A Brief Guide to the Great Equations (24 page)

How had Einstein, at that time still a patent clerk, come to take up this particular problem? The same way everyone else did: dissatisfaction.

Years later, Einstein wrote to a friend that, while only about 5 or 6 weeks elapsed between his conception of the idea of the special theory of relativity and a finished paper about it, ‘the arguments and building blocks were being prepared over a period of years.’
¹⁰

The earliest argument emerged in late 1895 or early 1896, when Einstein was sixteen. It came in the form of what he would call a ‘childlike thought-experiment.’ (The adjective ‘childlike’, in the sense of pure and direct, was often applied to Einstein.) What, the youth asked himself, would happen if he were traveling at the speed of light, and looked over at a light beam riding next to him?
¹¹
Newton said it could happen, Maxwell said it couldn’t.

This simple puzzle – can you or can’t you catch up to a light wave – had to have an answer, but none could be fashioned from the existing tools of physics. The puzzle focused the dissatisfaction of young Einstein, providing him with the mixture of bewilderment and curiosity needed for him to start fashioning the arguments and building blocks.

Einstein brooded for years. ‘I must confess’, he told a friend later, ‘that at the very beginning, when the Special Theory of Relativity began to germinate in me, I was visited by all sorts of nervous conflicts. When young, I used to go away for weeks in a state of confusion, as one who at that time had yet to overcome the stage of stupefaction in his first encounter with such questions.’
¹²
One day in 1905 he went to visit his close friend and patent office colleague Michele Besso, poured out the details of his ‘battle’ with the problem, and departed. But in the process of laying out the problem, Einstein found the solution. The next day, he dropped in on Besso again and greeted him with the words, ‘Thank you. I’ve completely solved the problem.’
¹³

The result was ‘On the Electrodynamics of Moving Bodies’, one of the most famous and momentous scientific papers ever written, sent to the journal
Annalen der Physik
in June 1905. Despite the angst behind the genesis of the paper, it follows a simple yet powerful logic – ‘a deep, almost childlike freshness of approach’
¹⁴
– that is relatively easy to understand.

‘It is well known’, Einstein begins, ‘that Maxwell’s electrodynamics – as usually understood at present – when applied to moving bodies, leads to asymmetries [eccentric results] that do not seem to attach to the phenomena [that is, they seem to be an artifact of our theories rather than a part of the world].’
¹⁵
He gives examples, and says that these, ‘and the failure of attempts to detect a motion of the earth relative to the ‘light medium’, ‘ lead to the postulate that there is no such thing as ‘absolute rest.’ He calls this conjecture ‘the principle of relativity’, and says that he will combine it with the postulate that in empty space ‘light is always propagated with a definite velocity V which is independent of the state of motion of the emitting body.’

Thus Einstein framed his paper around the logical requirement of reconciling the two key principles: relativity and the constancy of the speed of light. These are ‘seemingly incompatible’, Einstein says. Only seemingly. For he develops a reconciliation in the rest of the paper, claiming that, based on logic alone, he can produce a ‘simple and consistent electrodynamics of moving bodies’ with no need for the supposition of an ether or for an absolute rest frame. What would it take for observers on two different inertial frames to see light travel at the same speed? Einstein determines that it would require the same contraction factor for length in the direction of motion and time that Lorentz proposed.

But while Lorentz had based his work (as had FitzGerald) on the assumption that the ether existed and that the contraction was real (due to the effect of ether on molecular forces), Einstein based his only on the assumption of the validity of the principles of relativity and of the constancy of light. That is, while Lorentz and FitzGerald got their results by trying to save the ether, Einstein arrived at the same result by getting rid of it. As scientists said at the time, ‘There is no conspiracy of concealment, because there is nothing to conceal.’ Or as Feynman liked to say, a universal conspiracy is a law of nature.

Einstein refers to the contraction factor as ‘ß’ in his paper. Its deduction is most easily and frequently presented as a Pythagorean problem. Suppose two inertial reference frames,
A
and
B
, are moving at velocity
v
with respect to each other. In
A
, a beam of light is sent from a source, perpendicularly to the direction of motion, to bounce back off a mirror at a distance
d
away from the source. From the point of view of someone on
A
, the light simply travels a distance 2
d
. But to someone on
B
, for whom
A
– source, mirror, and all – is gliding past at a velocity
v
, the light travels a longer path; we’ll call it 2
d
’. Half of this path,
d
’, is the hypotenuse of a right-angled triangle whose other sides are
d
and
vt
’/2. Thus (
d
’)
²
= (
d
)
²
+ (
vt
’/2)
²
. Yet according to the second principle, the light has the same speed,
c
, covering the same distance in the same time, seen from
B
as it does seen from
A
. That is,
V
(the symbol Einstein is using for the speed of light) is equal to 2
d
/
t
in
A and
to 2
d
’
/
t
’ in B. How can that happen? Only if the distance and time of objects in
A
are shorter in
A
as seen from
B
. By how much? By just the amount that
d
is shorter than
d
’; that is,
d/d
’ or
t/t
’, or the contraction factor ß. If
V
= 2
d/t
, then
d
=
Vt
/2; and if
V
= 2
d
’
/
t
’, then
d
’ =
Vt
’
/
2. Substituting in the Pythagorean equation gives us ß (or the contraction factor
t
/
t
’ we are seeking) =
.

This, the seminal paper of what would become known as the ‘special theory of relativity’ (a usage that Einstein began in 1915, to distinguish it from his then-new ‘general theory of relativity’), was published in September 26, 1905. It introduced some radical changes in notions of space and time. A paper with such fundamental implications, however – especially one put together in such a short time by someone working feverishly on so many things – was bound to have more consequences than its author could foresee
while composing it. One struck him almost immediately. Sometime in fall 1905 he wrote to his friend Conrad Habicht,

A consequence of the study on electrodynamics did cross my mind. Namely, the relativity principle, in association with Maxwell’s fundamental equations, requires that the mass be a direct measure of the energy contained in a body; light carries mass with it. A noticeable reduction of mass would have to take place in the case of radium. The consideration is amusing and seductive; but for all I know, God Almighty might be laughing at the whole matter and might have been leading me around by the nose.
¹⁶

Being led ‘around by the nose’ – reminiscent of how Meno’s slave must have felt, learning something which appears to be true, yet which also must be further explored.

Einstein mailed a three-page paper outlining this consequence, entitled ‘Does the Inertia of a Body Depend Upon its Energy Content?’ to the
Annalen
the day after his relativity paper appeared, and it was published later that year. As historian of science John Rigden, among others, has pointed out, this paper does not break new ground, and simply draws a consequence that was logically implicit in the previous paper, and easily could have been its final section. If it had, Rigden says, ‘it would have made a spectacular conclusion.’
¹⁷

Einstein opens the ‘Energy Content’ paper in a disarmingly modest key, ‘The results of an electrodynamic investigation published by me recently in this journal lead to a very interesting conclusion.’ He reaches the conclusion via the following example. Suppose an object (an atom, say) of mass
m
at rest in reference frame
A
emits two beams of light – thus, it expends energy – in opposite directions. Let’s say the total amount of energy lost is
L
(as in the previous paper, Einstein uses the now-unfamiliar notation of
L
for energy and
V
for the speed of light), so each light beam carries away the energy
L/
2.
An observer on
A
sees the object as having no net change in kinetic energy; the atom is standing still, has shed some of the energy it had in an excited state, and continues to have the same mass that it was originally stamped with. But an observer in
B
, for whom
A
is moving, sees something different. The forward-moving light beam has more momentum than the backward one, meaning that the atom has had a net change – a decrease – in kinetic energy. This can happen only if the atom’s velocity or its mass decreases. But its velocity is the same; in the rest frame, there is no recoil. The only other possibility is that, from the perspective of the frame in which the atom is moving, the mass has decreased. The atom has not gained any mass from the point of view of its rest frame; its ‘inertial mass’ is the same. But its mass from the point of view of the laboratory, which views it as a moving object, changes. By how much? Applying the tools of the previous paper, Einstein finds that the conversion factor, once more, is ß.

Einstein continues – again, using the unfamiliar notation of
L
for energy and
V
for the speed of light – as follows:

If a body releases the energy L in the form of radiation, its mass decreases by L/V
²
. Since obviously here it is inessential that the energy withdrawn from the body happens to turn into energy of radiation rather than into some other kind of energy we are led to the more general conclusion: The mass of a body is a measure of its energy content.
¹⁸

This is the first appearance in print of the idea eventually to become famous as
E
=
mc
²
. It is not presented explicitly in the form of an equation, and it is not in its familiar symbols. However, the startling, even revolutionary mass-energy concept is fully articulated. The concept transformed some of the most fundamental notions of how the universe is assembled. It put together two things long thought to be utterly different: energy, whose conservation principle was a crowning achievement of nineteenth-century physics, and
mass, whose conservation principle was a crowning achievement of eighteenth-century science.
¹⁹
They can change into each other.

It also revolutionized the requirements for objectivity. On the Newtonian stage, energy and mass remained the same when observed from different inertial frames; on Einstein’s, they remain virtually the same at low speeds, but changes take place the closer the speed of the frames get to that of light. What is objective – really out there – is what changes in length and clock time by this amount when witnessed from another, sufficiently fast, inertial frame.
²⁰

Over the next several years, Einstein referred to this result several times, though again in the form of descriptions or in his original symbols, and not yet in his now-famous version. In a footnote to a 1906 paper, for instance, Einstein wrote that ‘the principle of the constancy of mass is a special case of the energy principle.’
²¹
Early in 1907, in another
Annalen
article, he refers to energy as e, mass as the Greek letter μ, and the speed of light as
V
, and he uses the equation