Annus MirabilisEinstein in 1905
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On the Electrodynamics of Moving Bodies

Zur Elektrodynamik bewegter Körper

by A. Einstein

It is known that Maxwell’s electrodynamics — as it is customarily understood at present — leads in its application to moving bodies to asymmetries which do not seem to be inherent in the phenomena. Think, for example, of the electrodynamic interaction between a magnet and a conductor. The observable phenomenon here depends only on the relative motion of conductor and magnet, whereas according to the customary view the two cases, in which one or the other of these bodies is the one in motion, are to be strictly separated from each other. For if the magnet moves and the conductor is at rest, there arises in the surroundings of the magnet an electric field with a certain energy value, which produces a current at the places where parts of the conductor are situated. But if the magnet is at rest and the conductor moves, no electric field arises in the surroundings of the magnet, but instead there arises in the conductor an electromotive force, to which no energy corresponds in itself, but which — assuming equality of the relative motion in the two cases considered — gives rise to electric currents of the same magnitude and the same course as those produced by the electric forces in the first case.

Examples of a similar kind, together with the unsuccessful attempts to detect a motion of the earth relative to the “light medium”, lead to the conjecture that the concept of absolute rest corresponds to no properties of the phenomena, not only in mechanics but also in electrodynamics, but rather that for all coordinate systems for which the equations of mechanics hold, the same electrodynamic and optical laws also hold, as has already been shown for the quantities of the first order. We shall raise this conjecture (whose content will be called the “principle of relativity” in what follows) to the status of a presupposition, and in addition introduce the presupposition, only apparently incompatible with it, that light is always propagated in empty space with a definite velocity VV which is independent of the state of motion of the emitting body. These two presuppositions suffice to arrive at a simple and consistent electrodynamics of moving bodies, taking as a basis Maxwell’s theory for bodies at rest. The introduction of a “light ether” will prove to be superfluous inasmuch as, according to the view to be developed, neither will an “absolutely resting space” endowed with special properties be introduced, nor will a velocity vector be assigned to a point of empty space in which electromagnetic processes take place.

The theory to be developed is based — like every other electrodynamics — on the kinematics of the rigid body, since the assertions of any theory concern relations between rigid bodies (coordinate systems), clocks and electromagnetic processes. Insufficient consideration of this circumstance is the root of the difficulties with which the electrodynamics of moving bodies currently has to contend.

I. Kinematical Part.

§ 1. Definition of Simultaneity.

Let there be given a coordinate system in which Newton’s mechanical equations hold. To distinguish it in words from coordinate systems to be introduced later, and to make the idea precise, we call this coordinate system the “stationary system”.

If a material point is at rest relative to this coordinate system, its position relative to the latter can be determined by rigid measuring rods, using the methods of Euclidean geometry, and expressed in Cartesian coordinates.

If we wish to describe the motion of a material point, we give the values of its coordinates as functions of time. Now it must be borne well in mind that a mathematical description of this kind has a physical meaning only when one has first become clear about what is to be understood here by “time”. We have to take into account that all our judgements in which time plays a part are always judgements about simultaneous events. If, for example, I say: “That train arrives here at 7 o’clock,” this means roughly: “The pointing of the small hand of my clock to 7 and the arrival of the train are simultaneous events.”[1)]

It might seem that all the difficulties concerning the definition of “time” could be overcome by my putting “the position of the small hand of my clock” in place of “time”. Such a definition does indeed suffice when it is a matter of defining a time exclusively for the place at which the clock is located; but the definition no longer suffices as soon as it is a matter of connecting with one another, with respect to time, series of events that occur at different places, or — what amounts to the same thing — of evaluating, with respect to time, events that occur at places remote from the clock.

We could, it is true, content ourselves with evaluating events with respect to time by having an observer, located together with the clock at the origin of coordinates, assign the corresponding position of the clock’s hands to each light signal that gives evidence of an event to be evaluated and reaches him through empty space. But such an assignment has the drawback that it is not independent of the standpoint of the observer provided with the clock, as we know from experience. We arrive at a far more practical stipulation through the following consideration.

If there is a clock at the point AA of space, an observer located at AA can evaluate, with respect to time, the events in the immediate vicinity of AA by finding the positions of the clock’s hands that are simultaneous with these events. If there is also a clock at the point BB of space — we wish to add, “a clock of exactly the same constitution as the one located at AA” — then an evaluation, with respect to time, of the events in the immediate vicinity of BB by an observer located at BB is also possible. But without a further stipulation it is not possible to compare, with respect to time, an event at AA with an event at BB; so far we have defined only an “AA-time” and a “BB-time”, but no “time” common to AA and BB. The latter time can now be defined by stipulating by definition that the “time” which light requires to travel from AA to BB is equal to the “time” which it requires to travel from BB to AA. For let a ray of light start at the “AA-time” tAt_A from AA towards BB, let it be reflected at the “BB-time” tBt_B at BB towards AA, and let it arrive back at the “AA-time” tA′t'_A at AA. The two clocks run synchronously by definition if tB−tA=tA′−tB.t_B - t_A = t'_A - t_B .Time in the stationary systemTime in the stationary systemTime in the stationary systemExplain this equation

We assume that this definition of synchronism is possible without contradiction, and indeed for arbitrarily many points, and that therefore the following relations hold in general:
1. If the clock at BB runs synchronously with the clock at AA, then the clock at AA runs synchronously with the clock at BB.
2. If the clock at AA runs synchronously both with the clock at BB and with the clock at CC, then the clocks at BB and CC also run synchronously relative to each other.

In this way, with the help of certain (imagined) physical experiences, we have established what is to be understood by synchronously running clocks at rest that are located at different places, and have thereby evidently obtained a definition of “simultaneous” and “time”. The “time” of an event is the reading, simultaneous with the event, of a clock at rest located at the place of the event, which runs synchronously with a specified clock at rest, and indeed with the same clock for all determinations of time.

In accordance with experience we further stipulate that the quantity 2AB‾tA′−tA=V\frac{2 \overline{AB}}{t'_A - t_A} = VDistance between two eventsTime in the stationary systemTime in the stationary systemSpeed of lightExplain this equation is a universal constant (the velocity of light in empty space).

It is essential that we have defined time by means of clocks at rest in the stationary system; because of this association with the stationary system we call the time just defined “the time of the stationary system”.

§ 2. On the Relativity of Lengths and Times.

The following considerations are based on the principle of relativity and on the principle of the constancy of the velocity of light, which two principles we define as follows.
1. The laws according to which the states of physical systems change are independent of which of two coordinate systems in uniform translational motion relative to each other these changes of state are referred to.
2. Every ray of light moves in the “stationary” coordinate system with the definite velocity VV, independently of whether this ray of light is emitted by a stationary or by a moving body. Here [velocity = light path / time interval] Geschwindigkeit=LichtwegZeitdauer,\text{Geschwindigkeit} = \frac{\text{Lichtweg}}{\text{Zeitdauer}} ,Explain this equation where “Zeitdauer” [time interval] is to be understood in the sense of the definition in § 1.

Let a stationary rigid rod be given; let it possess, measured with a measuring rod that is likewise stationary, the length ll. We now imagine the axis of the rod laid along the XX-axis of the stationary coordinate system, and the rod then given a uniform motion of parallel translation (velocity vv) along the XX-axis in the direction of increasing xx. We now inquire into the length of the moving rod, which we imagine to be ascertained by the following two operations:
a) The observer moves, together with the measuring rod just mentioned, with the rod to be measured, and measures the length of the rod directly by applying the measuring rod, just as if the rod to be measured, the observer and the measuring rod were at rest.
b) By means of clocks at rest set up in the stationary system and synchronous according to § 1, the observer ascertains at which points of the stationary system the beginning and the end of the rod to be measured are located at a definite time tt.
The distance between these two points, measured with the measuring rod already used, in this case at rest, is likewise a length, which may be called the “length of the rod”.

According to the principle of relativity, the length to be found by operation a), which we shall call “the length of the rod in the moving system”, must be equal to the length ll of the stationary rod.

The length to be found by operation b), which we shall call “the length of the (moving) rod in the stationary system”, we shall determine on the basis of our two principles, and we shall find that it differs from ll.

The kinematics in general use tacitly assumes that the lengths determined by these two operations are exactly equal to each other, or in other words, that a moving rigid body at the epoch tt may in geometrical respects be completely replaced by the same body when it is at rest in a definite position.

We imagine further that at the two ends of the rod (AA and BB) clocks are placed which are synchronous with the clocks of the stationary system, that is, whose readings correspond at any moment to the “time of the stationary system” at the places where they happen to be; these clocks are therefore “synchronous in the stationary system”.

We imagine further that with each clock there is an observer moving with it, and that these observers apply to the two clocks the criterion established in § 1 for the synchronous running of two clocks. At the time[1)] tAt_A let a ray of light go out from AA, let it be reflected at the time tBt_B at BB, and let it arrive back at the time tA′t'_A at AA. Taking into account the principle of the constancy of the velocity of light, we find: tB−tA=rABV−vt_B - t_A = \frac{r_{AB}}{V - v}Time in the stationary systemTime in the stationary systemLength of the moving rodSpeed of lightSpeed of the moving systemExplain this equation and tA′−tB=rABV+v,t'_A - t_B = \frac{r_{AB}}{V + v} ,Time in the stationary systemTime in the stationary systemLength of the moving rodSpeed of lightSpeed of the moving systemExplain this equation where rABr_{AB} denotes the length — measured in the stationary system — of the moving rod. Observers moving with the moving rod would thus find that the two clocks do not run synchronously, while observers located in the stationary system would declare the clocks to be running synchronously.

We see, then, that we must not attach any absolute meaning to the concept of simultaneity, but that two events which are simultaneous when viewed from one coordinate system are no longer to be regarded as simultaneous events when viewed from a system moving relative to that system.

§ 3. Theory of the Transformation of Coordinates and Times from the Stationary System to a System in Uniform Translational Motion Relative to It.

Let there be given in the “stationary” space two coordinate systems, that is, two systems each of three rigid material lines, perpendicular to one another and issuing from one point. Let the XX-axes of the two systems coincide, and their YY- and ZZ-axes be respectively parallel. Let each system be provided with a rigid measuring rod and a number of clocks, and let the two measuring rods, and all the clocks of the two systems, be exactly alike.

Now to the origin of one of the two systems (kk) let a (constant) velocity vv be imparted in the direction of increasing xx of the other, stationary system (KK), a velocity which is to be communicated also to the coordinate axes, the measuring rod in question and the clocks. To each time tt of the stationary system KK there then corresponds a definite position of the axes of the moving system, and for reasons of symmetry we are entitled to assume that the motion of kk can be such that the axes of the moving system are parallel to the axes of the stationary system at the time tt (by “tt” a time of the stationary system is always meant).

We now imagine space to be measured out both from the stationary system KK by means of the stationary measuring rod and from the moving system kk by means of the measuring rod moving with it, and the coordinates xx, yy, zz and ξ\xi, η\eta, ζ\zeta respectively to be thus obtained. Further, let the time tt of the stationary system be determined, by means of the clocks at rest in the stationary system and by light signals in the manner given in § 1, for all points of the latter at which there are clocks; likewise let the time τ\tau of the moving system be determined for all points of the moving system at which there are clocks at rest relative to it, by applying the method of light signals given in § 1 between the points at which these clocks are located.

To every system of values xx, yy, zz, tt which completely determines the place and time of an event in the stationary system there belongs a system of values fixing that event relative to the system kk, namely ξ\xi, η\eta, ζ\zeta, τ\tau, and the task now is to find the system of equations connecting these quantities.

In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time.

If we put x′=x−vtx' = x - v t, it is clear that to a point at rest in the system kk there belongs a definite system of values x′x', yy, zz, independent of time. We first determine τ\tau as a function of x′x', yy, zz and tt. To this end we have to express in equations that τ\tau is nothing other than the totality of the readings of clocks at rest in the system kk which have been made synchronous according to the rule given in § 1.

From the origin of the system kk let a ray of light be sent at the time τ0\tau_0 along the XX-axis to x′x', and from there be reflected at the time τ1\tau_1 towards the origin of coordinates, where it arrives at the time τ2\tau_2; we must then have: 12(τ0+τ2)=τ1\frac{1}{2}(\tau_0 + \tau_2) = \tau_1Time in the moving systemTime in the moving systemTime in the moving systemExplain this equation or, if we add the arguments of the function τ\tau and apply the principle of the constancy of the velocity of light in the stationary system: 12[τ(0,0,0,t)+τ(0,0,0,{t+x′V−v+x′V+v})]=τ(x′,0,0,t+x′V−v).\begin{aligned} &\frac{1}{2}\left[ \tau(0,0,0,t) + \tau\left( 0,0,0,\left\{ t + \frac{x'}{V-v} + \frac{x'}{V+v} \right\} \right) \right] \\ &= \tau\left( x',0,0,t + \frac{x'}{V-v} \right). \end{aligned}Time in the moving systemTime in the stationary systemAuxiliary Galilean coordinate: x'Speed of lightSpeed of the moving systemExplain this equation Hence, if x′x' is chosen infinitely small, it follows that: 12(1V−v+1V+v)∂τ∂t=∂τ∂x′+1V−v∂τ∂t,\frac{1}{2}\left( \frac{1}{V-v} + \frac{1}{V+v} \right) \frac{\partial \tau}{\partial t} = \frac{\partial \tau}{\partial x'} + \frac{1}{V-v} \frac{\partial \tau}{\partial t} ,Speed of lightSpeed of the moving systemTime in the moving systemTime in the stationary systemAuxiliary Galilean coordinate: x'Explain this equation or ∂τ∂x′+vV2−v2∂τ∂t=0.\frac{\partial \tau}{\partial x'} + \frac{v}{V^2 - v^2} \frac{\partial \tau}{\partial t} = 0 .Time in the moving systemAuxiliary Galilean coordinate: x'Speed of the moving systemSpeed of lightTime in the stationary systemExplain this equation

It is to be noted that instead of the origin of coordinates we could have chosen any other point as the starting point of the ray of light, and the equation just obtained therefore holds for all values of x′x', yy, zz.

An analogous consideration — applied to the HH- and ZZ-axes — gives, if it is borne in mind that light always propagates along these axes, viewed from the stationary system, with the velocity V2−v2\sqrt{V^2 - v^2}: ∂τ∂y=0∂τ∂z=0.\begin{aligned} \frac{\partial \tau}{\partial y} &= 0 \\ \frac{\partial \tau}{\partial z} &= 0 . \end{aligned}Time in the moving systemPosition in the stationary systemPosition in the stationary systemExplain this equation From these equations it follows, since τ\tau is a linear function, that: τ=a(t−vV2−v2x′),\tau = a\left( t - \frac{v}{V^2 - v^2} x' \right) ,Time in the moving systemUndetermined scaleTime in the stationary systemSpeed of the moving systemSpeed of lightAuxiliary Galilean coordinate: x'Explain this equation where aa is a function φ(v)\varphi(v) as yet unknown, and where for brevity it is assumed that at the origin of kk, for τ=0\tau = 0, t=0t = 0.

With the help of this result it is easy to determine the quantities ξ\xi, η\eta, ζ\zeta, by expressing in equations that light (as the principle of the constancy of the velocity of light in conjunction with the principle of relativity requires) also propagates with the velocity VV when measured in the moving system. For a ray of light emitted at the time τ=0\tau = 0 in the direction of increasing ξ\xi we have: ξ=Vτ,\xi = V \tau ,Position in the moving systemSpeed of lightTime in the moving systemExplain this equation or ξ=aV(t−vV2−v2x′).\xi = a V \left( t - \frac{v}{V^2 - v^2} x' \right) .Position in the moving systemUndetermined scaleSpeed of lightTime in the stationary systemSpeed of the moving systemAuxiliary Galilean coordinate: x'Explain this equation But the ray of light moves relative to the origin of kk, measured in the stationary system, with the velocity V−vV-v, so that we have: x′V−v=t.\frac{x'}{V-v} = t .Auxiliary Galilean coordinate: x'Speed of lightSpeed of the moving systemTime in the stationary systemExplain this equation If we substitute this value of tt in the equation for ξ\xi, we obtain: ξ=aV2V2−v2x′.\xi = a \frac{V^2}{V^2 - v^2} x' .Position in the moving systemUndetermined scaleSpeed of lightSpeed of the moving systemAuxiliary Galilean coordinate: x'Explain this equation In an analogous way, by considering rays of light moving along the two other axes, we find: η=Vτ=aV(t−vV2−v2x′),\eta = V \tau = a V \left( t - \frac{v}{V^2 - v^2} x' \right) ,Position in the moving systemSpeed of lightTime in the moving systemUndetermined scaleTime in the stationary systemSpeed of the moving systemAuxiliary Galilean coordinate: x'Explain this equation where yV2−v2=t;x′=0;\frac{y}{\sqrt{V^2 - v^2}} = t ; \quad x' = 0 ;Position in the stationary systemSpeed of lightSpeed of the moving systemTime in the stationary systemAuxiliary Galilean coordinate: x'Explain this equation hence η=aVV2−v2y\eta = a \frac{V}{\sqrt{V^2 - v^2}} yPosition in the moving systemUndetermined scaleSpeed of lightSpeed of the moving systemPosition in the stationary systemExplain this equation and ζ=aVV2−v2z.\zeta = a \frac{V}{\sqrt{V^2 - v^2}} z .Position in the moving systemUndetermined scaleSpeed of lightSpeed of the moving systemPosition in the stationary systemExplain this equation

If we substitute for x′x' its value, we obtain: τ=φ(v)β(t−vV2x),ξ=φ(v)β(x−vt),η=φ(v)y,ζ=φ(v)z,\begin{aligned} \tau &= \varphi(v) \beta \left( t - \frac{v}{V^2} x \right) , \\ \xi &= \varphi(v) \beta (x - v t), \\ \eta &= \varphi(v) y, \\ \zeta &= \varphi(v) z, \end{aligned}Time in the moving systemUnknown scale factor: phi(v)Speed of the moving systemLorentz factorTime in the stationary systemSpeed of lightPosition in the stationary systemPosition in the moving systemPosition in the moving systemPosition in the stationary systemPosition in the moving systemPosition in the stationary systemExplain this equation where β=11−(vV)2\beta = \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}}Lorentz factorSpeed of the moving systemSpeed of lightExplain this equation and φ\varphi is a function of vv as yet unknown. If no presupposition whatever is made about the initial position of the moving system and about the zero point of τ\tau, an additive constant is to be added to the right-hand side of each of these equations.

We now have to prove that every ray of light, measured in the moving system, propagates with the velocity VV if, as we have assumed, this is the case in the stationary system; for we have not yet furnished the proof that the principle of the constancy of the velocity of light is compatible with the principle of relativity.

At the time t=τ=0t = \tau = 0 let a spherical wave be emitted from the origin of coordinates, common at that time to both systems, a wave which propagates in the system KK with the velocity VV. If (x,y,z)(x, y, z) is a point just reached by this wave, we therefore have x2+y2+z2=V2t2.x^2 + y^2 + z^2 = V^2 t^2 .Position in the stationary systemPosition in the stationary systemPosition in the stationary systemSpeed of lightTime in the stationary systemExplain this equation

We transform this equation with the help of our transformation equations and obtain, after a simple calculation: ξ2+η2+ζ2=V2τ2.\xi^2 + \eta^2 + \zeta^2 = V^2 \tau^2 .Position in the moving systemPosition in the moving systemPosition in the moving systemSpeed of lightTime in the moving systemExplain this equation

The wave under consideration is therefore, viewed in the moving system too, a spherical wave with the velocity of propagation VV. It is thereby shown that our two fundamental principles are compatible with each other.

In the transformation equations developed there still occurs an unknown function φ\varphi of vv, which we now wish to determine.

For this purpose we introduce a third coordinate system K′K', which relative to the system kk is in such a motion of parallel translation, parallel to the Ξ\Xi-axis, that its origin moves with the velocity −v-v on the Ξ\Xi-axis. At the time t=0t = 0 let all three origins of coordinates coincide, and for t=x=y=z=0t = x = y = z = 0 let the time t′t' of the system K′K' be zero. We call x′x', y′y', z′z' the coordinates measured in the system K′K', and by twofold application of our transformation equations we obtain: t′=φ(−v)β(−v){τ+vV2ξ}=φ(v)φ(−v)t,x′=φ(−v)β(−v){ξ+vτ}=φ(v)φ(−v)x,y′=φ(−v)η=φ(v)φ(−v)y,z′=φ(−v)ζ=φ(v)φ(−v)z.\begin{aligned} t' &= \varphi(-v) \beta(-v) \left\{ \tau + \frac{v}{V^2} \xi \right\} &&= \varphi(v) \varphi(-v) t , \\ x' &= \varphi(-v) \beta(-v) \{ \xi + v \tau \} &&= \varphi(v) \varphi(-v) x , \\ y' &= \varphi(-v) \eta &&= \varphi(v) \varphi(-v) y , \\ z' &= \varphi(-v) \zeta &&= \varphi(v) \varphi(-v) z . \end{aligned}Time in the stationary systemUnknown scale factor: phi(v)Speed of the moving systemLorentz factorTime in the moving systemSpeed of lightPosition in the moving systemTime in the stationary systemPosition in the stationary systemPosition in the stationary systemPosition in the stationary systemPosition in the moving systemPosition in the stationary systemPosition in the stationary systemPosition in the moving systemPosition in the stationary systemExplain this equation

Since the relations between x′x', y′y', z′z' and xx, yy, zz do not contain the time tt, the systems KK and K′K' are at rest relative to each other, and it is clear that the transformation from KK to K′K' must be the identity transformation. We therefore have: φ(v)φ(−v)=1.\varphi(v) \varphi(-v) = 1 .Unknown scale factor: phi(v)Speed of the moving systemExplain this equation We now inquire into the meaning of φ(v)\varphi(v). We consider the portion of the HH-axis of the system kk which lies between ξ=0\xi = 0, η=0\eta = 0, ζ=0\zeta = 0 and ξ=0\xi = 0, η=l\eta = l, ζ=0\zeta = 0. This portion of the HH-axis is a rod moving relative to the system KK with the velocity vv perpendicular to its axis, whose ends have in KK the coordinates: x1=vt,y1=lφ(v),z1=0x_1 = v t, \quad y_1 = \frac{l}{\varphi(v)}, \quad z_1 = 0Position in the stationary systemSpeed of the moving systemTime in the stationary systemPosition in the stationary systemRest lengthUnknown scale factor: phi(v)Position in the stationary systemExplain this equation and x2=vt,y2=0,z2=0.x_2 = v t, \quad y_2 = 0, \quad z_2 = 0 .Position in the stationary systemSpeed of the moving systemTime in the stationary systemPosition in the stationary systemPosition in the stationary systemExplain this equation The length of the rod, measured in KK, is therefore l/φ(v)l/\varphi(v); this gives the meaning of the function φ\varphi. For reasons of symmetry it is now evident that the length of a given rod moving perpendicular to its axis, measured in the stationary system, can depend only on the velocity, and not on the direction and sense of the motion. Thus the length of the moving rod measured in the stationary system does not change if vv is interchanged with −v-v. From this it follows that: lφ(v)=lφ(−v),\frac{l}{\varphi(v)} = \frac{l}{\varphi(-v)} ,Rest lengthUnknown scale factor: phi(v)Speed of the moving systemExplain this equation or φ(v)=φ(−v).\varphi(v) = \varphi(-v) .Unknown scale factor: phi(v)Speed of the moving systemExplain this equation From this and the relation found earlier it follows that φ(v)=1\varphi(v) = 1 must hold, so that the transformation equations found become: τ=β(t−vV2x),ξ=β(x−vt),η=y,ζ=z,\begin{aligned} \tau &= \beta\left( t - \frac{v}{V^2} x \right), \\ \xi &= \beta(x - v t), \\ \eta &= y, \\ \zeta &= z, \end{aligned}Time in the moving systemLorentz factorTime in the stationary systemSpeed of the moving systemSpeed of lightPosition in the stationary systemPosition in the moving systemPosition in the moving systemPosition in the stationary systemPosition in the moving systemPosition in the stationary systemExplain this equation where β=11−(vV)2,\beta = \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} ,Lorentz factorSpeed of the moving systemSpeed of lightExplain this equation

§ 4. Physical Meaning of the Equations Obtained, Concerning Moving Rigid Bodies and Moving Clocks.

We consider a rigid sphere[1)] of radius RR, which is at rest relative to the moving system kk and whose centre lies at the origin of coordinates of kk. The equation of the surface of this sphere, which moves relative to the system KK with the velocity vv, is: ξ2+η2+ζ2=R2.\xi^2 + \eta^2 + \zeta^2 = R^2 .Position in the moving systemPosition in the moving systemPosition in the moving systemSphere radiusExplain this equation The equation of this surface, expressed in xx, yy, zz at the time t=0t = 0, is: x2(1−(vV)2)2+y2+z2=R2.\frac{x^2}{\left( \sqrt{1 - \left( \frac{v}{V} \right)^2} \right)^2} + y^2 + z^2 = R^2 .Position in the stationary systemSpeed of the moving systemSpeed of lightPosition in the stationary systemPosition in the stationary systemSphere radiusExplain this equation A rigid body which, measured in a state of rest, has the shape of a sphere therefore has in a state of motion — viewed from the stationary system — the shape of an ellipsoid of revolution with the axes R1−(vV)2,R,R.R \sqrt{1 - \left( \frac{v}{V} \right)^2} , \quad R , \quad R .Sphere radiusSpeed of the moving systemSpeed of lightExplain this equation

Thus, whereas the YY- and ZZ-dimensions of the sphere (and hence also of every rigid body of whatever shape) do not appear modified by the motion, the XX-dimension appears shortened in the ratio 1:1−(v/V)21 : \sqrt{1 - (v/V)^2}, that is, the more strongly the greater vv is. For v=Vv = V all moving objects — viewed from the “stationary” system — shrink into plane figures. For velocities greater than that of light our considerations become meaningless; we shall find, moreover, in the considerations that follow, that in our theory the velocity of light plays physically the part of infinitely great velocities.

It is clear that the same results hold for bodies at rest in the “stationary” system, viewed from a uniformly moving system. —

We imagine further one of the clocks which are fitted to indicate the time tt when at rest relative to the stationary system and the time τ\tau when at rest relative to the moving system, placed at the origin of coordinates of kk and so set that it indicates the time τ\tau. How fast does this clock go, viewed from the stationary system?

Between the quantities xx, tt and τ\tau, which refer to the position of this clock, the following equations evidently hold: τ=11−(vV)2(t−vV2x)\tau = \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} \left( t - \frac{v}{V^2} x \right)Time in the moving systemSpeed of the moving systemSpeed of lightTime in the stationary systemPosition in the stationary systemExplain this equation and x=vt.x = v t .Position in the stationary systemSpeed of the moving systemTime in the stationary systemExplain this equation We therefore have τ=t1−(vV)2=t−(1−1−(vV)2)t,\tau = t \sqrt{1 - \left( \frac{v}{V} \right)^2} = t - \left( 1 - \sqrt{1 - \left( \frac{v}{V} \right)^2} \right) t ,Time in the moving systemTime in the stationary systemSpeed of the moving systemSpeed of lightExplain this equation from which it follows that the reading of the clock (viewed in the stationary system) lags behind by (1−1−(v/V)2)(1 - \sqrt{1 - (v/V)^2}) sec. per second or — to within quantities of the fourth and higher order, by 12(v/V)2\frac{1}{2} (v/V)^2 sec.

From this the following peculiar consequence results. If at the points AA and BB of KK there are clocks at rest which, viewed in the stationary system, run synchronously, and if the clock at AA is moved with the velocity vv along the line joining it to BB, then on the arrival of this clock at BB the two clocks no longer run synchronously, but the clock moved from AA to BB lags behind the clock that has been at BB from the beginning by 12tv2/V2\frac{1}{2} t v^2 / V^2 sec. (to within quantities of the fourth and higher order), if tt is the time the clock needs to go from AA to BB.

One sees at once that this result still holds if the clock moves from AA to BB along any polygonal line, and that it holds even if the points AA and BB coincide.

If one assumes that the result proved for a polygonal line also holds for a continuously curved curve, one obtains the theorem: If there are two synchronously running clocks at AA, and one of them is moved along a closed curve with constant velocity until it returns to AA, which may take tt sec., then on its arrival at AA the latter clock lags behind the one that has remained unmoved by 12t(v/V)2\frac{1}{2} t (v/V)^2 sec. From this one concludes that a balance-wheel clock located at the earth’s equator must go more slowly, by a very small amount, than a precisely similar clock, subject to otherwise identical conditions, located at one of the earth’s poles.

§ 5. The Addition Theorem of Velocities.

In the system kk, moving along the XX-axis of the system KK with the velocity vv, let a point move according to the equations: ξ=wξτ,η=wητ,ζ=0,\begin{aligned} \xi &= w_\xi \tau , \\ \eta &= w_\eta \tau , \\ \zeta &= 0 , \end{aligned}Position in the moving systemVelocity along x, moving systemTime in the moving systemPosition in the moving systemVelocity along y, moving systemPosition in the moving systemExplain this equation where wξw_\xi and wηw_\eta denote constants.

The motion of the point relative to the system KK is sought. If, with the help of the transformation equations developed in § 3, we introduce the quantities xx, yy, zz, tt into the equations of motion of the point, we obtain: x=wξ+v1+vwξV2t,y=1−(vV)21+vwξV2wηt,z=0.\begin{aligned} x &= \frac{w_\xi + v}{1 + \frac{v w_\xi}{V^2}} t , \\ y &= \frac{\sqrt{1 - \left( \frac{v}{V} \right)^2}}{1 + \frac{v w_\xi}{V^2}} w_\eta t , \\ z &= 0 . \end{aligned}Position in the stationary systemVelocity along x, moving systemSpeed of the moving systemSpeed of lightTime in the stationary systemPosition in the stationary systemVelocity along y, moving systemPosition in the stationary systemExplain this equation The law of the parallelogram of velocities therefore holds, according to our theory, only to a first approximation. We put: U2=(dxdt)2+(dydt)2,w2=wξ2+wη2\begin{aligned} U^2 &= \left( \frac{d x}{d t} \right)^2 + \left( \frac{d y}{d t} \right)^2 , \\ w^2 &= w_\xi^2 + w_\eta^2 \end{aligned}Composed velocity (stationary system)Position in the stationary systemTime in the stationary systemPosition in the stationary systemVelocity in the moving frameVelocity along x, moving systemVelocity along y, moving systemExplain this equation and α=arctg⁡wywx;\alpha = \operatorname{arctg} \frac{w_y}{w_x} ;Velocity direction angle: alphaVelocity along y, moving systemVelocity along x, moving systemExplain this equation α\alpha is then to be regarded as the angle between the velocities vv and ww. After a simple calculation we obtain: U=(v2+w2+2vwcos⁡α)−(vwsin⁡αV)21+vwcos⁡αV2.U = \frac{\sqrt{(v^2 + w^2 + 2 v w \cos \alpha) - \left( \frac{v w \sin \alpha}{V} \right)^2}}{1 + \frac{v w \cos \alpha}{V^2}} .Composed velocity (stationary system)Speed of the moving systemVelocity in the moving frameVelocity direction angle: alphaSpeed of lightExplain this equation It is noteworthy that vv and ww enter the expression for the resultant velocity symmetrically. If ww also has the direction of the XX-axis (Ξ\Xi-axis), we obtain: U=v+w1+vwV2.U = \frac{v + w}{1 + \frac{v w}{V^2}} .Composed velocity (stationary system)Speed of the moving systemVelocity in the moving frameSpeed of lightExplain this equation From this equation it follows that from the composition of two velocities which are smaller than VV there always results a velocity smaller than VV. For if we put v=V−ϰv = V - \varkappa, w=V−λw = V - \lambda, where ϰ\varkappa and λ\lambda are positive and smaller than VV, then: U=V2V−ϰ−λ2V−ϰ−λ+ϰλV<V.U = V \frac{2 V - \varkappa - \lambda}{2 V - \varkappa - \lambda + \frac{\varkappa \lambda}{V}} < V .Composed velocity (stationary system)Speed of lightSpeed deficit from light speed: kappa, lambdaSpeed deficit from light speed: kappa, lambdaExplain this equation

It follows further that the velocity of light VV cannot be altered by composition with a “sub-light velocity”. For this case we obtain: U=V+w1+wV=V.U = \frac{V + w}{1 + \frac{w}{V}} = V .Composed velocity (stationary system)Speed of lightVelocity in the moving frameExplain this equation We could also have obtained the formula for UU, for the case in which vv and ww have the same direction, by composing two transformations according to § 3. If, besides the systems KK and kk figuring in § 3, we introduce a third coordinate system k′k', in parallel motion relative to kk, whose origin moves on the Ξ\Xi-axis with the velocity ww, we obtain between the quantities xx, yy, zz, tt and the corresponding quantities of k′k' equations which differ from those found in § 3 only in that the quantity v+w1+vwV2\frac{v + w}{1 + \frac{v w}{V^2}}Speed of the moving systemVelocity in the moving frameSpeed of lightExplain this equation takes the place of “vv”; one sees from this that such parallel transformations — as must be the case — form a group.

We have now derived the theorems, necessary for our purposes, of the kinematics corresponding to our two principles, and we proceed to show their application in electrodynamics.

II. Electrodynamical Part.

§ 6. Transformation of the Maxwell-Hertz Equations for Empty Space. On the Nature of the Electromotive Forces Occurring with Motion in a Magnetic Field.

Let the Maxwell-Hertz equations for empty space hold for the stationary system KK, so that we have: 1V∂X∂t=∂N∂y−∂M∂z,1V∂L∂t=∂Y∂z−∂Z∂y,1V∂Y∂t=∂L∂z−∂N∂x,1V∂M∂t=∂Z∂x−∂X∂z,1V∂Z∂t=∂M∂x−∂L∂y,1V∂N∂t=∂X∂y−∂Y∂x,\begin{aligned} \frac{1}{V} \frac{\partial X}{\partial t} &= \frac{\partial N}{\partial y} - \frac{\partial M}{\partial z} , &\qquad \frac{1}{V} \frac{\partial L}{\partial t} &= \frac{\partial Y}{\partial z} - \frac{\partial Z}{\partial y} , \\ \frac{1}{V} \frac{\partial Y}{\partial t} &= \frac{\partial L}{\partial z} - \frac{\partial N}{\partial x} , &\qquad \frac{1}{V} \frac{\partial M}{\partial t} &= \frac{\partial Z}{\partial x} - \frac{\partial X}{\partial z} , \\ \frac{1}{V} \frac{\partial Z}{\partial t} &= \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} , &\qquad \frac{1}{V} \frac{\partial N}{\partial t} &= \frac{\partial X}{\partial y} - \frac{\partial Y}{\partial x} , \end{aligned}Speed of lightElectric field, stationary systemTime in the stationary systemMagnetic field, stationary systemPosition in the stationary systemMagnetic field, stationary systemPosition in the stationary systemMagnetic field, stationary systemElectric field, stationary systemElectric field, stationary systemPosition in the stationary systemExplain this equation where (X,Y,Z)(X, Y, Z) denotes the vector of the electric force, and (L,M,N)(L, M, N) that of the magnetic force.

If we apply to these equations the transformation developed in § 3, by referring the electromagnetic processes to the coordinate system introduced there, moving with the velocity vv, we obtain the equations: 1V∂X∂τ=∂β(N−vVY)∂η−∂β(M+vVZ)∂ζ,1V∂β(Y−vVN)∂τ=∂L∂ζ−∂β(N−vVY)∂ξ,1V∂β(Z+vVM)∂τ=∂β(M+vVZ)∂ξ−∂L∂η,1V∂L∂τ=∂β(Y−vVN)∂ζ−∂β(Z+vVM)∂η,1V∂β(M+vVZ)∂τ=∂β(Z+vVM)∂ξ−∂X∂ζ,1V∂β(N−vVY)∂τ=∂X∂η−∂β(Y−vVN)∂ξ,\begin{aligned} \frac{1}{V} \frac{\partial X}{\partial \tau} &= \frac{\partial \beta \left( N - \frac{v}{V} Y \right)}{\partial \eta} - \frac{\partial \beta \left( M + \frac{v}{V} Z \right)}{\partial \zeta} , \\ \frac{1}{V} \frac{\partial \beta \left( Y - \frac{v}{V} N \right)}{\partial \tau} &= \frac{\partial L}{\partial \zeta} - \frac{\partial \beta \left( N - \frac{v}{V} Y \right)}{\partial \xi} , \\ \frac{1}{V} \frac{\partial \beta \left( Z + \frac{v}{V} M \right)}{\partial \tau} &= \frac{\partial \beta \left( M + \frac{v}{V} Z \right)}{\partial \xi} - \frac{\partial L}{\partial \eta} , \\ \frac{1}{V} \frac{\partial L}{\partial \tau} &= \frac{\partial \beta \left( Y - \frac{v}{V} N \right)}{\partial \zeta} - \frac{\partial \beta \left( Z + \frac{v}{V} M \right)}{\partial \eta} , \\ \frac{1}{V} \frac{\partial \beta \left( M + \frac{v}{V} Z \right)}{\partial \tau} &= \frac{\partial \beta \left( Z + \frac{v}{V} M \right)}{\partial \xi} - \frac{\partial X}{\partial \zeta} , \\ \frac{1}{V} \frac{\partial \beta \left( N - \frac{v}{V} Y \right)}{\partial \tau} &= \frac{\partial X}{\partial \eta} - \frac{\partial \beta \left( Y - \frac{v}{V} N \right)}{\partial \xi} , \end{aligned}Speed of lightElectric field, stationary systemTime in the moving systemLorentz factorMagnetic field, stationary systemSpeed of the moving systemElectric field, stationary systemPosition in the moving systemMagnetic field, stationary systemElectric field, stationary systemPosition in the moving systemMagnetic field, stationary systemPosition in the moving systemExplain this equation where β=11−(vV)2.\beta = \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} .Lorentz factorSpeed of the moving systemSpeed of lightExplain this equation

The principle of relativity now requires that the Maxwell-Hertz equations for empty space hold also in the system kk if they hold in the system KK, that is, that for the vectors of the electric and magnetic force ((X′,Y′Z′)(X', Y' Z') and (L′,M′,N′)(L', M', N')) of the moving system kk, which are defined in the moving system kk by their ponderomotive actions on electric or magnetic masses respectively, the equations hold: 1V∂X′∂τ=∂N′∂η−∂M′∂ζ,1V∂L′∂τ=∂Y′∂ζ−∂Z′∂η,1V∂Y′∂τ=∂L′∂ζ−∂N′∂ξ,1V∂M′∂τ=∂Z′∂ξ−∂X′∂ζ,1V∂Z′∂τ=∂M′∂ξ−∂L′∂η,1V∂N′∂τ=∂X′∂η−∂Y′∂ξ.\begin{aligned} \frac{1}{V} \frac{\partial X'}{\partial \tau} &= \frac{\partial N'}{\partial \eta} - \frac{\partial M'}{\partial \zeta} , &\qquad \frac{1}{V} \frac{\partial L'}{\partial \tau} &= \frac{\partial Y'}{\partial \zeta} - \frac{\partial Z'}{\partial \eta} , \\ \frac{1}{V} \frac{\partial Y'}{\partial \tau} &= \frac{\partial L'}{\partial \zeta} - \frac{\partial N'}{\partial \xi} , &\qquad \frac{1}{V} \frac{\partial M'}{\partial \tau} &= \frac{\partial Z'}{\partial \xi} - \frac{\partial X'}{\partial \zeta} , \\ \frac{1}{V} \frac{\partial Z'}{\partial \tau} &= \frac{\partial M'}{\partial \xi} - \frac{\partial L'}{\partial \eta} , &\qquad \frac{1}{V} \frac{\partial N'}{\partial \tau} &= \frac{\partial X'}{\partial \eta} - \frac{\partial Y'}{\partial \xi} . \end{aligned}Speed of lightElectric field, moving systemTime in the moving systemMagnetic field, moving systemPosition in the moving systemMagnetic field, moving systemPosition in the moving systemMagnetic field, moving systemElectric field, moving systemElectric field, moving systemPosition in the moving systemExplain this equation

Evidently the two systems of equations found for the system kk must now express exactly the same thing, since both systems of equations are equivalent to the Maxwell-Hertz equations for the system KK. Since, further, the equations of the two systems agree except for the symbols representing the vectors, it follows that the functions occurring at corresponding places in the systems of equations must agree up to a factor ψ(v)\psi(v) that is common to all the functions of the one system of equations, independent of ξ\xi, η\eta, ζ\zeta and τ\tau, and possibly dependent on vv. Thus the relations hold: X′=ψ(v)X,L′=ψ(v)L,Y′=ψ(v)β(Y−vVN),M′=ψ(v)β(M+vVZ),Z′=ψ(v)β(Z+vVM),N′=ψ(v)β(N−vVY).\begin{aligned} X' &= \psi(v) X , &\qquad L' &= \psi(v) L , \\ Y' &= \psi(v) \beta \left( Y - \frac{v}{V} N \right) , &\qquad M' &= \psi(v) \beta \left( M + \frac{v}{V} Z \right) , \\ Z' &= \psi(v) \beta \left( Z + \frac{v}{V} M \right) , &\qquad N' &= \psi(v) \beta \left( N - \frac{v}{V} Y \right) . \end{aligned}Electric field, moving systemUnknown scale factor: psi(v)Speed of the moving systemElectric field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, moving systemLorentz factorElectric field, stationary systemSpeed of lightMagnetic field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, stationary systemElectric field, moving systemMagnetic field, moving systemExplain this equation

If we now form the inverse of this system of equations, first by solving the equations just obtained, and second by applying the equations to the inverse transformation (from kk to KK), which is characterized by the velocity −v-v, it follows, taking into account that the two systems of equations so obtained must be identical: φ(v)⋅φ(−v)=1.\varphi(v) \cdot \varphi(-v) = 1 .Unknown scale factor: psi(v)Speed of the moving systemExplain this equation Further, it follows from reasons of symmetry[1)] that φ(v)=φ(−v);\varphi(v) = \varphi(-v) ;Unknown scale factor: psi(v)Speed of the moving systemExplain this equation hence φ(v)=1,\varphi(v) = 1 ,Unknown scale factor: psi(v)Speed of the moving systemExplain this equation and our equations take the form: X′=X,L′=L,Y′=β(Y−vVN),M′=β(M+vVZ),Z′=β(Z+vVM),N′=β(N−vVY).\begin{aligned} X' &= X , &\qquad L' &= L , \\ Y' &= \beta \left( Y - \frac{v}{V} N \right) , &\qquad M' &= \beta \left( M + \frac{v}{V} Z \right) , \\ Z' &= \beta \left( Z + \frac{v}{V} M \right) , &\qquad N' &= \beta \left( N - \frac{v}{V} Y \right) . \end{aligned}Electric field, moving systemElectric field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, moving systemLorentz factorElectric field, stationary systemSpeed of the moving systemSpeed of lightMagnetic field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, stationary systemElectric field, moving systemMagnetic field, moving systemExplain this equation For the interpretation of these equations we remark the following. Let there be a point-like quantity of electricity which, measured in the stationary system KK, is of magnitude “one”, that is, which, at rest in the stationary system, exerts the force of 1 dyne on an equal quantity of electricity at a distance of 1 cm. By the principle of relativity this electric mass is also of magnitude “one” when measured in the moving system. If this quantity of electricity is at rest relative to the stationary system, then by definition the vector (X,Y,Z)(X, Y, Z) is equal to the force acting on it. If the quantity of electricity is at rest relative to the moving system (at least at the instant in question), then the force acting on it, measured in the moving system, is equal to the vector (X′,Y′,Z′)(X', Y', Z'). Consequently the first three of the above equations can be put into words in the following two ways:

1. If a point-like electric unit pole moves in an electromagnetic field, there acts on it, besides the electric force, an “electromotive force”, which, neglecting terms multiplied by the second and higher powers of v/Vv/V, is equal to the vector product of the velocity of motion of the unit pole and the magnetic force, divided by the velocity of light. (Old mode of expression.)

2. If a point-like electric unit pole moves in an electromagnetic field, the force acting on it is equal to the electric force present at the location of the unit pole, which is obtained by transforming the field to a coordinate system at rest relative to the electric unit pole. (New mode of expression.)

Something analogous holds for the “magnetomotive forces”. One sees that in the theory developed here the electromotive force plays only the part of an auxiliary concept, which owes its introduction to the circumstance that the electric and magnetic forces have no existence independent of the state of motion of the coordinate system.

It is further clear that the asymmetry mentioned in the introduction, which arises when we consider the currents produced by the relative motion of a magnet and a conductor, disappears. The questions about the “seat” of electrodynamic electromotive forces (unipolar machines) also become pointless.

§ 7. Theory of Doppler’s Principle and of Aberration.

In the system KK, very far from the coordinate origin, let there be a source of electrodynamic waves which, in a part of space containing the coordinate origin, may be represented to a sufficient approximation by the equations: X=X0sin⁡Φ,L=L0sin⁡Φ,Y=Y0sin⁡Φ,M=M0sin⁡Φ,Φ=ω(t−ax+by+czV).Z=Z0sin⁡Φ,N=N0sin⁡Φ,\begin{aligned} X &= X_0 \sin \Phi , &\quad L &= L_0 \sin \Phi , \\ Y &= Y_0 \sin \Phi , &\quad M &= M_0 \sin \Phi , &\quad \Phi &= \omega \left( t - \frac{a x + b y + c z}{V} \right) . \\ Z &= Z_0 \sin \Phi , &\quad N &= N_0 \sin \Phi , \end{aligned}Electric field, stationary systemElectric field amplitude (stationary system)Wave phaseMagnetic field, stationary systemMagnetic field amplitude (stationary system)Electric field, stationary systemElectric field amplitude (stationary system)Magnetic field, stationary systemMagnetic field amplitude (stationary system)Wave angular frequency (stationary system)Time in the stationary systemDirection cosine (stationary system)Position in the stationary systemDirection cosine (stationary system)Position in the stationary systemDirection cosine (stationary system)Position in the stationary systemSpeed of lightElectric field, stationary systemElectric field amplitude (stationary system)Magnetic field, stationary systemMagnetic field amplitude (stationary system)Explain this equation Here (X0,Y0,Z0)(X_0, Y_0, Z_0) and (L0,M0,N0)(L_0, M_0, N_0) are the vectors that determine the amplitude of the wave train, and aa, bb, cc the direction cosines of the wave normal.

We ask about the nature of these waves when they are examined by an observer at rest in the moving system kk. — By applying the transformation equations for the electric and magnetic forces found in § 6, and the transformation equations for the coordinates and the time found in § 3, we obtain directly: X′=X0sin⁡Φ′,L′=L0sin⁡Φ′,Y′=β(Y0−vVN0)sin⁡Φ′,M′=β(M0+vVZ0)sin⁡Φ′,Z′=β(Z0+vVM0)sin⁡Φ′,N′=β(N0−vVY0)sin⁡Φ′,Φ′=ω′(τ−a′ξ+b′η+c′ζV),\begin{aligned} X' &= X_0 \sin \Phi' , &\qquad L' &= L_0 \sin \Phi' , \\ Y' &= \beta \left( Y_0 - \frac{v}{V} N_0 \right) \sin \Phi' , &\qquad M' &= \beta \left( M_0 + \frac{v}{V} Z_0 \right) \sin \Phi' , \\ Z' &= \beta \left( Z_0 + \frac{v}{V} M_0 \right) \sin \Phi' , &\qquad N' &= \beta \left( N_0 - \frac{v}{V} Y_0 \right) \sin \Phi' , \\ \Phi' &= \omega' \left( \tau - \frac{a' \xi + b' \eta + c' \zeta}{V} \right) , \end{aligned}Electric field, moving systemElectric field amplitude (stationary system)Wave phaseMagnetic field, moving systemMagnetic field amplitude (stationary system)Electric field, moving systemLorentz factorElectric field amplitude (stationary system)Speed of the moving systemSpeed of lightMagnetic field amplitude (stationary system)Magnetic field, moving systemMagnetic field amplitude (stationary system)Electric field amplitude (stationary system)Electric field, moving systemMagnetic field, moving systemWave angular frequency (moving system)Time in the moving systemDirection cosine (moving system)Position in the moving systemDirection cosine (moving system)Position in the moving systemDirection cosine (moving system)Position in the moving systemExplain this equation where ω′=ωβ(1−avV),\omega' = \omega \beta \left( 1 - a \frac{v}{V} \right) ,Wave angular frequency (moving system)Wave angular frequency (stationary system)Lorentz factorDirection cosine (stationary system)Speed of the moving systemSpeed of lightExplain this equation a′=a−vV1−avV,a' = \frac{a - \frac{v}{V}}{1 - a \frac{v}{V}} ,Direction cosine (moving system)Direction cosine (stationary system)Speed of the moving systemSpeed of lightExplain this equation b′=bβ(1−avV),b' = \frac{b}{\beta \left( 1 - a \frac{v}{V} \right)} ,Direction cosine (moving system)Direction cosine (stationary system)Lorentz factorDirection cosine (stationary system)Speed of the moving systemSpeed of lightExplain this equation c′=cβ(1−avV)c' = \frac{c}{\beta \left( 1 - a \frac{v}{V} \right)}Direction cosine (moving system)Direction cosine (stationary system)Lorentz factorDirection cosine (stationary system)Speed of the moving systemSpeed of lightExplain this equation have been set.

From the equation for ω′\omega' it follows: If an observer moves with the velocity vv relative to an infinitely distant light source of frequency ν\nu in such a way that the line joining “light source–observer” makes the angle φ\varphi with the velocity of the observer, referred to a coordinate system at rest relative to the light source, then the frequency ν′\nu' of the light perceived by the observer is given by the equation: ν′=ν1−cos⁡φvV1−(vV)2.\nu' = \nu \frac{1 - \cos \varphi \frac{v}{V}}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} .Frequency, moving systemFrequency, stationary systemAngle to the motion, stationary systemSpeed of the moving systemSpeed of lightExplain this equation This is Doppler’s principle for arbitrary velocities. For φ=0\varphi = 0 the equation takes the perspicuous form: ν′=ν1−vV1+vV.\nu' = \nu \sqrt{\frac{1 - \frac{v}{V}}{1 + \frac{v}{V}}} .Frequency, moving systemFrequency, stationary systemSpeed of the moving systemSpeed of lightExplain this equation One sees that — contrary to the usual view — for v=−∞v = -\infty, ν=∞\nu = \infty.

If we call φ′\varphi' the angle between the wave normal (direction of the ray) in the moving system and the line joining “light source–observer”, the equation for a′a' takes the form: cos⁡φ′=cos⁡φ−vV1−vVcos⁡φ.\cos \varphi' = \frac{\cos \varphi - \frac{v}{V}}{1 - \frac{v}{V} \cos \varphi} .Angle to the motion, moving systemAngle to the motion, stationary systemSpeed of the moving systemSpeed of lightExplain this equation This equation expresses the law of aberration in its most general form. If φ=π/2\varphi = \pi/2, the equation takes the simple form: cos⁡φ′=−vV.\cos \varphi' = - \frac{v}{V} .Angle to the motion, moving systemSpeed of the moving systemSpeed of lightExplain this equation

We still have to find the amplitude of the waves as it appears in the moving system. If AA or A′A' denotes the amplitude of the electric or magnetic force measured in the stationary or the moving system respectively, we obtain: A′2=A2(1−vVcos⁡φ)21−(vV)2,A'^2 = A^2 \frac{\left( 1 - \frac{v}{V} \cos \varphi \right)^2}{1 - \left( \frac{v}{V} \right)^2} ,Light amplitude (moving system)Light amplitude (stationary system)Speed of the moving systemSpeed of lightAngle to the motion, stationary systemExplain this equation an equation which for φ=0\varphi = 0 passes over into the simpler one: A′2=A21−vV1+vV.A'^2 = A^2 \frac{1 - \frac{v}{V}}{1 + \frac{v}{V}} .Light amplitude (moving system)Light amplitude (stationary system)Speed of the moving systemSpeed of lightExplain this equation

It follows from the equations developed that, for an observer approaching a light source with the velocity VV, this light source would have to appear infinitely intense.

§ 8. Transformation of the Energy of Light Rays. Theory of the Radiation Pressure Exerted on Perfect Mirrors.

Since A2/8πA^2/8\pi is equal to the light energy per unit volume, we have to regard A′2/8πA'^2/8\pi, by the principle of relativity, as the light energy in the moving system. Hence A′2/A2A'^2/A^2 would be the ratio of the “measured-in-motion” and the “measured-at-rest” energy of a given light complex, if the volume of a light complex were the same measured in KK and measured in kk. This, however, is not the case. If aa, bb, cc are the direction cosines of the wave normal of the light in the stationary system, then through the surface elements of the spherical surface moving with the velocity of light (x−Vat)2+(y−Vbt)2+(z−Vct)2=R2(x - V a t)^2 + (y - V b t)^2 + (z - V c t)^2 = R^2Position in the stationary systemSpeed of lightDirection cosine (stationary system)Time in the stationary systemPosition in the stationary systemDirection cosine (stationary system)Position in the stationary systemDirection cosine (stationary system)Light-sphere radiusExplain this equation no energy passes; we can therefore say that this surface permanently encloses the same light complex. We ask for the quantity of energy which this surface encloses, regarded in the system kk, that is, for the energy of the light complex relative to the system kk.

The spherical surface is — regarded in the moving system — an ellipsoidal surface, which at the time τ=0\tau = 0 has the equation: (βξ−aβvVξ)2+(η−bβvVξ)2+(ζ−cβvVξ)2=R2.\left( \beta \xi - a \beta \frac{v}{V} \xi \right)^2 + \left( \eta - b \beta \frac{v}{V} \xi \right)^2 + \left( \zeta - c \beta \frac{v}{V} \xi \right)^2 = R^2 .Lorentz factorPosition in the moving systemDirection cosine (stationary system)Speed of the moving systemSpeed of lightPosition in the moving systemDirection cosine (stationary system)Position in the moving systemDirection cosine (stationary system)Light-sphere radiusExplain this equation If SS is called the volume of the sphere and S′S' that of this ellipsoid, then, as a simple calculation shows: S′S=1−(vV)21−vVcos⁡φ.\frac{S'}{S} = \frac{\sqrt{1 - \left( \frac{v}{V} \right)^2}}{1 - \frac{v}{V} \cos \varphi} .Volume of the light packet, moving systemVolume of the light packet, stationary systemSpeed of the moving systemSpeed of lightAngle to the motion, stationary systemExplain this equation If, therefore, we call EE the light energy measured in the stationary system and E′E' the light energy measured in the moving system, enclosed by the surface under consideration, we obtain: E′E=A′28πS′A28πS=1−vVcos⁡φ1−(vV)2,\frac{E'}{E} = \frac{\frac{A'^2}{8\pi} S'}{\frac{A^2}{8\pi} S} = \frac{1 - \frac{v}{V} \cos \varphi}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} ,Energy of the light packet, moving systemEnergy of the light packet, stationary systemLight amplitude (moving system)Volume of the light packet, moving systemLight amplitude (stationary system)Volume of the light packet, stationary systemSpeed of the moving systemSpeed of lightAngle to the motion, stationary systemExplain this equation a formula which for φ=0\varphi = 0 passes over into the simpler one: E′E=1−vV1+vV.\frac{E'}{E} = \sqrt{\frac{1 - \frac{v}{V}}{1 + \frac{v}{V}}} .Energy of the light packet, moving systemEnergy of the light packet, stationary systemSpeed of the moving systemSpeed of lightExplain this equation

It is remarkable that the energy and the frequency of a light complex vary with the observer’s state of motion according to the same law.

Now let the coordinate plane ξ=0\xi = 0 be a perfectly reflecting surface, at which the plane waves considered in the last section are reflected. We ask for the light pressure exerted on the reflecting surface, and for the direction, frequency and intensity of the light after reflection.

Let the incident light be defined by the quantities AA, cos⁡φ\cos \varphi, ν\nu (referred to the system KK). Regarded from kk, the corresponding quantities are: A′=A1−vVcos⁡φ1−(vV)2,A' = A \frac{1 - \frac{v}{V} \cos \varphi}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} ,Light amplitude (moving system)Light amplitude (stationary system)Speed of the moving systemSpeed of lightAngle to the motion, stationary systemExplain this equation cos⁡φ′=cos⁡φ−vV1−vVcos⁡φ,\cos \varphi' = \frac{\cos \varphi - \frac{v}{V}}{1 - \frac{v}{V} \cos \varphi} ,Angle to the motion, moving systemAngle to the motion, stationary systemSpeed of the moving systemSpeed of lightExplain this equation ν′=ν1−vVcos⁡φ1−(vV)2.\nu' = \nu \frac{1 - \frac{v}{V} \cos \varphi}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} .Frequency, moving systemFrequency, stationary systemSpeed of the moving systemSpeed of lightAngle to the motion, stationary systemExplain this equation For the reflected light we obtain, when we refer the process to the system kk: A′′=A′,cos⁡φ′′=−cos⁡φ′,ν′′=ν′.\begin{aligned} A'' &= A' , \\ \cos \varphi'' &= - \cos \varphi' , \\ \nu'' &= \nu' . \end{aligned}Reflected light amplitude (moving system)Light amplitude (moving system)Reflected propagation angle (moving system)Angle to the motion, moving systemReflected light frequency (moving system)Frequency, moving systemExplain this equation Finally, by transforming back to the stationary system KK, we obtain for the reflected light: A′′′=A′′1+vVcos⁡φ′′1−(vV)2=A1−2vVcos⁡φ+(vV)21−(vV)2,A''' = A'' \frac{1 + \frac{v}{V} \cos \varphi''}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} = A \frac{1 - 2 \frac{v}{V} \cos \varphi + \left( \frac{v}{V} \right)^2}{1 - \left( \frac{v}{V} \right)^2} ,Reflected light amplitude (stationary system)Reflected light amplitude (moving system)Speed of the moving systemSpeed of lightReflected propagation angle (moving system)Light amplitude (stationary system)Angle to the motion, stationary systemExplain this equation cos⁡φ′′′=cos⁡φ′′+vV1+vVcos⁡φ′′=−(1+(vV)2)cos⁡φ−2vV1−2vVcos⁡φ+(vV)2,\cos \varphi''' = \frac{\cos \varphi'' + \frac{v}{V}}{1 + \frac{v}{V} \cos \varphi''} = - \frac{\left( 1 + \left( \frac{v}{V} \right)^2 \right) \cos \varphi - 2 \frac{v}{V}}{1 - 2 \frac{v}{V} \cos \varphi + \left( \frac{v}{V} \right)^2} ,Reflected propagation angle (stationary system)Reflected propagation angle (moving system)Speed of the moving systemSpeed of lightAngle to the motion, stationary systemExplain this equation ν′′′=ν′′1+vVcos⁡φ′′1−(vV)2=ν1−2vVcos⁡φ+(vV)2(1−vV)2.\nu''' = \nu'' \frac{1 + \frac{v}{V} \cos \varphi''}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} = \nu \frac{1 - 2 \frac{v}{V} \cos \varphi + \left( \frac{v}{V} \right)^2}{\left( 1 - \frac{v}{V} \right)^2} .Reflected frequencyReflected light frequency (moving system)Speed of the moving systemSpeed of lightReflected propagation angle (moving system)Frequency, stationary systemAngle to the motion, stationary systemExplain this equation

The energy (measured in the stationary system) incident on unit area of the mirror per unit time is evidently A2/8π(Vcos⁡φ−v)A^2/8\pi (V \cos \varphi - v). The energy leaving unit area of the mirror in unit time is A′′′2/8π(−Vcos⁡φ′′′+v)A'''^2/8\pi (- V \cos \varphi''' + v). By the energy principle, the difference of these two expressions is the work done by the light pressure in unit time. If we set the latter equal to the product P⋅vP \cdot v, where PP is the light pressure, we obtain: P=2A28π(cos⁡φ−vV)21−(vV)2.P = 2 \frac{A^2}{8\pi} \frac{\left( \cos \varphi - \frac{v}{V} \right)^2}{1 - \left( \frac{v}{V} \right)^2} .Radiation pressure on a mirrorLight amplitude (stationary system)Angle to the motion, stationary systemSpeed of the moving systemSpeed of lightExplain this equation To a first approximation we obtain, in agreement with experience and with other theories P=2A28πcos⁡2φ.P = 2 \frac{A^2}{8\pi} \cos^2 \varphi .Radiation pressure on a mirrorLight amplitude (stationary system)Angle to the motion, stationary systemExplain this equation

By the method employed here, all problems of the optics of moving bodies can be solved. The essential point is that the electric and magnetic force of the light which is influenced by a moving body are transformed to a coordinate system at rest relative to the body. In this way every problem of the optics of moving bodies is reduced to a series of problems of the optics of bodies at rest.

§ 9. Transformation of the Maxwell-Hertz Equations with Consideration of Convection Currents.

We start from the equations: 1V{uxϱ+∂X∂t}=∂N∂y−∂M∂z,1V∂L∂t=∂Y∂z−∂Z∂y,1V{uyϱ+∂Y∂t}=∂L∂z−∂N∂x,1V∂M∂t=∂Z∂x−∂X∂z,1V{uzϱ+∂Z∂t}=∂M∂x−∂L∂y,1V∂N∂t=∂X∂y−∂Y∂x,\begin{aligned} \frac{1}{V} \left\{ u_x \varrho + \frac{\partial X}{\partial t} \right\} &= \frac{\partial N}{\partial y} - \frac{\partial M}{\partial z} , &\qquad \frac{1}{V} \frac{\partial L}{\partial t} &= \frac{\partial Y}{\partial z} - \frac{\partial Z}{\partial y} , \\ \frac{1}{V} \left\{ u_y \varrho + \frac{\partial Y}{\partial t} \right\} &= \frac{\partial L}{\partial z} - \frac{\partial N}{\partial x} , &\qquad \frac{1}{V} \frac{\partial M}{\partial t} &= \frac{\partial Z}{\partial x} - \frac{\partial X}{\partial z} , \\ \frac{1}{V} \left\{ u_z \varrho + \frac{\partial Z}{\partial t} \right\} &= \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} , &\qquad \frac{1}{V} \frac{\partial N}{\partial t} &= \frac{\partial X}{\partial y} - \frac{\partial Y}{\partial x} , \end{aligned}Speed of lightCharge velocity (stationary system)Charge density, stationary systemElectric field, stationary systemTime in the stationary systemMagnetic field, stationary systemPosition in the stationary systemMagnetic field, stationary systemPosition in the stationary systemMagnetic field, stationary systemElectric field, stationary systemElectric field, stationary systemCharge velocity (stationary system)Position in the stationary systemCharge velocity (stationary system)Explain this equation where ϱ=∂X∂x+∂Y∂y+∂Z∂z\varrho = \frac{\partial X}{\partial x} + \frac{\partial Y}{\partial y} + \frac{\partial Z}{\partial z}Charge density, stationary systemElectric field, stationary systemPosition in the stationary systemElectric field, stationary systemPosition in the stationary systemElectric field, stationary systemPosition in the stationary systemExplain this equation denotes 4π4\pi times the density of electricity, and (ux,uy,uz)(u_x, u_y, u_z) the velocity vector of electricity. If we imagine the electric masses to be invariably bound to small rigid bodies (ions, electrons), these equations are the electromagnetic foundation of Lorentz’s electrodynamics and optics of moving bodies.

If these equations, which are to hold in the system KK, are transformed to the system kk with the help of the transformation equations of § 3 and § 6, we obtain the equations: 1V{uξϱ′+∂X′∂τ}=∂N′∂η−∂M′∂ζ,∂L′∂τ=∂Y′∂ζ−∂Z′∂η,1V{uηϱ′+∂Y′∂τ}=∂L′∂ζ−∂N′∂ξ,∂M′∂τ=∂Z′∂ξ−∂X′∂ζ,1V{uζϱ′+∂Z′∂τ}=∂M′∂ξ−∂L′∂η,∂N′∂τ=∂X′∂η−∂Y′∂ξ,\begin{aligned} \frac{1}{V} \left\{ u_\xi \varrho' + \frac{\partial X'}{\partial \tau} \right\} &= \frac{\partial N'}{\partial \eta} - \frac{\partial M'}{\partial \zeta} , &\qquad \frac{\partial L'}{\partial \tau} &= \frac{\partial Y'}{\partial \zeta} - \frac{\partial Z'}{\partial \eta} , \\ \frac{1}{V} \left\{ u_\eta \varrho' + \frac{\partial Y'}{\partial \tau} \right\} &= \frac{\partial L'}{\partial \zeta} - \frac{\partial N'}{\partial \xi} , &\qquad \frac{\partial M'}{\partial \tau} &= \frac{\partial Z'}{\partial \xi} - \frac{\partial X'}{\partial \zeta} , \\ \frac{1}{V} \left\{ u_\zeta \varrho' + \frac{\partial Z'}{\partial \tau} \right\} &= \frac{\partial M'}{\partial \xi} - \frac{\partial L'}{\partial \eta} , &\qquad \frac{\partial N'}{\partial \tau} &= \frac{\partial X'}{\partial \eta} - \frac{\partial Y'}{\partial \xi} , \end{aligned}Speed of lightCharge velocity (moving system)Charge density, moving systemElectric field, moving systemTime in the moving systemMagnetic field, moving systemPosition in the moving systemMagnetic field, moving systemPosition in the moving systemMagnetic field, moving systemElectric field, moving systemElectric field, moving systemCharge velocity (moving system)Position in the moving systemCharge velocity (moving system)Explain this equation where ux−v1−uxvV2=uξ,uyβ(1−uxvV2)=uη,ϱ′=∂X′∂ξ+∂Y′∂η+∂Z′∂ζ=β(1−vuxV2)ϱ.uzβ(1−uxvV2)=uζ.\begin{aligned} \frac{u_x - v}{1 - \frac{u_x v}{V^2}} &= u_\xi , \\ \frac{u_y}{\beta \left( 1 - \frac{u_x v}{V^2} \right)} &= u_\eta , &\qquad \varrho' = \frac{\partial X'}{\partial \xi} + \frac{\partial Y'}{\partial \eta} + \frac{\partial Z'}{\partial \zeta} = \beta \left( 1 - \frac{v u_x}{V^2} \right) \varrho . \\ \frac{u_z}{\beta \left( 1 - \frac{u_x v}{V^2} \right)} &= u_\zeta . \end{aligned}Charge velocity (stationary system)Speed of the moving systemSpeed of lightCharge velocity (moving system)Charge velocity (stationary system)Lorentz factorCharge velocity (moving system)Charge density, moving systemElectric field, moving systemPosition in the moving systemElectric field, moving systemPosition in the moving systemElectric field, moving systemPosition in the moving systemCharge density, stationary systemCharge velocity (stationary system)Charge velocity (moving system)Explain this equation Since — as follows from the addition theorem of velocities (§ 5) — the vector (uξ,uη,uζ)(u_\xi, u_\eta, u_\zeta) is nothing other than the velocity of the electric masses measured in the system kk, it is thereby shown that, on the basis of our kinematical principles, the electrodynamic foundation of Lorentz’s theory of the electrodynamics of moving bodies is in accordance with the principle of relativity.

It may also be briefly remarked that the following important theorem can easily be deduced from the equations developed: If an electrically charged body moves in any manner in space and its charge, regarded from a coordinate system moving with the body, does not change in the process, then its charge also remains — regarded from the “stationary” system KK — constant.

§ 10. Dynamics of the (Slowly Accelerated) Electron.

Let there move in an electromagnetic field a point-like particle provided with an electric charge ε\varepsilon (in what follows called “electron”), about whose law of motion we assume only the following:

If the electron is at rest at a given epoch, its motion in the next instant of time takes place according to the equations μd2xdt2=εXμd2ydt2=εYμd2zdt2=εZ,\begin{aligned} \mu \frac{d^2 x}{d t^2} &= \varepsilon X \\ \mu \frac{d^2 y}{d t^2} &= \varepsilon Y \\ \mu \frac{d^2 z}{d t^2} &= \varepsilon Z , \end{aligned}Electron massPosition in the stationary systemTime in the stationary systemElectron chargeElectric field, stationary systemPosition in the stationary systemElectric field, stationary systemPosition in the stationary systemElectric field, stationary systemExplain this equation where xx, yy, zz denote the coordinates of the electron, and μ\mu the mass of the electron, as long as it moves slowly.

Now, secondly, let the electron have the velocity vv at a certain epoch of time. We seek the law according to which the electron moves in the instant of time immediately following.

Without affecting the generality of the consideration, we may and will assume that the electron, at the moment we fix our attention on it, is at the origin of coordinates and is moving along the XX-axis of the system KK with the velocity vv. It is then evident that at the moment in question (t=0t = 0) the electron is at rest relative to a coordinate system kk moving parallel to itself along the XX-axis with the constant velocity vv.

From the assumption made above, in combination with the principle of relativity, it is clear that in the time immediately following (for small values of tt) the electron, regarded from the system kk, moves according to the equations: μd2ξdτ2=εX′,μd2ηdτ2=εY′,μd2ζdτ2=εZ′,\begin{aligned} \mu \frac{d^2 \xi}{d \tau^2} &= \varepsilon X' , \\ \mu \frac{d^2 \eta}{d \tau^2} &= \varepsilon Y' , \\ \mu \frac{d^2 \zeta}{d \tau^2} &= \varepsilon Z' , \end{aligned}Electron massPosition in the moving systemTime in the moving systemElectron chargeElectric field, moving systemPosition in the moving systemElectric field, moving systemPosition in the moving systemElectric field, moving systemExplain this equation where the symbols ξ\xi, η\eta, ζ\zeta, τ\tau, X′X', Y′Y', Z′Z' refer to the system kk. If we further stipulate that for t=x=y=z=0t = x = y = z = 0 we are to have τ=ξ=η=ζ=0\tau = \xi = \eta = \zeta = 0, then the transformation equations of §§ 3 and 6 hold, so that: τ=β(t−vV2x),ξ=β(x−vt),X′=X,η=y,Y′=β(Y−vVN),ζ=z,Z′=β(Z+vVM).\begin{aligned} \tau &= \beta \left( t - \frac{v}{V^2} x \right) , \\ \xi &= \beta (x - v t) , &\qquad X' &= X , \\ \eta &= y , &\qquad Y' &= \beta \left( Y - \frac{v}{V} N \right) , \\ \zeta &= z , &\qquad Z' &= \beta \left( Z + \frac{v}{V} M \right) . \end{aligned}Time in the moving systemLorentz factorTime in the stationary systemSpeed of the moving systemSpeed of lightPosition in the stationary systemPosition in the moving systemElectric field, moving systemElectric field, stationary systemPosition in the moving systemPosition in the stationary systemElectric field, moving systemElectric field, stationary systemMagnetic field, stationary systemPosition in the moving systemPosition in the stationary systemElectric field, moving systemElectric field, stationary systemMagnetic field, stationary systemExplain this equation

With the help of these equations we transform the above equations of motion from the system kk to the system KK, and obtain: {d2xdt2=εμ1β3X,d2ydt2=εμ1β(Y−vVN),d2zdt2=εμ1β(Z+vVM).\left\{ \begin{aligned} \frac{d^2 x}{d t^2} &= \frac{\varepsilon}{\mu} \frac{1}{\beta^3} X , \\ \frac{d^2 y}{d t^2} &= \frac{\varepsilon}{\mu} \frac{1}{\beta} \left( Y - \frac{v}{V} N \right) , \\ \frac{d^2 z}{d t^2} &= \frac{\varepsilon}{\mu} \frac{1}{\beta} \left( Z + \frac{v}{V} M \right) . \end{aligned} \right.Position in the stationary systemTime in the stationary systemElectron chargeElectron massLorentz factorElectric field, stationary systemPosition in the stationary systemElectric field, stationary systemSpeed of the moving systemSpeed of lightMagnetic field, stationary systemPosition in the stationary systemElectric field, stationary systemMagnetic field, stationary systemExplain this equation

Following the usual line of thought, we now ask for the “longitudinal” and “transverse” mass of the moving electron. We write the equations (A) in the form μβ3d2xdt2=εX=εX′,μβ2d2ydt2=εβ(Y−vVN)=εY′,μβ3d2zdt2=εβ(Z+vVM)=εZ′\begin{aligned} \mu \beta^3 \frac{d^2 x}{d t^2} &= \varepsilon X = \varepsilon X' , \\ \mu \beta^2 \frac{d^2 y}{d t^2} &= \varepsilon \beta \left( Y - \frac{v}{V} N \right) = \varepsilon Y' , \\ \mu \beta^3 \frac{d^2 z}{d t^2} &= \varepsilon \beta \left( Z + \frac{v}{V} M \right) = \varepsilon Z' \end{aligned}Electron massLorentz factorPosition in the stationary systemTime in the stationary systemElectron chargeElectric field, stationary systemElectric field, moving systemPosition in the stationary systemElectric field, stationary systemSpeed of the moving systemSpeed of lightMagnetic field, stationary systemElectric field, moving systemPosition in the stationary systemElectric field, stationary systemMagnetic field, stationary systemElectric field, moving systemExplain this equation and note first that εX′\varepsilon X', εY′\varepsilon Y', εZ′\varepsilon Z' are the components of the ponderomotive force acting on the electron, and indeed regarded in a system moving at this moment with the electron at the same velocity as it. (This force could be measured, for example, with a spring balance at rest in the latter system.) If we now call this force simply “the force acting on the electron”, and maintain the equation [mass number × acceleration number = force number] Massenzahl×Beschleunigungszahl=Kraftzahl\text{Massenzahl} \times \text{Beschleunigungszahl} = \text{Kraftzahl}Explain this equation and if we further stipulate that the accelerations are to be measured in the stationary system KK, we obtain from the above equations: Longitudinale Masse=μ(1−(vV)2)3,\text{Longitudinale Masse} = \frac{\mu}{\left( \sqrt{1 - \left( \frac{v}{V} \right)^2} \right)^3} ,Electron massSpeed of the moving systemSpeed of lightExplain this equation Transversale Masse=μ1−(vV)2.\text{Transversale Masse} = \frac{\mu}{1 - \left( \frac{v}{V} \right)^2} .Electron massSpeed of the moving systemSpeed of lightExplain this equation

Naturally, with a different definition of force and acceleration one would obtain other numbers for the masses; one sees from this that one must proceed very cautiously in comparing different theories of the motion of the electron.

We remark that these results about the mass also hold for ponderable material points; for a ponderable material point can be made into an electron (in our sense) by adding an arbitrarily small electric charge.

We determine the kinetic energy of the electron. If an electron moves from the origin of coordinates of the system KK, with initial velocity 0, steadily along the XX-axis under the action of an electrostatic force XX, it is clear that the energy withdrawn from the electrostatic field has the value ∫εXdx\int \varepsilon X d x. Since the electron is to be accelerated slowly and consequently is to give off no energy in the form of radiation, the energy withdrawn from the electrostatic field must be set equal to the energy of motion WW of the electron. Taking into account that during the whole process of motion considered the first of the equations (A) holds, we therefore obtain: W=∫εXdx=∫0vβ3vdv=μV2{11−(vV)2−1}.W = \int \varepsilon X d x = \int_0^v \beta^3 v d v = \mu V^2 \left\{ \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} - 1 \right\} .Kinetic energyElectron chargeElectric field, stationary systemPosition in the stationary systemSpeed of the moving systemLorentz factorElectron massSpeed of lightExplain this equation

WW thus becomes infinitely great for v=Vv = V. Velocities greater than that of light have — as in our earlier results — no possibility of existence.

By the argument given above, this expression for the kinetic energy must also hold for ponderable masses.

We shall now enumerate the properties of the electron’s motion that result from the system of equations (A) and are accessible to experiment.

1. From the second equation of the system (A) it follows that an electric force YY and a magnetic force NN have an equally strong deflecting action on an electron moving with the velocity vv when Y=N⋅v/VY = N \cdot v/V. One sees, therefore, that according to our theory the velocity of the electron can be determined, for arbitrary velocities, from the ratio of the magnetic deflectability AmA_m to the electric deflectability AeA_e, by applying the law: AmAe=vV.\frac{A_m}{A_e} = \frac{v}{V} .Magnetic deflectabilityElectric deflectabilitySpeed of the moving systemSpeed of lightExplain this equation

This relation is accessible to experimental test, since the velocity of the electron can also be measured directly, for example by means of rapidly oscillating electric and magnetic fields.

2. From the derivation of the kinetic energy of the electron it follows that between the potential difference traversed and the velocity vv acquired by the electron the relation must hold: P=∫Xdx=μεV2{11−(vV)2−1}.P = \int X d x = \frac{\mu}{\varepsilon} V^2 \left\{ \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} - 1 \right\} .Accelerating voltageElectric field, stationary systemPosition in the stationary systemElectron massElectron chargeSpeed of lightSpeed of the moving systemExplain this equation

3. We calculate the radius of curvature RR of the path when a magnetic force NN acting perpendicular to the velocity of the electron is present (as the only deflecting force). From the second of the equations (A) we obtain: −d2ydt2=v2R=εμvVN⋅1−(vV)2- \frac{d^2 y}{d t^2} = \frac{v^2}{R} = \frac{\varepsilon}{\mu} \frac{v}{V} N \cdot \sqrt{1 - \left( \frac{v}{V} \right)^2}Position in the stationary systemTime in the stationary systemSpeed of the moving systemRadius of the circular pathElectron chargeElectron massSpeed of lightMagnetic field, stationary systemExplain this equation or R=V2με⋅vV1−(vV)2⋅1N.R = V^2 \frac{\mu}{\varepsilon} \cdot \frac{\frac{v}{V}}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} \cdot \frac{1}{N} .Radius of the circular pathSpeed of lightElectron massElectron chargeSpeed of the moving systemMagnetic field, stationary systemExplain this equation

These three relations are a complete expression of the laws according to which, by the present theory, the electron must move.

In conclusion I remark that in working on the problem treated here my friend and colleague M. Besso stood loyally by my side, and that I owe him several valuable suggestions.

Bern, June 1905.

(Received 30 June 1905.)