
Time Travel Research Center
© 1998 Cetin BAL  GSM:+90
05366063183  Turkey / Denizli Einstein's Theory of Special Relativity
Relativistic Length Contraction
One of the peculiar aspects of Einstein's theory of special relativity is
that the length of objects moving at relativistic speeds undergo a
contraction along the dimension of motion. An observer at rest (relative to
the moving object) would observe the moving object to be shorter in length.
That is to say, that an object at rest might have be measured to be 200 feet
long; yet the same object when moving at relativistic speeds relative to the
observer/measurer would have a measured length which is less than 200 ft.
This phenomenon is not due to actual errors in measurement or faulty
observations; the object is actually contracted in length as seen from the
stationary reference frame. The amount of contraction of the object
is dependent upon the object's speed relative to the observer.
The animations below depict this phenomena of length contraction. In each
animation a spaceship is moving past Earth at a high speed. The spaceship
would be measured to be 200 feet in length when at rest relative to the
observer.
Spaceship Moving at the 10 % the Speed of Light
Spaceship Moving at the 86.5 % the Speed of Light
Spaceship Moving at the 99 % the Speed of Light
Spaceship Moving at the 99.99 % the Speed of Light
Note that the length contraction is only significant when the object is
moving at relativistic speeds  i.e., speeds which are a significant
fraction of the speed of light. Furthermore, note that the contraction only
occurs in the dimension of the object's motion. That is, if the object is
moving horizontally, then it is the horizontal dimension which is contracted;
there would be no contraction of the height of the object. This
information is summarized in the table below.
Speed of Spaceship

Observed Length

Observed Height

At rest

200 ft

40 ft

10 % the speed of light

199 ft

40 ft

86.5 % the speed of light

100 ft

40 ft

99 % the speed of light

28 ft

40 ft

99.99 % the speed of light

3 ft

40 ft

9.1 The Speed of Light
(Discovering the Universe, 5th ed., §3.1)
 Everything we know about the stars and other objects in the
universe comes to us through their release of some form of light, both
visible and invisible.
It is therefore important to understand light's basic properties to
determine what that reveals about the stars.
 Although light appears to travel instantaneously from its emission
to its detection, it does in fact have a finite speed, which must be
very, very fast.
 The first person known to attempt a determination of the speed of
light was Galileo.
Galileo and an assistant each took a shrouded lantern and stood within
site of each other on two widelyseparated hilltops.
Galileo opened his lantern, and when the assistant saw its light, he
opened his, too.
Galileo measured the time it took from when he opened his lantern to
when he saw his assistant's lamp, which should be the time for light
to travel back and forth between the two hilltops.
Question: how would this
information be used to determine light's speed?
Unfortunately the time Galileo measured could not be distinguished
from the time it took to open the lanterns, again due to light's high
speed.
 The first determination of the speed of light was an astronomical
one, since only in this circumstance will large distances result in a
measurable travel time.
In 1676, the Danish astronomer Ole Rømer (16441710) was
studying the orbits of Jupiter's Galilean satellites.
Over a period of many months, Rømer recorded the starting time of
eclipses of these moons by Jupiter.
These eclipses should occur periodically, so the nth eclipse
should begin after a time nP.
For example, the closest moon Io has an orbital period of P =
42.5 h, so the second eclipse takes place after 2P = 85.0 h,
the third after 3P = 127.5 h, etc.
However,
after a time equal to half of Jupiter's synodic period, the eclipses
were occuring after a time nP + 16.6 minutes!
And after another half synodic period, the variation had disappeared,
and the eclipses were again occuring after an integral number of
periods.
This correlation with the relative position of Earth and Jupiter could
only be due to the variation in their distance as they orbited the
Sun.
Rømer realized that if light had a finite speed, the time to travel
over an extra distance of two astronomical units would explain the
variation.
The speed of light must then be:
c = D/t = 2 AU/16.6 min = 7.2 AU/h
The size of the astronomical unit wasn't determined until the 19th
century, but once it was the speed of light could be expressed in
terrestrial units:
c = 2.998 x 10^{5} Km/s
Note the use of the special symbol c for the speed of light.
 By the middle of the 19th century improved technology allowed the
measurement of the speed of light in the laboratory, verifying this
value.
9.2 The Constancy of the Speed of Light
(Discovering the Universe, 5th ed., §3.1, §13.1)

One
prediction of Newtonian physics is that the speed of light c
should depend on the speed of its source v.
If light is emitted in the same direction that the source is moving,
it should travel faster, at a speed c + v.
If light is emitted in the direction opposite to the motion of the
source, it should travel slower, at a speed c  v.
 But there was a problem: a series of precise experiments in the
1880s found that the speed of light was always the same, no matter how
fast or in what direction the source was moving!
 Although physics is rooted in the scientific method, this conflict
between experiment and an otherwise wellfounded theory was very
difficult for physicists to overcome.
9.3 The Special Theory of Relativity
(Discovering the Universe, 5th ed., §13.1)
 While working in the Swiss Patent Office, Albert Einstein
(18791955) developed what is now known as the Special Theory of
Relativity, published in 1905.
Einstein began with the experimental observation that light always
has the same speed, and developed a number of unusual consequences:

The
faster an object moves, the shorter it becomes.
This length contraction is given by
.
The Lorentz factor
is equal to 1 when the object isn't moving (v = 0) and
approaches infinity as the object accelerates towards the speed of
light (v > c).

The
faster a clock moves, the slower it runs.
In other words, events in a moving system take longer to occur from
the perspective of a system at rest.
This time dilation is given by
.
In the picture on the right, the stopwatch in motion measures a fivesecond
event, while the stopwatch at rest has seen 6 seconds pass.
 The faster an object moves, the more massive it becomes.
This mass increase is given by
.
Question: how might one measure
this mass increase?
An important consequence of mass increase is a violation of the
principle of conservation of mass.
 Mass can be converted into energy and vice versa.
In addition to the violation of mass conservation above, it can be
shown that conservation of energy is violated, i.e. the energy you
put into accelerating an object doesn't all show up as the object's
energy.
But both the missing energy and the increased mass can be explained
if you allow mass to be a form of energy, according to the following
relation:
E = mc^{2}
Note that, because c is so large, mass is a very, very
concentrated form of energy.
This generalized principle of conservation of mass + energy
has profound consequences, as we will see later.
 Nothing can travel faster than light.
If you are accelerating an object towards the speed of light,
its mass increases without bound, making it harder and harder to
increase its speed.
As a consequence, a massive object must always travel at less than
the speed of light.
Since light itself travels at the speed of light, it must have no
mass!
 Newton's Laws of Motion are approximations to Special
Relativity that are applicable only at low speeds.
In other words, Special Relativity is a generalization of Newton's
Laws of Motion.
In practice, one must be traveling faster than ~0.1c (30,000
Km/s!) to observe these special relativistic effects.
Nevertheless, with sensitive experimental equipment such as atomic
clocks, the predictions of Special Relativity have been verified to
high precision.
Extra: Cship provides movies to help you visualize the effects of
special relativity at high speeds.
9.4 The General Theory of Relativity
(Discovering the Universe, 5th ed., §13.2)
 The equivalence of mass and energy soon led Einstein to develop
the General Theory of Relativity, published in 1916.
Light has no mass, but it does have energy, so it should be affected
by the presence of massive objects such as the Sun, just as the
planets are.
 The General Theory of Relativity is based on a fourdimensional
geometric framework of space and time called spacetime.
Spacetime is not an emptiness, but instead has a physical presence
that is affected by its contents.
The predictions of General Relativity include:

The
presence of mass or energy will curve space.
You can think of space as a threedimensional "rubber sheet" that is
usually flat.
But when a mass is placed on this "rubber sheet", it stretches and
curves in response.
When space is curved, a moving object must follow the curvature,
i.e. change its speed and direction, just as a car must turn when
the highway curves.
Curvature of space thereby replaces Newton's concept of force as a
means to accelerate objects.
The gravity well takes on a new physical meaning, with the
massive Sun sitting at its bottom, and the planets forced to curve
around the walls of the well.
Light must also curve around a massive object such as the Sun (see
the yellow line in the picture above).
Question: what shape will
light's path have?
The
deflection of light near the Sun was verified during the solar
eclipse of 1919, when a star that should have been invisible behind
the eclipsed Sun could be seen.
The observed amount of deflection, 1.75 arc sec, is the same as that
predicted by General Relativity.
 The presence of mass or energy will slow down time.
In other words, clocks run more slowly near a massive object,
another form of time dilation.
Atomic clocks can detect a time difference between the bottom and
top of a tall building!
Time
dilation near the Sun explains why Mercury's orbit precesses.
Recall that angular momentum is given by
AM ~ mr^{2}/P
AM is a conserved quantity, and the perihelion distance r
doesn't change, either.
So P, the time per orbit, must also be fixed.
But time must dilate in the vicinity of the Sun, which means the
orbit must become longer, i.e. the perihelion shifts its
position in the direction of motion.
The observed amount of perihelion shift, 43 arc sec/century, is the
same as that predicted by General Relativity.
 Newton's Laws of Motion and Law of Gravitation are
approximations to General Relativity that are only applicable away
from large masses or other concentrations of energy.
In other words, General Relativity is a generalization of Newton's
Laws of Motion and Gravitation.
In practice, one must have an escape velocity faster than ~0.1c
to observe these general relativistic effects.
Nevertheless, with sensitive experimental equipment such as atomic
clocks, the predictions of General Relativity have been verified to
high precision.
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