origin of relativity

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HPS 0410 Einstein for Everyone

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Einstein's Pathway to General Relativity

John D. Norton

Department of History and Philosophy of Science

University of Pittsburgh

The Starting Point

Adjusting Newton's Theory of Gravitation

"The Happiest Thought of My Life"

The Principle of EquivalenceRelativity of Inertia ("Mach's Principle")

Learning About Gravitation

Gravitational Slowing of Clocks

Gravitational Bending of Light

The Rotating Disk

Assembling the Pieces

What You Should Know

We have followed a simple pathway to the main ideas of thegeneral theory of relativity. We started with the geometrical notion

of the

curvature of space and saw how that geometrical notioncan be extended from space to spacetime. We then found the

resulting theory of curved spacetime not just to cover a curved

geometry of space, but gravitational phenomena as well.

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This pathway to the theory was not Einstein's. His was more

indirect, more inspired, more tortured and more fallible. The final

theory emerged after Einstein struggled for seven years withmany things: strong hunches about what the theory should say

physically, vivid thought experiments to support the hunches,

lengthy explorations into new mathematics, errors and confusions

that thoroughly derailed him and a final insight that rescued himfrom exhaustion and desperation.

The seven years of work divides loosely into two phases. The

earlier phase of his work was governed by powerful physicalintuitions that seemed as much rationally as instinctively based. He

felt a compelling need to generalize the principle of relativity from

inertial motion to accelerated motion. He was transfixed by the

ability of acceleration to mimic gravity and by the idea that inertia is

a gravitational effect. As Einstein struggled to incorporate these

ideas into a new physical theory, he was drawn to use the

mathematics of curvature as a means of formulating the new

theory.

As the mathematics of curvature took a more controlling position

in the later phase, his work began to change. The theorizing was governed increasingly by notions a mathematical simplicityand naturalness. When the theory was completed, Einstein's

starting point was quite distant. It remains a matter of controversy

today whether Einstein succeeded in realizing his original

ambitions.

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It is impractical in this chapter to review all these considerations.

Einstein's intricate mathematical struggles in the later years cannot

easily be described in informal terms. However some of his earlier

physical reflections are so famous and so characteristic ofEinstein, that they must be mentioned. You should treat these asinteresting reports on Einstein's intellectual biography. You may

well find it hard to connect some of the ideas to be laid out withthe final theory.

The Starting Point

Einstein's first concrete steps on his pathway to general relativitycame in 1907 when he was commissioned by Johannes Stark to

write a review article on relativity theory for Stark's journal Jahrbuch der Radioaktivitaet und Electronik . The exercise was,

apparently, quite straightforward. In his 1905 theory, Einstein had

offered a new account of space and time. Since the theories of

physics were all set in space and time, physicists needed to be

assured that these theories could be maintained; or, if not, shownhow they should be adjusted to fit with Einstein's new theory.



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The exercise proceeded well. Electrodynamics actually neededno adjustment. Einstein's 1905 theory of relativity had been

created to fit with the existing theory. The mechanics of bodies

required adjustments to the notions of energy, momentum andmass. The most prominent of these was the famous equivalence

E=mc2. Einstein also sketched a relativistic treatment of

thermodynamics, the theory of heat and work.

Then came gravity. Newton's celebrated theory of gravitationpresumed instantaneous action at a distance. The sun now exerts



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a gravitational force on the earth now with a magnitude set by

Newton's inverse square law. The key part was the "now." If the

sun were to move slightly, the resulting alteration in the force it

exerts on the earth would be felt by us instantaneously according

to Newtonian theory.

That means that Newton's theory depends upon a notion of

absolute simultaneity. A change there is felt here at the samemoment. However Einstein's 1905 theory had banished absolute

simultaneity from physics. Different observers would judge

different pairs of events to be simultaneous. Newton's theory hadto be adjusted to accommodate this new relativity.

Adjusting Newton's Theory of



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Gravitation

The change needed was, apparently, straightforward. In the

revised theory, a change in the sun should not be felt here on

earth instantly, but only after a time lag of around 8 1/3 minutes,the approximate time light takes to propagate from the sun to the

earth. Then absolute simultaneity would no longer be needed in

the theory.

This meant that Newton's theory needed to be adjusted to look

more like electrodynamics. In the latter theory, effects do not

propagate instantly in the electromagnetic field; they propagate in waves at the speed of light. There were many ways to make the

adjustments Newton's theory needed. All of them produced very

small changes in the predictions of the theory. While one might

not be sure precisely which of the many adjustments was the right

one to pick, there didn't seem to be any major problem. Rather the

issue was a surfeit of good solutions. Or so believed otherleading thinkers of Einstein's time, such as the great French

mathematician, Henri Poincaré, and the inventor of spacetime,

Hermann Minkowski.

Einstein, however, did not see it that way. He examined gravitation

theories, modified to allow for a finite time of propagation of

effects, and found a result that aroused great suspicions in him.In the modified theories, the distance fallen by a body varies



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according to its sideways motion. In the simplest case, the body

would fall a shorter distance if it has some sideways velocity.

The differences in the distances fallen were very small and notlikely to be detectable in an experiment. Nonetheless they

bothered Einstein. They contradicted the exact correctness of

Galileo's old observation that all bodies fall alike, even though thedifferences were far too small to be detectable by the methods

available to Galileo.

Other physicists of the time were aware of this effect, but

discounted it as too small to be of any concern. Einstein did not. It

meant that the way a body fell would depend on the energy of the



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body. We can only guess now why that bothered Einstein somuch. It might be that Einstein imagined that a hot body,

consisting of many small atoms in thermal motion, might fall

differently from a cold one according to these theories.

Einstein was still a clerk in the Bern patent office in 1907. Yet he

came to the extraordinary conclusion that an adequate theory ofgravitation could not be devised within the confines of hisexisting theory of relativity.

"The Happiest Thought of My

Life"

It was while pondering this problem that Einstein hit upon what he

later described as "the happiest thought of my life." If began when

he suddenly saw new significance in a commonplace of

Newtonian gravity. A body in free fall in Newtonian gravity doesnot feel its own weight. This effect is very familiar to us now.We have all watched space-walkers floating weightlessly outside

their capsules. They are in free fall above the earth, orbiting with

their space stations and that free fall cancels their weight.



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This effect came about from an apparently accidental agreement of two quantities

in Newtonian theory: the inertial mass of a body happens to equal its gravitational

mass exactly. Einstein now believed that this equality could be no accident. Heneeded to find a gravitation theory in which this equality is a necessity.

The inertial mass ofa body measures itsresistance toacceleration when aforce is applied to it.

The gravitation massof body measureshow it responds to agravitational field.

For more, see this.

The immediate outcome of this reflection was Einstein's

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"principle of equivalence." It formed the basis of the concluding

Part V of his 1907 Jahrbuch article. There he suggested that

gravitation required an extension of special relativity based on the

principle of equivalence.

The Principle of Equivalence

There are very many formulations of the principle ofequivalence in the literature. Most of them pick up directly on the

idea of weightlessness in free fall. They assert that free fall

transforms away a gravitational field in some tiny volume of space.

While this is a common formulation of the principle in text books, it

is troubled. Free fall transforms away gross effects of gravitation.

But, in Einstein's final theory, it does not transform away the

effects of spacetime curvature. In that sense, free fall does not

transform away gravity in the final theory.

Einstein later complained about this version of the principle,

objecting that one could not in general transform away an arbitrary

gravitational field over an extended region of space. His original

formulation and the one to which he adhered for his entire life

proceeded differently. He turned around the original idea offree fall eradicating gravitation. Acceleration can also produce a

For more, see John D. Norton,"What wasEinstein's Principle of Equivalence?" Studiesin History and Philosophy of Science, 16(1985) , pp. 203-246; reprinted in D. Howardand J. Stachel (eds.), Einstein and the History of General Relativity: Einstein Studies Vol. I,Boston: Birkhauser, 1989, pp.5-47.

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gravitational field.

More specifically, Einstein took the case of special relativity

without gravitation. He now imagined a uniformly accelerated

observer, in relation to whom all free objects would accelerate.

That state of space found by the observer, Einstein asserted in hisprinciple of equivalence, is a homogeneous gravitational field.In this case, uniform acceleration and homogeneous gravitation

are equivalent.

Einstein developed the idea in one of his best known thought

experiments. He asked us to imagine a physicist who awakens ina box. Unknown to the physicist, the box is in a distant part of thespace of special relativity and is being accelerated uniformly inone direction by the tug some agent. If the physicist were to

release objects in the box, they would be left behind by the

accelerating box; they would move inertially, while the box

accelerated. This figure shows this for two bodies of different

mass at rest and a third body that has a horizontal inertial motion.



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The physicist inside the box would find that the released masses

to accelerate in a direction opposite to the box's acceleration. The

physicist would judge there to be a field inside the box pulling on

all free bodies.



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Now comes the key point. All bodies released by the physicist

would fall exactly alike, no matter what their mass or composition.

So the field found by the physicist inside the box would manifest

the signature property of a gravitational field: it would accelerateall bodies exactly alike.

One might be tempted to say that the field inside the box is just an

"inertial field," some sort of fake gravitational field. Einstein'sassertion was otherwise. The field created by motion in the box

just is a full-blown, authentic homogeneous gravitational field.

Principle of Equivalence

The inertial effects inside a uniformly accelerated box in

gravitation free space are equivalent to those of a

homogeneous gravitational field; more tersely, uniform

acceleration creates a homogeneous gravitational field.

h i l j d b i j



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The equivalence just asserted may seem benign. It seems just to

codify an equivalence in the way bodies fall in two cases. In fact

the assertion is strong, for it asserts that the equivalence appliesto all processes, not just fall the bodies. That means that itapplies also to all processes involving fields, such as electric and

magnetic fields.

You will see why Einstein found this principle attractive. Hisefforts to produce a relativistic theory of gravity had failed since he

could find no theory in which all bodies fell alike, no matter what

their mass or composition. The gravitational field delivered by the

principle of equivalence was assured to have this property. In

particular, the sideways motion of a body would have no effect on

its rate of fall. The field generated in this thought experiment did

not have the defect of the earlier theories.

Relativity of Inertia ("Mach'sPrinciple")

What also attracted Einstein in this analysis was that it promised to

remedy a defect he perceived in both Newton's physics and inspecial relativity. In both, you will recall, it is just a brute fact that

certain motions are distinguished as inertial. This, in Einstein's

i i It b tt th th i i l id th t



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view, was worrisome. It was no better than the original idea that

there is an ether state of absolute rest. There seemed to Einstein

no good reason for why one state should be the absolute rest

state rather than another. Correspondingly, Einstein saw no good

reason for why some motions should be singled out as inertial and

others as accelerating.

In 1916, Einstein formulated this worry in a thought experiment.

He imagined two fluid bodies in a distant part of space. Thesebodies, the reader quickly infers, are like stars or planets, which

form roughly spherical shapes under their own gravity. Einstein

further imagines that there is relative rotation between the two

bodies about the axis that joins them. This relative rotation is

verifiable by observers on each body, who can trace out the

motion of the other body. Each would judge the other to berotating.

It can happen in ordinary Newtonian physics that one of these

bodies is not rotating with respect to an inertial frame and the other

one is. In that case, the second rotating body will bulge. Thiseffect arises on the earth. It rotates about the axis of its north and

south poles It bulges slightly at the equator as a result of



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south poles. It bulges slightly at the equator as a result of

centrifugal forces that seek to fling the matter of earth away from

this axis.

It would be entirely unacceptable, Einstein now asserted, were this

to happen to two spheres in an otherwise empty space. For there

is no difference in the observable relations between the two

spheres. Each rotates with respect to the other. So why should

just one bulge? The supposition of Newton's absolute space or of

inertial systems, Einstein protested, was an inadequateexplanation. Einstein demanded something observable to makethe difference.



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Einstein was an avid reader of the

physicist-philosopher Ernst Mach.

In Mach's writings, Einstein hadfound what seemed to be a

solution to the problem. Mach

seemed to be proposing, Einstein

thought, that the privileging of

certain states of motion is due tothe distribution of matter in the

universe. Why is our frame ofreference inertial? It is because

the stars are at rest in our frame.

Why is my wording so careful

here? it is not clear that whatEinstein reported Mach assaying is what Mach actuallysaid. For more, see John D.Norton, "Mach's Principlebefore Einstein." in J. Barbourand H. Pfister, eds., Mach'sPrinciple: From Newton'sBucket to Quantum Gravity:Einstein Studies, Vol. 6.Boston: B irkhäuser, 1995,

pp.9-57. Download.

When we try to accelerate, we feel inertial forces. These are theforces that make us dizzy when we spin in a fun fair; or they are

the forces that throw our coffee in the air when our airplane hits an

air pocket.

These forces, Einstein understood Mach to assert, arise from an

interaction between the mass of our body (and our coffee) and all

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interaction between the mass of our body (and our coffee) and all

the other masses of the universe, distributed in the stars. Einstein

first called this idea the "relativity of inertia" and later, in 1918,

"Mach's Principle."

In the case of Einstein's two fluid spheres, the bulge of one ofthem would now be explained by the fact that this bulging sphere

was rotating with respect to all the other masses of the universe,

whereas the other sphere was not. That would be the observable

difference between the two fluid bodies.

This analysis was clearly inspired by Mach's famous account of

Newton's bucket experiment. Newton had noted that water in aspinning bucket adopts a concave surface, as a result, Newton

urged, of its rotation with respect to absolute space. No, Mach had

responded several hundred years later all one has in the case of



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responded several hundred years later, all one has in the case of

Newton's bucket in rotation with respect to the stars. We cannot

know more than what our direct observations tell us. All they tell us

is that these inertial forces arise when we accelerate relative to the

stars.

The weakness of this analysis is that there is no account of how

rotation with respect to distant masses could produce these

inertial forces. In 1907, Einstein hoped that his emerging theoryof gravity would provide the mechanism. It could then satisfy

Mach's Principle and, through it, generalize the principle of

relativity to acceleration. For in a theory that satisfies Mach's

Principle, no state of motion is intrinsically inertial or accelerating.

When we see something accelerating, it is not accelerating

absolutely in such a theory; it is merely accelerating with respect tothe stars. Preferred inertial motions need not enter into the

account any more. All motion, accelerated or inertial, would be

relative.

To deliver this sort of account of inertial forces, Einstein's theory

would need to break down the strict division between inertial and

accelerated motion of his special theory of relativity. The

principle of equivalence promised to weaken this division.According to it, whether the physicist in the box was to be judged

accelerating or not depended on your point of view. An inertial

observer would judge the physicist to be accelerating uniformly in

a gravitation free space. The physicist would judge him or herself

to be unaccelerated in a gravitational field. It was a first step

towards generalizing the principle of relativity to acceleration



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towards generalizing the principle of relativity to acceleration,

Einstein believed.

Learning About Gravitation

By his own later judgment, Einstein did not, in the end, find a

theory that fully satisfied Mach's Principle. The immediate benefit

of his new principle of equivalence, however, was that it let

Einstein learn a lot about gravitation. For the principle delivered to

Einstein one special case of a gravitational field that, he believed,

conformed with relativity theory and in which all bodies truly fell

alike. Einstein's program of research on gravity in the five yearsfollowing 1907 was simply to examine the properties of this onespecial case and to try to generalize them to recover a full theory.

His early hope was that the generalization of the principle of

relativity would somehow emerge in the course of those

investigations.

Gravitational Slowing of Clocks

Two properties of this special case of the gravitational field were

noteworthy. First, Einstein recognized that clocks run at differentrates in the box of his thought experiment according to their

location. A clock placed lower in the created field runs slower.



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location. A clock placed lower in the created field runs slower.

Einstein immediately generalized that effect to all gravitational

fields. Clocks deeper in a gravitational field run slower. A clock in

the sun would run slower than one on earth--if only we could havea clock in the sun without it being destroyed by the heat of the sun.

It turns out we can find clocks in the sun. Radiating atoms radiate

in very definite frequencies of light according to which element

they are. That means that they behave like little clocks. Their

running slower is manifested in a slight reddening of the light they

emit. Einstein computed an effect on the wavelength of sunlight of

one part in two million.

While Einstein did not use spacetime diagrams in 1907, they

provide an easy way to see that clocks run at different rates

according to their position when they accelerate in a Minkowski

spacetime. The effect is driven almost entirely by the relativity ofsimultaneity.



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The spacetime diagram shows two clocks

A and B accelerating together towards the

right in a Minkowski spacetime. The

numbers show the proper time elapsed

along each clock's worldline and thus the

time each clock reads. The hypersurfacesof simultaneity are those of the inertialobserver on the left of the figure.According to that inertial observer, the two

clocks run at the same speed, at least for

the initial portion of their acceleration.

Why don't the two clocks run at exactly the same speed? This is an artifact ofhow uniform acceleration arises in a Minkowski spacetime. Observers on theclocks judge the distance between them to stay the same. Therefore an inertialobserver will judge this distance to contract. As a result, the inertial observer

will judge the two clocks to accelerate at slightly different rates; thedifference will be just enough to give the length contraction effect. This means

that, in the same time, the A clock will achieve a greater speed than the Bclock, according to the inertial observer's judgments of simultaneity. Hence theinertial observer will judge the A clock's reading to start to lag slightly behindthat of the B clock. This effect is shown in the figure, which has been drawncarefully to scale.

If you really have to see more details, see uniform acceleration in a Minkowskispacetime.

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Now consider an observer who accelerates with the rightmost "B"

clock, that is, the clock higher up in the created field. As the clock

changes speed, that observer's hypersurfaces of simultaneity will tilt so that the B observer will judge the A clock to be lagging

successively more behind. When B's clock reads 2, B will judgethe A clock to read 1; when B's clock reads 4, B will judge the A

clock to read 2. Overall, B will judge A's clock to be running at half

the B clock's speed. The effect, the figure shows, is entirely due

to the relativity of simultaneity.



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The geometry of uniform acceleration in a Minkowski spacetime

turns out to be especially simple. The hypersurfaces of

simultaneity of an observer accelerating with the B clock turn out to

coincide with the hypersurfaces of simultaneity of an observer

accelerating with the A clock. Hence the observer moving withclock A will agree that the A clock is running slower and the Bclock faster. When the A observer's clock reads 1, A will judge B's

clock to read 2. When the A observer's clock reads 2, A will judge

B's clock to read 4.




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The second important effect pertained to light. An unaccelerated

observer finds that light propagates in a straight line in Minkowskispacetime. Here, for example, is such a light flash propagating

across the box of Einstein's thought experiment.

For the physicist accelerating with the box, however, the light will

be judged to fall, just like everything else in the box. As a result,

the physicist will find the light's path to be bent downward by the



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gravitational field.

Einstein generalized this result to arbitrary gravitational fields. This

generalization enabled him to make one of the most celebratedpredictions of his theory. A ray of starlight grazing the sun

would be bent as the light fell into the sun's gravitational field. This

bending would be manifested as a displacement of the star's

apparent position in the sky and this displacement would be

visible at the time of solar eclipse.

In 1907, Einstein had predicted the gravitational bending of light.

But he did not realize that it might actually be tested at the time of

a solar eclipse. After his 1907 Jahrbuch article, Einstein's efforts

were redirected towards the puzzle of the quantum. In 1911,

however, he returned to theorize about gravity. He realized then

that his prediction of the gravitational bending of light could be

tested at a solar eclipse. He wrote another paper developing

this idea and also other aspects of his theory.



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Einstein was keen to see this test undertaken. The greatest

difficulty was that it required a solar eclipse and that meant that

astronomers must place themselves precisely in its path. In 1913,

Einstein wrote to the American astronomer G. E. Hale asking

whether the test could be undertaken without an eclipse.



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Hale responded that it could not. The brightness of the sky near an

uneclipsed sun is just too great.

In August 1914, there was a promising eclipse of the sun that

would be visible from the Crimea. Einstein's colleague, the

astronomer Erwin Freundlich, mounted an expedition to the

C i t b d h t h th li



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Crimea to observe and photograph the eclipse.

Unfortunately for Freundlich, the First World War broke out.Since he was German, the Russians interned him and confiscated

his equipment.

Fortunately for Einstein, no measurement was taken. Einstein'stheory of 1914 was not yet the complete general theory of

relativity. In his earlier theory, there was no curvature of ordinary

space in the vicinity of the sun. As a result, as we saw in another

chapter, his theory predicted the same deflection as Newtonian

gravitation theory (assuming light consists of massive corpuscles).

It was half the deflection predicted by the final theory. Had the test

been carried out successfully, it would have produced a result thatcontradicted Einstein's earlier theory.

Gravitational Slowing of Light

In 1907, Einstein had also concluded that the speed of light, andnot just its direction, would be affected by the gravitational field.

The effect was closely connected with the gravitational slowing of

clocks and is almost entirely a consequence of the relativity of

simultaneity. One can see how it comes about with a similar set of

spacetime diagrams. The clocks A, A', B and B' all accelerate

uniformly in a Minkowski spacetime and in a way that ensures that

the distance from A to A' remains the same as from B to B'. A light

signal propagates from A to A' and a second light signal

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signal propagates from A to A and a second light signal

propagates from B to B'.

The figure shows the hypersurfaces of simultaneity of an inertialobserver. Of course the inertial observer will judge the two lightsignals to propagate at the same speed. That is just familiar

special relativity.

We notice also that, initially, the four clocks A, A', B, B' run in

synchrony according to the judgments of simultaneity of the

inertial observer. Hence using the readings of these clocks

directly, we will infer that the two light signals propagate at the

same speed. In more detail, we note that the distance from A to A'

equals the distance from B to B'; and each light signal takes thesame time to traverse the distance. Both light signals leave when

the local clocks read 0 and arrive when the local clocks read 3.

Hence using these local clock readings, we infer that the twolight signals travel at the same speed.



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Now consider how these processes are judged by an observer

who accelerates with the clocks. All that changes in the analysis

that follows is that we use different judgments of simultaneity.That leads to the judgment of differing speeds for the propagation

of light.

Let us take the observer who accelerates with clock B. That

observer's hypersurfaces of simultaneity will tilt more and more as

clock B gains speed from the acceleration. This was the effect

that led observer B to judge that the A clock was running slower

than the B clock. This same tilting will lead observer B to judge

that the AA' light signal propagates at roughly half the speed ofthe BB' light signal Both signals traverse the same distance



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the BB light signal. Both signals traverse the same distance.

However the the AA' signal leaves A when the B clock reads 0 and

arrives at A' then the B clock reads 4. The BB' signal leaves B

when the B clock reads 0 and arrives at B' when the B clock reads

a little over 2.

Recall that the judgments of simultaneity of acceleratingobservers who move with the clocks agree, since they agree onthe hypersurfaces of simultaneity. So we can choose any one of

the accelerating observers and get the same outcome. Each of

the accelerating observers will judge the transit time for BB' to be

roughly half that of AA' They will agree that light propagates



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roughly half that of AA . They will agree that light propagates

slower on the left side of the figure, that is, deeper in the created

field.

Applying the principle of equivalence, we now conclude that thesame slowing manifests in a gravitational field. A light signal

deeper in the gravitational field at A propagates slower than a lightsignal higher in the gravitational field at B.

The conclusion that gravity slows the speed of light caused

Einstein some trouble with unkind contemporary critics.Einstein had first based his theory of 1905 of the striking idea of

the constancy of the speed of light, but he now seemed to be

retracting it.

By 1912, Einstein had developed all these ideas into a fairly

complete theory of static gravitational fields, that is gravitational

fields that do not vary with time and admit well defined spaces.

The most striking characteristic of the theory was that the

intensity of the gravitation field, the gravitational potential, wasgiven by the speed of light. So as one moved to different parts of

space, the intensity of the gravitational field would vary in concert with the changes in the speed of light. As late as 1912, some f ive

years after Minkowski's work, Einstein was loath to use spacetime

methods. While I have developed the clock slowing and light

slowing effects using spacetime diagrams, Einstein did not do

this. His method of analysis was algebraic. He represented the

processes by equations in which speeds and times appeared as

variables. He rarely if ever drew diagrams such as given above.



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variables. He rarely if ever drew diagrams such as given above.

What Einstein now needed was a way to extend these results tothe more general case of gravitational fields that vary with time.

That, it turned out, required Einstein to move well beyond the

mathematics he knew. Another thought experiment pointed the

way.

The Rotating Disk

If one has a circular disk at rest in some inertial reference system

in special relativity, the geometry of its surface is Euclidean. It willbe useful to spell out what that means in terms of the outcomes of

measuring operations. If the disk is ten feet in diameter, then it

means that we can lay 10 foot long rulers across a diameter.

Euclidean geometry tells us that the circumference is π x 10 feet,

which is about 31 feet. That means that we can traverse the full

circumference of the disk by laying 31 rulers around the outer

rim of the disk.



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What if we have a disk of the same diameterof 10 feet but in rapid uniform rotation withrespect to the first disk? Things will go rather

differently. Assume that this rotating disk is

covered with foot long rulers that move with it.

These rulers are just like the ones that were

used to survey the non-rotating disk. (That

means that an observer moving with the rod on

the rotating disk would find it to be identical to

one of the rulers used to survey the non-rotating

disk.) What will be the outcome of surveying the

geometry of this rotating disk with those rods?

Note what was not said in this account. It did not say that we takethe first disk and set it into rotation. The reason is that it is impossiblein relativity theory to take a disk made out of stiff material and set itinto rotation. If one were to try to do this, the disk would contract in thecircumferential direction but not in the radial direction. As a result, adisk made of stiff material would break apart. If we want a rotating diskmade of stiff material, we need to create it already rotating. Once in aletter on the subject, Einstein remarked that a way to get a disk of stiffmaterial into rotation is first to melt it, set the molten material intorotation and then allow it harden. The rotatin disk roblem has

An observer who is not rotating with the disk

would judge all these rulers to have shrunk in the

.created a rather large and fruitless literature that suggests some sortof paradox is at hand. Most of it derives from a failure to recognize thata stiff disk cannot be set into uniform rotation without destroying it



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j g

direction of their motion. That means that,

according to this new observer, the surveying of

the disk would proceed differently. Ten rulers

would still be needed to span the diameter of the

disk. Since the motion of the disk is

perpendicular to the rulers laid out along adiameter, the length of these rulers would be

unaffected by the rotation. That is not so for the

rulers laid along the circumference. They lie in

the direction of rapid motion. As a result, they

shorten. More rulers are needed to cover thefull circumference of the disk.

a stiff disk cannot be set into uniform rotation without destroying it.

Another little trap to avoid: While we have used the judgments of anobserver not on the disk to infer the outcome of the surveyingoperations on the disk, the outcomes of those operations areindependent of the observer's state of motion. Either a diameter can becovered with ten rods or it cannot; either the circumference can bespanned by 31 rulers or it cannot. Once one observer has found whichis the case, we know the result for all observers.

Thus we measure the circumference of the rotating disk to be

greater than 31 feet the Euclidean value In other words we find



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greater than 31 feet, the Euclidean value. In other words, we find

that the geometry of the disk is not Euclidean. The circumference

of the disk is more than the Euclidean value of π times its

diameter.

The significance of this thought experiment was great forEinstein. Through his principle of equivalence, Einstein had found

that linear acceleration produces a gravitational field. Now he

found that another sort of acceleration, rotation, produces

geometry that is not Euclidean.

Assembling the Pieces

Einstein had all this in place by the summer of 1912. He knewthat gravitation could bend light and slow clocks. He expected that

the final theory would somehow involve accelerations in a new way

and that such accelerations came with a breakdown of Euclidean

geometry. He also knew that the natural arena in which to conduct

relativity theory is Minkowski's spacetime.

To us, the

final step

does not

seem like

such a great



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leap.

Assemble the

pieces and

infer thatgravitation is a

curvature ofspacetime! All

that is needed

is nice

mathematical

clothing to

dress this

idea.

For Einstein in

1912 it was

far from easy.

He first

needed the

assistance of

his

mathematician

friend Marcel

Grossmann to

find his way in

the new and

Marcel Grossmann

difficult

mathematics



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the theory

required.

Then Einstein took a series of wrong turnings and ended up with

the wrong gravitational field equations--not the celebrated Einsteinequations that appear in all the modern textbooks. It required three

years of painful work first to recognize that something had gone

wrong and then to find the final equations.

The precise causes underpinning these wrong turning remain a

point of debate in the history of general relativity literature. Two

elements, however, played a role in misleading Einstein.

First, in 1912 and 1913, Einstein had recognized the need to

employ a geometry of variable curvature in spacetime in his theory

of gravity. However he was convinced that this curvature would not

be manifested in the space-space slices of spacetime in certain

simple cases. These were the cases of a static gravitational field

and also a very weak gravitational field. Both of these are realized

in the gravitational field of the sun. Einstein expected spacearound the sun to exactly Euclidean. Alas, as we have seen,Einstein's final theory required curvature in the space-space slices

even in this simple case. That meant that Einstein could not

accept the equations of the final theory for they would entail a

curvature of space when Einstein believed there was none.

Second, Einstein used a different style of theorizing to the one

largely used in these chapters. Here, we have used a geometrical

http://pdfcrowd.com/http://pdfcrowd.com/redirect/?url=http%3a%2f%2fwww.pitt.edu%2f~jdnorton%2fteaching%2fHPS_0410%2fchapters%2fgeneral_relativity_pathway%2f&id=ma-160420024343-c24926cchttp://pdfcrowd.com/customize/http://pdfcrowd.com/html-to-pdf-api/?ref=pdfhttp://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/index.html#Geometry


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approach, emphasizing the picturing of gravitational effects in

geometric diagrams. Einstein, however, labeled events in

spacetime with arbitrarily coordinate numbers and expressed all

his results in terms of equations relating these coordinates.

Einstein knew that this labeling of spacetime events with

coordinates was purely arbitrary and that all his results had to beindependent of the particular coordinate system used.However knowing this in the abstract and carrying through the

demand in all details are two different things. By his own later

admission, Einstein found it hard to purge his coordinate systems

of independent reality.

One the low points in his struggle with coordinate systems came when Einstein used an ingenious argument--the "holeargument"--to show that gravitational field equations like the onesof his final theory are inadmissible on physical grounds. While the

hole argument did not warrant that conclusion, it has been

rehabilitated in recent work in philosophy of space of time, where it

now lives a good life. (See, "The Hole Argument." Stanford Encyclopedia ofPhilosophy.)

For a glimpse into Einstein's private notebook to see his

calculations during the decisive phase of the discovery of general

relativity, see "A Peek into Einstein's Zurich Notebook." on my

Goodies page. Here's one page on which Einstein writes down

the Riemann curvature tensor for the first time and finds it hard to

see how it can be used in his gravitational field equations.

http://pdfcrowd.com/http://pdfcrowd.com/redirect/?url=http%3a%2f%2fwww.pitt.edu%2f~jdnorton%2fteaching%2fHPS_0410%2fchapters%2fgeneral_relativity_pathway%2f&id=ma-160420024343-c24926cchttp://pdfcrowd.com/customize/http://pdfcrowd.com/html-to-pdf-api/?ref=pdfhttp://plato.stanford.edu/entries/spacetime-holearg/http://www.pitt.edu/~jdnorton/Goodies/index.html#Zurich_Notebooklhttp://www.pitt.edu/~jdnorton/Goodies/index.html


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42/45


What made the last phase of this three years especially urgent

was the fact that David Hilbert, the greatest mathematician of the



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, g

era, had also become interested in the theory and had started to

formulate the gravitational field equations in a mathematically more

elegant formulation.

In November 1915, Einstein published his f inal version of the

theory, complete with the gravitational field equations sodistinctive of his theory. Here are those equations as he wrote

them at that time, in a 1916 review article:

Here he writes them later in the simple case of a matter free spacetime:



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What You Should Know

What first led Einstein to work on what became his general theory of relativity.

The principle of equivalence

How Einstein used it to infer the properties of gravitational fields.

The relativity of inertia.

Einstein's transition to the mathematics of spacetime curvature.

Copy right John D. Norton. February 2001; January 2, 2007, February 15, August 23, Oct ober 16, 27, 2008; February 5, 19, 2010. Minor edits February 26, 2013. December 29, 2015.



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origin of relativity

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