Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (112 page)

At first one might be inclined take these words as describing a horizontal broadening — an analogy-based extension from just
mechanics
to the
union
of mechanics with electromagnetism. (This should remind readers of Dr. Ellenbogen’s new way of curing an elbow disease, based on her
horizontal
analogical extension of the treatment of a knee disease, moving outwards from just
knees
to the
union
of knees and elbows, which are clearly close cousins.) But couldn’t one equally well hear Einstein’s sentence as declaring that he had such deep
a priori
faith in the uniformity of physics that he was willing to bet that Galileo’s principle holds
for all imaginable areas of physics
, not just for mechanics alone? (This alternative way of seeing Einstein’s act reminds us of Dr. Gerhard Gelenk’s
vertical
generalization of the treatment of knee diseases, wherein he changed perspective from just
knees
to the more abstract category of
joints.)
In sum, in this case, as in many others, we see that there is no sharp line of demarcation between vertical category leaps and horizontal category extensions.

In any case, whether it was a vertical or a horizontal mental move, Einstein’s extension of the Galilean principle of relativity wound up profoundly undermining much of the physics of the preceding three centuries. And yet this revolution emerged from the act of paying attention to the trivial-seeming similarity between experiments in a train (or any similar reference frame) that were limited to mechanics, and experiments that might also involve electromagnetism. Einstein’s intuition told him that any such distinction was unnatural, since, in the end, any conceivable experiment in any conceivable branch of physics belongs to the single unified tree of physics. Of course, this kind of sixth sense for how and when a category can be broadened is mysterious, and is one of the deepest of all arts.

The consequences of category-broadening by analogy, applied to the principle of Galilean relativity, were enormously deep and led Einstein to a rich network of ideas whose names are familiar to anyone interested in science today, such as the relativity of simultaneity, time dilation, the contraction of moving objects, the non-additivity of speeds, the twin paradox, and so on. But those ideas, fascinating though they are, are not our focus. We wish now to come back, as promised, to the equation
E = mc
²
, which, oddly enough, was nowhere to be found in Einstein’s first article on relativity. That thirty-page article, published in the summer of 1905, contained plenty of other equations whose consequences were unprecedented and revolutionary, but it lacked the little tiny equation that became the indisputable emblem of relativity.

For most people, experts and non-experts alike, the equation
E = mc
²
is so tightly linked with the notion of Einstein’s theory of relativity that imagining relativity with its signature equation completely absent would seem as strange as imagining the 1927 Yankees without Babe Ruth, or the town of Pisa in 1100 A.D., before its signature tower had ever been dreamt of. And yet the truth of the matter is that Einstein did not discover the now-celebrated equation until some months after his first relativity article appeared. He deemed his new finding interesting enough to warrant another article,
which appeared in November of that same year (thus just barely squeezing in under the wire of the
annus mirabilis
), and which was just two pages long.

Strictly speaking, the famous equation didn’t appear in that article, either, since the way Einstein saw fit to express his new discovery was through words rather than through an equation; however, those words were tantamount to saying “
E = mc
²
”. And then, two years further down the pike, he realized that his second relativity paper concealed some highly important implications that he hadn’t at all suspected when it was published. And so in 1907, he published yet a third article, at last spelling out the full meaning of the symbols “
E = mc
²
”. It was this article that grabbed the world’s imagination, because its conclusions not only were counterintuitive and scientifically far-reaching, but also had profound potential implications for society.

We will consider these developments in chronological order, starting with the very short article of November, 1905. In it, Einstein imagined an object that could simultaneously emit two flashes of light in opposite directions (say east and west). So let us imagine a flashlight with bulbs at both ends. Since a flash of light possesses some energy, and since energy is always perfectly conserved by all physical processes (mechanical, electromagnetic, and so on), our two-headed flashlight will necessarily
lose
some energy — namely, the total energy carried off by the two departing flashes. From the point of view of energy, one has to pay for producing light! All this is quite obvious.

The key step Einstein took here was the (nearly) trivial idea of looking at the two-headed flashlight from another frame of reference — specifically, a moving frame of reference, such as a train moving at 30 miles an hour, let’s say westwards. According to special relativity, observers sitting in the train have the right to consider themselves stationary and to claim that
the flashlight is moving eastwards
at 30 miles an hour (and
always
at that speed, since their frame of reference — the train — has a fixed speed). For the train’s passengers, the two flashes necessarily undergo the Doppler effect.

For those readers unfamiliar with it, the Doppler effect merits a brief digression. It holds for any kind of wave, including light and sound waves. In the case of sound, it’s the shift that one hears each time an ambulance approaches, passes by, and then recedes into the distance: just at the moment it drives by, its siren seems suddenly to sink to a lower pitch. To those inside the ambulance, nothing changes, of course, but for people standing on
terra firma
, it’s quite another story. Why does this surprising sonic shift take place?

Imagine a pond into which a stone has just been tossed. From the spot on the surface where the stone plunged into the pond and is now sinking, circular ripples go spreading out. Now toss in a cork floating somewhere on the pond’s surface. Soon enough the concentric ripples will reach the cork, one after the other, and they will start making it bob gently up and down at a regular frequency. This bobbing cork is analogous to the vibrating eardrum of a person who hears the siren from
within
the ambulance: the vibration clearly has a fixed frequency.

But now imagine, by contrast, a toy motorboat speeding across these same circular ripples, first heading straight toward the center of the concentric circles (the waves’ source), and then continuing onwards towards the far bank. While it is moving toward
the center as the ripples expand, the toy boat bobs up and down
more frequently
than the cork does (for the boat, the circles seem to be coming out to meet it), but once it has crossed the circles’ center (located just above the sinking stone), the toy boat has to catch up with the ripples that are now fleeing from it, and so it meets them
less frequently
than before, meaning that it bobs up and down less quickly than before. This is an aquatic Doppler effect: the felt frequency of the ripples suddenly falls, just at the moment when the boat passes their source.

Likewise, in the ambulance situation, the perceived frequency (i.e., the pitch) of the siren suddenly falls as the ambulance rushes by the observer on the street. The Doppler effect generally says that, if an observer is moving with respect to the source of some waves (or conversely, if the source is moving with respect to the observer), then the frequency of the waves, as perceived by the observer, will depend on the relative speed of the two reference frames. Of course it was a nontrivial analogical extension to generalize the original effect from sound waves to other types of waves, such as light waves and ripples on a pond — but that’s another story. Suffice it to say that the Doppler effect as applied to electromagnetic waves was a fairly new idea at the turn of the twentieth century, and the third-class patent clerk in Bern, though he didn’t invent the notion, took great advantage of it.

Indeed, Einstein calculated the Doppler effect for the double flashlight using his own theory of special relativity, freshly minted just a few months earlier. He imagined himself in the frame of reference where the flashlight was moving at a constant speed (in other words, the frame in which the train is stationary), and he carried out relativistic Doppler-effect calculations that gave him the energy of each of the two flashes of light that sped off simultaneously. By adding these energies together, he got the total energy lost by the flashlight. He was able to use this sum to calculate how much
kinetic
energy the moving flashlight had lost at the instant when the rays were emitted, which should have been exactly zero, since the flashlight had just kept on moving at a constant clip. But it wasn’t exactly zero — it was just a tiny bit different from zero. Einstein’s Doppler-shift calculations revealed to him that the moving flashlight had to have
lost
some kinetic energy by sending off two flashes of light.

This result was extremely peculiar. It was obvious that to produce light, the flashlight had to give up some
electrical
energy (in its battery), but why would it also give up some of its energy of
motion
(which is given by the standard formula “Mv
²
/2”, the capital
M
of course denoting the flashlight’s mass, and
v
denoting its velocity)? We know that the flashlight doesn’t slow up in the least! By fiat, it is moving at a
constant speed.
(Recall that in the first frame of reference it is perfectly stationary; it’s only the observers on board the train who see it as moving, because their frame of reference is gliding down the tracks. And given that their frame is gliding at a perfectly
constant
speed, and that the flashlight is stationary with respect to the ground, the “moving” flashlight never loses or gains a speck of speed, as seen by train-bound observers.) How then can the steadily-moving flashlight have lost even the tiniest fraction of its energy
of motion
? Let’s devote a moment’s thought to this humble riddle, a humble riddle whose solution shook the world.

If, upon releasing the two flashes of light that carry total energy E, the flashlight loses even the tiniest amount of its kinetic energy, then the just-cited formula for kinetic energy “Mv
²
/2” tells us that either the flashlight’s
mass M
or its
velocity v
must have suddenly diminished at the moment of emission. But as we just mentioned, the train has a constant speed, which means that the flashlight, as seen from the train, also has a constant speed. Thus
v
is unchanged. We therefore have no choice: the only thing that could possibly have become smaller is
M
, the flashlight’s mass, and according to Einstein’s Doppler-shift computations (which we will not spell out here), the tiny amount of mass that the flashlight loses, which we’ll denote by lowercase
m
, is equal to
E/c
²
. (It’s crucial not to confuse the flashlight’s
total
mass
M
with the negligible quantity of mass
m
that it loses when it gives off the two rays of light.)

Anyone who follows Einstein’s (rather simple) calculations must agree with him that an object that gives off electromagnetic radiation will necessarily lose some mass — namely, an inconceivably tiny quantity of mass that depends on the amount of energy
E
carried off by the radiation. Why tiny? Because the energy of the light itself (
E
, which is the fraction’s numerator) is negligible, and the fraction’s denominator
c
²
is incredibly huge — after all, it’s the square of the speed of light, which is to say, the square of 299,792 kilometers per second (that is, the square of 1,079,252,849 kilometers per hour). And when one divides an already microscopic energy — that of the two flashes — by this gigantic quantity, the result will necessarily be infinitesimal.

The fact that one is multiplying a mass by a speed squared here (
mc
²
) might surprise a nonscientist, but it doesn’t surprise physicists, for ever since Galileo, Kepler, and Newton, physicists have grown accustomed to the idea that the laws of nature involve algebraic expressions — often powers (most often squares or cubes) of quantities that are directly observed. Indeed, to anyone who has ever taken any physics at all, the formula “K.E. =
Mv
²
/2”, giving the kinetic energy of a moving object with mass
M
and velocity
v
, is both familiar and unsurprising.

So let’s come back to the quantity
E/c
²
, which Einstein had just identified as being relevant to this situation. What is surprising in this quantity, then, is not the nature of the algebraic expression itself, featuring an energy divided by a velocity squared, the result of which will necessarily have the units of mass — but its
meaning.
The amazing thing is firstly that this
m
represents the mass lost by our energy-emitting object, no matter what its original mass
M
was, and secondly that the relationship between the sizes of
m
and
E
is mediated by a special and universal constant of nature — namely, the speed of light.
This
is what was truly new and strange, not the algebraic structure of the formula (an energy divided by a velocity squared), which in itself contains no surprises. In summary, it’s the
idea
suggested by this formula — the idea of
energy possessing mass
— that should catch people totally off guard, not its mere
algebraic form
, which is rather ho-hum to anyone who realizes, as physicists already had realized for three centuries, that energy always has the units of mass times velocity squared.