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3 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
5 “On the Electrodynamics of Moving Bodies,” (June 1905; received 30 June 1905) Annalen der Physik, 17(1905), pp
6 What is the special theory of relativity? Rapidly moving bodies shrink in direction of their motion. Speed of light is the same (“c”) for all inertially moving observers. Rapidly moving clocks slow. We discard the ether state of rest of 19th century electrodynamics.
7 Its importance to the philosophy of science It is the first of the new theories of modern physics. Factually… I. Principle of relativity : all states of uniform motion are equivalent. II. Light postulate : the speed of light is a constant c. It has a simple axiomatic foundation. Methodologically: Everyone wants to do again what Einstein did…. but even now our understanding of how Einstein discovered special relativity is incomplete. It gives the most important conceptual analysis of the 20 th century Einstein’s operational analysis of distant simultaneity using light signals.
8 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.” The Pathway… 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.”
9 A new proposal for what Einstein really meant when he related the story of this thought experiment in his 1946 Autobiographical Notes. 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.” This Talk 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.”
10 Chasing the Light
11 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.”
12 The Thought A frozen waveform!
13 …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. “…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?” The thought experiment generates no trouble for an ether based Maxwell electrodynamics.
14 W hy does the thought experiment merit pride of place in Einstein’s defining autobiography? I s 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? O r does it have a cogency that extends beyond Einstein’s final high school year? Einstein (16yrs) in 1896 in the cantonal school of Aarau
15 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. The constancy of the speed of light is abandoned. All speeds are possible. Einstein to Ehrenfest, June, 1912 and elsewhere “…Ritz’s conception, which incidentally was also mine before rel. theory.” E instein 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. c v c+v
16 Einstein’s Objections to All Emission Theories of Light. T he 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 Problems with shadow formation by a moving screen. Collected from remarks in many places. e.g. To Mario Viscardini, April 1922 T he theory cannot be formulated in terms of differential equations. e.g. Einstein to Shankland, 1950s Different velocities entail that light can back up on itself (later parts overtakes earlier). e.g. Einstein to Shankland, 1950s; to Hines Feb. 1952
17 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.
18 “…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.
19 “…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”? O therwise the theory is indeterministic! The present does not fix the future.
20 “…For how should the first observer know or be able to determine, that he is in a state of fast uniform motion?” An extra property is needed for the instantaneous state to determine the future. But color, amplitude and polarization are the only properties light has. A light wave of definite color, amplitude, polarization at an instant. What happens next? Is it a propagating wave? Observer at rest with respect to source. Or a frozen wave? Observer moves with respect to source.
21 Present state of field Rate of change of field Future time development of field. An emission theory of light cannot be formulated in terms of differential field equations. x H = (1/c)(∂E/∂t) x E = - (1/c)(∂H/∂t) Example: Maxwell’s theory 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.
22 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.”
23 Rejecting the Absoluteness of Simultaneity
24 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”
25 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.
26 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 Principle of relativity Relativity of simultaneity and Light postulate are compatible.
27 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.
28 … Relativity of simultaneity rotates objects moving transversely. This effect also rotates a propagating plane wave. 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.
29 How did Einstein Take “The Step”?
30 Did Einstein actually discover the relativity of simultaneity by reflecting on clocks and their synchronization by light signals? Einstein’s earlier recollections are of problems in electrodynamics, electromagnetic waveforms and not spatially localized signals. Was the celebrated analysis of clock synchronization a convenient way to present a result already found by other means? Stellar aberration and Fizeau’s measurement of the speed of light in moving water are experimental manifestations of the relativity of simultaneity. Or
31 Stellar Aberration: apparent position of star displaced due to relative motion of star and earth. velocity of light c with respect to star velocity of star v with respect to earth v resultant gives apparent direction of light propagation as judged on earth Maximum aberration angle v/c when the direction of the star and the earth’s motion are perpendicular. 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
32 Star at rest in the ether. Earth moves. Analogy: Catching raindrops in a tall hat while running. Telescope must be tilted at the aberration angle v/c so that the starlight can reach the eyepiece
33 Galilean transform to the earth’s frame of reference The principle of relativity is not respected. A telescope at rest should no longer be tilted to intercept the starlight.
34 Star moves. Earth at rest in the ether. H. A. Lorentz, Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern 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/c 2 x Aberration angle is v/c whether star moves or earth moves.
35 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, “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,
36 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 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, Notes by Jun Ishiwara
37 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. “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, star moves star at rest …so we transform between inertial frames using Lorentz’s local time t --> t - v/c 2 x Relativity of simultaneity to first order v/c is expressed directly in rotation of wavefronts.
38 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? Conclude “ ‘local time’ can simply be defined as ‘time’.” Assume Stellar aberration conforms to the principle of relativity Hence read relativity of simultaneity from observation. Exactly analogous reasoning: Read the relativity of simultaneity from Fizeau’s experimental result of the speed of light in moving water.
40 Einstein’s recounting of this thought experiment in Autobiographical Notes makes most sense as a recounting of his objections to emission theories of light. 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.” This Talk 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.”
44 Albert Einstein, “Autobiographical Sketch” As recounted to Max Wertheimer in 1916 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.” “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.’”
45 Propagation Ritz imagined that charges emit fictitious particles that are projected by ordinary rules of Galilean kinematics. 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. versus Projection (Ritz) (Maxwell)
46 “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, The obvious escape… A field theory in which the color of a wave fixes its velocity of propagation. Example: The differential field equation (∂ 2 /∂t 2 - ∂ 2 /∂x 2 -m 2 ) (x,t)= 0 admits waves (x,t)= exp i ( t-kx) where m 2 =k 2 - 2 Color (wave number k) fixes velocity v = /k = (1- m 2 / k 2 ) 1/2 k = m --> v=0
47 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, Notes by Jun Ishiwara; translation Akira Ukawa; revised John Stachel.
48 Experimental Manifestations of the Relativity of Simultaneity First order Lorentz transformation t t - v/c 2 x x x - vt Wave propagates in y-direction f( t-ky) where c= /k. Wave deflected by aberration angle v/c f( t-k(v/c x + y)) v/c x + y = b.r where b=(v/c,1) is a vector normal to the wavefront. Stellar aberration Wave propagates in x-direction f( t-kx) at c/n, where c/n= /k. Wave propagates in x-direction as f( (1+vn/c)t-k(1+v/cn)x) at speed c/n + v(1-1/n 2 ) (1+vn/c) k(1+v/cn) Motion of Light in Moving Water (Fizeau’s Experiment)