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Einstein’s Discovery of the
Special Theory of Relativity:
His First and Final Steps
John D. Norton
Department of History and Philosophy of Science
Center for Philosophy of Science
University of Pittsburgh
Pitt-Tsinghua Summer School for Philosophy of Science
Institute of Science, Technology and Society, Tsinghua University
Center for Philosophy of Science, University of Pittsburgh
At Tsinghua University, Beijing
June 27- July 1, 2011
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“On the Electrodynamics of Moving Bodies,”
(June 1905; received 30 June 1905)
Annalen der Physik, 17(1905), pp. 891-921.
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What is the special theory of relativity?
Rapidly moving bodies shrink in
direction of their motion.
Rapidly moving clocks slow.
Speed of light is the same (“c”) for all
inertially moving observers.
We discard the ether state of rest of 19th century
electrodynamics.
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Its importance to the philosophy of science
Factually…
It is the first of the new
theories of modern physics.
Methodologically:
It has a simple
axiomatic
foundation.
I. Principle of relativity: all states of
uniform motion are equivalent.
II. Light postulate: the speed of
light is a constant c.
It gives the most
important conceptual
analysis of the 20th
century
Einstein’s operational
analysis of distant
simultaneity using light
signals.
Everyone wants to do again what Einstein did….
but even now our understanding of how Einstein
discovered special relativity is incomplete.
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The Pathway…
At the age of 16, Einstein imagined himself chasing a beam of light.
“One sees in this paradox the germ of the special relativity theory is already
contained.”
Einstein hit upon the magnet and conductor thought experiment.
“The phenomenon of magneto-electric induction compelled me to postulate
the (special) principle of relativity.”
Einstein considered replacing Maxwell’s electrodynamics by an
emission theory of light, in which the velocity of the emitter is added
vectorially to the velocity of the light emitted.
Einstein decided that all
emission theories of light are inadmissible.
Five to six weeks prior to completing the special relativity paper, Einstein
discovered the relativity of simultaneity.
He called this moment “the step.”
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This Talk
At the age of 16, Einstein imagined himself chasing a beam of light.
“One sees in this paradox the germ of the special relativity theory is already
contained.”
A new proposal for what Einstein really
meant when he related the story of this
thought experiment in his 1946
Autobiographical Notes.
Perhaps Einstein did not make “The
Step” by reflecting on clocks and the
signals that synchronize them.
Five to six weeks prior to completing the special relativity paper, Einstein
discovered the relativity of simultaneity.
He called this moment “the step.”
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Chasing the Light
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Einstein, Autobiographical Notes, 1946
“After ten years of reflection such a principle
resulted from a paradox upon which I had
already hit at the age of sixteen:
If I pursue a beam of light with the velocity c
(velocity of light in a vacuum),
I should observe such a beam of light as an electromagnetic field at rest
though spatially oscillating. There seems to be no such thing, however,
neither on the basis of experience nor according to Maxwell’s equations.
From the very beginning it appeared to me intuitively clear that, judged
from the standpoint of such an observer, everything would have to happen
according to the same laws as for an observer who, relative to the earth,
was at rest. For how should the first observer know or be able to
determine, that he is in a state of fast uniform motion?
One sees in this paradox the germ of the special relativity theory is
already contained.”
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The Thought
A frozen waveform!
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The thought experiment generates no trouble for an ether based
Maxwell electrodynamics.
“…I should observe such a beam of light
as an electromagnetic field at rest though
spatially oscillating.
There seems to be no such thing,
however, neither on the basis of
experience
nor according to Maxwell’s equations.
From the very beginning it appeared to me
intuitively clear that, judged from the standpoint of
such an observer, everything would have to happen
according to the same laws as for an observer who,
relative to the earth, was at rest. For how should
the first observer know or be able to
determine, that he is in a state of fast
uniform motion?”
…but only because we have
no experience of moving at
the speed light in the ether.
…but it is allowed by
Maxwell’s equations through
the simplest transformation.
…but the observer would know
he is moving rapidly because the
light would appear frozen.
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Why does the thought experiment merit pride of
place in Einstein’s defining autobiography?
Is it merely the recording of the
visceral hunches of a precocious
sixteen year old, who did not study
Maxwell’s theory until two years later?
Or does it have
a cogency that
extends beyond
Einstein’s final
high school
year?
Einstein (16yrs) in 1896 in the cantonal school of Aarau
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Ritz’s 1908
Emission “Theory”
of light
A modified electrodynamics in which the
velocity of the emitter is added vectorially to
the velocity of light and electrodynamic action.
The new theory conforms to the
principle of relativity without
modifying Newtonian notions of
space and time.
c
v
The constancy of the speed of
light is abandoned. All speeds
are possible.
c+v
Einstein later reported that he had given long and serious
consideration to a Ritz-like electrodynamics during the seven years prior
to 1905 in which he struggled to reconcile electrodynamics and the
principle of relativity.
“…Ritz’s conception, which incidentally
was also mine before rel. theory.”
Einstein to Ehrenfest,
June, 1912 and elsewhere
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Einstein’s Objections to All Emission Theories of Light.
Collected from remarks in many places.
The physical state of a light ray is determined
completely by its intensity and color [and polarization].
“ I decided [against an emission theory], since I was convinced that each
light [ray] should be defined by frequency and intensity alone, quite
independently of whether it comes from a moving or a resting light source.”
Einstein to Ehrenfest, mid June 1912
The theory cannot be formulated in
Different velocities entail that light
can back up on itself (later parts
overtakes earlier).
terms of differential equations.
e.g. Einstein to Shankland, 1950s
e.g. Einstein to Shankland, 1950s; to Hines Feb.
1952
Problems with shadow
formation by a moving screen.
e.g. To Mario Viscardini, April 1922
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The Thought Experiment succeeds against an emission theory of light.
i.e. a theory that conforms to the principle of relativity
using Newtonian notions of space and time.
Frozen lightwaves? “…There seems
to be no such thing, however, neither
on the basis of experience...”
A light source receding at c
leaves a frozen wave behind.
We should expect to experience these frozen waves
if there are rapidly receding light sources. There is
no need for us to move at c.
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“…or according to Maxwell’s equations…”
Frozen electromagnetic
waves are possible in
any inertial frame of
reference.
Frozen electromagnetic
waves must be admissible
in electrostatics and
magnetostatics.
Electrostatics and magnetostatics
of an emission theory should
agree with the electrostatics and
magnetostatics of Maxwell’s
theory. (Oldest and most secure part of
theory.)
BUT Maxwell’s equations
prohibit frozen waves.
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“…For how should the first observer know or
be able to determine, that he is in a state of fast
uniform motion?”
i.e., fast uniform motion with respect to the source.
Why should “the first
observer know or be able to
determine , that he is in a state of
fast uniform motion”?
Otherwise
the theory is indeterministic!
The present does not fix the
future.
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“…For how should the first observer know or
be able to determine, that he is in a state of fast
uniform motion?”
A light wave
of definite color,
amplitude, polarization
at an instant.
Is it a
propagating
wave?
Or a frozen
wave?
An extra property is needed for
the instantaneous state to
determine the future.
What
happens
next?
Observer
at rest with
respect to
source.
Observer
moves with
respect to
source.
But color, amplitude and
polarization are the only
properties light has.
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An emission theory of light cannot be formulated in terms of
differential field equations.
Field theory formulated with
differential equations: present, local
state of the field determines its future
time development.
Precluded in an emission theory of
light. An extra property is needed to
distinguish frozen from propagating
waves.
Example: Maxwell’s theory
xH =
(1/c)(∂E/∂t)
xE = - (1/c)(∂H/∂t)
Present
state of
field
Rate of
change of
field
Future time
development
of field.
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Einstein concludes…
“One sees that in this paradox the germ of the special relativity
theory is already contained. Today everyone knows, of course, that
all attempts to clarify this paradox satisfactorily were condemned
to failure as long as the axiom of the absolute character of
time, or of simultaneity, was rooted unrecognized in the
unconscious. To recognize clearly this axiom and its arbitrary
character already implies the essentials of the solution of the
problem.”
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Rejecting the
Absoluteness of
Simultaneity
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The “apparent incompatibil[ity]” of the principle of relativity
and the light postulate…
Chasing after a beam of light
does not slow it down?!
…is resolved by abandoning the
absoluteness of simultaneity.
“The Step”
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Einstein’s analysis in his 1905 “On the Electrodynamics of Moving Bodies”
(simplified):
The platform observer judges the two flashes to be
simultaneous and the two clocks to be properly synchronized.
The moving observer judges the A flash to happen earlier
and the two clocks not to be properly synchronized.
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Relativity of Simultaneity. Observers in relative motion
disagree on the simultaneity of spatially separated events
(and on the synchrony of clocks).
Relativity of simultaneity deduced
Principle of
relativity
+
Light
postulate
Relativity of
simultaneity
The deduction reversed
Relativity of
simultaneity
Principle of
relativity
and
Light
postulate
are compatible.
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Unexpected consequences…
A rod moves transversely to the direction of motion of a second observer.
We deem the rod to be parallel to second observer’s direction of motion
because we judge the two flashes to be simultaneous.
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… Relativity of simultaneity rotates objects moving transversely.
Transforming to the frame of reference of the second observer rotates the rod,
since the second observer does not judge the two flashes to be simultaneous.
This effect also rotates a propagating plane wave.
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How did
Einstein Take
“The Step”?
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Did Einstein actually discover
the relativity of simultaneity by
reflecting on clocks and their
synchronization by light signals?
Or
Was the celebrated analysis of
clock synchronization a convenient
way to present a result already
found by other means?
Einstein’s earlier recollections are
of problems in electrodynamics,
electromagnetic waveforms and
not spatially localized signals.
Stellar aberration and
Fizeau’s measurement of
the speed of light in moving water
are experimental manifestations of
the relativity of simultaneity.
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Stellar Aberration: apparent position of star displaced due to relative motion
of star and earth.
resultant gives apparent
velocity
of light
direction of light
propagation as
judged on earth
c with
respect
to star
velocity of star
Maximum
aberration angle
v/c
when the direction of
the star and the
earth’s motion are
perpendicular.
v with respect to
v
earth
All velocities are relative
velocities, so the effect conforms
to the principle of relativity.
How can this effect be
recovered in an ether based
electrodynamics?
Lorentz 1895 Versuch
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Star at rest in the ether.
Earth moves.
Telescope must be tilted
at the aberration angle v/c
so that the starlight can
reach the eyepiece
Analogy: Catching
raindrops in a tall
hat while running.
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Galilean transform to the earth’s frame of reference
A telescope at rest
should no longer be
tilted to intercept the
starlight.
The principle of
relativity is not
respected.
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Star moves.
Earth at rest in the ether.
H. A. Lorentz, Versuch
einer Theorie der electrischen und optischen
Erscheinungen in bewegten Körpern. 1895
Solve Maxwell’s equations for this
case by transforming the case of
the star at rest in the ether to its
corresponding state.
Wavefronts rotated due to
dislocation of temporal processes in
space by means of “local time”
t  t - v/c2 x
Aberration angle is v/c whether
star moves or earth moves.
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Einstein studied Lorentz’s Versuch and then worked on Fizeau’s
experiment and stellar aberration before discovering special relativity.
“… Lorentz’s path breaking investigation on the electrodynamics of moving bodies
(1895), which I knew before the establishment of the special theory of relativity. …
My direct path to the sp. th. rel. was mainly determined by the conviction that the
electromotive force induced in a conductor moving in a magnetic field is nothing
other than an electric field. But the results of Fizeau’s experiment and phenomenon
of aberration also guided me.”
Einstein, 1952 , In Memory of Albert A. Michelson…
“…the experimental results which had influenced him most were the observations of stellar
aberration and Fizeau’s measurements on the speed of light in moving water…”
Einstein reported by Shankland, 1950.
“Prof. Einstein volunteered a rather strong statement that he had been more influenced by
the Fizeau experiment on the effect of moving water on the speed of light, and by
astronomical aberration, especially Airy’s observation with a water filled telescope, than
by the Michelson-Morley experiment.”
Einstein reported by Shankland, 1950-54.
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Einstein studied Lorentz’s Versuch and then worked on Fizeau’s
experiment and stellar aberration before discovering special relativity.
“…I had the chance to read Lorentz’s monograph of 1895. There, Lorentz dealt with the
problems of electrodynamics and was able to solve them completely in the first
approximation…
… Then I dealt with Fizeau’s experiment and tried to approach it with the hypothesis that
the equations for electrons given by Lorentz held just as well for the system of coordinates
fixed in the moving body as for that fixed in the vacuum…
…Why are these two things [constancy velocity of light and classical velocity addition]
inconsistent with each other? I felt that I was facing an extremely difficult problem. I
suspected that Lorentz’s ideas had to be modified somehow, but spent almost a year
on fruitless thoughts. And I felt that was puzzle not to be easily solved.”
From a lecture given in Kyoto, Dec. 14, 1922. Notes by Jun Ishiwara
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Lorentz’s two cases without an ether state of rest
Einstein (I propose):
These are simply the same
process viewed from two
different frames of reference.
…so we transform
between inertial frames
using Lorentz’s local time
t --> t - v/c2 x
Relativity of
simultaneity
star at rest
to first order v/c
is expressed directly in rotation
of wavefronts.
star moves
“One needed only to realize that an auxiliary quantity that was introduced by H. A. Lorentz
and that he called ‘local time’ can simply be defined as ‘time’.”
Einstein, 1907.
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I propose Einstein inverted Lorentz’s reasoning
and freed it from dependence on electrodynamics.
Lorentz
Assume
Maxwell’s
electrodynamics
Theorem of
corresponding
states. Local time
Conclude
Stellar aberration
conforms to the
principle of relativity
Einstein?
Hence read
relativity of
simultaneity
from observation.
Exactly
analogous
reasoning:
Conclude
“ ‘local time’ can
simply be defined as
‘time’.”
Assume
Stellar aberration
conforms to the
principle of relativity
Read the relativity of simultaneity from Fizeau’s
experimental result of the speed of light in moving water.
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Conclusion
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This Talk
At the age of 16, Einstein imagined himself chasing a beam of light.
“One sees in this paradox the germ of the special relativity theory is already
contained.”
Einstein’s recounting of this thought
experiment in Autobiographical Notes
makes most sense as a recounting of his
objections to emission theories of light.
Einstein could read the relativity of
simultaneity from the observational
results of stellar aberration and Fizeau’s
experiment.
Five to six weeks prior to completing the special relativity paper, Einstein
discovered the relativity of simultaneity.
He called this moment “the step.”
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Finis
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Appendices
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Albert Einstein, “Autobiographical Sketch”
published 1956
“During this year in Aarau the following question came to me:
if one chases a light wave with the speed of light, then one
would have before one a time independent wave field. But
such a thing appears not to exist! This was the first child-like
thought experiment related to the special theory of relativity.
Discovery is not a work of logical thought, even if the final
product is bound in logical form.”
As recounted to Max Wertheimer in 1916
“The problem began when Einstein was sixteen years old, a pupil in the Gymnasium
(Aarau, Kantonschule)…
The process started in a way that was not very clear, and is therefore difficult to
describe—in a certain state of being puzzled. First came such questions as: What if one were to run
after a ray of light? What if one were riding on the beam? If one were to run after a ray of light as it
travels, would its velocity thereby be decreased? If one were to run fast enough, would it no longer
move at all?…[W’s ellipses] To young Einstein this seemed strange.
…When I asked him whether, during this period, he had already had some idea of the
constancy of light velocity, independent of the movement of the reference system, Einstein answered
decidedly: ‘No, it was just curiosity. That the velocity of light could differ depending upon the
movement of the observer was somehow characterized by doubt. Later developments increased that
doubt.’”
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Propagation
versus
(Maxwell)
Projection
(Ritz)
Light and
electromagnetic
action propagates
from fixed point in
space that is left
behind by a moving
source.
The apparent
source of light and
electromagnetic
action is boosted,
and moves with
uniformly moving
source.
Ritz imagined that charges emit fictitious particles that are
projected by ordinary rules of Galilean kinematics.
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The obvious
escape…
A field theory in which the color of a
wave fixes its velocity of propagation.
Example:
The differential field equation
(∂2/∂t2 - ∂2/∂x2 -m2) j(x,t)= 0
admits waves
j(x,t)= exp i (wt-kx)
where m2=k2-w2
Color (wave number k)
fixes velocity
v = w/k = (1- m2/ k2) 1/2
k = m --> v=0
“But the strongest argument [against an emission theory] seemed
to me: If there is no fixed velocity for light at all, then why should
it be that all light emitted by “stationary” bodies has a velocity
completely independent of the color? This seemed absurd to me.
Therefore I rejected this possibility as a priori improbable.”
Einstein to Hines, Feb. 1952,
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Taking The Step: “One beautiful day…”
“Why are these two things inconsistent with each other? I felt that I was
facing an extremely difficult problem. I suspected that Lorentz’s ideas had
to be modified somehow, but spent almost a year on fruitless thoughts.
And I felt that was puzzle not to be easily solved.
But a friend of mine living in living in Bern (Switzerland) [Michele Besso] helped me by
chance. One beautiful day, I visited him and said to him: ‘I presently have a problem
that I have been totally unable to solve. Today I have brought this “struggle” with me.’ We
then had extensive discussions, and suddenly I realized the solution. The very next day, I
visited him again and immediately said to him: ‘Thanks to you, I have completely
solved my problem.’
… After I had this inspiration, it took only five weeks to complete what is now known as
the special theory of relativity.”
From a lecture given in Kyoto, Dec. 14, 1922. Notes by Jun
Ishiwara; translation Akira Ukawa; revised John Stachel.
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Experimental Manifestations of the Relativity of Simultaneity
Stellar aberration
Wave propagates
in y-direction
Wave deflected by aberration angle v/c
f(wt-k(v/c x + y))
f(wt-ky)
where c= w/k.
First order Lorentz
transformation
t  t - v/c2 x
v/c x + y = b.r
where b=(v/c,1) is a vector normal to the
wavefront.
x  x - vt
Wave propagates in
x-direction
f(wt-kx)
at c/n,
where c/n= w/k.
Wave propagates in x-direction as
f(w(1+vn/c)t-k(1+v/cn)x)
at speed
w(1+vn/c)  c/n + v(1-1/n2)
k(1+v/cn)
Motion of Light in Moving Water (Fizeau’s Experiment)
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