
cause was a force. Nothing happened in the universe without one of the basic forces
making it happen. Light and the wave nature of particles seem to violate the “cause and
effect” nature of both Newton’s and Einstein’s theories.
Einstein was probably more aware than anyone of the few problems concerning general
relativity. In 1915 when Einstein finished formulating this theory of gravity and general
relativity, Einstein set his sights on developing a unified field theory that would unify his
theory of gravity with Maxwell’s theory of electromagnetism. His goal was to
accomplish this unification by expanding the general theory of relativity so that it would
“swallow up” and incorporate the theory of electromagnetism. By incorporating the two
he hoped to eliminate what he saw as problems of electromagnetism.
One problem that relativity had was working with very small particles such as electrons.
These particles were treated mathematically as points of mass and charge. This resulted
in point particles being represented by a “singularity” which is a point where field
strength goes to infinity. Having infinite field strength is an impossibility; therefore this
point representation of particles had to be wrong. Einstein wanted to replace this
singularity with a smooth deformation of space and time. If he could do this, subatomic
particles, such as the electron, would emerge as kinks or as some kind of small wrinkle
on the surface of space-time.
Einstein was back in the hunt, searching for “big game”. The “big game” he was seeking
was an even deeper understanding of the universe and he intended to use the few existing
problems of relativity as the starting point of his search. The challenges Einstein
accepted when he developed relativity were truly immense; however, the challenge he
was now accepting was insurmountable. He was now working at least 50 years ahead of
his time. The nucleus of the atom had only been discovered by Rutherford in 1911 and
was shrouded in mystery. The strong and weak nuclear forces were not really
recognized, let alone understood. These were later to prove an important part of the
puzzle.
Einstein was to spend the rest of his life in the quest for a unification theory. What kept
him going, however, were the clues he saw everywhere and that unification was one of
the grand schemes of the universe. He wrote, “Nature shows us only the tail of the lion.
But I do not doubt that the lion belongs to it even though he cannot at once reveal himself
because of his enormous size.”
The goal of ultimate unification (that is, a single equation to describe everything in the
universe) is still alive and well. As you shall see in the material on quantum mechanics
and string theory, there has been progress. Consider the dramatic changes in our concept
of the universe as a result of Einstein’s relativity. As science moves forward, the
surprises keep coming. It has been appropriately said that reality is not only stranger than
we imagine, it is stranger than we can imagine.