TO UNDERSTAND RELATIVITY
==========================
You are travelling in a rocket at 200,000 km/sec, and you send, in your movement's
direction, a projectile that, seen from the earth, propagates at 250,000
km/sec.
Does it mean that, if you measure the speed of the projectile from the
rocket, your measures will give 50,000 km/sec?
It's not sure at all.
To calculate the speed of the projectile, that is the distance covered by unit
of time, you will have to use some instruments to measure the distance (a ruler)
and the time (a clock).
Let's take a simple clock, made of a light ray going on and back between two
mirrors, distant of 1.5 meter. One second corresponds to 100,000,000
rebounds.
You install this clock in the rocket so that the light ray propagates
perpendicularly to the rocket's trajectory.
For the ruler, it's a little more complicate, since the projectile moves
outside the rocket.
You find the following solution: before sending the projectile, you send a
mirror (M) provided with a small motor before the rocket. You follow
the progression of the mirror with the help of a light signal. When the
signal will take 1/3 second to go to the mirror and come back, you will know that
the mirror will be at 50,000 km. 1/6 of second before this moment, you send a
signal starting the motor of the mirror in order to stop it (we suppose the motor
reacts instantaneously).
Then you control that the mirror remains stationary at this distance.
Now you are convinced to have perfect instruments which will allow you to
measure the speed of the projectile.
But it is not the opinion of the observer who follows you on the earth!
Seen from the earth, the mirrors of the clock move while the light ray
propagates from a mirror to the other. Thus the distance covered by light is
more than 1.5m! But for you, in the rocket, the distance between the mirrors is still 1.5m
and you still consider that 1 second = 100,000,000 rebounds, whereas
for the terrestrial observer, this number corresponds to more that one second.
In other words, your clock slows down (note 1).
And it is not better, with regard to the "ruler".
During the displacement of the light signal to the mirror (M), the latter moves
away from the signal; in the return, the rocket advances toward the signal.
The signal "wastes time" in the first half and "wins" some in the return, but
since the first part is longer than the second, the total distance is longer
than if you were immobile. Therefore, when you place the mirror so
that the round trip of the signal takes 1/3 second, the mirror's distance is
much less than 50,000 km.
In other words, your "ruler" contracted (note 2).
It is clear that, when you use this type of instruments, the speed of light
will remain constant whatever is your own speed. This isn't mysterious: it is
the consequence of the use of light to define the units of time and space.
However, if you took a chronometer and a rigid ruler, you could expect that
these instruments would give a different result; and this difference would
permit you to measure the speed of your rocket.
Well, this is not true.
Numerous experiences have been made toward 1900 to find such a difference.
Indeed, the earth is not immobile (unless to consider it as the center of the
universe) and its speed should have been detectable. But all results
were negative.
With our present knowledge of matter, the reason of these failures is
very comprehensible.
How to imagine, for example, a perfectly rigid ruler?
Matter is made mainly of emptiness between the particles that are joined by
electromagnetic forces. These forces propagate like light.
The structure of an object results from the balance between forces in all
directions. When the object moves, a new balance gets settled between the
transverse and longitudinal forces, what results, all as in our "light ruler"
in a contraction in the direction of the movement.
In the same way, when the chronometer is in movement, the distance the
interactions must pass between the atoms constituting the pendulum and the
hands increases, and this slows down the internal movement, just like in our
clock made of mirrors.
To the limit, if the rocket reached the speed of light, the chronometer would
stop, since the electromagnetic interactions could not go from one particle
to the other.
It is evidently valid not only for the measuring tools,but for any object
(including your own body, which thus ages less when it is in movement).
In other words, in your frame of reference, time dilates and lengths decrease.
Let's go back to our projectile. What will be its speed, when measured from the
rocket?
At first sight, since the transformation formulas show that your clock slows
down at 3/4, but that your ruler contracts in the same proportion, one
could conclude that the measures of speed remain unaltered.
But here another problem appears.
To measure the speed of the projectile, it will be necessary to compare the
time when you throw it and when it arrives at the end of the "ruler".
It will thus be necessary to provide the mirror (M) with a clock (C). But
we know that the displacement will influence this clock. It will therefore
be necessary to adjust the clock (C) before sending the
projectile. One could calculate with the Lorentz formula the time "lost" by
the mirror (M) and take account of it in the calculation of the speed of
the projectile.
This method however has big disadvantages. On the one hand, it is not
objective: the correction of the clock (C) depends of its displacement speed.
The terrestrial observer will have another opinion than you as for the
correction to make. On the other hand, the method is only applicable if one
knows precisely the way all clocks moved.
There is another method, general and objective.
You send a light signal to the mirror (M) and you agree that the instant
t(M) when the signal rebounds on (M) is located in the middle of the interval
between the broadcast and the receipt of the signal by the rocket.
If you send the signal at 12h00 and it comes back 1/3sec. later, the
instant t(M) = 12h00 + 1/6sec.
In this way, the synchronization is an objective event, even though different
observers will have (literally speaking) a different point of view of this
event.
And it is this difference of point of view that will make that the speed of
the projectile measured in the rocket will be different from the one
measured on the earth.
Seen from the earth, the instant t(M) is not the middle (t(m))
between the departure and the arrival of the signal, since (M) "escapes" the
signal and that the rocket approaches to it during the return. t(M) comes
after t(m).
If you adjust C to 12h00+1/6sec at the moment t(M), whereas you should have
done it at the instant t(m) (on the point of view of the earth), C will delay
with respect to your clock.
According to your calculations, the projectile will take less than one second
to reach (M), that is, according to you, 50,000km away.
The more the speed of the projectile increases (and needs less time to reach
(M)), the more the shift of t(M) will influence your calculations.
To the limit, if the " projectile " is an electromagnetic signal, you will
observe, in spite of your own velocity, that its speed equals
300.000km/sec. This is the logical consequence of the fact you synchronized
your clocks using electromagnetic signals.
So far, we considered the earth as being immobile.
However, as we already indicated, this choice is quite arbitrary.
It is even much more normal to suppose that we are not the center of the
universe (provided that this center exists).
But, actually, it doesn't have any importance. All observers can consider
themselves immobile; their measures will give coherent results.
The (special) theory of relativity is a mathematical model which takes
account of this fact. It gives us the formulas that permit to compare
the measures made in different frames of reference, given that all observers,
in spite of their relative movement, consider the speed of light as a constant.
Unfortunately, the lack of clarity of the logical base of the theory
is the reason of many misunderstandings (note 3).
The twins paradox illustrates the difficulty to understand the theory as it
is usually presented.
It the well known story of Peter and Paul, two twin brothers.
Paul decides to make a journey in the space at very high speed, while
Peter stays at home.
According to the laws of relativity, Paul's time slows down during his
journey. Upon his return, he is therefore younger than his brother.
The story is usually intended to impress the reader, and the
explanations remain very vague.
Yet, as we showed before, there isn't any paradox here:
as soon as Paul moves, he ages effectively less than if he was
immobile.
Nevertheless, as long as he moves uniformly, he will measure a
slowing of his brother's time (see the explanations
by exchange of signals
and by clocks indications
).
One could object however that, since the earth is also moving, it could be
that, when Paul begins his journey, his "absolute" speed would
decrease instead of increase, and in this case, his local
time would run faster than on the earth.
It 's true, but in that case, Paul should travel twice as fast to join his brother,
and, since the time slows down according to the square of the speed, slowing will
win at the end of the trip.
Besides, all measures made on the earth will indicate Paul's clock
slowing during the whole journey. I showed it
in this example.
It's thus clear that, although in my opinion the existence of waves which
propagate undependently from the movement of the source automatically leads
to the idea of a privileged reference frame where the centers of the spherical
wave fronts are motionless wrt each other (- I do not make assumptions about
possible physical properties of that frame -), its existence is
a theoretical question, insofar as no experience permits (currently)
to determine it.
Whatever the speed of a reference frame with respect to another, one can at
any time take the first frame as reference, and the measures made in
that frame will indicate a time dilation in all frames moving relative to
this one.
Let's notice however that the discovery of a signal propagating in empty space
independently of the source with a speed different from c (lower or superior,
no matter) would permit to determine the absolute speed of all reference frames.
Maybe such a signal cannot exist, but this remains to be proved.
It doesn't however seem acceptable to me to found the physics on such an
hypothesis, especially as it is not necessary.
In fact, everything can be easily explained if, in Newton's theory, one
remplaces the instantanous remote actions by forces which propagate at the
speed c.
That speed is here a data provided by experiment, and not a constant which is
necessary to the theory.
This approach has the big advantage that it does not require to question our
intuitive vision of the world.
It's by no means incompatible with the theory of special relativity.
The propagation of forces and information at finite speed implies that
the measures of time and space are indissociable.
For all signals which propagate at the same speed (which is, in empty space,
the case for all the known signals), this connection can be represented in a
mathematical construct, called spacetime.
Bruno Van Rossum
january 2001
last update: june 2003
-copyright-
note 1:
The calculation of this slowing is very simple: the signal moves on
the hypotenuse (h) of the rectangular triangle whose right legs are the
distance between the mirrors (b1) and the distance covered by the rocket
during the journey (b2).
h = ct (c=the speed of light; t=the measured time on the earth, needed
by the signal to cover the distance between the mirrors)
b1 = ct' (t'=the time measured in the rocket needed by the signal to
cover the distance between the mirrors)
b2 = vt (v=the speed of the rocket)
b1xb1 = hxh - b2xb2
(1)
(2)
(3)
(4)
note 2:
The time taken by the signal to go forth and back is the same as
to make the same trip perpendicularly to the sense of the movement (1/3sec).
Let's take the distance between the mirrors (A et B) of the light clock as unit of
length (l') in the rocket, and let's place a new mirror (C) horizontally.
The distance AC will also be equal to the length unit if a signal sent from A to C comes
back in A at the same time as a signal sent from A to B. But, when in the vertical
direction the time to go forth is the same as to come back (t'(1)=l'/c), it is not
the case in the horizontal direction. Here, C moves away from the signal going from A TO C,
during time t'(2)=l'/(c-v). When coming back, A gets closer and time needed by the signal
to reach A is t'(3)=l'/(c+v).
2t'(1) = t'(2)+t'(3)
(5)
(6)
l'(1) however is not the perpendicular distance (b1), that corresponds to
l, but the hypotenuse (h).
After (4) the rate between l and l'(1) is:
(7)
(6) becomes:
(8)
(9)
(10)
note 3:
The fundamental problem is that the postulate of the equivalence of all
(inertial) reference frames on which the Special Relativity is founded
needs a reciprocity of the observed changes (time dilation and length
contraction) in different frames.
Is this reciprocity real or apparent? If it is real (which is the current
accepted thesis), the changes are apparent, and vice versa.
But in the case of time, the change is indeed real, so "something happens"
which is not explained by the theory.
And concerning the lengths, a real reciprocity gives logical problems,
as shown by the following example:
Two space travelers are moving toward each other.
They want to know whose ship is the longest. They decide to do the following
experiment: they place a clock at each end of the ships and each of them
synchronizes its two clocks. They take place in the middle of the ship.
When they will cross the other, they 'll spit a paint spot to the other
ship on each end (This moment can be calculated in advance, so the two shoots
can be done at the same moment).
After that they will compare the impacts.
But they are estonished about the experiment and they decide to meet to
compare the spaceships.
The two ships have the same length. Where are the paint spots?
These are the two versions of the experiment as viewed by A and B:
(Note that each of them consider himself at rest)
There are two important facts:
- the synchronization is done by light signals. Therefor, each observer
considers that his clocks indicate the same time, but that the other's
clocks are not synchronized (the front clock runs slow and the back one
runs fast wrt the middle of the ship).
- the bodies are contracted in the direction of the movement.
VERSION A VERSION B
1. A is at rest. B is at rest
B's clocks are not synchronized. A's clocks are not synchronized.
B shoots first at the back and A shoots first at the back and
misses A. misses B.
v====B====| --> |=====B=====|
|=====A=====| <-- |====A====^
2. B is smaller than A, so A doesn't A is smaller than B, so B doesn't
hit B. hit A.
|====B====| --> v=====B=====v
^=====A=====^ <-- |====A====|
3. Now B shoots at the front and Now A shoots at the front and
misses A. misses B.
|====B====v --> |=====B=====|
|=====A=====| <-- ^====A====|
4. When they meet, the lengths are the same and the result corresponds
with both 's experience (they both missed the other).
|=====B=====| |=====B=====|
|=====A=====| |=====A=====|
The crucial moment is of course point 2.
At the same time, A pretends that B is smaller, but B states the contrary.
Are they both right?
Obviously, both versions are consistent, so both can be true.
But logically, the two versions are mutually exclusive, so they can't be
true together, at the event E, which is the moment A and B are
at the same place (point 2).
At that moment, the reason why the spots miss the target is
- either because one of the ships is smaller than the other (1)
- or because the clocks are not synchronous (2)
(1) can not be right for both ships, but (2) implies that the ship in which
the clocks are not synchronized is moving, so at least one of the observers has
to accept that he is not really stationary.