EINSTEIN'S
POSTULATES
In the previous chapters I have shown
you how to deduce the postulates of Relativity as theorems from the observation
that we exist. At the beginning of the Twentieth Century, Albert Einstein was
not so fortunate: he was obliged to devise the postulates by intuition. Before I
show you how he used those postulates to deduce the features of the Lorentz
Transformation, I want to show you how he may have conceived them and why it was
important that he did so.
One of the great projects of
Nineteenth Century physics was that of working out the nature of light. Though
the prospect of that project's success was dim at the beginning of the century,
by century's end physicists had succeeded well beyond their expectations, not
only showing how light is related to electricity and magnetism, but also
deducing and verifying experimentally the existence of related radiations. But
the physics underlying a clearer understanding of light also revealed deeper,
more subtle problems with physics as a whole. In spite of having the fundamental
equations of electromagnetism in hand and in spite of performing sophisticated
experiments suggested by those equations, physicists were unable to locate the
foundations of the Universe. That failure gave physicists a serious problem,
because the laws of electromagnetism make explicit reference to velocity in a
way that implied to the physicists of the Nineteenth Century the existence of an
experimentally verifiable state of absolute rest, a state that the best
experiments of the day could not find.
Albert Einstein did not actually set
out to solve that problem. By his own account, he had, at the age of sixteen (in
1895), devised a fantasy of flying alongside a ray of light. That fantasy was no
mere daydream: he was using it to help him to imagine what the light in the ray
would look like to him if he could pace it on its flight through space. He knew
that light, as Maxwell had shown, consists of an electric field and a magnetic
field, both so formed (specifically, in the shape of a wave) that when they move
past an observer they appear to that observer to vibrate. The vibrations are
important because a changing (i.e. vibrating) electric field generates a
magnetic field (by Maxwell's addition to Ampere's law) and a changing magnetic
field generates an electric field (by Faraday's law), so the moving fields in
light regenerate each the other as the light propagates. But if you could travel
with a ray of light, moving at the same speed at which it moves, the fields
would not appear to you to vibrate, so would the light exist for you? Einstein
reasoned that it would not, that for light to exist it had to move past all
possible observers at a certain speed, and showed as a consequence (in 1905)
that the Universe has no solid foundation.
That hypothetical foundation was
called the æther and it was a contemplation of light that led physicists to
postulate its existence in the first place. In 1801 Thomas Young demonstrated
conclusively that light is a wave-like phenomenon: he achieved that
demonstration by showing that light displays both constructive and destructive
interference, just as waves in a pan of water do. Reasoning from that
demonstration and making an analogy with sound, which is a vibration of an
elastic medium (i.e. air), physicists hypothesized that light is the vibration
of some hugely dense, intensely stiff, superfluid medium, which they named after
the celestial substance that the Ancient Greeks believed gave the sky its blue
color, the æther. In Maxwell's original work on electromagnetic theory in 1861
the æther was given another use: Maxwell conceived electric and magnetic fields
as strains induced in the æther by the presence of electric charges and
currents. In 1864, though, when he reworked his theory, he abandoned his use of
the æther, though he doesn't seem to have considered the consequences of doing
so. And there were consequences.
In Newtonian dynamics the effect of a
force is a change in a velocity. That means that in a calculation related to
describing the path of a body affected by a force we are using the difference
between the body's velocities measured before and after the application of the
force, so it makes no difference in the calculation whether we use absolute
velocities that are defined relative to some state of Absolute Rest that's the
same for everyone or whether we use relative velocities that are defined
relative to some convenient reference marker. But in the equations of
electromagnetism velocity appears in the description of the cause of the force
that electric currents exert and that force is an absolute quantity, so
Maxwell's contemporaries were convinced that velocity must be absolute as well.
They believed that all of the velocities used in the equations of
electromagnetism had to be measured relative to a frame of Absolute Rest, which
frame was occupied and marked by the æther, the foundation stone of Reality, the
Prime Meridian of velocity. Maxwell's abandoning the use of the æther in his
theorizing was tantamount to claiming that relative velocities can be used in
the laws of electromagnetic force, but that claim couldn't actually be true to
Reality, could it? Well, as always, there were experiments that would reveal the
truth of the matter.
Alas for those who preferred to
believe in the existence of the æther, the Universe seems somewhat perversely to
be designed specifically to hide the existence of the æther from physicists
conducting clever experiments. And the experiments were quite clever. In 1887
Albert Abraham Michaelson (1852-1931) and Edward Williams Morley (1838-1923)
attempted to measure Earth's motion through the æther with an interferometer, a
device that splits light into two beams, makes those two beams follow two
different paths, and then brings them together so that they will create an
interference pattern. They hypothesized that light traveling across the
direction of the æther's flow through their laboratory would move at a speed
different from the speed at which light would travel in the direction with or
against the ætherial flow. To test that hypothesis, they passed light of a
single frequency through a tilted beam splitter (a partly transparent mirror
that reflects half the light striking it and allows the other half to pass
through it) that sent the two resulting rays down paths oriented perpendicular
to each other. Reflected off mirrors at the ends of the paths, the rays came
back through the beam splitter and were projected onto a small screen in such a
way that they interfered with each other; that is, the electromagnetic fields in
the rays so canceled or augmented each other that the resulting light formed a
pattern of alternating dark and bright bands on the screen. Those bands were the
clever part of Michaelson and Morley's experiment.
In accordance with their hypothesis,
Michaelson and Morley figured that the time that the light required to traverse
one of the paths in their interferometer would change as the orientation of the
path was changed from being parallel to the direction of the æther wind to being
perpendicular to it. They didn't have the means to measure the traversal times
directly, so they designed their experiment to measure them indirectly. That's
why they split one beam of light into two beams that then traversed paths
oriented at a right angle to each other. They had in mind the idea that as the
apparatus was rotated (and it was mounted on a granite slab floating on a pool
of mercury to enable it to be rotated smoothly) the traversal time along one
path would increase slightly and the traversal time along the other path would
decrease slightly. Those changes would change the way in which the two rays
interfered with each other on the screen and thus make the pattern of dark and
bright bands appear to shift sideways. Measurement of that shift, correlated
with the orientation of the apparatus, would enable Michaelson and Morley to
calculate the speed and direction at which the hypothetical æther wind blew
through their laboratory. So they switched on their apparatus, took their data,
and discovered that the speed of the æther through their apparatus was precisely
zero. It was always zero. Whenever they made their measurements, regardless of
time of day or time of year, it was always zero. And they did conduct their
experiment at different times of the year, just in case Earth periodically slips
into some ætherial doldrum. But Earth changes its velocity by sixty kilometers
per second every six months, so a zero result throughout the year was a truly
strange result to obtain.
In 1892 the Irish physicist George F.
Fitzgerald (1851-1901) offered a suitably strange hypothesis to explain
Michaelson and Morley's result. He reasoned that if electromagnetic fields are
strains in the æther, as Maxwell had initially suggested, then when the æther
blows through a body it might alter the electromagnetic fields that hold the
body's atoms together; specifically, he hypothesized that the body would be made
to shrink in the direction parallel to the direction in which the æther was
moving. When he used that idea in its mathematical form to analyze Michaelson
and Morley's experiment, he found that the presumed shrinkage of the
interferometer would, as the apparatus was rotated, cancel the difference
between the changes in the times that the two rays took to traverse their paths
through the apparatus, thereby producing the null result.
Three years later the Dutch physicist
Hendrik Antoon Lorentz (1853-1928) elaborated Fitzgerald's hypothesis, in part
by noting that a clock moving through the æther would tell a time different from
the time told by a clock at rest in the æther. He summed up his elaboration of
Fitzgerald's work in four equations that we call the Lorentz Transformation,
though the interpretation that we give those equations differs remarkably from
the one that Lorentz and his contemporaries gave them. And, although we no
longer believe that it is caused by the æther wind, we call the shrinkage of
objects due to their relative motions the Lorentz-Fitzgerald contraction in
remembrance of the two men's contribution to Relativity.
As disappointing as their results
were, Michaelson and Morley did not pronounce the last word on clever
experiments to find the æther. If the Lorentz-Fitzgerald contraction prevented
interferometers from detecting the effects of the moving æther, then someone
would have to devise an experiment that did not depend upon the length of their
apparatus. In 1903 the obscure team of Trouton and Noble did just that. They
were inspired by the notion that an electric current imposes upon the æther a
stress that strains the æther into manifesting a magnetic field. In order to be
the source of such a stress, the current must consist of electric charges moving
through the æther, so an electric charge that's stationary in the laboratory
will nonetheless generate a magnetic field if the æther blows over it. To test
that hypothesis, Trouton and Noble attached two flat metal plates to each other
face to face with a narrow gap between them, suspended them from a thin quartz
fiber, and put equal and opposite electric charges on the plates. If their
hypothesis was correct, the magnetic forces generated by the æther wind blowing
through the apparatus would exert a torque that would act to turn the plates
parallel to the direction in which the æther was moving. The apparatus and its
successors were all sensitive enough to detect the expected torque, but
regardless of how the apparati were turned or when the experiments were
performed, the measured torque was always zero. And in this case there was no
analogue of the Lorentz-Fitzgerald contraction to provide theoretical cover for
the null result.
Do you see what I mean about the
Universe appearing to be so designed that the æther eludes the grasp of clever
experimentalists? Apparently none of our experiments can manipulate the laws of
physics in a way that will reveal the existence or the motion of the æther. What
that means, though, is that in our theorizing about those laws of physics we can
ignore the æther, pretend that it doesn't exist, and yet enjoy the confidence
that our theories won't go wrong as a result. But how can we ignore the æther in
the laws of electromagnetic force? The velocities that appear in those laws must
be measured relative to something. If that something is not the æther, then what
is it? According to Einstein, it's everyone's favorite inertial frame - their
own.
Einstein first presented his theory of Special Relativity in the 1905 Sep 26 issue of Annalen der Physik, in a paper titled "On the Electrodynamics of Moving Bodies". In that paper he showed that the laws of electromagnetic force are not invalidated if the velocities used in the calculations are referred to arbitrarily chosen inertial frames of reference. He showed that the calculation of the electromagnetic forces acting among several bodies observed from one inertial frame can be translated into a correct description of the forces as they would be measured by an observer in another frame, the translation being made with formulae based upon the Lorentz Transformation. His first move was, of course, to deduce the Lorentz Transformation from two postulates; that is, from two statements asserted as if they were axioms, even though they both lack the axiom's necessary property of being self-evident. The first of these postulates, as translated into English from the original German, is
"1. The laws by which the states of physical systems undergo change are not affected, whether those changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion."
Putting that statement another way, we have
COSMIC THEOREM 1: The
results of any given experiment or observation and the laws of physics derived
from them are the same for all experimenters or observers, regardless of how
those experimenters or observers are positioned, oriented, or moving relative to
each other.
That postulate, the Principle of
Relativity, did not originate with Einstein. The Italian mathematician and
natural philosopher Galileo Galilei (1564-1642) first described the principle in
1633 in his book "Dialogue on the Two Chief World Systems". He described the
principle indirectly by stating that certain phenomena (such as the flight of
insects, the fall of objects, and the motion of a man jumping about as though
playing hopscotch) observed aboard a ship sailing across a smooth sea would
appear no different from the same phenomena observed on land. Later that same
century (in 1687) Isaac Newton (1642-1725) offered much the same description in
his Principia.
Einstein's version of the principle, the statement that the laws of physics must be expressed in the same mathematical form, regardless of which inertial frame of coordinates is chosen as the frame of reference, is equivalent to claiming that there is no experiment that will reveal which frame is that of absolute rest. Thus did Einstein indirectly dismiss the æther as being of any relevance to his theory. But if all inertial frames are perfectly equivalent to each other, then the physical constants (the proportionality factors that appear in the equations of physics) must also be the same for all observers. In particular, the electric permittivity of vacuum and the magnetic permeability of vacuum must be the same in all inertial frames. Because the product of those two numbers is inversely proportional to the square of the speed at which electromagnetic waves propagate through vacuum, we are led to Einstein's second postulate, the one that makes Special Relativity truly special and that marks the place in the theory where Einstein's stroke of genius struck hardest. That postulate is
"2. Any ray of light moves in the 'stationary' system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body."
We can restate that postulate in a perfectly equivalent form that refers to the scientists who observe the ray rather than to the motion or nonmotion of the ray's source. We have, then
COSMIC THEOREM 2: Any phenomenon that moves at the same speed at which the boundary of space moves passes all observers at that same speed, regardless of how those observers are positioned, oriented, or moving relative to each other.
The "determined velocity c" to which
Einstein referred is just the speed of light, which we hypothesized is the same
as the speed at which the boundary of space seems to recede from us and it has
the value 299,792.458 kilometers per second or 186,234.709 miles per second.
With those two postulates in mind,
Einstein performed a series of imaginary experiments, involving fast-moving
trains, in which experiments he worked out the features of the Lorentz
Transformation before going on to work out the dynamical laws that govern the
interactions among electromagnetic fields and moving bodies. Now I'm going to
guide you through a repetition of those thought experiments.
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