©Robert DiSalle 2001
I. Background to Einstein: The Michelson-Morely Experiment and the
Lorentz Contraction
1. The wave theory of light. In the 17th century, the mechanical philosophers assumed that white light is a "pure" kind of light, produced by a uniform pulse in the ether; they hypothesized that colours arise from the disturbance or "modification" of white light by refraction or reflection. Newton introduced a "New Theory of Light and Colours" in 1672, according to which white light is actually a heterogeneous aggregate of rays of different refrangibilities that produce the different colours-i.e. rays that are disposed to be refracted at different angles, so that each degree of refrangibility corresponds to a different colour. (E.g. red and blue are both present in white light, but are refracted at different angles by a glass prism, and this is why the prism "separates" the white light into the colours of the rainbow. The mechanical philosophers, in contrast, assumed that the prism "modifies" the pure white light to produce the spectrum of colours.) Newton's theory was criticised on the grounds that it contained no explanation for the nature of light, such as the wave theories seemed to supply.
But Newton suggested to his opponents that they could accept his "new theory" of light and colours without giving up the hypothesis that light is some kind of wave. One would only have to assume that a ray of light of a given degree of refrangibility, and of the corresponding colour, is produced by a wave of a given wavelength. Thus the infinite gradations of colour that make up the visible spectrum, from red to violet, would correspond to a gradation of wavelengths. In the 19th century, this suggestion was taken up by physicists, and eventually led to a unification of optics with electromagnetism. On this theory, light is just one kind of electromagnetic radiation, corresponding to one small part of a spectrum of possible wavelengths, which includes waves shorter than those of "violet" light ("ultraviolet" light) and longer than those of "red" light ("infrared" light).
According to Newton's conception of the role of mathematical laws in
physics, A is a model of B if A can be seen to obey the same mathematical
laws as B. So, for example, the force holding the moon in orbit is the
same as gravity just in case the former follows the same mathematical law
as gravity; a fluid vortex is a poor model for planetary motion because
planetary motions don't obey the same mathematical laws as motions of particles
in fluid vortices. To regard wave motion as a model for light propagation,
we have to be able to show that light follows the same mathematical laws
that wave motion follows. Newton thought this was impossible, because light
appears to follow the simple mathematical law of travelling in a straight
line, and he couldn't see how a wave would always do this. But 19th
century physicists developed the theory of the transverse wave,
or a wave whose vibration is perpendicular to the direction of motion;
such a wave can indeed propagate in a straight line. By the end of the
19th century, the wave theory of light was empirically extremely
well established.
2. The ether. If light is a wave, it is natural to assume that it is a wave propagating in some medium; the waves must be the vibrations of something. Since electromagnetic radiation appears to be everywhere, and can even pass through solid bodies, this medium must fill all of space, including the interstitial spaces of the fundamental parts of matter. This medium became known as the "luminiferous" (light-bearing) ether. Measurements of the velocity of light showed that it travels at a constant velocity (c), which was assumed to be a velocity relative to the ether.
If this is true, however, measurements of the velocity of light ought
to differ when the measuring apparatus is moving relative to the ether
as well. Consider the velocity of a wave in water, and assume that the
wave propagates through standing water with velocity w. If you stand
still in the water and measure the velocity, you should get the result
w, but, obviously, if you are on a boat moving with velocity v,
your result should reflect the difference between v and w.
If the velocity of sound in still air is s, then if you are moving
through the air with velocity v, your measurement of the speed of
sound should depend on the difference between s and w. By
the same reasoning, your measurement of the velocity of light should depend
on the difference between the velocity of light (c) and the velocity
at which you are moving through the ether.
Since we have no idea of the velocity of the earth relative to the ether,
it might have seemed impossible to calculate this difference. But it was
possible to calculate how the measured velocity of light in the direction
of motion through the ether might differ from its velocity in a different
direction. The calculation was done, and in the 1880's A.A. Michelson devised
an experiment to test it.
3. The Michelson-Morley experiment. The Michelson interferometer
(see diagram) is supposed to measure variations in the velocity of light
that depend on the motion of the interferometer itself through the ether.
Therefore it compares the speed of light in the direction of the earth's
motion through the ether with its speed in other directions. In other words,
the experiment is supposed to show a physical effect of the motion of
the earth relative to the ether.
A light source at A sends a beam of light to a half-silvered
mirror at B, so that the beam is split: the reflected part goes
through a tube toward a mirror at D, and the transmitted part goes
through another tube toward a mirror at E; the paths BD and
BE have the same length (L). The parted beams are reflected
at D and E and then rejoin. If the times of the two trips
BE and BD are equal, the rejoined waves will be "in phase,"
and if the times are unequal they will be "out of phase" (i.e., the "crests"
of one wave don't correspond to the "crests" of the other).
Now suppose that the whole apparatus is moving through the ether at velocity v. with the direction BE parallel to the direction of motion v. Will the times of light travel along the two arms still be equal? A simple calculation shows that they should not be, and that the time from B to D and back should be shorter than the time from B + E and back.
4. The result. The experimental result is null! The calculated
difference between the two velocities is enough to produce a detectable
interference between the recombined beams of light, but none was observed
(even in many repeated experiments over several decades). In other words,
the motion of the earth relative to the ether is undetectable. We saw that
from the point of view of Galilean relativity, the motion of the earth
through space is undetectable, because only change of motion (acceleration)
makes a physical difference, and acceleration has the same value for observers
in different states of uniform motion. (It is an "invariant" of Newtonian
mechanics.) But from the point of view of the wave theory of light, motion
relative to the ether must be detectable. To accept that motion
relative to the ether really makes no physical difference, we would have
to accept the notion that there is actually a velocity (the
velocity of light) that has the same value for observers in different
states of uniform motion-i.e. a velocity that appears the same to
observers with different velocities. And this seems absurd.
5. The Lorentz contraction. H.A. Lorentz, in the 1890's, developed
a solution to this problem. Instead of supposing the "absurdity" that the
speed of light is really the same in both arms of the Michelson apparatus,
he asked: Assuming that the beam of light from B to E really
takes longer than the beam from B to D, as the calculation
shows, what could possibly account for the fact that it arrives back at
the source at exactly the same? One crucial assumption of the calculation
must be wrong: we assumed that the length BE is the same as the
length BD, and this must be the problem. Recall the familiar equation,
rate x time = distance. This means that if the rate or time isn't
what we expect, it could be that we were mistaken about the distance. And
if the rate and time don't change as a result of motion through the ether,
as we expected, it could be that it was the distance that changed. Hence
Lorentz's interpretation: The light did not travel at the same speed in
both directions; rather, the times were the same because the path BE
contracted. In other words, the speed of light appears to be the same because
the measuring apparatus contracted in the direction of motion. All objects
moving through the ether contract in the dimension parallel to their motion
through the ether, and this explains why motion through the ether cannot
be detected.
II. Special Relativity: Lorentz vs. Einstein..
1. Lorentz's interpretation of the Michelson-Morley result follows these simple steps:
a. The velocity of light is a constant relative to the ether. Therefore it should appear to differ to observers moving relative to the ether, and to each other.
b. The Michelson-Morley device is sufficiently delicate to register such differences: the light beam moving parallel to the motion of the device through the ether should take longer to return to the detector than the beam moving perpendicular to the motion. [Note: How do we know which arm of the device is moving parallel to the motion through the ether? There is no way to know this. But the experiment is very cleverly designed: we know that, as the earth turns, and the device turns in the laboratory, sooner or later the arms will be aligned, one parallel and one perpendicular to the motion. All we need to do is watch the interferometer for some indication (interference) that the two beams of light do not arrive at the same time.]
c. The device fails to detect such a difference, so the beams are arriving back at the same time.
d. Since the beams take the same amount of time, the difference in the velocity of light in the two directions must be hidden somehow. It must be the distance that has been affected.
e. Therefore the arm of the apparatus that is moving parallel to the motion through the ether must have contracted, and the contraction must be exactly enough to hide the difference in the speed of light. That is, the relative velocity of light is actually slower in this direction, but the effect of this is hidden because the distance has contracted.
f. Generally: The velocity of light cannot be the same for observers whose relative velocities are not the same. Therefore, if light appears to travel at the same velocity in every reference frame, it must be because the distance it travels is shortened by motion. All objects are shortened in the dimension parallel to their motion through the ether. Because the contraction depends on the ratio of their velocity to the velocity of light, and because this is usually extremely small, the contraction is too small to notice in most experiments. [Note: This is not really as bizarre as it may sound. As Lorentz pointed out, we assume that bodies are held together by electromagnetic forces-which are, again, transmitted by waves in the ether. So it is not unreasonable to think that these same forces can be affected by motion through the ether.]
2. Einstein's interpretation: The failure to detect the difference in the velocity of light is an indication that there really is no difference-that the velocity of light really is invariant, or the same for observers whose relative velocities are different. In other words, just as there is no physical difference between uniform motion and rest in Newtonian mechanics, there is no difference between uniform motion and rest in electrodynamics. Lorentz had a logical argument that the velocity cannot be invariant, but Einstein examined the premises of that argument. What do I mean when I say that I am at rest in the ether? I mean that, if I explode a firecracker, the light spreads out in all directions around me, like an expanding sphere, while I remain in the centre of the sphere. More simply, suppose that I flash two beams of light in opposite directions, toward two mirrors located at equal distances from me; I will remain in the centre of these two beams. If I remain at rest, the beam moving to my right will reach the mirror on my right at exactly the same time that the beam moving left will reach the mirror on my left, and they will return to me (i.e. I will see the reflections in the mirrors) at exactly the same time. If you are rolling by me (say, to my left) while I do this experiment, passing me just as I flash the lights, you won't stay in the centre of the two beams, but will chase the one moving left. In other words: I suppose I am at rest, because the beams reach equal distances from me at the same time; I suppose you are in motion because the beams do not reach equal distances from you at the same time. (When the beams reach the mirrors, you are much closer to the left beam than to the right.)
Now, suppose what seems to be absurd: that you, too, think that you
are at rest. What does this mean? Again, it can only mean that you think
that the two beams reached equal distances from you at the same time, i.e.
that you remain in the centre of the two beams (or at the centre of the
expanding sphere of light). I, from your perspective, must be much closer
to the right beam when the two beams reached their mirrors, while you at
this moment are equidistant from them.
In this simple example we already see the solution to Lorentz's puzzle. How can observers whose velocities are different agree on the velocity of light? By disagreeing on which events are simultaneous. I think that the two events, when the light beams reached equal distances from me, are simultaneous; you think the two events, when the beams reached equal distances from you, are simultaneous. To determine what time on my watch is simultaneous with the beam reaching the mirror, I calculate the midpoint between the time the beam left and the time it returns; you do the same to calculate what time on your watch is simultaneous with the reflection. The velocity of light can be invariant only if simultaneity is relative.
Einstein gives a simple definition of simultaneity. Send a light signal to a mirror; measure the time elapsed before you see the reflection; when you divide the elapsed time in half, you get the time on your watch that was simultaneous with the reflection. Obviously this definition assumes that light takes the same amount of time to travel from you to the mirror as from the mirror back to you. Now, by the reasoning given above, two observers in relative motion who use this criterion will get different results about which events are simultaneous.
Comparison. Both Lorentz's and Einstein's interpretations rest
on assumptions that that certain physical quantities are invariant. If
the fundamental quantities of Newtonian mechanics are invariant, the velocity
of light must be relative. Einstein's revolutionary idea was that if the
velocity of light is invariant, the fundamental quantities of Newtonian
mechanics (all of which, as we know, presuppose the invariance of simultaneity)
must be relative. Assuming absolute simultaneity, the relativity principle
and the principle of the invariance of the velocity of light contradict
one another; the two principles can be reconciled only if we accept the
principle that simultaneity is relative. Why should we accept such a theory?
Einstein's answer is essentially a philosophical one: Lorentz's theory
presupposes that simultaneity is absolute, but provides no way to measure
it; what we are sacrificing, then, is an idea of simultaneity for which
we had no good physical definition in the first place. Light signals provide
a robust definition based on something that appears to be invariant (as
far as experiments can detect). By this criterion, simultaneity will necessarily
depend on the motion of the observer.
Lorentz:
1. Force, mass, and acceleration are invariant.
2. Therefore length and time must be invariant.
3. Therefore simultaneity is invariant.
4. Therefore the velocity of light cannot be invariant.
5. Therefore the apparent invariance of the velocity of light
can be explained by the real contraction of the measuring device.
Einstein:
5. Therefore force, mass, and acceleration are relative.
4. Therefore length and time must be relative.
3. Therefore simultaneity is relative.
2. Therefore the velocity of light is really invariant.
1. The apparent contraction of the measuring device can be explained
by the invariance of the velocity of light.
Everything that is strange about Einstein's theory follows from the relativity of simultaneity. For example:
"Bodies in motion contract in the dimension parallel to their motion." The fact is that the measurement of length depends on the state of motion of the observer. If I am at rest and you are moving, you appear to me to have contracted; but you will say the same thing about me. The truth is that there is no invariant measure of length: we agree on the velocity of light, but we disagree on simultaneity, so we disagree on length.
"Clocks in motion slow down." Again, because relatively-moving observers disagree on simultaneity, they disagree on the measurement of time-intervals. Each will think that the other's clock is moving more slowly.
"Mass increases with velocity." Again, because there is no invariant measure of acceleration, there is no invariant measure of mass. The faster you go relative to me, the greater your mass will seem-i.e. the harder it will be to accelerate you (whereas by Newton's laws, acceleration depends only on the applied force, not on the initial velocity). But you will think the same of me.
"No mass can travel at the speed of light." This follows from the relativity of mass: as your velocity approaches the velocity of light relative to me, the amount of force required to accelerate you-the measure of your mass--becomes infinite. (Hence the causal structure illustrated above, Part C.)
Moral of the story: The structure of spacetime is defined by the laws of electrodynamics, not the laws of mechanics.