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Read · Special relativity: from clock operations to electrodynamics · Results
Special relativity: from clock operations to electrodynamics
Clocks, rods and light first, then electricity and magnetism: follow the relativity paper through its introduction and all ten sections, including the later ones most retellings skip.
The paper's results
Pages 891, 892
The principle of relativity and the constancy of the velocity of light
The paper raises two presuppositions: that the same electrodynamic and optical laws hold in every coordinate system in which the equations of mechanics hold, which it calls the principle of relativity, and that light in empty space always travels with a definite velocity V, independent of the state of motion of the emitting body; together, it says, they suffice for a simple and consistent electrodynamics of moving bodies, in which, it says, a light ether will prove superfluous, insofar as no absolutely resting space is introduced and no point of empty space is given a velocity.
As printed
Beispiele ähnlicher Art, sowie die mißlungenen Versuche, eine Bewegung der Erde relativ zum „Lichtmedium“ zu konstatieren, führen zu der Vermutung, daß dem Begriffe der absoluten Ruhe nicht nur in der Mechanik, sondern auch in der Elektrodynamik keine Eigenschaften der Erscheinungen entsprechen, sondern daß vielmehr für alle Koordinatensysteme, für welche die mechanischen Gleichungen gelten, auch die gleichen elektrodynamischen und optischen Gesetze gelten, wie dies für die Größen erster Ordnung bereits erwiesen ist. Wir wollen diese Vermutung (deren Inhalt im folgenden „Prinzip der Relativität“ genannt werden wird) zur Voraussetzung erheben und außerdem die mit ihm nur scheinbar unverträgliche Voraussetzung einführen, daß sich das Licht im leeren Raume stets mit einer bestimmten, vom Bewegungszustande des emittierenden Körpers unabhängigen Geschwindigkeit V fortpflanze. Diese beiden Voraussetzungen genügen, um zu einer einfachen und widerspruchsfreien Elektrodynamik bewegter Körper zu gelangen unter Zugrundelegung der Maxwellschen Theorie für ruhende Körper. Die Einführung eines „Lichtäthers“ wird sich insofern als überflüssig erweisen, als nach der zu entwickelnden Auffassung weder ein mit besonderen Eigenschaften ausgestatteter „absolut ruhender Raum“ eingeführt, noch einem Punkte des leeren Raumes, in welchem elektromagnetische Prozesse stattfinden, ein Geschwindigkeitsvektor zugeordnet wird.
Premise: Both are presuppositions, and the paper does not derive them. Together with Maxwell's theory for bodies at rest, it says, they suffice for its electrodynamics of moving bodies.
Premise: The paper cites the magnet and the conductor, and the unsuccessful attempts to detect the earth's motion through the light medium, as what suggests the principle, not as a proof of it.
A comparison, not a premise: Lorentz's theory predicts the same currents at the speeds of real apparatus, so the introduction objects to an asymmetry in the explanation, not to a failed prediction.
Decoder
"Prinzip der Relativität"
The principle of relativity: the conjecture, raised to a presupposition, that absolute rest corresponds to no property of the phenomena, in electrodynamics as in mechanics.
V
The velocity of light in empty space, printed V; today it is written c.
"nur scheinbar unverträglich"
Only apparently incompatible: light keeping one velocity seems to contradict the principle of relativity, and § 3 shows that the two agree.
"Lichtmedium", "Lichtäther"
The light medium, or light ether: the unsuccessful attempts the paper mentions tried to detect the earth's motion relative to it. The paper introduces no absolutely resting space and assigns no velocity to a point of empty space.
Where this stops: A simulation built on the postulates cannot prove them. The model of a magnet and a conductor loop does not cover every induction experiment. Read the argument
Two presuppositions, from which the paper derives the rest
Model
The theory's starting points, not results of it
Evidence
A conjecture raised to a presupposition, suggested by examples such as the magnet and the conductor and by the failed attempts to detect the earth's motion through the light medium
History
The magnet-and-conductor case was a textbook example, discussed in Föppl's 1894 introduction to Maxwell's theory; the two presuppositions are introduced in this paper
Pages 894, 895
Synchronous clocks and the time of a distant event
Two clocks at rest at A and B run synchronously, by definition, when a ray of light that leaves A at tA, is reflected at B at tB and returns to A at tA′ takes as long out as back, tB−tA=tA′−tB; the time of an event is then the reading of such a clock at its place, and the quantity 2AB/(tA′−tA) is set, in accordance with experience, to be a universal constant, the speed of light in empty space.
As printed
Die letztere Zeit kann nun definiert werden, indem man durch Definition festsetzt, daß die „Zeit“, welche das Licht braucht, um von A nach B zu gelangen, gleich ist der „Zeit“, welche es braucht, um von B nach A zu gelangen. Es gehe nämlich ein Lichtstrahl zur „A-Zeit“ tA von A nach B ab, werde zur „B-Zeit“ tB in B gegen A zu reflektiert und gelange zur „A-Zeit“ tA′ nach A zurück. Die beiden Uhren laufen definitionsgemäß synchron, wenn
Die „Zeit“ eines Ereignisses ist die mit dem Ereignis gleichzeitige Angabe einer am Orte des Ereignisses befindlichen, ruhenden Uhr, welche mit einer bestimmten, ruhenden Uhr, und zwar für alle Zeitbestimmungen mit der nämlichen Uhr, synchron läuft.
Wir setzen noch der Erfahrung gemäß fest, daß die Größe eine universelle Konstante (die Lichtgeschwindigkeit im leeren Raume) sei.
Wesentlich ist, daß wir die Zeit mittels im ruhenden System ruhender Uhren definiert haben; wir nennen die eben definierte Zeit wegen dieser Zugehörigkeit zum ruhenden System „die Zeit des ruhenden Systems“.
Premise: The paper assumes that the definition can be applied without contradiction to any number of points: if the clock at B is synchronous with the clock at A, then A's is synchronous with B's, and two clocks synchronous with a third are synchronous with each other.
Premise: Two different things are laid down. The equality of the two travel times is a definition; that the round-trip quantity is a universal constant is stipulated in accordance with experience.
A comparison, not a premise: Whether the equal-times rule is a free choice was argued later, beginning with Reichenbach in 1928, and the question is still debated. The paper calls it a definition.
Decoder
tA, tB, tA′
The departure of the ray from A, read on A's clock; its reflection at B, read on B's clock; and its return to A, read on A's clock.
AB
The distance between A and B.
V
The speed of light in empty space, printed V; today it is written c.
"durch Definition"
By definition, letter-spaced in the print: that light takes as long from A to B as from B to A is stipulated, and nothing in the procedure measures it.
"die Zeit des ruhenden Systems"
The time of the stationary system: the paper stresses that the time so defined belongs to clocks at rest in that system.
Where this stops: A round trip read on one clock cannot measure the two one-way times separately. Setting clocks does not move them or change their rates. Read the argument
A definition, with an assumption that it is consistent and a stipulation from experience
Model
A definition for clocks at rest in one system
Evidence
A definition, not evidence; the constancy of the round-trip speed is taken from experience
History
In 1900 Poincaré had described observers moving through the ether who set their clocks by exchanging light signals and so obtain Lorentz's local time, to first order; here the procedure defines the time of the stationary system
Pages 896, 897
The relativity of simultaneity
Clocks at the two ends of a moving rod, which the stationary system calls synchronous, are not synchronous by the moving observers' own test, so two events that are simultaneous in one coordinate system are in general not simultaneous in a system moving relative to it.
Mit dem bewegten Stabe bewegte Beobachter würden also die beiden Uhren nicht synchron gehend finden, während im ruhenden System befindliche Beobachter die Uhren als synchron laufend erklären würden.
Wir sehen also, daß wir dem Begriffe der Gleichzeitigkeit keine absolute Bedeutung beimessen dürfen, sondern daß zwei Ereignisse, welche, von einem Koordinatensystem aus betrachtet, gleichzeitig sind, von einem relativ zu diesem System bewegten System aus betrachtet, nicht mehr als gleichzeitige Ereignisse aufzufassen sind.
Premise: The result rests on § 1's definition of synchronous clocks, by a ray of light sent out and reflected, and on the principle of the constancy of the velocity of light. Simultaneity at a distance is defined by that procedure; the paper does not observe it.
Decoder
tA, tB, tA′
The times, read on the stationary system's clocks, at which a ray of light leaves the end A of the rod, is reflected at the end B, and arrives back at A.
rAB
The length of the moving rod, measured in the stationary system.
v
The speed of the rod along the X-axis of the stationary system.
V
The speed of light, printed V; today it is written c.
why the two times differ
In the stationary system the ray chases the end B, which moves away at v, and meets the end A coming toward it, so the way out takes rAB/(V−v) and the way back rAB/(V+v). The test of § 1 that the moving observers apply calls the clocks synchronous only when the two times are equal.
Where this stops: The measured length is not what a photograph of the moving rod would show. Choosing a different pair of end events for each frame is the point of the measurement, not a correction to it. Read the argument
A consequence of the definition of synchronous clocks and the two principles
Model
Exact within the paper's kinematics
Evidence
A derivation, not evidence
History
Introduced in this paper. In 1900 Poincaré had described observers moving through the ether who set their clocks by light signals and so obtain Lorentz's local time, to first order
Page 901
The two principles are compatible
A spherical light wave sent out from the common origin at the time zero spreads with the speed V in the stationary system and, transformed by the equations of § 3, also with the speed V in the moving system, which shows that the principle of relativity and the constancy of the speed of light are compatible.
As printed
Zur Zeit t=τ=0 werde von dem zu dieser Zeit gemeinsamen Koordinatenursprung beider Systeme aus eine Kugelwelle ausgesandt, welche sich im System K mit der Geschwindigkeit V ausbreitet.
Die betrachtete Welle ist also auch im bewegten System betrachtet eine Kugelwelle von der Ausbreitungsgeschwindigkeit V. Hiermit ist gezeigt, daß unsere beiden Grundprinzipien miteinander vereinbar sind.
Premise: The transformation equations were built from the constancy of the speed of light in the stationary system and the definition of time; the spherical wave checks that the same holds in the moving system, which is what the principle of relativity requires.
Decoder
x2+y2+z2=V2t2
The points the wave has just reached at the time t of the stationary system K: a sphere of radius Vt.
ξ2+η2+ζ2=V2τ2
The same wave front in the coordinates of the moving system k, at its time τ: again a sphere, growing at the speed V.
ξ,η,ζ,τ
The coordinates and the time of the moving system k, as § 3 defines them; τ is k's time coordinate, not a proper time.
Where this stops: Keeping one light ray at speed c does not fix every coefficient of the map. The spacetime interval and matrices check the result later; the 1905 argument does not assume them. Read the argument
Argument
A check that the two postulates do not contradict each other
Model
Exact within the paper's kinematics
Evidence
A consistency argument, not evidence
History
Introduced in this paper
Page 902
The transformation equations
An event with coordinates x,y,z and time t in the stationary system has, in a system moving along x with speed v, the time τ=β(t−vx/V2) and the coordinates ξ=β(x−vt), η=y, ζ=z, once the unknown factor φ(v) of the derivation is shown to be 1.
As printed
Aus dieser und der vorhin gefundenen Relation folgt, daß φ(v)=1 sein muß, so daß die gefundenen Transformationsgleichungen übergehen in: wobei
Premise: The equations follow from the two principles, from § 1's definition of time and from the symmetry arguments the paper states; the factor φ(v) is fixed by symmetry, not by an observation.
Decoder
β
Einstein's β is 1/1−(v/V)2, the factor written γ today. It is not today's β, which means v/c.
τ
The time of the moving system k at the event, as its own clocks, set by § 1's rule, read it. It is k's time coordinate, not a proper time.
ξ,η,ζ
The event's coordinates in k, whose axes are parallel to those of the stationary system K.
V
The speed of light, printed V; today it is written c.
φ(v)=1
The derivation leaves a factor φ(v) open. A third system moving back with −v gives φ(v)φ(−v)=1, and a rod moving across its length, which cannot depend on the sense of the motion, gives φ(v)=φ(−v); so φ(v)=1.
Where this stops: Keeping one light ray at speed c does not fix every coefficient of the map. The spacetime interval and matrices check the result later; the 1905 argument does not assume them. Read the argument
Where this stops: Checking a ray along the motion cannot fix the sideways scale. Checks by determinant or interval confirm the result; they do not replace the physical premises it rests on. Read the argument
The derivation of § 3, completed by a symmetry argument
Model
Exact within the paper's kinematics
Evidence
A derivation, not evidence
History
Lorentz had published equivalent coordinate equations in 1904, as a change of variables with a local time, and Poincaré named them the Lorentz transformation in June 1905; this paper derives them from its two principles and cites neither
Page 903
A moving sphere measured as an ellipsoid
A rigid body that is a sphere of radius R when measured at rest has, measured from the stationary system while it moves with speed v, the shape of an ellipsoid whose axis along the motion is R1−(v/V)2 and whose two other axes stay R.
As printed
Ein starrer Körper, welcher in ruhendem Zustande ausgemessen die Gestalt einer Kugel hat, hat also in bewegtem Zustande — vom ruhenden System aus betrachtet — die Gestalt eines Rotationsellipsoides mit den Achsen
Während also die Y- und Z-Dimension der Kugel (also auch jedes starren Körpers von beliebiger Gestalt) durch die Bewegung nicht modifiziert erscheinen, erscheint die X-Dimension im Verhältnis 1:1−(v/V)2 verkürzt, also um so stärker, je größer v ist. Für v=V schrumpfen alle bewegten Objekte — vom „ruhenden“ System aus betrachtet — in flächenhafte Gebilde zusammen. Für Überlichtgeschwindigkeiten werden unsere Überlegungen sinnlos; wir werden übrigens in den folgenden Betrachtungen finden, daß die Lichtgeschwindigkeit in unserer Theorie physikalisch die Rolle der unendlich großen Geschwindigkeiten spielt.
Premise: The shape is a set of positions read at one time of the stationary system. It is not what a camera would record, which also depends on when light from each point left it.
Decoder
R1−(v/V)2
The length of the axis along the motion, measured in the stationary system: the sphere is shortened in the ratio 1:1−(v/V)2 in that direction only.
R,R
The two axes across the motion, which the motion leaves unchanged.
"vom ruhenden System aus betrachtet"
Viewed from the stationary system: the shape is found from the positions of the body's points at one time t=0 of that system.
v=V
At the speed of light every moving object would be flat in the stationary system; for speeds greater than light the paper says its considerations become meaningless.
Where this stops: Each of two clocks in relative motion runs slow as judged from the other, because the two judgments use different distant clocks. Transforming one pair of events does not by itself give a length or a clock reading; the pair has to be the right one. Read the argument
Exact within the paper's kinematics, for a body the paper calls rigid
Evidence
A derivation, not evidence
History
FitzGerald in 1889 and Lorentz in 1892 had proposed that bodies moving through the ether contract along their motion, as a physical effect; here the same factor follows from the definition of simultaneity and the two principles, and the paper cites no one for it
Pages 903, 904, 905
A moving clock falls behind
A clock at rest in the moving system, viewed from the stationary system, loses 1−1−(v/V)2 seconds each second, so a clock moved from A to B, or around a closed curve back to A, lags behind the clock that stayed, by 21tv2/V2 seconds to second order.
As printed
Wir denken uns ferner eine der Uhren, welche relativ zum ruhenden System ruhend die Zeit t, relativ zum bewegten System ruhend die Zeit τ anzugeben befähigt sind, im Koordinatenursprung von k gelegen und so gerichtet, daß sie die Zeit τ angibt. Wie schnell geht diese Uhr, vom ruhenden System aus betrachtet?
Sind in den Punkten A und B von K ruhende, im ruhenden System betrachtet, synchron gehende Uhren vorhanden, und bewegt man die Uhr in A mit der Geschwindigkeit v auf der Verbindungslinie nach B, so gehen nach Ankunft dieser Uhr in B die beiden Uhren nicht mehr synchron, sondern die von A nach B bewegte Uhr geht gegenüber der von Anfang an in B befindlichen um 21tv2/V2 Sek. (bis auf Größen vierter und höherer Ordnung) nach, wenn t die Zeit ist, welche die Uhr von A nach B braucht.
Nimmt man an, daß das für eine polygonale Linie bewiesene Resultat auch für eine stetig gekrümmte Kurve gelte, so erhält man den Satz: Befinden sich in A zwei synchron gehende Uhren und bewegt man die eine derselben auf einer geschlossenen Kurve mit konstanter Geschwindigkeit, bis sie wieder nach A zurückkommt, was t Sek. dauern möge, so geht die letztere Uhr bei ihrer Ankunft in A gegenüber der unbewegt gebliebenen um 21t(v/V)2 Sek. nach. Man schließt daraus, daß eine am Erdäquator befindliche Unruhuhr um einen sehr kleinen Betrag langsamer laufen muß als eine genau gleich beschaffene, sonst gleichen Bedingungen unterworfene, an einem Erdpole befindliche Uhr.
Approximation: The lag 21tv2/V2 is printed to within quantities of the fourth and higher order; the exact loss follows from the square root.
Premise: The theorem for a closed curve assumes that the result proved for a polygonal line also holds for a continuously curved one, and the paper says it assumes this.
An inference beyond the argument: The equator remark applies the closed-curve theorem to the earth's rotation, and it holds in this paper's kinematics alone. The card keeps the remark as printed.
A comparison, not a premise: On the real earth the difference in gravitational potential cancels the effect, and clocks at sea level keep the same rate everywhere, which the later general theory of relativity explains.
Decoder
τ
The reading of the clock at the origin of the moving system k: the time of k there. It is k's time coordinate, not a proper time.
τ=t1−(v/V)2
At the place of that clock, x=vt, the transformation gives this: while the stationary system's clocks advance by t, the moving clock advances by less.
21tv2/V2
The lag after a trip lasting t seconds of the stationary system, to within quantities of the fourth and higher order in v/V.
Unruhuhr
A balance-wheel clock, the paper's example for the clock at the equator.
Where this stops: Each of two clocks in relative motion runs slow as judged from the other, because the two judgments use different distant clocks. Transforming one pair of events does not by itself give a length or a clock reading; the pair has to be the right one. Read the argument
Where this stops: Comparisons on the real Earth also need gravity. Acceleration changes the path a clock takes; it adds no extra slowing of its own to the formula. Read the argument
A consequence of the transformation equations, and an assumption for curved paths
Model
Exact for a straight path, second order as printed, and assumed for a curved one
Evidence
A derivation, not evidence
History
Introduced in this paper
Pages 905, 906
The addition theorem of velocities
A point that moves with the velocity w in the system k, while k moves with v relative to K, has in K a velocity U that for two velocities along one line is U=(v+w)/(1+vw/V2), so that two velocities smaller than V always compose to one smaller than V, and the velocity of light composed with a smaller one stays V.
As printed
Das Gesetz vom Parallelogramm der Geschwindigkeiten gilt also nach unserer Theorie nur in erster Annäherung.
Aus dieser Gleichung folgt, daß aus der Zusammensetzung zweier Geschwindigkeiten, welche kleiner sind als V, stets eine Geschwindigkeit kleiner als V resultiert.
Es folgt ferner, daß die Lichtgeschwindigkeit V durch Zusammensetzung mit einer „Unterlichtgeschwindigkeit“ nicht geändert werden kann.
Premise: The theorem applies the transformation equations of § 3 to a point moving uniformly in k.
Conditional: The bound is proved for two velocities each smaller than V; § 4 says that for velocities greater than that of light the paper's considerations become meaningless.
A comparison, not a premise: Laue showed in 1907 that the formula for velocities along one line gives Fresnel's drag coefficient 1−1/n2, which Fizeau had measured in flowing water in 1851. That is a later use of the theorem, not part of the paper.
Decoder
U
The velocity of the point relative to the stationary system K.
w, α
The velocity of the point relative to the moving system k, and the angle between the velocities v and w.
ϰ, λ
The amounts by which v and w fall short of V, in v=V−ϰ and w=V−λ, both positive and smaller than V. They are speeds; λ is not a wavelength here.
"nur in erster Annäherung"
Only to a first approximation: when v and w are small compared with V, the terms divided by V2 are small, and U is close to what the parallelogram of velocities gives.
the modern layer
The theorem finds the velocity in K from the velocity in k, with a plus sign. The modern equations run the other way, from K to k, with a minus sign, and write c for V.
Where this stops: The plus-sign formula for combining speeds and the minus-sign change of frame run in opposite directions. Two boosts in different directions combine into a boost and a rotation. Read the argument
Argument
A consequence of the transformation equations
Model
Exact within the paper's kinematics
Evidence
A derivation, not evidence
History
Introduced in this paper
Pages 909, 910
The electric and magnetic forces in the moving system
Carried into the moving system by the equations of § 3, the Maxwell-Hertz equations for empty space keep their form when the electric and magnetic forces in that system are the printed combinations, in which the components along the motion are unchanged and those across it mix electric with magnetic, so that, as the paper concludes, the two forces have no existence independent of the state of motion of the coordinate system.
1. Ist ein punktförmiger elektrischer Einheitspol in einem elektromagnetischen Felde bewegt, so wirkt auf ihn außer der elektrischen Kraft eine „elektromotorische Kraft“, welche unter Vernachlässigung von mit der zweiten und höheren Potenzen von v/V multiplizierten Gliedern gleich ist dem mit der Lichtgeschwindigkeit dividierten Vektorprodukt der Bewegungsgeschwindigkeit des Einheitspoles und der magnetischen Kraft. (Alte Ausdrucksweise.)
2. Ist ein punktförmiger elektrischer Einheitspol in einem elektromagnetischen Felde bewegt, so ist die auf ihn wirkende Kraft gleich der an dem Orte des Einheitspoles vorhandenen elektrischen Kraft, welche man durch Transformation des Feldes auf ein relativ zum elektrischen Einheitspol ruhendes Koordinatensystem erhält. (Neue Ausdrucksweise.)
Man sieht, daß in der entwickelten Theorie die elektromotorische Kraft nur die Rolle eines Hilfsbegriffes spielt, welcher seine Einführung dem Umstande verdankt, daß die elektrischen und magnetischen Kräfte keine von dem Bewegungszustande des Koordinatensystems unabhängige Existenz besitzen.
Es ist ferner klar, daß die in der Einleitung angeführte Asymmetrie bei der Betrachtung der durch Relativbewegung eines Magneten und eines Leiters erzeugten Ströme verschwindet. Auch werden die Fragen nach dem „Sitz“ der elektrodynamischen elektromotorischen Kräfte (Unipolarmaschinen) gegenstandslos.
Premise: The derivation assumes the Maxwell-Hertz equations for empty space, not a theory of matter, and the principle of relativity, which requires them to hold in the same form in k.
Premise: That the transformed quantities are the forces a charge at rest in k would feel is a further step. § 6 takes it by measuring force on a unit charge, and by taking, from the principle of relativity, that a charge of one unit in K is one unit in k; that the equations keep their form does not by itself prove it.
Approximation: The old manner of expression is printed with terms in the second and higher powers of v/V neglected. The transformation of the forces is exact.
Decoder
X,Y,Z and L,M,N
The components of the electric and of the magnetic force in K; primed, the same components in k. Here L and N are magnetic components, not the L or the N of the other papers.
β
1/1−(v/V)2, the factor written γ today, as in § 3.
Y−VvN
The paper uses Gaussian units, in which electric and magnetic forces have the same dimension. In SI the same combination is Ey−vBz, and the modern layer writes the SI form.
"Alte" and "Neue Ausdrucksweise"
The old and the new manner of expression. In the old, a charge moving in a field feels an electromotive force besides the electric force; in the new, the force on it is the electric force in a system in which it is at rest.
Where this stops: Einstein’s Gaussian equations become SI ones by a change of units, not by renaming symbols. A common factor on all the new fields is fixed by a separate argument, going there and back and the symmetry of space; rearranging the equations does not fix it. Read the argument
Where this stops: An accelerating charge has a different momentary rest frame at each instant, not one frame for its whole path. Fields, forces and electromotive force each transform by their own rule. Read the argument
Argument
A consequence of the principle of relativity applied to the Maxwell-Hertz equations, with the forces in k defined by their action on a charge at rest there
Model
Exact within the electrodynamics of empty space
Evidence
A derivation, not evidence
History
Lorentz's 1904 theory had already transformed the free-space fields, as mathematical aids in an ether at rest; here they are the forces measured in the moving system
Pages 911, 912
Doppler's principle and aberration for any velocity
An observer who moves with the velocity v relative to a very distant source of frequency ν, at the angle φ between the line from source to observer and the observer's velocity, perceives the frequency ν′=ν(1−cosφVv)/1−(v/V)2 and finds the ray at the angle φ′ given by cosφ′=(cosφ−Vv)/(1−Vvcosφ): Doppler's principle and the law of aberration for arbitrary velocities.
As printed
Aus der Gleichung für ω′ folgt: Ist ein Beobachter relativ zu einer unendlich fernen Lichtquelle von der Frequenz ν mit der Geschwindigkeit v derart bewegt, daß die Verbindungslinie „Lichtquelle–Beobachter“ mit der auf ein relativ zur Lichtquelle ruhendes Koordinatensystem bezogenen Geschwindigkeit des Beobachters den Winkel φ bildet, so ist die von dem Beobachter wahrgenommene Frequenz ν′ des Lichtes durch die Gleichung gegeben:
Premise: The source is so distant that near the observer its light is represented, to a sufficient approximation, by a plane wave, as § 7 sets it up; the frequency and the direction follow from transforming that wave's phase and fields.
A comparison, not a premise: Stellar aberration had been observed since Bradley's report of 1729, and Ives and Stilwell measured the second-order Doppler shift in canal rays in 1938, reading it within Lorentz's theory. Both are measurements of the world, separate from this calculation.
Decoder
ν, ν′
The frequency of the light in the system of the source, and the frequency the moving observer perceives.
φ
The angle between the line from source to observer and the observer's velocity, measured in the system at rest relative to the source. At φ=0 the observer moves away from the source along that line.
φ′
The angle between the ray and the same line, as the moving observer measures it.
v=−∞, ν=∞
As printed. The equation above it grows without bound as v approaches −V, an observer approaching the source at nearly the speed of light, and has no real value beyond; the older relation ν′=ν(1−v/V) grows without bound only as v goes to −∞, which is the usual view the sentence sets itself against. So the sentence reads as v=−V, and its ν most likely as the perceived ν′. Every face keeps the printed reading; the correction is offered here only.
Where this stops: Light that crosses the motion at right angles in one frame does not, in general, do so in the other. No observer can move at the speed of light, so the formulas’ limit at v = c describes no frame. Read the argument
Argument
A consequence of the transformation of a plane wave's phase and fields
Model
Exact within the paper's electrodynamics, for a plane wave from a very distant source
Evidence
A derivation, not evidence
History
Doppler's principle and aberration were known before, and the paper names the first after Doppler; their forms for arbitrary velocities are derived in this paper
Pages 913, 914
The energy of a light complex in the moving system
Measured in the moving system k, the light enclosed by a sphere that moves with it at the speed of light has its energy per unit volume multiplied by q2 and its volume divided by q, where q=(1−Vvcosφ)/1−(v/V)2, so its energy changes by the factor q, the same factor by which its frequency changes.
As printed
Es wäre daher A′2/A2 das Verhältnis der „bewegt gemessenen“ und „ruhend gemessenen“ Energie eines bestimmten Lichtkomplexes, wenn das Volumen eines Lichtkomplexes in K gemessen und in k gemessen das gleiche wäre. Dies ist jedoch nicht der Fall.
Nennt man S das Volumen der Kugel, S′ dasjenige dieses Ellipsoides, so ist, wie eine einfache Rechnung zeigt:
Nennt man also E die im ruhenden System gemessene, E′ die im bewegten System gemessene Lichtenergie, welche von der betrachteten Fläche umschlossen wird, so erhält man: welche Formel für φ=0 in die einfachere übergeht:
Es ist bemerkenswert, daß die Energie und die Frequenz eines Lichtkomplexes sich nach demselben Gesetze mit dem Bewegungszustande des Beobachters ändern.
Premise: That A′2/8π is the light energy per unit volume in k is taken from the principle of relativity, as the first sentence of § 8 says.
Premise: The light is a plane wave in empty space, and the complex is the part of it inside the moving sphere; because no energy crosses that surface, the same light is compared in both systems.
A comparison, not a premise: The volume's factor 1−(v/V)2/(1−Vvcosφ) depends on the direction of the light: with φ=0 and v at 0.6 of the speed of light it is 2, where a rigid body at rest in K is shortened to 0.8 of its length along the motion.
A comparison, not a premise: § 8 remarks that the energy and the frequency of a light complex change by the same law, and does not connect this with the light quanta of the March paper; reading a complex's energy as hν in every system came later.
Decoder
"Lichtkomplex"
A light complex: the part of a plane wave inside a spherical surface whose centre moves with the speed of light along the wave normal. No energy crosses that surface, so it encloses the same light at every moment.
A2/8π, A′2/8π
The energy of the light per unit volume, in the paper's Gaussian units, measured in K and in k. A and A′ are the amplitudes of its electric or magnetic force, transformed in § 7.
S, S′
The volume the surface encloses, measured in K, where it is a sphere, and in k at the time τ=0, where it is an ellipsoid.
E, E′
The light energy the surface encloses, measured in K and in k.
φ
The angle between the wave normal of the light and the direction in which k moves, measured in K. At φ=0 the light travels the way k moves.
q
The modern layer's name for the factor by which the light's frequency changes, § 7's Doppler factor for this direction of the light.
Where this stops: A light complex is not a rod, and light has no rest frame. The result needs neither light quanta nor the mass-energy relation it later supports. Read the argument
A consequence of the transformed amplitude of § 7 and of the volume of a surface that moves with the light
Model
Exact within the paper's electrodynamics, for a plane wave in empty space
Evidence
A derivation, not evidence
History
Introduced in this paper; the remark that energy and frequency transform alike is Einstein's, and the paper does not connect it with light quanta
Pages 914, 915
Light reflected by a moving mirror, and its pressure
Light of frequency ν and energy per unit volume A2/8π, falling at the angle φ on a perfect mirror that moves with the velocity v along its normal, comes back with the frequency ν(1−2Vvcosφ+(v/V)2)/(1−(v/V)2) by the paper's own steps, and presses on the mirror with P=28πA2(cosφ−Vv)2/(1−(v/V)2), which to a first approximation is 28πA2cos2φ, the pressure on a mirror at rest, obtained, the paper says, "in agreement with experience and with other theories".
As printed
Es sei nun die Koordinatenebene ξ=0 eine vollkommen spiegelnde Fläche, an welcher die im letzten Paragraph betrachteten ebenen Wellen reflektiert werden. Wir fragen nach dem auf die spiegelnde Fläche ausgeübten Lichtdruck und nach der Richtung, Frequenz und Intensität des Lichtes nach der Reflexion.
Die Differenz dieser beiden Ausdrücke ist nach dem Energieprinzip die vom Lichtdrucke in der Zeiteinheit geleistete Arbeit. Setzt man die letztere gleich dem Produkt P⋅v, wobei P der Lichtdruck ist, so erhält man:
Premise: The mirror is ideal: perfectly reflecting, its motion prescribed, with no recoil, absorption or thickness.
An inference beyond the argument: The pressure comes from the energy principle: the difference between the energy arriving on the mirror and the energy leaving it is the work done by the pressure of the light.
Conditional: The light reaches the mirror only while Vcosφ exceeds v.
Approximation: The form 28πA2cos2φ neglects the terms in v/V. The formula before it is exact within the model.
A comparison, not a premise: The first approximation is the pressure Maxwell's theory gave for a mirror at rest, which Bartoli had also argued for from thermodynamics; Lebedev, and independently Nichols and Hull, measured light pressure in 1901. Those are measurements of the world, separate from this calculation.
Decoder
A, φ, ν
The amplitude, the angle of incidence and the frequency of the incident light, measured in K. The mirror is the plane ξ=0 of k, so its normal is the direction of motion, and φ is the angle between the wave normal and that direction.
A′′′, φ′′′, ν′′′
The same quantities for the reflected light, measured in K. The double-primed quantities describe it in k, where the mirror is at rest and reflects it as a mirror at rest does.
(1−Vv)2 in the ν′′′ display
As printed. The paper's own steps give 1−(v/V)2, the denominator of the A′′′ display above it; with the printed square, a mirror moving along the line of the light would return it with its frequency unchanged. Every face keeps the printed reading, and the modern layer has 1−(v/V)2.
P
The pressure of the light on the mirror, force per unit area. It is found from the work it does: the energy arriving on unit area of the mirror in unit time, minus the energy leaving it, equals P⋅v.
28πA2
Twice the energy per unit volume. With the energy per unit volume written u, in any system of units the pressure is 2u(cosφ−v/c)2/(1−v2/c2).
Where this stops: Where the light only just fails to catch the mirror, a number from the calculation is not a reflection. A reflected ray can still move towards +x in the laboratory, and a mirror receding faster moves away from it. Read the argument
Argument
A consequence of transforming the light to the mirror's rest system, reflecting it there, transforming it back, and applying the energy principle
Model
Exact within the paper's electrodynamics, for a plane wave and an ideal mirror; the \(\cos^2\varphi\) form is a first approximation
Evidence
A derivation, not evidence
History
The first approximation was Maxwell's pressure on a mirror at rest; the exact formula for a moving mirror is derived in this paper
Pages 916, 917
The Maxwell-Hertz equations with convection current keep their form
Carried into the moving system by the transformations of § 3 and § 6, the Maxwell-Hertz equations with convection current, which § 9 states for K, keep their form, with the charge density ϱ′=β(1−V2vux)ϱ and the velocity of the electricity transformed by the addition theorem of § 5; so the electrodynamic basis of Lorentz's theory of moving bodies conforms to the principle of relativity, and a body whose charge is constant in a system moving with it has a constant charge in K as well.
As printed
Transformiert man diese Gleichungen, welche im System K gelten mögen, mit Hilfe der Transformationsgleichungen von § 3 und § 6 auf das System k, so erhält man die Gleichungen: wobei
Da — wie aus dem Additionstheorem der Geschwindigkeiten (§ 5) folgt — der Vektor (uξ,uη,uζ) nichts anderes ist als die Geschwindigkeit der elektrischen Massen im System k gemessen, so ist damit gezeigt, daß unter Zugrundelegung unserer kinematischen Prinzipien die elektrodynamische Grundlage der Lorentzschen Theorie der Elektrodynamik bewegter Körper dem Relativitätsprinzip entspricht.
Es möge noch kurz bemerkt werden, daß aus den entwickelten Gleichungen leicht der folgende wichtige Satz gefolgert werden kann: Bewegt sich ein elektrisch geladener Körper beliebig im Raume und ändert sich hierbei seine Ladung nicht, von einem mit dem Körper bewegten Koordinatensystem aus betrachtet, so bleibt seine Ladung auch — von dem „ruhenden“ System K aus betrachtet — konstant.
Premise: The electric masses are bound unchangeably to small rigid bodies (ions, electrons); on that supposition the equations are, as § 9 says, the basis of Lorentz's electrodynamics and optics of moving bodies.
An inference beyond the argument: What § 9 shows is that the transformed equations have the same form, with charge density and velocity transformed as stated, and it draws from them the constancy of a body's charge. It does not write a four-current or an invariant, and it does not discuss the continuity equation.
A comparison, not a premise: Joining charge and current density into one four-vector, with (cρ)2−∣J∣2 unchanged, is the later language of Poincaré and Minkowski. That a wire carrying a current appears charged to a moving observer is a modern consequence the paper does not discuss.
Decoder
ϱ, ϱ′
4π times the density of electricity, as § 9 defines it, in K and in k.
ux,uy,uz and uξ,uη,uζ
The velocity of the electricity in K and in k. § 9 notes that the second is the first transformed by the addition theorem of § 5, so it is the velocity measured in k.
β
1/1−(v/V)2, the factor written γ today, as in § 3.
uϱ
The convection current: electricity carried along by moving charges, in the equations beside the rate of change of the electric force. In SI the charge density transforms as ρ′=γ(ρ−vJx/c2), the modern layer's form.
∂τ∂L′, ∂τ∂M′, ∂τ∂N′, as printed
The three magnetic equations of k are printed without the factor 1/V before the time derivative, which the same equations in K carry, as do those of § 6. They contain no convection term and transform as in § 6, so the factor belongs there. Every face keeps the printed reading.
Where this stops: Charge density is not total charge: a density can change between frames while the body’s charge stays fixed. In a neutral wire the positive and negative charges move at different speeds, so there is no single velocity of the charge. Read the argument
A consequence of the transformations of §§ 3, 5 and 6 applied to the Maxwell-Hertz equations with convection current
Model
Exact within Lorentz's electrodynamics of charges bound to small rigid bodies
Evidence
A derivation, not evidence
History
The equations are the basis of Lorentz's theory; that they conform to the principle of relativity under these kinematics is shown in this paper, and the four-current is the later language of Poincaré and Minkowski
Pages 918, 919
The longitudinal and transverse masses of the electron
Calling the force on the electron the force measured in a system that moves with it at that moment, and measuring its acceleration in K, the paper finds the longitudinal mass μ/(1−(v/V)2)3 and the transverse mass μ/(1−(v/V)2), and warns that other definitions of force and acceleration give other numbers for the masses.
As printed
Wir fragen nun in Anlehnung an die übliche Betrachtungsweise nach der „longitudinalen“ und „transversalen“ Masse des bewegten Elektrons. Wir schreiben die Gleichungen (A) in der Form und bemerken zunächst, daß εX′, εY′, εZ′ die Komponenten der auf das Elektron wirkenden ponderomotorischen Kraft sind, und zwar in einem in diesem Moment mit dem Elektron mit gleicher Geschwindigkeit wie dieses bewegten System betrachtet. (Diese Kraft könnte beispielsweise mit einer im letzten System ruhenden Federwage gemessen werden.)
Wenn wir nun diese Kraft schlechtweg „die auf das Elektron wirkende Kraft“ nennen und die Gleichung aufrechterhalten, und wenn wir ferner festsetzen, daß die Beschleunigungen im ruhenden System K gemessen werden sollen, so erhalten wir aus obigen Gleichungen:
Natürlich würde man bei anderer Definition der Kraft und der Beschleunigung andere Zahlen für die Massen erhalten; man ersieht daraus, daß man bei der Vergleichung verschiedener Theorien der Bewegung des Elektrons sehr vorsichtig verfahren muß.
Premise: The force is the one measured in the system moving with the electron at that moment, and the acceleration is measured in K.
A comparison, not a premise: With force and acceleration both measured in K, the transverse ratio is μβ, today's convention, while the longitudinal ratio is μβ3 under both. The paper says as much: other definitions give other numbers, and theories of the motion of the electron must be compared with great care.
An inference beyond the argument: § 10 extends the masses to ponderable material points, since one becomes an electron in the paper's sense when an arbitrarily small charge is added to it.
Decoder
μ, ε
The electron's mass, as long as it moves slowly, and its charge.
β
1/1−(v/V)2, the factor written γ today. The longitudinal mass is μβ3 and the transverse mass μβ2.
μβ3 in the third row of (A)
As printed. The paper's own steps give μβ2 for the z row, as for the y row: both directions are across the motion, and the transverse mass two displays below is μβ2. Every face keeps the printed β3, and the modern layer has γ2.
εX′, εY′, εZ′
The force on the electron measured in the system that moves with it at that moment, which a spring balance at rest in that system could measure, as § 10 says.
"Massenzahl × Beschleunigungszahl = Kraftzahl"
Mass number times acceleration number equals force number: the equation the paper keeps, with force and acceleration defined as it has just stated.
Where this stops: The two masses are ratios of force to acceleration under stated conventions; the electron’s own mass does not change with the observer’s speed. The electron is treated as a point driven slowly, without the effects of its own radiation or field. Read the argument
Argument
A consequence of the equations of motion (A), with force and acceleration defined as the paper states
Model
Exact within the paper's dynamics of a charged point; the numbers depend on the definition of force
Evidence
A derivation, not evidence
History
Longitudinal and transverse mass were the usual terms of the theory of the electron, as the paper's question says; these values follow from its kinematics
Pages 919, 920
The kinetic energy of the electron
An electron accelerated slowly from rest along the X-axis by an electrostatic force takes from the field the energy W=μV2{1/1−(v/V)2−1}, which grows without bound as v approaches V, so that, as with the paper's earlier results, speeds greater than that of light have no possibility of existence; the same expression must hold for ponderable masses.
As printed
Wir bestimmen die kinetische Energie des Elektrons. Bewegt sich ein Elektron vom Koordinatenursprung des Systems K aus mit der Anfangsgeschwindigkeit 0 beständig auf der X-Achse unter der Wirkung einer elektrostatischen Kraft X, so ist klar, daß die dem elektrostatischen Felde entzogene Energie den Wert ∫εXdx hat. Da das Elektron langsam beschleunigt sein soll und infolgedessen keine Energie in Form von Strahlung abgeben möge, so muß die dem elektrostatischen Felde entzogene Energie gleich der Bewegungsenergie W des Elektrons gesetzt werden. Man erhält daher, indem man beachtet, daß während des ganzen betrachteten Bewegungsvorganges die erste der Gleichungen (A) gilt:
Premise: The electron starts from rest at the origin of K and moves along the X-axis under an electrostatic force. It is accelerated slowly, so that it gives up no energy as radiation, and the energy taken from the field is then its kinetic energy.
Premise: The first equation of (A), with the longitudinal mass, holds throughout the motion.
Approximation: For small v/V the expression becomes 21μv2, the kinetic energy of Newton's mechanics.
An inference beyond the argument: It holds for ponderable masses by the argument the paper gives for the masses: a small charge makes a ponderable point an electron in its sense.
Decoder
W
The kinetic energy of the electron, set equal to the energy taken from the electrostatic field.
∫0vβ3vdv, as printed
The middle member lacks the factor μ, which the first equation of (A) carries; with it, the three members are equal. The result on the right is correct as printed, and every face keeps the printed middle member.
μV2
The electron's mass times the square of the speed of light. The paper does not read it as an energy the electron has at rest; the relation between a body's energy and its inertia is the subject of the paper of September 1905.
Where this stops: Losses by radiation and interactions with matter are left out. The kinetic energy formula alone does not give the September paper’s result that a body’s mass changes with its energy. Read the argument
A consequence of the longitudinal equation of motion and the energy principle
Model
Exact within the paper's dynamics, for slow acceleration without radiation
Evidence
A derivation, not evidence
History
Introduced in this paper; the relation between a body's energy and its inertia is taken up in the mass-energy paper of September 1905
Pages 920, 921
Three relations for experiment on the electron
From the equations of motion (A) the paper draws three relations open to experiment: the ratio of an electron's magnetic to its electric deflectability is v/V at any speed, the potential difference it has passed through fixes its speed by P=εμV2{1/1−(v/V)2−1}, and a magnetic force N at right angles to its motion bends its path with the radius R=V2εμ1−(v/V)2v/VN1; these three, it says, express completely the laws by which the electron must move on its theory.
As printed
Wir wollen nun die aus dem Gleichungssystem (A) resultierenden, dem Experimente zugänglichen Eigenschaften der Bewegung des Elektrons aufzählen.
1. Aus der zweiten Gleichung des Systems (A) folgt, daß eine elektrische Kraft Y und eine magnetische Kraft N dann gleich stark ablenkend wirken auf ein mit der Geschwindigkeit v bewegtes Elektron, wenn Y=N⋅v/V. Man ersieht also, daß die Ermittelung der Geschwindigkeit des Elektrons aus dem Verhältnis der magnetischen Ablenkbarkeit Am und der elektrischen Ablenkbarkeit Ae nach unserer Theorie für beliebige Geschwindigkeiten möglich ist durch Anwendung des Gesetzes:
Diese Beziehung ist der Prüfung durch das Experiment zugänglich, da die Geschwindigkeit des Elektrons auch direkt, z. B. mittels rasch oszillierender elektrischer und magnetischer Felder, gemessen werden kann.
2. Aus der Ableitung für die kinetische Energie des Elektrons folgt, daß zwischen der durchlaufenen Potentialdifferenz und der erlangten Geschwindigkeit v des Elektrons die Beziehung gelten muß:
3. Wir berechnen den Krümmungsradius R der Bahn, wenn eine senkrecht zur Geschwindigkeit des Elektrons wirkende magnetische Kraft N (als einzige ablenkende Kraft) vorhanden ist.
Premise: The first and third relations follow from the second equation of (A); the second follows from the kinetic energy derived before it.
Premise: The electron has passed through the potential difference from rest, slowly enough to radiate nothing, as the kinetic energy requires.
A comparison, not a premise: The paper names the test: the speed of an electron can also be measured directly, for example with rapidly oscillating electric and magnetic fields. Kaufmann's deflection measurements of fast electrons, made from 1901 to 1906, seemed at first to favour Abraham's rival theory, while Bucherer's of 1908 favoured the Lorentz-Einstein formulas, and the question stayed open for some years. Those are measurements of the world, separate from this calculation.
Decoder
Am, Ae
The magnetic and the electric deflectability of the electron. An electric force Y and a magnetic force N deflect it equally when Y=Nv/V.
P
Here the potential difference the electron has passed through from rest. In § 8 the same letter is the pressure of light.
R, N
The radius of curvature of the path, and a magnetic force at right angles to the electron's velocity, the only deflecting force. N is a magnetic component here, not a number of molecules.
εμV2
In the paper's Gaussian units. In SI, with the size of the charge ∣q∣ and of the voltage U, the second relation reads ∣q∣U=mc2(γ−1) and the third R=γmv/(∣q∣B), the modern layer's forms.
Where this stops: Losses by radiation and interactions with matter are left out. The kinetic energy formula alone does not give the September paper’s result that a body’s mass changes with its energy. Read the argument
Argument
Consequences of the equations of motion (A) and the kinetic energy, stated by the paper as open to experiment
Model
Exact within the paper's dynamics of a charged point in given fields
Evidence
Predictions for experiment; the paper names the tests and reports no data
History
Kaufmann's measurements of 1901 to 1906 and Bucherer's of 1908 tested such relations, and the question between the theories of the electron stayed open for some years
Introduction · magnet, conductor, and the postulates
One apparatus, two descriptions
Move the magnet past a resting conductor, or the conductor past a resting magnet: the current is the same, yet the physics of 1905 explained the two cases differently. Einstein's paper begins from that asymmetry and asks for one description that depends only on the relative motion.
To set a distant clock, send light to it and back, and give the moment of reflection the time halfway between sending and receiving. That rule defines the time of a distant event; it is not the moment its image arrives.
To measure a moving rod, mark both of its ends at the same time in your own frame. Two marks that are simultaneous in another frame are made at different times in yours, so they do not measure the rod.
Ask for a linear map that keeps light at speed c in both directions along the motion, and space and time must mix. One overall factor is left over; going there and back, and the symmetry of space, fix it at gamma.
Light along the motion says nothing about lengths across it. Send light across instead and require it to move at c in the moving frame too; with going there and back, that leaves the sideways coordinates unchanged: y' = y, z' = z.
A clock moving at constant speed runs slow: between two of its ticks, less time passes on it than on the resting clocks it passes. A moving rod, with both ends marked at the same time, is shorter than at rest. Both are results of measurement, not of how things look in a photograph.
Two clocks that part and meet again can be compared directly, and the one that travelled shows less elapsed time. That comparison adds up each clock's whole path. Einstein's remark about a clock at the equator needs more than this model, because a real Earth has gravity.
A velocity is a distance divided by a time, and both change between frames. Transform both before dividing, and two speeds below c combine to a speed that is still below c, while light stays at c.
In the moving observer’s coordinates the derivatives in time and along x mix. Grouping the result so that the Maxwell-Hertz equations keep their form points to the moving observer’s electric and magnetic components; the form alone does not prove that identification, which rests on the principle of relativity.
For an observer moving along x, the field components along x stay the same, and the electric and magnetic components across the motion mix into each other. Where one observer finds only a magnetic field, the other can find an electric one.
Rewriting the phase of a light wave in the moving observer’s coordinates gives the new frequency and the new direction in one step. At 0.6c, light travelling the same way as the observer has half its frequency in the new frame; light that crosses the motion at right angles in the original frame has 1.25 times its frequency.
In the new frame the light’s energy density changes by the factor q squared, while the volume of the same light complex, measured at one instant of the new time, changes by 1/q. Its total energy therefore changes by q, the same factor as its frequency.
First check that the light can catch the mirror: its speed along the mirror’s normal, c cos φ, must exceed the mirror’s speed away from it. Then change to the mirror’s rest frame, reflect the light there, and change back.
Charge density and the current along the motion mix when the frame changes, so a wire that is neutral while it carries a current has a charge density in a frame moving along it. There is no single charge velocity to find by dividing current by charge density: in a neutral wire that ratio divides by zero.
A force-to-acceleration ratio needs two frame labels
Einstein’s transverse mass, m times gamma squared, divides the force measured in the electron’s momentary rest frame by the acceleration measured in the laboratory. Measure both in the laboratory and the ratio is m times gamma. The two numbers answer two different questions; the electron has one mass m.
Adding up the work of the force along the motion, from rest to speed v, gives the electron’s kinetic energy, mc²(γ − 1). Einstein then gives two more relations a laboratory can test: the voltage that accelerates the electron, and how sharply electric and magnetic fields bend its path.