If you are a fan of science fiction, then
you know that "relativity" is a fairly common part of the
genre. For example, people on Star Trek are always talking
about the spacetime continuum, worm holes, time dilations and
all sorts of other things that are based on the principle of
relativity in one way or another. If you are a fan of science
you know that relativity plays a big part there as well,
especially when talking about things like black holes and
astrophysics.
If you have ever wanted to understand the fundamentals of
relativity, then this edition of How Stuff
Works will be incredibly interesting to you. In this
edition the major principles of the theory are discussed in an
accessible way so that you can understand the lingo and the
theories involved. Once you understand these concepts, you
will find that scientific news articles and science fiction
stories are much more interesting! The links section offers
three additional sources of information that you can tap into
if you want to learn more.
1.0  The Fundamental Properties of the
Universe
If you want to describe the universe as we
know it in its most basic terms, you could say that it
consists of a handful of properties. We are all familiar with
these properties  so familiar, in fact, that we take them
completely for granted. However, under special relativity many
of these properties behave in very unexpected ways! Let's
review the fundamental properties of the universe so that we
are clear about them.
Space
Space is the three dimensional
representation of everything we observe and everything that
occurs. Space allows objects to have lengths in the
left/right, up/down, and forward/backward directions.
Time
Time is a fourth dimension. In normal
life, time is a tool we use to measure the procession of
events of space. But time is something more. Yes, we use time
as a "tool", but time is essential for our physical existence.
Space and time when used to describe events can't be clearly
separated. Therefore, space and time are woven together in a
symbiotic manner. Having one without the other has no meaning
in our physical world. To be redundant, without space, time
would be useless to us and without time, space would be
useless to us. This mutual dependence is known as the
Spacetime Continuum. It means that any occurrence in our
universe is an event of Space and Time. In Special Relativity,
spacetime does not require the notion of a universal time
component. The time component for events that are viewed by
people in motion with respect to each other will be different.
As you will see later, spacetime is the death of the concept
of simultaneity.
Matter
Matter in the most fundamental
definition is anything that takes up space. Any object you can
see, touch, or move by applying a force is matter. Most people
probably remember from school that matter is made up of
millions of billions of tightly packed atoms. Water, for
example, is the compound H2O, meaning two hydrogen atoms
combined with one oxygen atom forms one molecule of water.
To fully understand matter let's look at the atom. It is
now generally accepted that atoms are made up of three
particles called neutrons, protons, and electrons. The
neutrons and protons are found in the nucleus (center) of the
atom and the electrons reside in a shell surrounding the
nucleus. Neutrons are heavy particles, but they have no charge
 they are neutral. Protons are also heavy particles and they
have a positive charge. Electrons are light particles and they
are negatively charged. There are many important features that
arise from considering the number of these particles in each
atom. For example, the number of protons an atom has will
determine the atom's place on the periodic table, and it will
determine how the atom behaves in the physical universe. (See
the HSW
article entitled "How Nuclear Radiation Works" for a
further discussion of atoms and subatomic particles.)
Motion
Anything that is in the act of changing
its location in space is said to be in motion. As you will see
later, consideration of "motion" allows for or causes some
very interesting concepts.
Mass
Mass has two definitions that are
equally important. One is a general definition that most high
school students are taught and the other is a more technical
definition that is used in physics.
Generally, mass is defined as the measure of how much
matter an object or body contains  the total number of
subatomic particles (electrons, protons and neutrons) in the
object. If you multiply your mass by the pull of earth's
gravity, you get your weight. So if your body weight is
fluctuating, by eating or exercising, it is actually your mass
that is changing. It is important to understand that mass is
independent of your position in space. Your body's mass on the
moon is the same as its mass on the earth. The earth's
gravitational pull, on the other hand, decreases as you move
farther away from the earth. Therefore, you can lose weight by
changing your elevation, but your mass remains the same. You
can also lose weight by living on the moon, but again your
mass is the same.
In physics, mass is defined as the amount of force required
to cause a body to accelerate. Mass is very closely related to
energy in physics. Mass is dependent on the body's motion
relative to the motion of an observer. If the body in motion
measured its mass, it is always the same. However, if an
observer that is not in motion with the body measures the
body's mass, the observer would see an increase in mass when
the object speeds up. This is called relativistic mass.
It should be noted that physics has actually stopped using
this concept of mass and now deals mostly in terms of energy
(see the section on the unification of mass and energy) . At
this stage, this definition of mass may be a little cloudy,
but it is important to know the concept. It should become
clearer in the special relativity discussion. The important
thing to understand here is that there is a relationship
between mass and energy.
Energy
Energy is the measure of a system's
ability to perform "work". It exists in many forms…potential,
kinetic, etc. The law of conservation of energy tells us that
energy can neither be created nor destroyed; it can only be
converted from one form to another. These separate forms of
energy are not conserved, but the total amount of energy is
conserved. If you drop a baseball from your roof, the ball has
kinetic energy the moment it starts to move. Just before you
dropped the ball, it had only potential energy. As the ball
moves, the potential energy is converted into kinetic energy.
Likewise, when the ball hits the ground, some of its energy is
converted to heat (sometimes called heat energy or heat
kinetic energy). If you go through each phase of this scenario
and totaled up the energy for the system, you will find that
the amount of energy for the system is the same at all times.
Light
Light is a form of energy, and
exists in two conceptual frameworks: light exhibits properties
that have characteristics of discrete particles (eg. energy is
carried away in "chunks") and characteristics of waves (eg.
diffraction). This split is known as duality. It is important
to understand that this is not an "either/or" situation.
Duality means that the characteristics of both waves and
particles are present at the same time. The same beam of light
will behave as a particle and/or as a wave depending on the
experiment. Furthermore, the particle framework (chunks) can
have interactions which can be described in terms of wave
characteristics and the wave framework can have interactions
that can be described in terms of particle characteristics.
The particle form is known as a photon, and the waveform is
known as electromagnetic radiation. First the photon…
A photon is the light we see when an atom emits energy. In
the model of an atom, electrons orbit a nucleus made of
protons and neutrons. There are separate electron levels for
the electrons orbiting the nucleus. Picture a basketball with
several sizes of hulahoops around it. The basketball would be
the nucleus and the hulahoops would be the possible electron
levels. These surrounding levels can be referred to as
orbitals. Each of these orbitals can only accept a
discrete amount of energy. If an atom absorbs some energy, an
electron in an orbital close to the nucleus (a lower energy
level) will jump to an orbital that is farther away from the
nucleus (a higher energy level). The atom is now said to be
excited. This excitement generally will not last very
long, and the electron will fall back into the lower shell. A
packet of energy, called a photon or quanta, will be released.
This emitted energy is equal to the difference between the
high and low energy levels, and may be seen as light depending
on its wave frequency, discussed below.
The wave form of light is actually a form of energy that is
created by an oscillating charge. This charge consists of an
oscillating electric field and an oscillating magnetic field,
hence the name electromagnetic radiation. We should note that
the two fields are oscillating perpendicular to each other.
Light is only one form of electromagnetic radiation. All forms
are classified on the electromagnetic spectrum by the number
of complete oscillations per second that the electric and
magnetic fields undergo, called frequency. The
frequency range for visible light is only a small portion of
the spectrum with violet and red being the highest and lowest
frequencies respectively. Since violet light has a higher
frequency than red, we say that it has more energy. If you go
all the way out on the electromagnetic spectrum, you will see
that gamma rays are the most energetic. This should come as no
surprise since it is commonly known that gamma rays have
enough energy to penetrate many materials. These rays are very
dangerous because of the damage they can do to you
biologically (See the HSW
article entitled "How Nuclear Radiation Works" for a
further discussion of gamma radiation.). The amount of energy
is dependent on the frequency of the radiation. Visible
electromagnetic radiation is what we commonly refer to as
light, which can also be broken down into separate frequencies
with corresponding energy levels for each color.
As light travels its path, through space, it often
encounters matter in one form or another. We should all be
familiar with reflection since we see bright reflections when
a light hits a smooth shiny surface like a mirror. This is an
example of light interacting with matter in a certain way.
When light travels from one medium to another, the light
bends. This is called refraction. If the medium, in the path
of the light, bends the light or blocks certain frequencies of
it, we can see separate colors. A rainbow,
for example, occurs when the sun's light becomes separated by
moisture in the air. The moisture bends the light, thus
separating the frequencies and allowing us to see the unique
colors of the light spectrum. Prisms also provide this effect.
When light hits a prism at certain angles, the light will
refract (bend), causing it to be separated into its individual
frequencies. This effect occurs because of the shape of the
prism and the angle of the light.
If you look closely at what happens as the light wave
enters the prism in the second diagram, you will notice that
it bends down. This bending occurs because the light travels
faster through the air than it does through the prism. When
the lower portion of the wave enters the prism, it slows down.
Since the upper portion of the wave (still in the air) is
traveling faster than the lower portion, the wave bends.
Similarly, as the wave exits the prism, the upper portion
exits first and begins travelling faster than the lower
portion that is still in the prism. This speed differential
causes the wave to bend once again. Think of a skateboard
rider going down the driveway. If the rider turns and goes
into the grass, his body will lunge forward and actually fly
off of the board if he is traveling fast enough originally.
This is analogous to light bending as it goes through
different mediums. The skateboard and the rider are moving at
the same speed until the wheels hit the grass. Now suddenly,
the skateboard is traveling slower than the rider is, so the
rider begins to bend forward (the rider is trying to continue
traveling at the same speed he was before the wheels hit the
grass).
Now that we have a little understanding of the composition
of light, we can begin to resolve the oft under explained
concept of "the speed of light". Since light itself is just a
form of electromagnetic radiation, the speed of light is just
an easy way of talking about the speed of electromagnetic
radiation in general. If you think about it, the speed of
light is the "speed of information". We can not acknowledge
that an event has occurred until the information about that
event reaches us. The information is contained in the
electromagnetic radiation from the event via a radio signal, a
flash of light etc. Any event is just an occurrence of space
and time, and any information that can be transmitted about an
event is emitted outward as radiation of some sort. The
information (electromagnetic radiation) from the event travels
at 186,000 miles/second in a vacuum. If you picture a long
train that begins to move forward from a stopped position, you
do not expect the very last car to begin moving
instantaneously. There is an amount of time that passes before
the last car begins to get pulled. Thus, there is an expected
delay for last car to "receive" the information that the first
car is moving and pulling. This delay is analogous to the
transfer of information in special relativity, but SR only
imposes an upper limit on the speed of the information; the
speed of light. You can make the train example as detailed as
you like, but regardless, you will always find that there can
be no reaction without a time delay of at least the speed of
light between the action and reaction. In the special
relativity section we will further discuss the importance of
this speed.
2.0  Special Relativity
You are now familiar
with the major players in the universe: space, time, matter,
motion, mass, gravity, energy and light. The neat thing about
Special Relativity is that many of the simple properties
discussed in section 1 behave in very unexpected ways in
certain specific "relativistic" situations. The key to
understanding special relativity is understanding the effects
that relativity has on each property.
Frames of Reference
Lorentz Transformations
The Lorentz Transformations are mathematical
equations that allow us to transform from one coordinate
system to another. Why would we want to do this? Because
special relativity deals with frames of reference. When
you analyze properties from one frame to another, it is
necessary to first transform from one coordinate system
to another. Thus, we can utilize the Lorentz Transforms
to convert length and time from one frame of reference
to another. For example, if you are flying in an
airplane and I am standing still on the ground, you
could apply the transformations to transform my frame of
reference into your frame of reference and I could do
the same for you in my frame of reference. The previous
statements imply that lengths and times are not the same
for objects that are in motion with respect to each
other. As unbelievable as this may seem, it is a result
of SR. Einstein utilized the transformations because
they provide a method of translating the properties from
one frame of reference to another when the speed of
light is held constant in both.

Einstein's special theory of
relativity is based on the idea of reference frames. A
reference frame is simply "where a person (or other observer)
happens to be standing". You, at this moment, are probably
sitting at your computer. That is your current reference
frame. You feel like you are stationary, even though you know
the earth is revolving on its axis and orbiting around the
sun. Here is an important fact about reference frames:
There is no such thing as an absolute frame of reference in
our universe. By saying absolute, what is actually
meant is that there is no place in the universe that is
completely stationary. This statement says that since
everything is moving, all motion is relative. Think about it 
the earth itself is moving, so even though you are standing
still, you are in motion. You are moving through both space
and time at all times. Because there is no place or object in
the universe that is stationary, there is no single place or
object on which to base all other motion. Therefore, if John
runs toward Hunter, it could be correctly viewed two ways.
From Hunter's perspective, John is moving towards Hunter. From
John's perspective, Hunter is moving towards John. Both John
and Hunter have the right to observe the action from their
respective frames of reference. All motion is relative to your
frame of reference. Another example: If you throw a ball, the
ball has the right to view itself as being at rest relative to
you. The ball can view you as moving away from it, even though
you view the ball as moving away from you. Keep in mind that
even though you are not moving with respect to the earth's
surface, you are moving with the earth.
The First Postulate of the Special Theory of
Relativity
The first postulate of the theory of
special relativity is not too hard to swallow: The laws of
physics hold true for all frames of reference. This is the
simplest of all relativistic concepts to grasp. The physical
laws help us understand how and why our environment reacts the
way it does. They also allow us to predict events and their
outcomes. Consider a yardstick and a cement block. If you
measure the length on the block, you will get the same result
regardless of whether you are standing on the ground or riding
a bus. Next, measure the time it takes a pendulum to make 10
full swings from a starting height of 12 inches above its
resting point. Again, you will get the same results whether
you are standing on the ground or riding a bus. Note that we
are assuming that the bus is not accelerating, but traveling
along at a constant velocity on a smooth road. Now if we take
the same examples as above, but this time measure the block
and time the pendulum swings as they ride past us on the bus,
we will get different results than our previous results. The
difference in the results of our experiments occurs because
the laws of physics remain the same for all frames of
reference. The discussion of the Second Postulate will explain
this in more detail. It is important to note that just because
the laws of physics are constant, it does not mean that we
will get the same experimental results in differing frames.
That depends on the nature of the experiment. For example, if
we crash two cars into each other, we will find that the
energy was conserved for the collision regardless of whether
we were in one of the cars or standing on the sidewalk.
Conservation of energy is a physical law and therefore, must
be the same in all reference frames.
The Second Postulate of the Special Theory of
Relativity
The second postulate of the special
theory of relativity is quite interesting and unexpected
because of what it says about frames of reference. The
postulate is: The speed of light is measured as constant in
all frames of reference. This can really be described as
the first postulate in different clothes. If the laws of
physics apply equally to all frames of reference, then light
(electromagnetic radiation) must travel at the same speed
regardless of the frame. This is required for the laws of
electrodynamics to apply equally for all frames.
This postulate is very odd if you think about it for a
moment. Here is one fact you can derive from the postulate:
Regardless of whether you are flying in an airplane or sitting
on the couch, the speed of light would measure the same to you
in both situations. The reason that is unexpected is because
most physical objects that we deal with in the world add their
speeds together. Consider a convertible approaching you at a
speed of 50 miles/hour. The passenger pulls out a slingshot
and shoots a rock 20 miles/hour at you. If you measured the
speed of the rock, you would expect it to be traveling at 70
miles/hour (the speed of the car plus the speed of the rock
from the slingshot). That is, in fact, what happens. If the
driver measured the speed of the rock, he would only measure
20 miles/hour, since he is already moving at 50 miles/hour
with the car. Now if that same car is approaching you at 50
miles/hour and the driver turns on the headlights, something
different happens? Since the speed of light is known to be
669,600,000 miles/hour, common sense tells us that the car's
speed plus the headlight beam speed gives a total of
669,600,050 miles/hour (50 miles/hour + 669,600,000
miles/hour). The actual speed would measure 669,600,000
miles/hour, exactly the speed of light. To understand why this
happens, we must look at our notion of speed.
Speed is the distance traveled in a given amount of time.
For example, if you travel 60 miles in one hour, your speed is
60 miles per hour. We can easily change our speed by
accelerating and decelerating. In order for the speed of light
to be constant, even if the light is "launched" from a moving
object, only two things can be happening. Either something
about our notion of distance and/or something about our notion
of time must be skewed. As it turns out, both are skewed.
Remember, speed is distance divided by time.
In the headlight example, the distance that you are using
in your measurement is not the same as the distance that the
light is using. This is a very difficult concept to grasp, but
it is true. When an object (with mass) is in motion, its
measured length shrinks in the direction of its motion. If the
object reaches the speed of light, its measured length shrinks
to nothing. Only a person that is in a different frame of
reference from the object would be able to detect the
shrinking  as far as the object is concerned, in its frame of
reference, its size remains the same. This phenomenon is
referred to as "length contraction". It means, for example,
that as your car approaches the speed of light, the length of
the car measured by a stationary observer would be smaller
than if the car was measured as it stood still. Look at Fig 2
and Fig 3 below.
In Fig 2 the car is stopped at the stop sign. In Fig 3 the
same car is moving past you. You will readily notice that the
moving car in the figure is shorter than the stopped car. Note
that the car would only be shorter in the direction it is
traveling, its height and width are not affected  only its
length. Length contraction only affects the length in the
direction you are traveling. Imagine that you are running
super fast toward an open door. From your perspective, the
distance from the front of the door opening to the back of the
door opening would decrease. From the doors perspective the
width of your body  the distance from your chest to your back
 would decrease.
Scientists feel that they have actually proved this notion
of length contraction. Therefore, in reality, all objects are
perceived to shorten in the direction they are traveling, if
they are viewed by someone who is not in motion with them. If
you are in a moving car and measure the length of the armrest,
you will never notice the change regardless of how fast you
are going, because your tape measure would also be shortened
from the motion.
In our lives we do not ever perceive length contraction
because we move at speeds that are very small with respect to
the speed of light. The change is too small for us to notice.
Remember the speed of light is 669,600,000 miles/hour or
186,400 miles/sec, so it is easy to see why our everyday
speeds are negligible.
The Lorentz Transforms allow us to calculate the length
contraction. How much contraction occurs is dependent on how
fast an object is traveling with respect to the observer. Just
to put some numbers to this, assume that a 12inch football
flies past you and it is moving at a rate of 60% the speed of
light. You would measure the football to be 9.6 inches long.
So at 60% the speed of light, you measure the football to be
80% of its original length (original 12 inch measurement was
made at rest with respect to you). Keep in mind that all
measurements are in the direction of the motion  The diameter
of the ball is not changed by the ball's forward motion. Here
are two points to keep in mind:
 if you ran beside the football at the same speed, 60%
the speed of light, you would always measure the length to
be 12 inches. This is no different than you standing still
and measuring the football while holding it.
 if a lady running with the football measured a ruler
that you are holding, she would measure you and your ruler
to be length contracted as well. Remember, she has equal
right to view you as being in motion with respect to her.
The Effect of Motion on Time
I mentioned that
time also changes with different frames of reference (motion).
This is known as "time dilation". Time actually slows with
motion but it only becomes apparent at speeds close to the
speed of light. Similar to length contraction, if the speed
reaches that of light, time slows to a stop. Again, only an
observer that is not in motion with the time that is
being measured would notice. Like the tape measure in length
contraction, a clock in motion would also be affected so it
would never be able to detect that time was slowing down
(remember the pendulum). Since our everyday motion does not
approach anything remotely close to the speed of light, the
dilation is completely unnoticed by us, but it is there. In
order to attempt to prove this theory of time dilation, two
very accurate atomic clocks were synchronized and one was
taken on a highspeed trip on an airplane. When the plane
returned, the clock that took the plane ride was slower by
exactly the amount Einstein's equations predicted. Thus, a
moving clock runs more slowly when viewed by a frame of
reference that is not in motion with it. Keep in mind that
when the clock returned, it had recorded less time than the
ground clock. Once reunited with the ground clock, the slow
clock will again record time at the same rate as the ground
clock (obviously, it will remain behind by the amount of time
it slowed on the trip unless resynchronized). It is only when
the clock is in motion with respect to the other clock that
the time dilation occurs. Take a look at Fig 4 and Fig 5
below.

Let's assume that the object under the sun in Fig 4 is a
light clock on wheels. A light clock measures time by sending
a beam of light from the bottom plate to the top plate where
it is then reflected back to the bottom plate. A light clock
seems to be the best measure of time since its speed remains
constant regardless of motion. So in Fig 4, we walk up to the
light clock and find that it takes 1 sec for the light to
travel from the bottom to the top and back to the bottom
again. Now look at Fig 5. In this example, the light clock is
rolling to the right, but we are standing still. If we could
see the light beam as the clock rolled past us, we would see
the beam travel at angles to the plates. If you are confused,
look at Fig 4 and you'll see that both the sent beam and
received beam occur under the sun, thus the clock is not
moving. Now look at fig 5, the sent beam occurs under the sun,
but the reflected beam returns when the clock is under the
lightning bolt, thus the clock is rolling to the right. What
is this telling us? We know that the clock standing still
sends and receives at 1second intervals. We also know that
the speed of light is constant. Regardless of where we are, we
would measure the light beam in fig 4 and fig 5 to be the
exact same speed. But Fig 5 looks like the light traveled
farther because the arrows are longer. And guess what, it did.
It took the light longer to make one complete send and receive
cycle, but the speed of the light was unchanged. Because the
light traveled farther and the speed was unchanged, this could
only mean that the time it took was longer. Remember speed is
distance / time, so the only way for the speed to be unchanged
when the distance increases is for the time to also increase.
Using the Lorentz Transform, let's put numbers to this
example. Let's say the clock in Fig 5 is moving to the right
at 90% of the speed of light. You, standing still, would
measure the time of that clock as it rolled by to be
2.29seconds. It is important to note that anyone in motion
with the clock in Fig 5 would only measure 1second, because
it would be no different than him standing beside the clock in
Fig 4. Hence, the rider aged by 1 second but you aged by 2.29
seconds. This is a very important concept. If we look closely
at the clocks, we find that they do not really measure what we
think they do. Clocks record the interval between two spatial
events. This interval may differ depending on what coordinate
system the clock is in (ie. what frame of reference). If the
speed of light is held constant (has the same measured value
regardless of frame of reference), time is no longer "just" a
tool to measure the procession of space. It is a property that
is required for the defining and existence of the event.
Remember from earlier, any occurrence is an event of space and
time (hence, the SpaceTime Continuum).
[Note: If the reader decides to learn more about time
dilation, it is absolutely imperative that strong emphasis be
put on "proper time". This concept is not discussed in this
article, but "proper time" is the foundation of the frame
geometry of SR. This topic is clearly derived and discussed in
the book Spacetime
Physics by Taylor and Wheeler.]
The Unification of Energy and Mass
Undoubtedly
the most famous equation ever written is E=mc^2. This equation
says that energy is equal to the rest mass of the object times
the speed of light squared (c is universally accepted as the
speed of light). What is this equation actually telling us?
Mathematically, since the speed of light is constant, an
increase or decrease in the system's rest mass is proportional
to an increase or decrease in the system's energy. If this
relationship is then combined with the law of conservation of
energy and the law of conservation of mass, an equivalence can
be formed. This equivalence results in one law for the
conservation of energy and mass. Let's now take a look at a
couple examples of this relationship...
You should readily understand how a system with very little
mass has the potential to release a phenomenal amount of
energy (in E=mc^2, c^2 is an enormous number). In nuclear
fission, an atom splits to form two more atoms. At the same
time, a neutron is released. The sum of the new atoms' masses
and the neutron's mass are less than the mass of the initial
atom. Where did the missing mass go? It was released in the
form of heat  kinetic energy. This energy is exactly what
Einstein's E=mc^2 predicts. Another nuclear event that
corresponds with Einstein's equation is fusion. Fusion occurs
when lightweight atoms are subjected to extremely high
temperatures. The temperatures allow the atoms to fuse
together to form a heavier atom. Hydrogen fusing into helium
is a typical example. What is critical is the fact that the
mass of the new atom is less than the sum of the lighter
atoms' masses. As with fission, the "missing" mass is released
in the form of heat  kinetic energy.
One oftenmisinterpreted aspect of the energymass
unification is that a system's mass increases as the system
approaches the speed of light. This is not correct. Let's
assume that a rocket ship is streaking through space. The
following occurs:
 Energy must be added to the system to increase the
ship's speed.
 More of the added energy goes towards increasing the
system's resistance to acceleration.
 Less of the added energy goes into increasing the
system's speed.
 Eventually, the amount of added energy required to reach
the speed of light would become infinite.
In step 2,
the system's resistance to acceleration is a measurement of
the system's energy and momentum. Take notice that in the
above 4 steps, there is no reference to mass. Nor does there
need to be.
Simultaneous Events
There is no such thing as
simultaneity between two events when viewed in different
frames of reference. If you understand what we have talked
about so far, this concept will be a breeze. First let's
clarify what this concept is stating. If Meagan sees two
events happen at the same time for her frame of reference,
Garret, who is moving with respect to Meagan, will not see the
events occur at the same time. Let's use another example.
Imagine that Meagan is standing outside and notices that there
are two identical cannons 100 yards apart and facing each
other. All of the sudden, both cannons fire at the same time
and the cannonballs smash into each other at exactly half
their distance, 50 yards. This is no surprise since, the
cannons are identical and they fire cannonballs at the same
speed. Now, suppose that Garret was riding his skateboard
super fast towards one of the cannons, and he was directly in
the line of fire for both. Also suppose he was exactly half
way between the two cannons when they fired. What would
happen? The cannonball that Garret was moving towards would
hit him first. It had less distance to travel since he was
moving towards it.
Now, let's replace the cannons with light bulbs that turn
on at the same time in Meagan's frame of reference. If Garret
rides his skateboard in the same fashion as he did with the
cannonballs, when he reaches the halfway mark, he sees the
light bulb he is moving towards turn on first and then he sees
the light bulb he is moving away from turn on last. See Fig 6
below for clarification.
In Fig 6, the bulb on the right turns on first. I have
shown Garret to be moving in the same direction of the
distance line between the bulbs, and he is looking towards the
moon. As stated earlier, when the bulbs turn on in Meagan's
frame of reference, Garret will see the bulb on the right turn
on before the bulb on the left does. Since he is moving toward
the bulb on the right, its light has a shorter distance to
travel to reach him. Garret would argue with Meagan that the
bulbs did not turn on at the same time, but in Meagan's
perspective they did. Hopefully, you can see how different
frames of reference will not allow events to be observed as
simultaneous.
3.0  Fun with the Special Theory of Relativity
The Infamous Twin Paradox
Since SR dictates that
two different observers each have equal right to view an event
with respect to their frames of reference, we come to many
notsoapparent paradoxes. With a little patience, most of the
paradoxes can be shown to have logical answers that agree with
both the predicted SR outcome and the observed outcome. Let's
look the most famous of these paradoxes  The Twin Paradox.
Suppose two twins, John and Hunter, share the same
reference frame with each other on the earth. John is sitting
in a spaceship and Hunter is standing on the ground. The twins
each have identical watches that they now synchronize. After
synchronizing, John blasts off and speeds away at 60% the
speed of light. As John travels away, both twins have the
right to view the other as experiencing the relativistic
effects (length contraction and time dilation). For the sake
of simplicity, we will assume that they have an accurate
method with which to measure these effects. If John never
returns, there will never be an answer to the question of who
actually experienced the effects. But what happens if John
does turn around and return to the earth? Both would agree
that John aged more slowly than Hunter did, thus time for John
was slower than it was for Hunter. To prove this, all they
have to do is look at their watches. John's watch will show
that it took less time for him to go and return than Hunter's
watch shows. As Hunter stood there waiting, time passed faster
for him than it did for John. Why is this the case if both
were traveling at 60% the speed of light with respect to one
another?
The first point to understand is that acceleration in SR is
a little tricky (it's actually handled better in Einstein's
Theory of General Relativity  GR). I don't mean to say that
SR can't handle acceleration, because it can. In SR, you can
describe the acceleration in terms of locally "comoving"
inertial frames. This allows SR to view all motion to be
uniform, meaning constant velocity (nonaccelerating). The
second point is that SR is a "special" theory. By this, I mean
that it is applicable in situations where there is no gravity,
hence where spacetime is flat. In GR, Einstein unifies
acceleration and gravity so actually my previous statement is
redundant. Anyway, the lack of gravity in SR is why it is
called "Special Relativity". Now, back to the paradox… While
both did view the other as shrinking and slowing down, the
person that actually underwent the acceleration to reach the
high speed is the one that aged less. If you dig deeper into
the world of SR, you will realize that it's not really the
acceleration that is important; it's the change of frame.
Until John and Hunter returned to a frame of reference where
their relative motion was zero (where they are standing beside
each other) they would always disagree with what the other
said he saw. As strange as this seems, there really isn't a
conflict  both did observe that the other was experiencing
the relativistic effects. One technique that is used to show
the dynamics of the Twin Paradox is a concept is called the
Relativistic Doppler Effect.
The Doppler Effect basically says that there is an observed
frequency shift in electromagnetic waves due to motion. The
direction of the shift is dependent on whether the relative
motion is traveling towards you or away from you (or vice
versa). Also, the amplitude of the shift is dependent on the
speed of the source (or the speed of the receiver). A good
place to start in understanding the Doppler effect would be to
first look at sound waves. There is a Doppler Shift associated
with sound waves that you should recognize easily. When a
sound source approaches you, the frequency of the sound
increases and likewise, when the sound source moves away from
you, the frequency of the sound decreases. Think about an
approaching train blowing its whistle. As the train
approaches, you hear the whistle tone as a high note. When the
train passes you, you can hear the whistle tone change to a
lower note. Another example occurs when cars race around a
racetrack. You can hear a definite shift in the sound of the
car as it passes where you are standing. One last example is
the change in tone you hear when a police car passes you with
its siren on. I'm sure that at some point in our lives, all of
us have imitated the sound of a passing car or passing police
car; we imitated the Doppler Shift. This Doppler shift also
affects light (electromagnetic radiation) in the same manner
with one critical exception; the shift will not allow you to
determine if the light source is approaching you or if you are
approaching the source and vice versa for moving away. This
being said, let's look a fig 7 below.
In the top part of fig 7 you can see a stationary light
source is emitting light in all directions. In the second
part, you can see that source "S" is moving to the right and
the light waves are shifted (they look as though they are
being compressed in the front and dragged in the rear). If you
approach the light source or the light source approaches you,
the frequency of the light will appear to increase (notice
that the waves in the front are closer together than in the
rear). The opposite is true for a light source that is moving
away from you or that you are moving away from. The importance
of the frequency change is that if the frequency increases,
then the time it takes for one complete cycle (oscillation) is
less. Likewise, if the frequency decreases, the time it takes
for one complete cycle is more.
Now let's apply this information to the Twin Paradox.
Recall that John sped away from Hunter at 60% the speed of
light. I picked this speed, because the corresponding
relativistic Doppler shift ratio is "2 times" for an
approaching source and "1/2" for a source that is moving away.
This means that if the source is approaching you, the
frequency will appear doubled (time is then halved) and if the
source is moving away from you, the frequency will appear
halved (time is then doubled). (similarly I could have used
any speed for the paradox; for example, 80% the speed of light
would have led to a Doppler shift of "3" and "1/3" for
approaching and moving away respectively). Remember, the
direction of the shift is dependent on the direction of the
source, while the amplitude of the shift increases with the
speed of the source.
Let's take another trip with the twins, but this time John
will travel 12 hours away and 12 hours back, as measured by
his clock. Every hour he will send a radio signal to Hunter
telling him the hour. A radio signal is just another form of
electromagnetic radiation; therefore, it also travels at the
speed of light. What do we get as John travels away from
Hunter? When John's clock reads "1 hour" he sends the first
signal. Because he is moving away from Hunter at 60% of the
speed of light, the relativistic Doppler Effect causes Hunter
to observe John's transmission to be ½ the source value. From
our discussion above, ½ the frequency means the time it takes
is twice as long, therefore, Hunter receives the John's "1
hour" signal when his clock reads "2 hours". When John sends
his "2 hour" signal, Hunter receives it at hour 4 for him. So
you can see the relationship developing. For every 1hour
signal by John's watch, the elapsed time for Hunter is 2
hours. When John's clock reads "12 hours" he has sent 12
signals. Hunter, on the other hand, has received 12 signals,
but they were all 2 hours apart…thus 24 hours have passed for
Hunter. Now John turns around and comes back sending signals
every hour in the same manner as before. Since he is
approaching Hunter, the Doppler shift now causes Hunter to
observe the frequency to be twice the source value. Twice the
frequency is the same as ½ the time, so Hunter receives John's
"1 hour" signals at 30min intervals. When the 12hour return
trip is over, John has sent 12 signals. Hunter has received 12
signals, but they were separated by 30 minutes, thus 6 hours
have pasted for Hunter. If we now total up the elapsed time
for both twins, we see that 24 hours (12 + 12) have elapsed
for John, but 30 hours (24 + 6) have elapsed for Hunter. Thus,
Hunter is now older than his identical twin, John. If John had
traveled farther and faster, the time dilation would have been
even greater. Look at the twins again, but this time let John
travel 84 hours out and 84 hours back (by his clock) at 80%
the speed of light. The total trip for John will be 168 hours,
and the total time elapsed for Hunter will be 280 hours; John
was gone for 1 week by his clock, but Hunter waited for 1 week
4 days and 16 hours by his clock. Remember that Hunter will
receive John's outgoing signals at half the frequency which
means twice the time. Therefore, Hunter receives John's 84
hourly signals every 3 hours for a total of 252 hours (3 is
the Relativistic Doppler shift for 80% the speed of light).
Likewise, Hunter receives John's return trip 84 hourly signals
every 20 minutes for a total of 28 hours (20 minutes is the
1/3 Relativistic Doppler shift for the return). Now you know
the total round trip from Hunter's perspective, 252 + 28 = 280
hours or 1 week 4 days and 16 hours. John, on the other hand,
traveled 84 hours out and 84 hours back for a total of 168
hours or 1 week.
Now let's look at the twins again, but this time Hunter
will send a signal every hour by his clock. What will John
see? When Hunter sees the outgoing leg of John's trip end, his
clock reads 15 hours and he has sent 15 signals. John,
however, will say that he received 6 signals separated by
2hours (relativistic Doppler shift) for a total of 12 hours.
What happened to the other 9 signals? They are still in
transit to John. Therefore, when John changes to his return
leg, he will now encounter the missing 9 signals plus the 15
signals Hunter sent for the 15 hours his clock recorded for
the return leg. So John receives 24 signals that are 30
minutes apart for a total of 12 hours. Like the previous
example, these 24 signals have all been doppler shifted to a
higher frequency because John is now approaching them. Now if
we total the whole trip, Hunter sent one signal every hour for
thirty hours, but John received 6 signals that were 2 hours
apart and 24 signals that were 30 minutes apart. Hunter sent
30 signals in 30 hours; John received 30 signals in 24 hours.
The result is the same as before, but the twins do not agree
on when the first leg ended and the last leg began. So from
this we can conclude that the change of frame for John (from
outgoing to return) is what distinguishes him from Hunter. For
Hunter, nothing changes at all. Anyway you look at it; he
waits 30 hours without a change. John, however, does change.
He changes from a frame in which he is moving away to a frame
in which he is moving back. It is this change that breaks the
symmetry between John and Hunter, thus removing the paradox as
well.
Before going on to the next concept, I want to make sure
that a couple things about SR and the speed of light are
properly understood. First, SR predicts doom for anything with
mass approaching the speed of light from a slower speed due to
length contraction and time dilation, but it does allow for
speeds greater than the speed of light. Consider the speed of
light as a barrier. SR allows for existence on both sides of
the barrier, but neither side can cross over to the other. As
of yet, nothing has been discovered on the fasterthanlight
side, and all that we have are theories on particles
(tachyons) that may have the ability to exist there. Maybe one
day someone will discover their existence.
Secondly, velocities from a different frame of reference
can not be summed. For example, if I run 5 miles/hour and at
the same time, throw a rock 5 miles/hour, the only reason you
(standing still) can say the rock is travelling 10 miles/hour
is because the speed is so small with respect to the speed of
light. We use the Lorentz Transformations to transform from
one frame to another using the relative velocity of the
frames. These transformations tell us mathematically that
while at slow speeds the error in straight addition is much
too small for us to detect, at very fast speeds, the error
would become quite large. So classical mechanics, which
teaches us to sum these velocities, is actually incorrect. We
can do it, but it's a case of getting the right answer for the
wrong reason.
The Twin Paradox using Simultaneous
Events
simultaneity (or lack thereof) is a terrific
tool for understanding many of the paradoxes associated with
SR. And, if I am to be thorough, simultaneity must be
considered for all SR events between separate frames of
reference. Let's revisit the twin paradox (John travels out
12 hours at 60% the speed of light and returns at the same
speed). Basically, there are three frames of reference to
consider. First, the twins are on the earth with no relative
velocity between them. Second, John embarks on the outgoing
leg of his trip. Thirdly, John (after instantaneously turning
around) embarks on his return leg of his trip. I am using the
same example as before, except I am using numbers from the
Lorentz Transforms as opposed to the Relativistic Doppler
Shift to explain the observed phenomena.
1st frame:
Hunter and John each agree on everything
they observe. This should be easy to understand since there is
no relative velocity between the two twins. They are in motion
together.
2nd frame:
John travels out 12 hours by his clock.
With the two postulates in mind, we realize that Hunter
observes time dilation for John's outgoing trip. Thus, if John
records 12 hours, Hunter will record 15 hours. Remember that
at 60% the speed of light, the time dilation will be 80%.
Therefore, if John records his time to be 12 hours, this is
80% of what Hunter records  15 hours. But what does John
observe for Hunter's time? He observes the time dilation as
effecting Hunter; therefore, he measures his trip to be 12
hours, but he observes 9.6 hours (80% of his clock's time) for
Hunter's time.
2nd frame totals:
Hunter measures his time to be 15
hours, but John's time to be 12 hours. John measures his time
to be 12 hours, but Hunter's time to be 9.6 hours.
Obviously, the event, which is the end of the outgoing
trip, is not simultaneous. John thinks Hunter's time is 9.6
hours but Hunter thinks his time is 15 hours. On top of that,
they both think that John's time is 12 hours, which doesn't
agree with either of the first two times.
3rd frame:
From Hunter's perspective, nothing new
has happened. He remained in his initial frame of reference
and John returned at the same velocity he left with.
Therefore, Hunter measured the return trip to take 15 hours
for his frame (same as the outgoing trip) and observes the
trip to take 12 hours for John. From John's perspective, he
encountered a major change. He actually changed frames from
one of traveling out to one of traveling back. Now, at the
start of the return trip, when John looks at his clocks, he
observes his clock to read 12 hours and Hunter's clock to read
20.4 hours. Think about this. John now shows that Hunter's
clock has jumped ahead from 9.6 hours to 20.4 hours. How can
this be???? When John changed from the 2nd frame to the 3rd
frame, the established symmetry between Hunter and John was
broken. Thus, each views their own time as having no change.
And since John was the one that actually changed frames, he
showed more elapsed time for Hunter. From here on out, it is
business as usual. The return trip is clocked at 12 hours by
John, but he observes 9.6 hours for Hunter. Again, let's clean
this up…
3rd frame totals:
Hunter measures his time to be 15
hours, but he measures John's time to be 12 hours. John
measures his time to be 12 hours, but he measures Hunter's
time to be 9.6 hours. Remember, this 9.6 is only for the
return trip after the frame change.
Trip totals:
Hunter measured his time to be 15 hours
for the outgoing trip + 15 hours for the return trip…30
hours.
Hunter observed John's time to be 12 hours outgoing
+ 12 hours return …24 hours.
John measured his time to be
12 hours outgoing + 12 hours return…24 hours.
John observed
Hunter's time to be 20.4 hours (after outgoing trip and frame
change) + 9.6 hours for the return trip…20.4 + 9.6 = 30 hours.
Can you find any events in which both John and Hunter agree
on the time for both themselves and the other? No, you can't.
The lack of simultaneity is the key to the paradox. Both twins
are measuring and observing. Unfortunately, they are not
measuring and observing the same events. It is impossible for
them to consider something like the end of the first leg as
simultaneous when they each view it occurring at different
times for Hunter. It's interesting to note that the results
are the same as the Relativistic Doppler shift results. Is
there a pattern here? SR allows for various methods to be
employed to resolve the problems. For this case, use of
spacetime diagrams (there's those words again) would clearly
show every point that we have talked about. I have merely used
the Lorentz transforms in combination with the Relativistic
Doppler effect.
Many people have trouble with the twin paradox because of
the way in which the frame change is handled. In this case,
the jump on John's clock for Hunter after the frame change
(9.6 to 20.4 hours) is the problem. There really is no problem
here. If you want to integrate the acceleration to use various
inertial frames during the turn around, it can be done (with
the same results). Another common approach is to imagine
someone else in space that passes John just when he reaches
the point of his turnaround. This person is heading towards
Hunter at the same speed that John was travelling, so there is
no need to consider John any further. The key fact is that if
we then went back in the substitute's frame and looked at his
clock for Hunter, it would show that some amount of time had
already been recorded when the substitute began his trip
towards Hunter. How far back should we go? Since John traveled
out 12 hours on the outgoing trip, we should go back 12 hours
in the substitute's frame. At this starting point for the
substitute, his clock for Hunter would read 10.8 hours. This
is extremely important. It clearly shows that both twins or
the twin and the substitute observe the other as having slower
times. The big shift occurs when the frame of reference is
changed. This means that both observe the other to have a
slower time during the actual outgoing and return trips, but
there is a shift during the frame change that more than makes
up for John's account of Hunter's slowly running clock. After
the frame change, the damage has been done. John will still
observe Hunter's clock to run slow, but it will never slow
down enough to compensate for the 10.8 hours that were
perceived during the frame change. Is this time jump a
physical occurrence? No. The time jump occurs because when
John changes frames, he is no longer using the same event as a
reference. When John made his turnaround, the event in
Hunter's frame that John thought was simultaneous with his
turnaround changed. John's frame change caused this confusion
because his new frame uses a different time for the event in
Hunter's frame. More clearly, the turnaround event in Hunter's
frame has a different time value for the outgoing leg and the
return leg, as perceived by John. Keep in mind that in the
above references to Hunter's frame, I'm really talking about
what John thinks Hunter's frame time would be. This time
difference is only apparent to John because it is his frame
change that causes the discrepancy. In Hunter's frame, nothing
changes for Hunter when John changes frames. Here again, by
realizing that the two events are not simultaneous, the
paradox is resolved. The point I am trying to emphasis is that
there are a variety of ways to handle the paradox. All of the
methods yield the same result, but if you actually consider
the simultaneity of the situation, then the how's and why's
become more clear.
Time Travel
Now that you have been introduced to
the concepts of the theory, let's take a quick look at the
relation between time travel and Special Relativity. If you
remember the result from the twin paradox, you should agree
that traveling into the future is possible, even at the speeds
that our astronauts travel. Granted they would probably only
be gaining a few nanoseconds, but when they return, the time
on earth is ahead of their system time. Thus, they have
returned to the future. As far as travelling back in time,
Special Relativity is not as gracious as it is with moving
forward. Let's take a look at this approach…
Many creative minds have wondered that since time slows
down as you approach the speed of light, if you could find a
way to travel faster than the speed of light, could you travel
back in time? If I am to believe that special relativity is
correct, then I am also to believe that the following events
would occur. In order to travel faster than the speed of
light, I assume that you would at some point have to travel at
exactly the speed of light. For example, you can not travel 51
miles/hour without having traveled 50 miles/hour at some
point, of course, this is providing that you were traveling 50
miles/hour or less to begin with. Now SR tells us that at the
speed of light, time stops, your length contracts to nothing,
and your resistance to acceleration becomes infinite requiring
infinite energy (as observed by a frame of reference that is
not in motion with the system). These conditions do not sound
very conducive to life. Thus, I conclude that time travel into
the past, using the concepts of SR, has some severe issues to
overcome.
Conclusion
SR deals with contractions and
dilations that are not in agreement with our commonsense views
of the universe. In fact, they almost appear ludicrous. Yet,
there have been several observations that agree with the
predictions of SR. So, until the theory is proved wrong or a
simpler theory produces the same results, SR will maintain its
position as the best theory out there.
Here are five concepts you have discovered in this article:
 There is no such thing as an absolute (completely
stationary) frame of reference.
 The laws of physics apply equally to all frames of
reference.
 The speed of light is constant in all frames of
reference.
 There is no simultaneity of events between separate
frames of reference.
 You are never too old to learn.
As you pursue a better understanding of SR, Do Not
fall prey to these errant statements:
 Time slows as speed increases. (Only when viewed by
another frame of reference)
 Objects shorten as speed increases. (Same as above)
 SR can't handle acceleration. (Biggest misconception
about SR)
 Mass increases with speed. (Energy increases, not the
rest mass)
 Nothing can travel faster than the speed of light.
Crossing the speed of light barrier from either a faster or
a slower speed is disallowed.
The beauty in the theory of special relativity is that it
gives us laws from which we can unite space and time and also
energy and mass. Special relativity is definitely a thinking
person's playground.
Special thanks to John M. Zavisa for contributing this
article.