The above conclusions can be very confusing so it might be worthwhile to
discuss the a bit.
Take for example length contraction: the Principle of Relativity implies
that if we measure the length some rod while at rest with respect to it,
and then we measure it when it is moving along its length, the second measurement
yields a smaller value. The crucial point to keep in mind is the condition
that the first measurement is made at rest with respect to the rod.
Similarly suppose we have two clocks labeled 1 and 2. which are in perfect
agreement when they are at rest with respect to each other. Suppose now
these clocks are endowed with a relative velocity. Then when we look at
clock 2 in the frame of reference in which clock 1 is at rest, clock 2 will
be measured to tick slower compared to clock 1. Similarly, in the frame
of reference in which clock 2 is stationary, clock 1 will run slower compared
to clock 2.
These results can be traced back to the fact that simultaneous events are
not preserved when we go from one reference frame to another.
There are many ``paradoxes'' which appear to imply that the Principle of
Relativity is wrong. The do not, of course, but it is interesting to see
how the Principle of Relativity defends itself.
Consider a man running with a ladder of length l
(measured at rest) and a barn also of length l
(again, when measured at rest).
The barn has two doors and there are two persons
standing at each of them; the door nearer to the ladder is open the farthest
is closed. Now the man with the ladder runs fast towards the barn while
the door persons have agreed to close the first door and open the second
door as soon as the rear of the ladder goes through the first door.
This is a paradox for the following reason. The ladder guy is in a frame
of reference in which the ladder is at rest but the barn is moving toward
him, hence he will find the length of the barn shortened (shorter
than his ladder), and will conclude that the front of the ladder will hit
the second door before the first door is closed.
The barn people in contrast
find the ladder shortened and will conclude that it will fit comfortably.
There will even be a short lapse between the closing of the first door and
the opening of the second, there will be no crash and the ladder guy will
So who is right?
The answer can be found by remembering that
an even simultaneous for the barn
people (the closing and opening of the doors) will not be simultaneous for
the ladder guy. So, while for the door person the opening of the rear door
and closing of the front occur at the same time, the ladder guy will see
the person at the second door open it before
the person at his rear closes
that door and so he will sail through but only because, he would argue,
the door guards were not synchronized.
There is an astronaut whose length is 6 ft and he sees a big slab of metal
with which he/she is going to crash. This piece of metal has a square hole
of length 6 ft. (measured at rest with respect to the slab). From the point
of view of the astronaut the hole is shrunk and so he will be hit...and
die! From the point of view of an observer on the shuttle the plate is falling
toward earth and the astronaut moving at right angles toward it, hence this
observer would measure a short astronaut (5 ft) and conclude that he/she will not be harmed (see Fig. 6.14). What does really happen?
Figure 6.14: An astronaut's close encounter with a metal plate
The problem is solved in the same way as above. For the astronaut to be
hit a simultaneous coincidence of his head and legs with the two extremes
of the slab's hole should occur. In fact he is not hit. What is more peculiar
is what he sees: he will see the slab tilt in such a way that he goes through
the hole with no problem!
This story illustrates the peculiar look which big objects acquire at very
large speeds. For example, a kettle moving close to the speed of light with
respect to, say, the Mad Hatter will be observed to twist in a very curious
way indeed, see Fig. 6.15.
Figure 6.15: A relativistic kettle. The top view shows how the three dimensional view
is distorted due to relativistic effects. The bottom view shows the corresponding
behavior of a flat kettle which exhibits only length contraction.
Just as for the case of length contraction and time dilation, the effect
on the kettle is not an optical illusion, but any unbiased observer (such as a photographic
camera) would detect the above images precisely as shown. If the relative
velocity between the observer and the kettle is known, one can use the formulas
of special relativity to determine the shape of the object when at rest
with respect to it...and we would obtain the first of the images: a nice
Consider two identical twins. One goes to space on a round trip to Alpha-Centauri
(the star nearest to the Sun) traveling at speeds very close to c. The round trip takes 10 years as clocked on Earth . As seen by the twin remaining on Earth all clocks on the ship slow down,
including the biological clocks. Therefore he expects his traveling twin
to age less than 10 years (about 4.5 years for these speeds; the difference
is large since the speed is close to c).
On the other hand the twin in the spacecraft sees his brother (a together
with the rest of the solar system) traveling backwards also at speeds close
to c and he argues that Einstein requires the twin on Earth to age less than
10 years. Thus each one states that the other will be younger when they
The solution lies in the fact that the traveling twin is not always in
an inertial frame of reference: he must decelerate as he reaches Alpha-Centauri
and then accelerate back. Because of this the expressions for time dilation
as measured by the traveling twin will not coincide with the ones given
above (which are true only for observers in different inertial frames).
It is the traveling twin that will be younger.