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Special Relativity
Special relativity is the theory developed by Albert Einstein
in 1905 to explain the observed fact that the speed of light is
a constant regardless of the direction or velocity of one’s motion.
Einstein laid down two simple postulates to explain this strange
fact, and, in the process, derived a number of results that are
even stranger. According to his theory, time slows down for objects
moving at near light speeds, and the objects themselves become shorter
and heavier. The wild feat of imagination that is special relativity
has since been confirmed by experiment and now plays an important
role in astronomical observation.
The Michelson-Morley Experiment
As we discussed in the chapter on waves, all waves travel
through a medium: sound travels through air, ripples travel across
water, etc. Near the end of the nineteenth century, physicists were
still perplexed as to what sort of medium light travels through.
The most popular answer at the time was that there is some sort
of invisible ether through which light travels. In 1879, Albert
Michelson and Edward Morley made a very precise measurement to determine
at what speed the Earth is moving relative to the ether. If the
Earth is moving through the ether, they reasoned, the speed of light
should be slightly different when hitting the Earth head-on than
when hitting the Earth perpendicularly. To their surprise, the speed
of light was the same in both directions.
For people who believed that light must travel through
an ether, the result of the Michelson-Morley experiment was
like taking a ride in a boat and discovering that the boat crossed
the wave crests at the same rate when it was driving against the
waves as when it was driving in the same direction as the waves.
No one was sure what to make of the Michelson-Morley experiment
until 1905, when Albert Einstein offered the two basic postulates
of special relativity and changed forever the way we think about
space and time. He asked all sorts of unconventional questions, such
as, “What would I see if I were traveling at the speed of light?”
and came up with all sorts of unconventional answers that experiment
has since more or less confirmed.
The Basic Postulates of Special Relativity
Special relativity is founded upon two basic postulates,
one a holdover from Newtonian mechanics and the other a seeming
consequence of the Michelson-Morley experiment. As we shall see,
these two postulates combined lead to some pretty counterintuitive
results.
First Postulate
The laws of physics are the same in all inertial
reference frames.
An inertial reference frame is one where
Newton’s First Law, the law of inertia, holds. That means that if
two reference frames are moving relative to one another at a constant
velocity, the laws of physics in one are the same as in the other.
You may have experienced this at a train station when the train
is moving. Because the train is moving at a slow, steady velocity,
it looks from a passenger’s point of view that the station is moving
backward, whereas for someone standing on the platform, it looks
as if the train is moving forward.
Einstein’s first postulate tells us that neither the passenger
on the train nor the person on the platform is wrong. It’s just
as correct to say that the train is still and the Earth is moving as
it is to say that the Earth is still and the train is moving. Any
inertial reference frame is as good as any other.
Second Postulate
The speed of light in a vacuum is a constant—
m/s—in every reference frame, regardless
of the motion of the observer or the source of the light.
m/s—in every reference frame, regardless
of the motion of the observer or the source of the light.This postulate goes against everything we’ve learned about
vector addition. According to the principles of vector addition,
if I am in a car moving at 20 m/s and collide with
a wall, the wall will be moving at 20 m/s relative
to me. If I am in a car moving at 20 m/s and collide with
a car coming at me at 30 m/s, the other car will be
moving at 50 m/s relative to me.
By contrast, the second postulate says that, if I’m standing
still, I will measure light to be moving at 
m/s, or c,
relative to me, and if I’m moving toward the source of light at
one half of the speed of light, I will still observe the light to
be moving at c relative to me.
m/s, or c,
relative to me, and if I’m moving toward the source of light at
one half of the speed of light, I will still observe the light to
be moving at c relative to me.By following out the consequences of this postulate—a
postulate supported by the Michelson-Morley experiment—we can derive
all the peculiar results of special relativity.
Time Dilation
One of the most famous consequences of relativity is time
dilation: time slows down at high speeds. However, it’s important
to understand exactly what this means. One of the consequences of
the first postulate of special relativity is that there is no such
thing as absolute speed: a person on a train is just as correct
in saying that the platform is moving backward as a person on the
platform is in saying that the train is moving forward. Further, both
the person on the train and the person on the platform are in inertial
reference frames, meaning that all the laws of physics are totally
normal. Two people on a moving train can play table tennis without
having to account for the motion of the train.
The point of time dilation is that, if you are moving
relative to me in a very highspeed train at one-half the speed of
light, it will appear to me that time is moving slower on board the
train. On board the train, you will feel like time is moving at
its normal speed. Further, because you will observe me moving at
one-half the speed of light relative to you, you
will think time is going more slowly for me.
What does this all mean? Time is relative.
There is no absolute clock to say whether I am right or you are
right. All the observations I make in my reference frame will be
totally consistent, and so will yours.
We can express time dilation mathematically. If I were
carrying a stopwatch and measured a time interval, 
, you would get a different measure, t,
for the amount of time I had the stopwatch running.
, you would get a different measure, t,
for the amount of time I had the stopwatch running. The relation between these measures is:
So suppose I am moving at one-half the speed of light
relative to you. If I measure 10 seconds on my stopwatch, you will
measure the same time interval to be:
This equation has noticeable effects only at near light
speeds. The difference between t and 
is only a factor of 
. This factor—which comes up so frequently
in special relativity that it has been given its own symbol, 
—is very close to 1 unless v is
a significant fraction of c. You don’t
observe things on a train moving at a slower rate, since even on
the fastest trains in the world, time slows down by only about 0.00005%.
is only a factor of . This factor—which comes up so frequently
in special relativity that it has been given its own symbol, —is very close to 1 unless v is
a significant fraction of c. You don’t
observe things on a train moving at a slower rate, since even on
the fastest trains in the world, time slows down by only about 0.00005%.Time Dilation and Simultaneity
Normally, we would think that if two events occur at the
same time, they occur at the same time for all observers, regardless
of where they are. However, because time can speed up or slow down
depending on your reference frame, two events that may appear simultaneous to
one observer may not appear simultaneous to another. In other words,
special relativity challenges the idea of absolute simultaneity
of events.
Example
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Because the Earth is moving at such a high speed relative
to the alien spaceship, time appears to move slower on Earth from
the aliens’ vantage point. To be precise, a minute of Earth time
seems to last:
Length Contraction
Not only would you observe time moving more slowly on
a train moving relative to you at half the speed of light, you would
also observe the train itself becoming shorter. The length of an
object, 
, contracts in the direction of
motion to a length 
when observed from a
reference frame moving relative to that object at a speed v.
, contracts in the direction of
motion to a length when observed from a
reference frame moving relative to that object at a speed v.
Example
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What is its length?
We can determine the length of the train using the equation
above:
.gif)
What is its width?
The width of the train remains at 5 m, since
length contraction only works in the direction of motion.
Addition of Velocities
If you observe a person traveling in a car at 20 m/s,
and throwing a baseball out the window in the direction of the car’s
motion at a speed of 10 m/s, you will observe the baseball
to be moving at 30 m/s. However, things don’t quite
work this way at relativistic speeds. If a spaceship moving toward
you at speed u ejects something in
the direction of its motion at speed 
relative to the spaceship, you will observe
that object to be moving at a speed v:
relative to the spaceship, you will observe
that object to be moving at a speed v:
Example
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The most precise way to solve this problem is simply to
do the math. If we let the speed of the spaceship be u = 0.5c and
the speed of the rocket relative to the spaceship be 
= 0.8c, then
the speed, v, of the rocket relative
to the Earth is
= 0.8c, then
the speed, v, of the rocket relative
to the Earth is
As we can see, the answer is (C). However, we could also
have solved the problem by reason alone, without the help of equations.
Relative to the Earth, the rocket would be moving faster than 0.8c,
since that is the rocket’s speed relative to a spaceship that is
speeding toward the Earth. The rocket cannot move faster than the
speed of light, so we can safely infer that the speed of the rocket
relative to the Earth must be somewhere between 0.8c and c.
Mass and Energy
Mass and energy are also affected by relativistic speeds.
As things get faster, they also get heavier. An object with mass 
at rest will have a mass m when
observed to be traveling at speed v:
at rest will have a mass m when
observed to be traveling at speed v:.gif)
Kinetic Energy
Because the mass increases, the kinetic energy of objects
at high velocities also increases. Kinetic energy is given by the
equation:
You’ll notice that as v approaches c,
kinetic energy approaches infinity. That means it would take an
infinite amount of energy to accelerate a massive object to the
speed of light. That’s why physicists doubt that anything will ever
be able to travel faster than the speed of light.
Mass-Energy Equivalence
Einstein also derived his most famous equation from the
principles of relativity. Mass and energy can be converted into
one another. An object with a rest mass of 
can be converted into an amount of energy,
given by:
can be converted into an amount of energy,
given by:
We will put this equation to work when we look at nuclear
physics.
Relativity and Graphs
One of the most common ways SAT II Physics tests your
knowledge of special relativity is by using graphs. The key to remember
is that, if there is a dotted line representing the speed of light,
nothing can cross that line. For instance, here are two graphs of
kinetic energy vs. velocity: the first deals with normal speeds
and the second deals with relativistic speeds:
In the first graph, we get a perfect parabola. The second
graph begins as a parabola, but as it approaches the dotted line
representing c, it bends so that it
constantly approaches c but never
quite touches it, much like a y = 1/x graph
will constantly approach the x-axis but never quite
touch it.
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