HPS 0410  Einstein for Everyone 
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Special Theory of Relativity: Adding Velocities
John
D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
Special relativity contains many results that are, on first look, odd and unexpected. Rapidly moving rods shrink; rapidly moving clocks slow; and, as we will soon find, many more oddities. It is important to recognize that these results are not merely a grab bag of oddities thrown together at our whim. They all live within a definite logical structure. We cannot overturn one of them without overturning the whole structure. That structure is fixed by Einstein's two principles: the principle of relativity and the light postulate. If we start with them, we arrive by straightforward reasoning at all the effect we have seen and will shortly see. Let us explore these results further.
The speed of light clearly has a special place in this theory. If something is traveling at the speed of light c, then all observers will find it to be traveling at exactly same speed.
A similar thing happens to things
traveling at less than the speed of light. If one observer finds
an object to be traveling at less than the speed of light, say, then so
must every other. There is no way that observers can change their states
of motion so as to find the object traveling at faster than the speed of
light. And there is a similar result for objects traveling at faster than
the speed of lightif such things exist. If one observer finds them
traveling at faster than the speed of light, then so must all.
All observers agree on which are: Things that travel slower than light. Pebbles, bullets, rocketships, planets, protons, electrons, neutrons, ... 
Things that travel at the speed of
light. 
All observers agree on which are: Things that travel faster than light. Tachyons?...

This division is a logical consequence of the theory's two principles. To see how it arises, we will imagine otherwise. That is, we will imagine that inertial observers disagree on whether some particle is traveling faster or slower than the speed of light. A contradiction with the principle of relativity will follow.
To make things concrete, imagine that we have a machine capable of emitting a particle at high speed, but at less than the speed of light. That means that a light signal will overtake the particle. To be specific imagine that a light signal passes the particle emitting machine at the moment that the particle is emitted. The observer moving with the machine would (obviously) judge that the light signal overtakes the particle. 
We transport the particle emitting machine and its observer from
the earth to a rapidly moving spaceship and repeat the entire
experiment. The machine observer will find the passing light signal
still to overtake the particle. It is the same experiment,
duplicated in a different inertial frame of reference, so the
principle of relativity requires the same outcome. Now consider how this same emission process will be judged by the Earthbound observer. That observer must also judge the light to signal overtake the particle. It is just the one experiment conducted locally, examined by two observers. Both observers must judge the same outcome. 
What else would you expect? Might it be that the light signal would overtake a particle emitted by the machine, when the machine is on earth. But when the machine emits on a rapidly moving spaceship, then the particle overtakes the light?
That is exactly what the principle of relativity prohibits! For then we have an experiment that can detect absolute motion. The resting machine emits particles that do not overtake light; the rapidly moving machine emits particle that do overtake light. Observers with the particle emitting machine could then tell whether they are "at rest" or rapidly moving just by checking whether light does or does not overtake the particle emitted. The principle of relativity demands that the experiment must proceed in the same way when carried out on earth or a rapidly moving spaceship.  (For experts) Those
who have read ahead might worry that each observer might find a
different outcome, perhaps as an artefact of the relativity of
simultaneity (to
be described soon). That won't happen. Whether light overtakes
the particle or not can be reduced to local facts independent of
judgments of simultaneity. Imagine that the light signal and the
particle are to traverse the same interval in space AB. Both depart
A at the same momentjudged locally. If light outstrips the
particle, it will arrive at B before the particle. That earlier
arrival is once again a local fact that obtains just at point B. This situation will be complicated if there are tachyons, preview here. For then the direction of propagation of the tachyon will vary according to the frame of reference of the inertial observer. But what will not change is that all inertial observers will find tachyons to move faster than light. 
One of light's most important roles as a limiting velocity
now follows from this: no matter how hard we try, it is impossible
to accelerate something through the speed of light. More
generally, the speeds of things are divided into three groups:
things that travel slower than light,
things that travel at exactly the speed of light,
and things that travel faster than the speed of light.
We cannot slow down or speed up anything so that it crosses the barrier of
the speed of light.
Yet it looks
like it would be pretty easy to violate the limiting character of
the speed of light by accelerating something through the speed of light.
We might have a gun that can fire particles at, say, 2,000 miles per
second. That is well below the speed of light. We put the gun on a
spaceship that we accelerate up to 185,000 miles per seconda mere 1,000
miles per second short of the speed of light. If we fire the gun in the
direction of motion, would it not accelerate the particle through the
speed of light?
Is not 185,000 + 2,000 = 187,000?
Let us see how this limiting character of the speed of
light follows from the principle of relativity.
To see it, let us set up the challenge quite solidly. Imagine that a machine that can fire particles at 100,000 miles per second, which is more than half the speed of light, 186,000 miles per second. 
Now we will try to push things past the speed of light. Imagine that the machine is placed on a spaceship that also moves at 100,000 miles per second in the direction that the machine fires the particles; that is, it moves at this speed with respect to a second observer on the earth. 
So, let us ask the obvious question. What will the
earth bound observer find for the speed of the particle? The calculation seems irresistible. The spaceship moves at 100,000 miles per second with respect to the earthbound observer; and the particle moves at 100,000 miles per second with respect to the spaceship. So... 
100,000 + 100,000 = 200,000 ??
But that would be faster than the
speed of light, 186,000 miles per second. We have just seen that
the principle of relativity prohibits exactly this. If the spaceship
observer finds the particle to move at less than the speed of light, then
so also must the earthbound observer.
What this shows is that the principle of relativity prohibits us adding velocities in the usual way. We cannot add velocities by the ordinary rule 100,000 + 100,000 = 200,000. More generally, the classical rule for the composition of velocities fails:
Velocity of A with respect to C 
=  Velocity of A with respect to B 
+  Velocity of B with respect to C 
In its place we need a new rule for the composition of velocities. It ought to look like the ordinary rule as long as velocities are smallwe do know that the ordinary rule works for slow moving things like cars on freeways and trains. But it must look very different at high speeds. If we use it to add two velocities close to light, we must get a resultant that is still less than the velocity of light. Einstein found that the principle of relativity forces a particular rule. For the case of velocities oriented in the same direction in space, the relativistic rule for composition of velocities is:
Velocity of A with respect to C 
=  Velocity of A with respect to B 
+  Velocity of B with respect to C 
__________________________________  
reduction factor 
All the work is done in this new rule by the reduction factor. When the velocities are small, this factor is close to 1. So it is as if it isn't really there and Einstein's rule just behaves like the classical rule. But when the velocities get to be close to that of light, the factor starts to get larger and larger and in just the right way to prevent any composition of velocities less than light exceeding that of light.  (For experts only) Click here to see the complete formula. 
If we use the rule to add 100 mph to 100 mph, the reduction factor is almost exactly one, so the ordinary rule works: 100 + 100 = 200.
If we use the rule for adding
100,000 miles per second to 100,000 miles per second, we are now
dealing with velocities that are 100,000/186,000 = 0.54 the speed of
light. For that sum, the reduction factor is 1.29, so the composition
yields:
(100,000 +
100,000)/1.29 = 200,000/1.29 = 155,000
which is still less than the speed of light.
What is most instructive is to see what happens if we start with a velocity of 100,000 miles per second; and add 100,000 miles per second to it; and add it again; and again; and again.
To picture physically what we are doing, imagine that we start with our base machine "I" that happens already to be moving at 100,000 miles per second with respect to our original observer. From machine I, we shoot out a second smaller version of the same machinecall it "II"at 100,000 miles per second with respect to "I."
Now let's repeat the operation. From the smaller machine "II," we'll shoot out a yet smaller version of the same machine at 100,000 miles per second with respect to "II." Call it "III."
Then machine "III" will shoot out machine "IV"; and so on; and so on. As we pass through the series of machines "I," "II,", "III," "IV," etc., we are boosting each with a speed of 100,000 miles per second with respect to the one before.
In sum, we have
I moves at 100,000 miles per second with respect to the original observer.
II moves at 100,000 miles per second with respect to I.
III moves at 100,000 miles per second with respect to II.
IV moves at 100,000 miles per second with respect to III.
V moves at 100,000 miles per second with respect to IV.
... and so on.
The cumulative effect of the repeating boosting by 100,000 miles per second is shown below. The figures in the boxes are the total speeds of the numbered machines with respect to the original observer. The total speed of the last boosted machine increases as we proceed along the sequence "I," "II," etc. But the increases become smaller and smaller.
No matter how often we add 100,000 miles per second, we never get past the speed of lighthere set at exactly 186,000 miles per second. We get closer and closer to it. But never past it.
One way to think of it is as an "Einstein tax," that copies the way a very severe progressive taxation might increase the amount of tax paid as we get more income. We keep adding 100,000 miles per second to the speed, but the Einstein taximplemented through the reduction factorprecludes our total speed ever exceeding that of light. 
Public domain image from http://jet.wikia.com/wiki/File:Sm_stack_coins_gold.png
Adding velocities by Einstein's rule keeps all speeds partitioned into three sets: speeds less that that of light; those equal to light and those greater than light. An interesting case arises when we try to boost something already moving at the speed of light. To make this concrete, let us imagine that our machine emits something that moves at the speed of light.
Such machine is not hard to imagine. An ordinary flashlight does it!
What happens when we set this machine on a spaceship moving at, say, 1% the speed of light, that is, at 0.01c? For this case, the reduction factor is 1.01. Applying Einstein's rule, the final speed for the light will be:
(0.01c + c)/1.01 = (1.01 c)/1.01 = c
That is, the motion of the emitting machine will not affect the speed of the light at all. It will remain exactly c.
What about other cases? What if the machine sped up to move at:
10%, 20%, 50% or 70% the speed of light;
that is at
0.1c, 0.2c, 0.5c, 0.7c?
The reductions factors for these four cases are:
1.1, 1.2, 1.5, 1.7.
I'm sure you see the simple pattern used to find these reduction factors! (For experts only, here's how they were computed.) So the final speeds are:
(0.1c + c)/1.1 = (1.1 c)/1.1 = c
(0.2c + c)/1.2 = (1.2 c)/1.2 = c
(0.3c + c)/1.3 = (1.3 c)/1.3 = c
(0.4c + c)/1.4 = (1.4 c)/1.4 = c
We recover exactly the result we should expect. No matter how fast the emitter is moving, if we add speeds by Einstein's rule, the motion of the emitter fails to alter the speed of the emitted light from its constant value c.
That the ordinary addition rule fails follows from the principle of relativity. Why does the ordinary rule fail? Here's a way to get comfortable with the the failure. In the original example, the spaceship observer uses rods and clocks that move with the spaceship to measure the speed of the emitted particle as 100,000 miles per second. The earthbound observer now wants to find the speed of the emitted particle. That observer, however, cannot directly use measurements made with the spaceship rods and clocks, for the earthbound observer thinks that they have shrunk and slowed. The earthbound observer must correct the spaceship observer's measurements for effects such as these. The result of the these corrections is Einstein's formula!
Let's try to put this together more explicitly. If a body is moving at 100,000 miles per second, then it will traverse a 100,000 mile rod in one second.
Now imagine that measurement is made by a spaceship observer who moves at 100,000 miles per second with respect to a planet. We are tempted to picture things as that body moving over two 100,000 mile rods laid end to end in the one second:
So would mean that the body covers 100,000+100,000=200,000 miles in one second.
What is wrong with this picture is that the two measuring rods belong to observers in relative motion. One belongs to the spaceship observer and the other to the planet observer. Each would judge the other's rod as not 100,000 miles long. Similarly the one second traversal time is measured by clocks belonging to observers in relativity motion. Each would judge the other's clock as running slow. And there are more differences as we shall soon see: the two observers will disagree on judgments of which events are simultaneous.
The upshot is that the attractive sum 100,000+100,000=200,000 requires us to mix quantities from different frames of reference that simply do not belong together.
http://en.wikipedia.org/wiki/Sky#mediaviewer/File:Sky_over_Washington_Monument.JPG
This special role for the speed of light sometimes arouses special wonder. What is so special about light, we may be drawn to ask, that everything else takes such special note of it? Once one starts along this path, all sorts of confusions may arise. Is it that light is used for communication and finding things out? Does everything somehow respond to how we find things out? Does special relativity still work in the dark?
Wellyou can forget all this mystical mumbojumbo, if ever it attracted you. There is nothing special about light. It's space and time that is special. They have properties we don't expect. Space and time are such that rapidly moving objects shrink and their processes slow down. For a long time, we didn't notice these effects because we did not have a thorough account of a probe of space and time that moves very fast. That changed in the nineteenth century when we developed good theories of light. It is the probe that moves very fast and, for the first time, begins to reveal to us that space and time are not quite what we thought.
There is one further fact about space and time. It harbors a special velocity, one that is the same for all inertial observers. It is an invariant (="unchanging") velocity. Light is just something that happens to go as fast as it possibly can and thereby ends up going at that speed.
There's nothing special about light. What is special is the speed at which it goes.