A Brief Guide to the Great Equations (23 page)

The inconsistency involved an idea called
invariance
. In its loosest sense, invariance simply means that something can appear in two different ways but nonetheless be the same thing. Two people standing in different parts of a room, for instance, can see a chair as to the right or to the left of the television – but once we take the difference in positions into account, it becomes clear not only that they are seeing the same chair, but also how and why that chair appears differently to each. If we cannot account for the difference in appearance, then one or both of these people is hallucinating or seeing some sort of illusion. Real things, we might say, are
supposed
to look different
from different perspectives. Reality therefore necessarily involves a difference between how something appears to us, and what it really is. We can put this point another way, in terms of the difference between local effects and global properties. When I see an object I see only one profile of it – one that changes if I move, if the light changes, and so forth. As I change my position, so does this ‘local’ effect. Yet all the time I am seeing the same object. Invariance therefore involves understanding unity as it shows itself in changing appearances. Philosophers call this the noetic-noematic correlation; physicists call it invariance under transformation, or covariance. Covariance is simply part of the definition of objectivity; to say that something is a real part of the world is to say that it looks different from different perspectives, though the descriptions flow together in an orderly way when described by the right set of transformations.

Newton’s mechanics supposed the existence of an absolute time and an absolute space as the arena in or stage on which events happen, and that there is no privileged time or space on that stage. This is called translation invariance, and all it means is that if we move about in time or space the laws are still the same. But Newtonian mechanics implies a further kind of invariance, that according to a ‘principle of relativity of motion’, there is no privileged movement, whether in motion or at rest. The laws of physics are the same for anyone moving at constant speed regardless of direction, no matter how fast or slow. This is a familiar experience. So long as a train, say, is traveling smoothly and doesn’t jostle, anything we do – drink a glass of water, play cards or handball, dance – happens exactly the same as if the train were at rest in the station. The water stays in the glass and doesn’t slosh, the ball bounces at the equivalent point on the floor, and the dancer confidently executes the same gestures and winds up in the same spot as if the train were still. We could perform no experiment to tell how fast the train was moving, or even whether it was moving. Even people on a second train at rest in the station down the track would see the same laws of physics in play on ours, once the difference in speed between the trains was taken into
account. And that difference in speed is a matter of simple addition and subtraction.

Scientists would call such a train, from whose position we describe events, a
reference frame
, and one moving at uniform speed an
inertial reference frame
. They call
transformations
the equations used to change the mathematical description of an event – its
x
,
y
, and
z
of position and its time
t
– from one reference frame into another. They call the equations that connect the properties of a description in one inertial reference frame with those of another
Galilean transformations
, for these express a principle of relativity of motion already present before Newton in Galileo’s mechanics, in the latter’s thought experiments involving dropping cannonballs from the masts of sailing ships. The Galilean transformations are quite simple. On board the moving train, for instance, the only thing about events that changes is their distance down the track (let’s make that the
x
-axis). Any
x
position on that train, call it
x
’, differs from the
x
position for an observer on the ground by the distance the train has traveled in a time
t
:
x
’ =
x
−
vt
. All the other coordinates –
y
and
z
– remain the same, and things continue to happen at the same time
t
.

A physicist’s definition of reality and objectivity depends on Galilean transformations. A ‘real’ thing or event is one with the same physical description in different inertial frameworks, once you use the appropriate transformations to take the differences in speeds and directions into account. The notion of reality
requires
drawing a difference between how something appears to us, and how we describe it; the variability to observation is built into the objectivity of the object that I see. In developing the notion of transformations, scientists were merely elaborating the conditions of objectivity – of what is the same regardless of which inertial frame it is seen from.

Thus the principle of the relativity of motion was at the core of Newtonian mechanics. But according to a ‘principle of the constancy of the velocity of light’ central to Maxwellian mechanics, light introduces a new element into this neat picture. Light acts more like sound. Sound always travels at the same speed (about
1,100 feet a second in air), regardless of how fast its source travels. The reason has to do with the properties of the medium (air molecules, say) that propagate the sound waves that make it impossible to push sound waves any faster than a certain speed. According to Maxwell’s equations, light also always travels at the same speed (about 186,000 miles a second) regardless of the speed of the source. Physicists assumed this stemmed from the fact that light moves in a medium called ether, whose properties governed how fast light could travel. If so, this principle implied that there
was
a favored inertial reference frame in the ‘stage’ of absolute space and time, provided by the ether. In moving around the sun, the earth moves through the ether, and while it might ‘drag’ some small amount with it, its speed with respect to the ether could be detected by measuring the speed of light in different directions. For the ether moves the light, by an amount that involves the Pythagorean theorem.

Imagine a speedboat heading toward the opposite bank of a 400-yard river, whose flow carries it 300 yards downstream in the process. The water, of course, moves the boat. The speedboat winds up moving at an angle along the hypotenuse of a right triangle: 400 yards across and 300 yards downstream; it has traveled (because we’ve conveniently based the example on a Pythagorean triplet) a total of 500 yards in crossing the river. To end up directly across the river, it would have to point itself upstream while traveling by the same angle, and will in effect travel a longer distance (the hypotenuse of the above triangle) in crossing the bank directly. For the same reason – so the argument ran – the ether would move the light traveling in it, and the light would travel at a different speed when crossing the ether’s direction of motion.

In 1881 and 1887, two American physicists, Albert Michelson and Edward Morley, carried out an extremely sensitive experiment to detect what scientists were calling the ‘ether drift.’ Their instrument consisted of two ‘arms’, one pointing in the presumed direction of the ether’s motion and the other perpendicular to it, along which beams of light would travel back and forth. Mirrors were
mounted on a bed of mercury, then rotated 90 degrees so that the light would be traveling in a different direction with respect to the ether’s motion. By bringing the beams together to create an interference pattern, Michelson and Morley would be able to detect any slight difference in their velocities. But the experiment failed to detect any such difference.

Physicists were baffled. Something was clearly wrong either with Newton’s or Maxwell’s equations.

At first they assumed the problem lay with Maxwell. He was the Johnny-come-lately. Maxwell’s equations had been around for only a few decades, while Newton’s laws had been around for two hundred years and successfully accounted for everything except a few disagreeable but minor discrepancies that there was little reason to think would not prove to be due to some experimental error or overlooked effect. Some of the brightest scientists of the day attempted to modify Maxwell’s equations to fit the Galileo transformations.
³
But these equations proved remarkably resistant. They were embedded in an elaborate network of interrelated concepts, and any change in one rippled through the others with undesirable results.

As the nineteenth century drew to a close, many physicists interested in electrodynamics felt a deep dissatisfaction. There
had
to be an explanation – the constancy of the speed of light regardless of direction of motion had to be reconcilable with Maxwell and Newton – but none could be found. ‘The most incomprehensible thing about the world is that it is comprehensible’, Einstein once declared. The unstated corollary, is that, to a scientist, the most frustrating thing about the world is not being able to comprehend it.

Dissatisfaction led to desperation. In 1889, Irish physicist George FitzGerald wrote a short, one-paragraph article – a mere five sentences, no equations – stating that ‘almost the only hypothesis’ that can reconcile the Michelson-Morley experiment with Maxwell and
Newton ‘is that the length of material bodies changes, according as they are moving through the ether or against it by an amount depending on the square of the ratio of their velocities to that of light.’
⁴
Suppose, FitzGerald thought, the arm of Michelson and Morley’s instrument pointing in the direction of motion shrank due to the impact of the ether on its molecules. If it shrank by just the right amount, it would ‘measure’ the light beam as going up and down in the direction of the ether at the same speed as it measured the light traversing the perpendicular arm. Still, the idea – objects shrink in size when moving at high speeds? – seemed too bizarre to take seriously.

Another desperate soul was Dutch theorist Hendrik Lorentz, who wrote to his friend Lord Rayleigh in 1892 about the predicament created by the Michelson-Morley experiment, ‘I am utterly at a loss to clear away this contradiction.’
⁵
That year, he independently proposed the same idea that FitzGerald had, writing that ‘I can think of only one idea’ to explain the experiment, namely, that the ether causes some contraction effect in the length of a solid body. When he learned of FitzGerald’s idea and contacted him, FitzGerald was overjoyed to learn of a fellow champion of contraction, writing back that he had been ‘laughed at’ for his ideas.
⁶
Lorentz then went on to work out in detail the set of transformations that would have to occur for this contraction to work. Lorentz found that time, too, would be affected. For while FitzGerald was only trying to save the Michelson-Morley experiment, which determined the light in two directions to be traveling at the same speed, Lorentz, more ambitiously, wanted to make sure that the speed of light stayed constant, and appeared the same to moving and nonmoving observers. To accomplish this, clocks would have to tick slower. He then produced a set of formulas now known as the Lorentz transformations, which gave compensations in length and time between stationary and moving systems that would preserve the possibility of the light moving at a constant speed in the ether as detected by the Michelson-Morley experiment, and thus the agreement between Maxwell and Newton. The compensation factor
for both space and time was
.
⁷
Notice that when there is no relative motion (and
v
is 0), there is no correction. At low speeds, the correction is so small it would not be noticed. But the closer the object approached the speed of light, the larger the corrective factor grew – the more the object shrank in the direction of motion, and the slower clocks ticked. Meanwhile, however, most scientists continued to regard the idea as too strange to take seriously. But the fact that it was said at all shows the lengths to which scientists were prepared to go to save the ether.

It seemed, one scientist would say later, that ‘all the forces of Nature had entered on a conspiracy’ with the goal of ‘preventing us from measuring or even detecting our motion through the ether.’
⁸

The consternation mounted. Great scientists began reaching for fantastic ideas. In 1898, French mathematician Henri Poincaré toyed with the idea of giving up absolute time in favor of ‘local time’, and soon tried to use it to explain the puzzle regarding the speed of light in ether. In a public lecture at the 1904 World’s Fair in St. Louis, Poincaré remarked almost whimsically, ‘Perhaps we should construct a whole new mechanics, of which we only succeed in catching a glimpse…in which the velocity of light would become an impassible limit.’
⁹

Thus the problem that Lord Kelvin had called ‘Cloud No. 1’ obscuring the ‘beauty and clearness’ of nineteenth-century dynamical theory was only getting more and more difficult. The year after Poincaré spoke, in 1905, all the fantastic ideas that had been cited to try to banish it – the contraction of space and time at high velocities, the nonexistence of absolute space and time, and the speed of light as an absolute upper limit – were shown, in one form or another, to be true.