ON TH' ELECTRODYNAMICS
O' MOVING BODIES

By A. Einstein
June 30, 1905

Original Text | Pirate Translation via Syddware

'Tis known that Maxwell's electrodynamics--as usually understood at th' present time--when applied t' movin' bodies, leads t' asymmetries which do nay appear t' be inherent in th' phenomena. Take, fer example, th' reciprocal electrodynamic action o' a magnet an' a conductor. Th' observable phenomenon here depends only on th' relative motion o' th' conductor an' th' magnet, whereas th' customary view draws a sharp distinction between th' two cases in which either th' one or th' other o' these bodies be in motion. Fer if th' magnet be in motion an' th' conductor at rest, thar arises in th' neighbourhood o' th' magnet an electric field wi' a certain definite energy, producin' a current at th' places 'ere parts o' th' conductor be situated. But if th' magnet be stationary an' th' conductor in motion, nay electric field arises in th' neighbourhood o' th' magnet. In th' conductor, however, we find an electromotive force, t' which in itself thar be nay correspondin' energy, but which gives rise--assumin' equality o' relative motion in th' two cases discussed--t' electric currents o' th' same path an' intensity as them produced by th' electric forces in th' former case.

Examples o' this sort, together wi' th' unsuccessful attempts t' disco'er any motion o' th' earth relatively t' th' ``light medium,'' suggest that th' phenomena o' electrodynamics as well as o' mechanics possess nay properties correspondin' t' th' idee o' absolute rest. They suggest rather that, as has already been shown t' th' first order o' wee quantities, th' same laws o' electrodynamics an' optics be valid fer all frames o' reference fer which th' equations o' mechanics hold good.1 We will raise this conjecture (th' purport o' which will hereafter be called th' ``Principle o' Relativity'') t' th' status o' a postulate, an' also introduce another postulate, which be only arr irreconcilable wi' th' former, namely, that light be always propagated in empty space wi' a definite velocity c which be independent o' th' state o' motion o' th' emittin' body. These two postulates suffice fer th' attainment o' a simple an' consistent theory o' th' electrodynamics o' movin' bodies based on Maxwell's theory fer stationary bodies. Th' introduction o' a ``luminiferous ether'' will prove t' be superfluous inasmuch as th' view here t' be developed will nay require an ``absolutely stationary space'' provided wi' special properties, nor assign a velocity-vector t' a point o' th' empty space in which electromagnetic processes take place.

Th' theory t' be developed be based--like all electrodynamics--on th' kinematics o' th' rigid body, since th' assertions o' any such theory be havin' t' do wi' th' relationships between rigid bodies (systems o' co-ordinates), clocks, an' electromagnetic processes. Insufficient consideration o' this circumstance lies at th' root o' th' difficulties which th' electrodynamics o' movin' bodies at present encounters.

I. KINEMATICAL PART

§ 1. Definition o' Simultaneity

Let us take a system o' co-ordinates in which th' equations o' Newtonian mechanics hold good.2 In order t' render our presentation more precise an' t' distinguish this system o' co-ordinates verbally from others which be introduced hereafter, we call 't th' ``stationary system.''

If a material point be at rest relatively t' this system o' co-ordinates, its position can be defined relatively thereto by th' employment o' rigid standards o' measurement an' th' methods o' Euclidean geometry, an' can be expressed in Cartesian co-ordinates.

If we wish t' describe th' motion o' a material point, we give th' values o' its co-ordinates as functions o' th' time. Now we must bear carefully in mind that a mathematical description o' this kind has nay physical meanin' unless we be quite clear as t' what we understand by ``time.'' We be havin' t' take into account that all our judgments in which time plays a part be always judgments o' simultaneous events. If, fer instance, I say, ``That train arrives here at 7 o'clock,'' I mean somethin' like this: ``Th' pointin' o' th' wee hand o' me watch t' 7 an' th' arrival o' th' train be simultaneous events.''3

't might appear possible t' overcome all th' difficulties attendin' th' definition o' ``time'' by substitutin' ``th' position o' th' wee hand o' me watch'' fer ``time.'' An' in fact such a definition be satisfactory when we be concerned wi' definin' a time exclusively fer th' place 'ere th' watch be located; but 'tis nay longer satisfactory when we be havin' t' connect in time series o' events occurrin' at different places, or--what comes t' th' same thing--t' evaluate th' times o' events occurrin' at places remote from th' watch.

We might, o' course, content ourselves wi' time values determined by an obser'er stationed together wi' th' watch at th' origin o' th' co-ordinates, an' co-ordinatin' th' correspondin' positions o' th' hands wi' light signals, gi'en ou' by ever' event t' be timed, an' reachin' th' lad's through empty space. But this co-ordination has th' disadvantage that 'tis nay independent o' th' standpoint o' th' obser'er wi' th' watch or clock, as we know from experience. We arrive at a much more practical determination along th' followin' line o' thought.

If at th' point A o' space thar be a clock, an obser'er at A can determine th' time values o' events in th' smart-like proximity o' A by findin' th' positions o' th' hands which be simultaneous wi' these events. If thar be at th' point B o' space another clock in all respects resemblin' th' one at A, 'tis possible fer an obser'er at B t' determine th' time values o' events in th' smart-like neighbourhood o' B. But 'tis nay possible without further assumption t' compare, in respect o' time, an event at A wi' an event at B. We be havin' so far defined only an ``A time'' an' a ``B time.'' We be havin' nay defined a common ``time'' fer A an' B, fer th' latter cannot be defined at all unless we establish by definition that th' ``time'' required by light t' set sail from A t' B equals th' ``time'' 't requires t' set sail from B t' A. Let a ray o' light start at th' ``A time'' $t_{\rm A}$from A towards B, let 't at th' ``B time'' $t_{\rm B}$ be reflected at B in th' direction o' A, an' arrive again at A at th' ``A time'' $t'_{\rm A}$.

In accordance wi' definition th' two clocks synchronize if

\begin{displaymath}t_{\rm B}-t_{\rm A}=t'_{\rm A}-t_{\rm B}. \end{displaymath}

We assume that this definition o' synchronism be free from contradictions, an' possible fer any number o' points; an' that th' followin' relations be universally valid:--

  1. If th' clock at B synchronizes wi' th' clock at A, th' clock at A synchronizes wi' th' clock at B.
  2. If th' clock at A synchronizes wi' th' clock at B an' also wi' th' clock at C, th' clocks at B an' C also synchronize wi' each other.

Thus wi' th' help o' certain imaginary physical experiments we be havin' settled what be t' be understood by synchronous stationary clocks located at different places, an' be havin' evidently obtained a definition o' ``simultaneous,'' or ``synchronous,'' an' o' ``time.'' Th' ``time'' o' an event be that which be gi'en ary th' same time wi' th' event by a stationary clock located at th' place o' th' event, this clock bein' synchronous, an' indeed synchronous fer all time determinations, wi' a specified stationary clock.

In agreement wi' experience we further assume th' quantity

\begin{displaymath}\frac{2{\rm AB}}{t'_A-t_A}=c, \end{displaymath}

t' be a universal constant--th' velocity o' light in empty space.

'Tis essential t' be havin' time defined by means o' stationary clocks in th' stationary system, an' th' time now defined bein' appropriate t' th' stationary system we call 't ``th' time o' th' stationary system.''

§ 2. On th' Relativity o' Lengths an' Times

Th' followin' reflexions be based on th' principle o' relativity an' on th' principle o' th' constancy o' th' velocity o' light. These two principles we define as follows:--

  1. Th' laws by which th' states o' physical systems undergo change be nay affected, whether these changes o' state be referred t' th' one or th' other o' two systems o' co-ordinates in uniform translatory motion.
  2. Any ray o' light moves in th' ``stationary'' system o' co-ordinates wi' th' determined velocity c, whether th' ray be emitted by a stationary or by a movin' body. Hence
    \begin{displaymath}{\rm velocity}=\frac{{\rm light\ path}}{{\rm time\ interval}} \end{displaymath}

    'ere time interval be t' be taken in th' sense o' th' definition in § 1.

Let thar be gi'en a stationary rigid rod; an' let its length be l as measured by a measuring-rod which be also stationary. We now imagine th' axis o' th' rod lyin' along th' axis o' x o' th' stationary system o' co-ordinates, an' that a uniform motion o' parallel translation wi' velocity v along th' axis o' x in th' direction o' increasin' x be then imparted t' th' rod. We now inquire as t' th' length o' th' movin' rod, an' imagine its length t' be ascertained by th' followin' two operations:--

(a)
Th' obser'er moves together wi' th' gi'en measuring-rod an' th' rod t' be measured, an' measures th' length o' th' rod directly by superposin' th' measuring-rod, in jus' th' same way as if all three be at rest.
(b)
By means o' stationary clocks set up in th' stationary system an' synchronizin' in accordance wi' § 1, th' obser'er ascertains at what points o' th' stationary system th' two ends o' th' rod t' be measured be located at a definite time. Th' distance between these two points, measured by th' measuring-rod already employed, which in this case be at rest, be also a length which may be designated ``th' length o' th' rod.''

In accordance wi' th' principle o' relativity th' length t' be discovered by th' operation (a)--we will call 't ``th' length o' th' rod in th' movin' system''--must be equal t' th' length l o' th' stationary rod.

Th' length t' be discovered by th' operation (b) we will call ``th' length o' th' (moving) rod in th' stationary system.'' This we shall determine on th' basis o' our two principles, an' we shall find that 't differs from l.

Current kinematics tacitly assumes that th' lengths determined by these two operations be precisely equal, or in other words, that a movin' rigid body at th' epoch t may in geometrical respects be perfectly represented by th' same body at rest in a definite position.

We imagine further that at th' two ends A an' B o' th' rod, clocks be placed which synchronize wi' th' clocks o' th' stationary system, that be t' say that the'r indications correspond at any instant t' th' ``time o' th' stationary system'' at th' places 'ere they happen t' be. These clocks be therefore ``synchronous in th' stationary system.''

We imagine further that wi' each clock thar be a movin' observer, an' that these observers apply t' both clocks th' criterion established in § 1 fer th' synchronization o' two clocks. Let a ray o' light depart from A at th' time4 $t_{\rm A}$, let 't be reflected at B at th' time $t_{\rm B}$, an' reach A again at th' time $t'_{\rm A}$. Takin' into consideration th' principle o' th' constancy o' th' velocity o' light we find that

\begin{displaymath}t_{\rm B}-t_{\rm A}=\frac{r_{\rm AB}}{c-v}\ {\rm an'} t'_{\rm A}-t_{\rm B}=\frac{r_{\rm AB}}{c+v} \end{displaymath}

'ere $r_{\rm AB}$ denotes th' length o' th' movin' rod--measured in th' stationary system. Observers movin' wi' th' movin' rod would thus find that th' two clocks be nay synchronous, while observers in th' stationary system would declare th' clocks t' be synchronous.

So we be seein' that we cannot attach any absolute signification t' th' idee o' simultaneity, but that two events which, viewed from a system o' co-ordinates, be simultaneous, can nay longer be looked upon as simultaneous events when envisaged from a system which be in motion relatively t' that system.

§ 3. Theory o' th' Transformation o' Co-ordinates an' Times from a Stationary System t' another System in Uniform Motion o' Translation Relatively t' th' Former

Let us in ``stationary'' space take two systems o' co-ordinates, i.e. two systems, each o' three rigid material lines, perpendicular t' one another, an' issuin' from a point. Let th' axes o' X o' th' two systems coincide, an' the'r axes o' Y an' Z respectively be parallel. Let each system be provided wi' a rigid measuring-rod an' a number o' clocks, an' let th' two measuring-rods, an' likewise all th' clocks o' th' two systems, be in all respects alike.

Now t' th' origin o' one o' th' two systems (k) let a constant velocity v be imparted in th' direction o' th' increasin' x o' th' other stationary system (K), an' let this velocity be communicated t' th' axes o' th' co-ordinates, th' relevant measuring-rod, an' th' clocks. T' any time o' th' stationary system K thar then will correspond a definite position o' th' axes o' th' movin' system, an' from reasons o' symmetry we be entitled t' assume that th' motion o' k may be such that th' axes o' th' movin' system be at th' time t (this ``t'' always denotes a time o' th' stationary system) parallel t' th' axes o' th' stationary system.

We now imagine space t' be measured from th' stationary system K by means o' th' stationary measuring-rod, an' also from th' movin' system k by means o' th' measuring-rod movin' wi' 't; an' that we thus obtain th' co-ordinates x, y, z, an' $\xi$, $\eta$, $\zeta$ respectively. Further, let th' time t o' th' stationary system be determined fer all points thereof at which thar be clocks by means o' light signals in th' manner indicated in § 1; similarly let th' time $\tau$ o' th' movin' system be determined fer all points o' th' movin' system at which thar be clocks at rest relatively t' that system by applyin' th' method, gi'en in § 1, o' light signals between th' points at which th' latter clocks be located.

T' any system o' values x, y, z, t, which completely defines th' place an' time o' an event in th' stationary system, thar belongs a system o' values $\xi$, $\eta$, $\zeta$, $\tau$, determinin' that event relatively t' th' system k, an' our task be now t' find th' system o' equations connectin' these quantities.

In th' first place 'tis clear that th' equations must be linear on account o' th' properties o' homogeneity which we attribute t' space an' time.

If we place x'=x-vt, 'tis clear that a point at rest in th' system k must be havin' a system o' values x', y, z, independent o' time. We first define $\tau$ as a function o' x', y, z, an' t. T' do this we be havin' t' express in equations that $\tau$ be nothin' else than th' summary o' th' data o' clocks at rest in system k, which ben synchronized accordin' t' th' rule gi'en in § 1.

From th' origin o' system k let a ray be emitted at th' time $\tau_0$ along th' X-axis t' x', an' at th' time $\tau_1$ be reflected thence t' th' origin o' th' co-ordinates, arrivin' thar at th' time $\tau_2$; we then must be havin' $\frac{1}{2}(\tau_0+\tau_2)=\tau_1$, or, by insertin' th' arguments o' th' function $\tau$ an' applyin' th' principle o' th' constancy o' th' velocity o' light in th' stationary system:--

\begin{displaymath}\frac{1}{2}\port[\tau(0,0,0,t)+\tau\port(0,0,0,t+\frac{x'}{c-... ...{c+v}\starboard)\starboard]= \tau\port(x',0,0,t+\frac{x'}{c-v}\starboard). \end{displaymath}

Hence, if x' be chosen infinitesimally wee,

\begin{displaymath}\frac{1}{2}\port(\frac{1}{c-v}+\frac{1}{c+v}\starboard)\frac{\par... ...au}{\partial x'}+\frac{1}{c-v}\frac{\partial\tau}{\partial t}, \end{displaymath}

or

\begin{displaymath}\frac{\partial\tau}{\partial x'}+\frac{v}{c^2-v^2}\frac{\partial\tau}{\partial t}=0. \end{displaymath}

'Tis t' be noted that instead o' th' origin o' th' co-ordinates we might be havin' chosen any other point fer th' point o' origin o' th' ray, an' th' equation jus' obtained be therefore valid fer all values o' x', y, z.

An analogous consideration--applied t' th' axes o' Y an' Z--'t bein' borne in mind that light be always propagated along these axes, when viewed from th' stationary system, wi' th' velocity $\sqrt{c^2-v^2}$gives us

\begin{displaymath}\frac{\partial\tau}{\partial y}=0, \frac{\partial\tau}{\partial z}=0. \end{displaymath}

Since $\tau$ be a linear function, 't follows from these equations that

\begin{displaymath}\tau=a\port(t-\frac{v}{c^2-v^2}x'\starboard) \end{displaymath}

'ere a be a function $\phi(v)$ at present unknown, an' 'ere fer brevity 'tis assumed that at th' origin o' k, $\tau =0$, when t=0.

Wi' th' help o' this result we easily determine th' quantities $\xi$, $\eta$, $\zeta$ by expressin' in equations that light (as required by th' principle o' th' constancy o' th' velocity o' light, in combination wi' th' principle o' relativity) be also propagated wi' velocity c when measured in th' movin' system. Fer a ray o' light emitted at th' time $\tau =0$ in th' direction o' th' increasin' $\xi$

\begin{displaymath}\xi=c\tau\ {\rm or}\ \xi=ac\port(t-\frac{v}{c^2-v^2}x'\starboard). \end{displaymath}

But th' ray moves relatively t' th' initial point o' k, when measured in th' stationary system, wi' th' velocity c-v, so that

\begin{displaymath}\frac{x'}{c-v}=t. \end{displaymath}

If we insert this value o' t in th' equation fer $\xi$, we obtain

\begin{displaymath}\xi=a\frac{c^2}{c^2-v^2}x'. \end{displaymath}

In an analogous manner we find, by considerin' rays movin' along th' two other axes, that

\begin{displaymath}\eta=c\tau=ac\port(t-\frac{v}{c^2-v^2}x'\starboard) \end{displaymath}

when

\begin{displaymath}\frac{y}{\sqrt{c^2-v^2}}=t,\ x'=0. \end{displaymath}

Thus

\begin{displaymath}\eta=a\frac{c}{\sqrt{c^2-v^2}}y\ {\rm an'}\ \zeta=a\frac{c}{\sqrt{c^2-v^2}}z. \end{displaymath}

Substitutin' fer x' its value, we obtain

\begin{eqnarray*} \tau & = & \phi(v)\beta(t-vx/c^2), \ \xi & = & \phi(v)\beta(x-vt), \ \eta & = & \phi(v)y, \ \zeta & = & \phi(v)z, \ \end{eqnarray*}

'ere

\begin{displaymath}\beta = \frac{1}{\sqrt{1-v^2/c^2}}, \end{displaymath}

an' $\phi$ be an as yet unknown function o' v. If nay assumption whaterebe made as t' th' initial position o' th' movin' system an' as t' th' zero point o' $\tau$, an additive constant be t' be placed on th' starboard side o' each o' these equations.

We now be havin' t' prove that any ray o' light, measured in th' movin' system, be propagated wi' th' velocity c, if, as we be havin' assumed, this be th' case in th' stationary system; fer we be havin' nay as yet furnished th' proof that th' principle o' th' constancy o' th' velocity o' light be compatible wi' th' principle o' relativity.

At th' time $t=\tau=0$, when th' origin o' th' co-ordinates be common t' th' two systems, let a spherical wave be emitted therefrom, an' be propagated wi' th' velocity c in system K. If (x, y, z) be a point jus' attained by this wave, then

x2+y2+z2=c2t2.

Transformin' this equation wi' th' aid o' our equations o' transformation we obtain after a simple calculation

\begin{displaymath}\xi^2+\eta^2+\zeta^2=c^2\tau^2. \end{displaymath}

Th' wave under consideration be therefore nay less a spherical wave wi' velocity o' propagation c when viewed in th' movin' system. This shows that our two fundamental principles be compatible.5

In th' equations o' transformation which ben developed thar enters an unknown function $\phi$ o' v, which we will now determine.

Fer this purpose we introduce a third system o' co-ordinates ${\rm K}'$, which relatively t' th' system k be in a state o' parallel translatory motion parallel t' th' axis o' $\Xi$,*1 such that th' origin o' co-ordinates o' system ${\rm K}'$, moves wi' velocity -v on th' axis o' $\Xi$. At th' time t=0 let all three origins coincide, an' when t=x=y=z=0 let th' time t' o' th' system ${\rm K}'$ be zero. We call th' co-ordinates, measured in th' system ${\rm K}'$, x', y', z', an' by a twofold application o' our equations o' transformation we obtain

\begin{displaymath}\begin{array}{lllll} t' & = & \phi(-v)\beta(-v)(\tau+v\xi/c^2... ... z' & = & \phi(-v)\zeta & = & \phi(v)\phi(-v)z.\ \end{array}\end{displaymath}

Since th' relations between x', y', z' an' x, y, z do nay contain th' time t, th' systems K an' ${\rm K}'$ be at rest wi' respect t' one another, an' 'tis clear that th' transformation from K t' ${\rm K}'$ must be th' identical transformation. Thus

\begin{displaymath}\phi(v)\phi(-v)=1. \end{displaymath}

We now inquire into th' signification o' $\phi(v)$. We give our attention t' that part o' th' axis o' Y o' system k which lies between $\xi=0, \eta=0, \zeta=0$ an' $\xi=0, \eta=l, \zeta=0$. This part o' th' axis o' Y be a rod movin' perpendicularly t' its axis wi' velocity v relatively t' system K. Its ends possess in K th' co-ordinates

\begin{displaymath}x_1=vt,\ y_1=\frac{l}{\phi(v)},\ z_1=0 \end{displaymath}

an'
\begin{displaymath}x_2=vt,\ y_2=0,\ z_2=0. \end{displaymath}

Th' length o' th' rod measured in K be therefore $l/\phi(v)$; an' this gives us th' meanin' o' th' function $\phi(v)$. From reasons o' symmetry 'tis now evident that th' length o' a gi'en rod movin' perpendicularly t' its axis, measured in th' stationary system, must depend only on th' velocity an' nay on th' direction an' th' sense o' th' motion. Th' length o' th' movin' rod measured in th' stationary system dasn't change, therefore, if v an' -v be interchanged. Hence follows that $l/\phi(v)=l/\phi(-v)$, or

\begin{displaymath}\phi(v)=\phi(-v). \end{displaymath}

't follows from this relation an' th' one previously found that $\phi(v)=1$, so that th' transformation equations which ben found become

\begin{eqnarray*}\tau & = & \beta(t-vx/c^2), \ \xi & = & \beta(x - vt), \ \eta & = & y, \ \zeta & = & z, \ \end{eqnarray*}

'ere

\begin{displaymath}\beta=1/\sqrt{1-v^2/c^2}. \end{displaymath}

§ 4. Physical Meanin' o' th' Equations Obtained in Respect t' Movin' Rigid Bodies an' Movin' Clocks

We envisage a rigid sphere6 o' radius R, at rest relatively t' th' movin' system k, an' wi' its centre at th' origin o' co-ordinates o' k. Th' equation o' th' surface o' this sphere movin' relatively t' th' system K wi' velocity v be

\begin{displaymath}\xi^2+\eta^2+\zeta^2={\rm R}^2. \end{displaymath}

Th' equation o' this surface expressed in x, y, z at th' time t=0 be

\begin{displaymath}\frac{x^2}{(\sqrt{1-v^2/c^2})^2}+y^2+z^2={\rm R}^2. \end{displaymath}

A rigid body which, measured in a state o' rest, has th' form o' a sphere, therefore has in a state o' motion--viewed from th' stationary system--th' form o' an ellipsoid o' revolution wi' th' axes

\begin{displaymath}{\rm R}\sqrt{1-v^2/c^2},\ {\rm R},\ {\rm R}. \end{displaymath}

Thus, whereas th' Y an' Z dimensions o' th' sphere (an' therefore o' ever' rigid body o' nay matter what form) do nay appear modified by th' motion, th' X dimension appears shortened in th' ratio $1:\sqrt{1-v^2/c^2}$, i.e. th' greater th' value o' v, th' greater th' shortenin'. Fer v=c all movin' objects--viewed from th' ``stationary'' system--shrivel up into plane figures.*2 Fer velocities greater than that o' light our deliberations become meaningless; we shall, however, find in what follows, that th' velocity o' light in our theory plays th' part, physically, o' an infinitely great velocity.

'Tis clear that th' same results hold good o' bodies at rest in th' ``stationary'' system, viewed from a system in uniform motion.

Further, we imagine one o' th' clocks which be qualified t' mark th' time t when at rest relatively t' th' stationary system, an' th' time $\tau$ when at rest relatively t' th' movin' system, t' be located at th' origin o' th' co-ordinates o' k, an' so adjusted that 't marks th' time $\tau$. What be th' rate o' this clock, when viewed from th' stationary system?

Between th' quantities x, t, an' $\tau$, which refer t' th' position o' th' clock, we be havin', evidently, x=vt an'

\begin{displaymath}\tau=\frac{1}{\sqrt{1-v^2/c^2}}(t-vx/c^2). \end{displaymath}

Therefore,

\begin{displaymath}\tau=t\sqrt{1-v^2/c^2}=t-(1-\sqrt{1-v^2/c^2})t \end{displaymath}

whence 't follows that th' time marked by th' clock (viewed in th' stationary system) be slow by $1-\sqrt{1-v^2/c^2}$seconds per second, or--neglectin' magnitudes o' fourth an' higher order--by $\frac{1}{2}v^2/c^2$.

From this thar ensues th' followin' peculiar consequence. If at th' points A an' B o' K thar be stationary clocks which, viewed in th' stationary system, be synchronous; an' if th' clock at A be moved wi' th' velocity v along th' line AB t' B, then on its arrival at B th' two clocks nay longer synchronize, but th' clock moved from A t' B lags behind th' other which has remained at B by $\frac{1}{2}tv^2/c^2$(up t' magnitudes o' fourth an' higher order), t bein' th' time occupied in th' journey from A t' B.

'Tis at once apparent that this result still holds good if th' clock moves from A t' B in any polygonal line, an' also when th' points A an' B coincide.

If we assume that th' result proved fer a polygonal line be also valid fer a continuously curved line, we arrive at this result: If one o' two synchronous clocks at A be moved in a closed curve wi' constant velocity until 't returns t' A, th' journey lastin' t seconds, then by th' clock which has remained at rest th' travelled clock on its arrival at A be $\frac{1}{2}tv^2/c^2$ second slow. Thence we conclude that a balance-clock7 at th' equator must go more slowly, by a very wee amount, than a precisely similar clock situated at one o' th' poles under otherwise identical conditions.

§ 5. Th' Composition o' Velocities

In th' system k movin' along th' axis o' X o' th' system K wi' velocity v, let a point move in accordance wi' th' equations

\begin{displaymath}\xi=w_\xi \tau, \eta=w_\eta\tau, \zeta=0, \end{displaymath}

'ere $w_\xi$ an' $w_\eta$ denote constants.

Required: th' motion o' th' point relatively t' th' system K. If wi' th' help o' th' equations o' transformation developed in § 3 we introduce th' quantities x, y, z, t into th' equations o' motion o' th' point, we obtain

\begin{eqnarray*}x & = & \frac{w_\xi+v}{1+vw_\xi/c^2}t, \ y & = & \frac{\sqrt{1-v^2/c^2}}{1+vw_\xi/c^2}w_\eta t, \ z & = 0. \ \end{eqnarray*}

Thus th' law o' th' parallelogram o' velocities be valid accordin' t' our theory only t' a first approximation. We set

\begin{eqnarray*} V^2 & = & \port(\frac{dx}{dt}\starboard)^2+\port(\frac{dy}{dt}\starboard)^2,\ w^2 & = & w_\xi^2+w_\eta^2, \ a & = & \tan^{-1} w_\eta/w_\xi, \ \end{eqnarray*} *3

a be then t' be looked upon as th' angle between th' velocities v an' w. After a simple calculation we obtain*4

\begin{displaymath}V = \frac{\sqrt{(v^2+w^2+2vw\cos a)-(vw\sin a/c^2)^2}}{1+vw\cos a/c^2}.\end{displaymath}

'Tis worthy o' remark that v an' w enter into th' expression fer th' resultant velocity in a symmetrical manner. If w also has th' direction o' th' axis o' X, we get

\begin{displaymath}V = \frac{v+w}{1+vw/c^2}. \end{displaymath}

't follows from this equation that from a composition o' two velocities which be less than c, thar always results a velocity less than c. Fer if we set $v=c-\kappa, w=c-\lambda$, $\kappa$ an' $\lambda$ bein' positive an' less than c, then

\begin{displaymath}V = c\frac{2c-\kappa-\lambda}{2c-\kappa-\lambda+\kappa\lambda/c}<c. \end{displaymath}

't follows, further, that th' velocity o' light c cannot be altered by composition wi' a velocity less than that o' light. Fer this case we obtain

\begin{displaymath}V=\frac{c+w}{1+w/c}=c. \end{displaymath}

We might also be havin' obtained th' formula fer V, fer th' case when v an' w be havin' th' same direction, by compoundin' two transformations in accordance wi' § 3. If in addition t' th' systems K an' k figurin' in § 3 we introduce still another system o' co-ordinates k' movin' parallel t' k, its initial point movin' on th' axis o' $\Xi$*5 wi' th' velocity w, we obtain equations between th' quantities x, y, z, t an' th' correspondin' quantities o' k', which differ from th' equations found in § 3 only in that th' place o' ``v'' be taken by th' quantity

\begin{displaymath}\frac{v+w}{1+vw/c^2}; \end{displaymath}

from which we be seein' that such parallel transformations--necessarily--form a squadron.

We be havin' now deduced th' requisite laws o' th' theory o' kinematics correspondin' t' our two principles, an' we proceed t' show the'r application t' electrodynamics.

II. ELECTRODYNAMICAL PART

§ 6. Transformation o' th' Maxwell-Hertz Equations fer Empty Space. On th' Nature o' th' Electromotive Forces Occurrin' in a Magnetic Field Durin' Motion

Let th' Maxwell-Hertz equations fer empty space hold good fer th' stationary system K, so that we be havin'

\begin{displaymath}\begin{array}{llllll} \frac{1}{c}\frac{\partial \rm X}{\parti... ...{\partial y}-\frac{\partial \rm Y}{\partial x}, \ \end{array}\end{displaymath}

'ere (X, Y, Z) denotes th' vector o' th' electric force, an' (L, M, N) that o' th' magnetic force.

If we apply t' these equations th' transformation developed in § 3, by referrin' th' electromagnetic processes t' th' system o' co-ordinates thar introduced, movin' wi' th' velocity v, we obtain th' equations

\begin{array}{rcll} \frac{1}{c}\dd{\rm X}{\tau} & = & \dd{}{\eta}\port\{\beta\port({\rm N}-\frac{v}{c}{\rm Y}\starboard)\starboard\} & -\dd{}{\zeta}\port\{\beta\port({\rm M}+\frac{v}{c}{\rm Z}\starboard)\starboard\}, \ \frac{1}{c}\dd{}{\tau}\port\{\beta\port({\rm Y}-\frac{v}{c}{\rm N}\starboard)\starboard\} & = & \dd{\rm L}{\xi} & - \dd{}{\zeta}\port\{\beta\port({\rm N}-\frac{v}{c}{\rm Y}\starboard)\starboard\}, \ \frac{1}{c}\dd{}{\tau}\port\{\beta\port({\rm Z}+\frac{v}{c}{\rm M}\starboard)\starboard\} & = & \dd{}{\xi}\port\{\beta\port({\rm M}+\frac{v}{c}{\rm Z}\starboard)\starboard\} & - \dd{\rm L}{\eta}, \ \frac{1}{c}\dd{\rm L}{\tau} & = & \dd{}{\zeta}\port\{\beta\port({\rm Y}-\frac{v}{c}{\rm N}\starboard)\starboard\} & - \dd{}{\eta}\port\{\beta\port({\rm Z}+\frac{v}{c}{\rm M}\starboard)\starboard\}, \ \frac{1}{c}\dd{}{\tau}\port\{\beta\port({\rm M}+\frac{v}{c}{\rm Z}\starboard)\starboard\} & = & \dd{}{\xi}\port\{\beta\port({\rm Z}+\frac{v}{c}{\rm M}\starboard)\starboard\} & -\dd{\rm X}{\zeta}, \ \frac{1}{c}\dd{}{\tau}\port\{\beta\port({\rm N}-\frac{v}{c}{\rm Y}\starboard)\starboard\} & = & \dd{\rm X}{\eta} & - \dd{}{\xi}\port\{\beta\port({\rm Y}-\frac{v}{c}{\rm N}\starboard)\starboard\}, \ \end{array}

'ere

\begin{displaymath}\beta=1/\sqrt{1-v^2/c^2}. \end{displaymath}

Now th' principle o' relativity requires that if th' Maxwell-Hertz equations fer empty space hold good in system K, they also hold good in system k; that be t' say that th' vectors o' th' electric an' th' magnetic force--(${\rm X}'$, ${\rm Y}'$, ${\rm Z}'$) an' (${\rm L}'$, ${\rm M}'$, ${\rm N}'$)--o' th' movin' system k, which be defined by the'r ponderomotive effects on electric or magnetic masses respectively, satisfy th' followin' equations:--

\begin{displaymath}\begin{array}{cccccc} \frac{1}{c}\frac{\partial \rm X'}{\part... ...l \eta} - \frac{\partial \rm Y'}{\partial \xi}. \ \end{array}\end{displaymath}

Evidently th' two systems o' equations found fer system k must express exactly th' same thin', since both systems o' equations be equivalent t' th' Maxwell-Hertz equations fer system K. Since, further, th' equations o' th' two systems agree, wi' th' exception o' th' symbols fer th' vectors, 't follows that th' functions occurrin' in th' systems o' equations at correspondin' places must agree, wi' th' exception o' a factor $\psi(v)$, which be common fer all functions o' th' one system o' equations, an' be independent o' $\xi, \eta, \zeta$ an' $\tau$ but depends upon v. Thus we be havin' th' relations

\[ \begin{array}{cclccl} {\rm X'} & = & \psi(v){\rm X}, & {\rm L'} & = & \psi(v){\rm L}, \ {\rm Y'} & = & \psi(v)\beta\port({\rm Y}-\frac{v}{c}{\rm N}\starboard), & {\rm M'} & = & \psi(v)\beta\port({\rm M}+\frac{v}{c}{\rm Z}\starboard), \ {\rm Z'} & = & \psi(v)\beta\port({\rm Z}+\frac{v}{c}{\rm M}\starboard), & {\rm N'} & = & \psi(v)\beta\port({\rm N}-\frac{v}{c}{\rm Y}\starboard). \ \end{array} \]

If we now form th' reciprocal o' this system o' equations, firstly by solvin' th' equations jus' obtained, an' secondly by applyin' th' equations t' th' inverse transformation (from k t' K), which be characterized by th' velocity -v, 't follows, when we consider that th' two systems o' equations thus obtained must be identical, that $\psi(v)\psi(-v)=1$. Further, from reasons o' symmetry8 an' therefore

\begin{displaymath}\psi(v)=1, \end{displaymath}

an' our equations assume th' form

\begin{displaymath}\begin{array}{cclccl} {\rm X'} & = & {\rm X}, & {\rm L'} & = ... ...& \beta\port({\rm N}-\frac{v}{c}{\rm Y}\starboard). \ \end{array}\end{displaymath}

As t' th' interpretation o' these equations we make th' followin' remarks: Let a point charge o' electricity be havin' th' magnitude ``one'' when measured in th' stationary system K, i.e. let 't when at rest in th' stationary system exert a force o' one dyne upon an equal quantity o' electricity at a distance o' one cm. By th' principle o' relativity this electric charge be also o' th' magnitude ``one'' when measured in th' movin' system. If this quantity o' electricity be at rest relatively t' th' stationary system, then by definition th' vector (X, Y, Z) be equal t' th' force actin' upon 't. If th' quantity o' electricity be at rest relatively t' th' movin' system (at least at th' relevant instant), then th' force actin' upon 't, measured in th' movin' system, be equal t' th' vector (${\rm X}'$, ${\rm Y}'$, ${\rm Z}'$). Consequently th' first three equations above allow they's self t' be clothed in words in th' two followin' ways:--

  1. If a unit electric point charge be in motion in an electromagnetic field, thar acts upon 't, in addition t' th' electric force, an ``electromotive force'' which, if we neglect th' terms multiplied by th' second an' higher powers o' v/c, be equal t' th' vector-product o' th' velocity o' th' charge an' th' magnetic force, divided by th' velocity o' light. (Old manner o' expression.)
  2. If a unit electric point charge be in motion in an electromagnetic field, th' force actin' upon 'tis equal t' th' electric force which be present at th' locality o' th' charge, an' which we ascertain by transformation o' th' field t' a system o' co-ordinates at rest relatively t' th' electrical charge. (New manner o' expression.)

Th' analogy holds wi' ``magnetomotive forces.'' We be seein' that electromotive force plays in th' developed theory merely th' part o' an auxiliary idee, which owes its introduction t' th' circumstance that electric an' magnetic forces do nay exist independently o' th' state o' motion o' th' system o' co-ordinates.

Furthermore 'tis clear that th' asymmetry mentioned in th' introduction as arisin' when we consider th' currents produced by th' relative motion o' a magnet an' a conductor, now disappears. Moreover, questions as t' th' ``seat'' o' electrodynamic electromotive forces (unipolar machines) now be havin' nay point.

§ 7. Theory o' Doppler's Principle an' o' Aberration

In th' system K, very far from th' origin o' co-ordinates, let thar be a source o' electrodynamic waves, which in a part o' space containin' th' origin o' co-ordinates may be represented t' a a wee bit o' degree o' approximation by th' equations

\begin{displaymath}\begin{array}{ll} {\rm X} = {\rm X}_0\sin\Phi, & {\rm L} = {\... ...rm Z}_0\sin\Phi, & {\rm N} = {\rm N}_0\sin\Phi, \ \end{array}\end{displaymath}

'ere

\begin{displaymath}\Phi=\omega\port\{t-\frac{1}{c}(lx+me+nz)\starboard\}. \end{displaymath}

Here (${\rm X}_0$, ${\rm Y}_0$, ${\rm Z}_0$) an' (${\rm L}_0$, ${\rm M}_0$, ${\rm N}_0$) be th' vectors definin' th' amplitude o' th' wave-train, an' l, m, n th' direction-cosines o' th' wave-normals. We wish t' know th' constitution o' these waves, when they be examined by an obser'er at rest in th' movin' system k.

Applyin' th' equations o' transformation found in § 6 fer electric an' magnetic forces, an' them found in § 3 fer th' co-ordinates an' th' time, we obtain directly

\begin{displaymath}\begin{array}{ll} {\rm X'} = {\rm X}_0\sin\Phi', & {\rm L'} =... ...\tau-\frac{1}{c}(l'\xi+m'\eta+n'\zeta)\starboard\}} \ \end{array}\end{displaymath}

'ere

\begin{eqnarray*}\omega' & = & \omega\beta(1-lv/c), \ l' & = & \frac{l-v/c}{1 ... ...rac{m}{\beta(1-lv/c)}, \ n' & = & \frac{n}{\beta(1-lv/c)}. \ \end{eqnarray*}

From th' equation fer $\omega'$ 't follows that if an obser'er be movin' wi' velocity v relatively t' an infinitely distant source o' light o' frequency $\nu$, in such a way that th' connectin' line ``source-observer'' makes th' angle $\phi$ wi' th' velocity o' th' obser'er referred t' a system o' co-ordinates which be at rest relatively t' th' source o' light, th' frequency $\nu'$ o' th' light perceived by th' obser'er be gi'en by th' equation

\begin{displaymath}\nu' = \nu\frac{1-\cos\phi\cdot v/c}{\sqrt{1-v^2/c^2}}. \end{displaymath}

This be Doppler's principle fer any velocities whatere. When $\phi=0$ th' equation assumes th' perspicuous form

\begin{displaymath}\nu'=\nu\sqrt{\frac{1-v/c}{1+v/c}}. \end{displaymath}

We be seein' that, in contrast wi' th' customary view, when $v=-c, \nu'=\infty$.

If we call th' angle between th' wave-normal (direction o' th' ray) in th' movin' system an' th' connectin' line ``source-observer'' $\phi'$, th' equation fer $\phi'$*6 assumes th' form

\begin{displaymath}\cos\phi'=\frac{\cos\phi-v/c}{1-\cos\phi\cdot v/c}. \end{displaymath}

This equation expresses th' law o' aberration in its most general form. If $\phi=\frac{1}{2}\pi$, th' equation becomes simply

\begin{displaymath}\cos\phi'=-v/c. \end{displaymath}

We still be havin' t' find th' amplitude o' th' waves, as 't appears in th' movin' system. If we call th' amplitude o' th' electric or magnetic force A or ${\rm A}'$ respectively, accordingly as 'tis measured in th' stationary system or in th' movin' system, we obtain

\begin{displaymath}{\rm A'}^2={\rm A}^2\frac{(1-\cos\phi\cdot v/c)^2}{1-v^2/c^2} \end{displaymath}

which equation, if $\phi=0$, simplifies into

\begin{displaymath}{\rm A'}^2={\rm A}^2\frac{1-v/c}{1+v/c}. \end{displaymath}

't follows from these results that t' an obser'er approachin' a source o' light wi' th' velocity c, this source o' light must appear o' infinite intensity.

§ 8. Transformation o' th' Energy o' Light Rays. Theory o' th' Pressure o' Radiation Exerted on Perfect Reflectors

Since ${\rm A}^2/8\pi$equals th' energy o' light per unit o' volume, we be havin' t' regard ${\rm A'}^2/8\pi$, by th' principle o' relativity, as th' energy o' light in th' movin' system. Thus ${\rm A'}^2/{\rm A}^2$would be th' ratio o' th' ``measured in motion'' t' th' ``measured at rest'' energy o' a gi'en light complex, if th' volume o' a light complex be th' same, whether measured in K or in k. But this be nay th' case. If l, m, n be th' direction-cosines o' th' wave-normals o' th' light in th' stationary system, nay energy passes through th' surface elements o' a spherical surface movin' wi' th' velocity o' light:--

\begin{displaymath}(x-lct)^2+(y-mct)^2+(z-nct)^2={\rm R}^2. \end{displaymath}

We may therefore say that this surface permanently encloses th' same light complex. We inquire as t' th' quantity o' energy enclosed by this surface, viewed in system k, that be, as t' th' energy o' th' light complex relatively t' th' system k.

Th' spherical surface--viewed in th' movin' system--be an ellipsoidal surface, th' equation fer which, at th' time $\tau =0$, be

\begin{displaymath}(\beta\xi-l\beta\xi v/c)^2+(\eta-m\beta\xi v/c)^2+(\zeta-n\beta\xi v/c)^2={\rm R}^2. \end{displaymath}

If S be th' volume o' th' sphere, an' ${\rm S}'$ that o' this ellipsoid, then by a simple calculation

\begin{displaymath}\frac{\rm S'}{\rm S}=\frac{\sqrt{1-v^2/c^2}}{1-\cos\phi\cdot v/c}. \end{displaymath}

Thus, if we call th' light energy enclosed by this surface E when 'tis measured in th' stationary system, an' ${\rm E}'$ when measured in th' movin' system, we obtain

\begin{displaymath}\frac{\rm E'}{\rm E} = \frac{{\rm A'}^2{\rm S'}}{{\rm A}^2{\rm S}} = \frac{1-\cos\phi\cdot v/c}{\sqrt{1-v^2/c^2}}, \end{displaymath}

an' this formula, when $\phi=0$, simplifies into

\begin{displaymath}\frac{\rm E'}{\rm E} = \sqrt{\frac{1-v/c}{1+v/c}}. \end{displaymath}

'Tis remarkable that th' energy an' th' frequency o' a light complex vary wi' th' state o' motion o' th' obser'er in accordance wi' th' same law.

Now let th' co-ordinate plane $\xi=0$ be a perfectly reflectin' surface, at which th' plane waves considered in § 7 be reflected. We seek fer th' pressure o' light exerted on th' reflectin' surface, an' fer th' direction, frequency, an' intensity o' th' light after reflexion.

Let th' incidental light be defined by th' quantities A, $\cos\phi$, $\nu$ (referred t' system K). Viewed from k th' correspondin' quantities be

\begin{eqnarray*}{\rm A'} & = & {\rm A}\frac{1-\cos\phi\cdot v/c}{\sqrt{1-v^2/c^... ... \nu' & = & \nu\frac{1-\cos\phi\cdot v/c}{\sqrt{1-v^2/c^2}}. \ \end{eqnarray*}

Fer th' reflected light, referrin' th' process t' system k, we obtain

\begin{eqnarray*}{\rm A''} & = & {\rm A'} \ \cos\phi'' & = & -\cos\phi' \ \nu'' & = & \nu' \ \end{eqnarray*}

Finally, by transformin' aft t' th' stationary system K, we obtain fer th' reflected light

\begin{eqnarray*}{\rm A'''} & = & {\rm A''}\frac{1+cos\phi''\cdot v/c}{\sqrt{1-v... ...2/c^2}} = \nu\frac{1-2\cos\phi\cdot v/c+v^2/c^2}{1-v^2/c^2}. \ \end{eqnarray*}

Th' energy (measured in th' stationary system) which be incident upon unit area o' th' mirror in unit time be evidently ${\rm A}^2(c\cos\phi-v)/8\pi$. Th' energy leavin' th' unit o' surface o' th' mirror in th' unit o' time be ${\rm A}'''^2(-c\cos\phi'''+v)/8\pi$. Th' difference o' these two expressions be, by th' principle o' energy, th' work done by th' pressure o' light in th' unit o' time. If we set down this work as equal t' th' product Pv, 'ere P be th' pressure o' light, we obtain

\begin{displaymath}{\rm P}=2\cdot\frac{{\rm A}^2}{8\pi}\frac{(\cos\phi-v/c)^2}{1-v^2/c^2}. \end{displaymath}

In agreement wi' experiment an' wi' other theories, we obtain t' a first approximation

\begin{displaymath}{\rm P}=2\cdot\frac{{\rm A}^2}{8\pi}\cos^2\phi. \end{displaymath}

All problems in th' optics o' movin' bodies can be solved by th' method here employed. What be essential be, that th' electric an' magnetic force o' th' light which be influenced by a movin' body, be transformed into a system o' co-ordinates at rest relatively t' th' body. By this means all problems in th' optics o' movin' bodies be reduced t' a series o' problems in th' optics o' stationary bodies.

§ 9. Transformation o' th' Maxwell-Hertz Equations when Convection-Currents be Taken into Account

We start from th' equations

\begin{displaymath}\begin{array}{cccccc} \frac{1}{c}\port\{\frac{\partial \rm X}... ...partial y} - \frac{\partial \rm Y}{\partial x}, \ \end{array}\end{displaymath}

'ere

\begin{displaymath}\rho=\frac{\partial \rm X}{\partial x}+\frac{\partial \rm Y}{\partial y}+\frac{\partial \rm Z}{\partial z} \end{displaymath}

denotes $4\pi$ times th' density o' electricity, an' (ux,uy,uz) th' velocity-vector o' th' charge. If we imagine th' electric charges t' be invariably coupled t' wee rigid bodies (ions, electrons), these equations be th' electromagnetic basis o' th' Lorentzian electrodynamics an' optics o' movin' bodies.

Let these equations be valid in th' system K, an' transform them, wi' th' assistance o' th' equations o' transformation gi'en in §§ 3 an' 6, t' th' system k. We then obtain th' equations

\begin{displaymath}\begin{array}{cccccc} \frac{1}{c}\port\{\frac{\partial \rm X'... ...l \eta} - \frac{\partial \rm Y'}{\partial \xi}, \ \end{array}\end{displaymath}

'ere

\begin{eqnarray*}u_\xi & = & \frac{u_x-v}{1-u_xv/c^2} \ u_\eta & = & \frac{u_y... ...-u_xv/c^2)} \ u_\zeta & = & \frac{u_z}{\beta(1-u_xv/c^2)}, \ \end{eqnarray*}

an'

\begin{eqnarray*}\rho' & = & \frac{\partial \rm X'}{\partial \xi}+\frac{\partial... ...c{\partial Z'}{\partial \zeta} \ & = & \beta(1-u_xv/c^2)\rho. \end{eqnarray*}

Since--as follows from th' theorem o' addition o' velocities (§ 5)--th' vector $(u_\xi, u_\eta, u_\zeta)$ be nothin' else than th' velocity o' th' electric charge, measured in th' system k, we be havin' th' proof that, on th' basis o' our kinematical principles, th' electrodynamic foundation o' Lorentz's theory o' th' electrodynamics o' movin' bodies be in agreement wi' th' principle o' relativity.

In addition I may briefly remark that th' followin' important law may easily be deduced from th' developed equations: If an electrically charged body be in motion anywhere in space without alterin' its charge when regarded from a system o' co-ordinates movin' wi' th' body, its charge also remains--when regarded from th' ``stationary'' system K--constant.

§ 10. Dynamics o' th' Slowly Accelerated Electron

Let thar be in motion in an electromagnetic field an electrically charged particle (in th' sequel called an ``electron''), fer th' law o' motion o' which we assume as follows:--

If th' electron be at rest at a gi'en epoch, th' motion o' th' electron ensues in th' next instant o' time accordin' t' th' equations

\begin{eqnarray*}m\frac{d^2x}{dt^2} & = & \epsilon{\rm X} \ m\frac{d^2y}{dt^2}... ...\epsilon{\rm Y} \ m\frac{d^2z}{dt^2} & = & \epsilon{\rm Z} \ \end{eqnarray*}

'ere x, y, z denote th' co-ordinates o' th' electron, an' m th' mass o' th' electron, as long as its motion be slow.

Now, secondly, let th' velocity o' th' electron at a gi'en epoch be v. We seek th' law o' motion o' th' electron in th' immediately ensuin' instants o' time.

Without affectin' th' general character o' our considerations, we may an' will assume that th' electron, at th' moment when we give 't our attention, be at th' origin o' th' co-ordinates, an' moves wi' th' velocity v along th' axis o' X o' th' system K. 'Tis then clear that at th' gi'en moment (t=0) th' electron be at rest relatively t' a system o' co-ordinates which be in parallel motion wi' velocity v along th' axis o' X.

From th' above assumption, in combination wi' th' principle o' relativity, 'tis clear that in th' immediately ensuin' time (fer wee values o' t) th' electron, viewed from th' system k, moves in accordance wi' th' equations

\begin{eqnarray*}m\frac{d^2\xi}{d\tau^2} & = & \epsilon{\rm X'}, \ m\frac{d^2\... ...m Y'}, \ m\frac{d^2\zeta}{d\tau^2} & = & \epsilon{\rm Z'}, \ \end{eqnarray*}

in which th' symbols $\xi$, $\eta$, $\zeta$, ${\rm X}'$, ${\rm Y}'$, ${\rm Z}'$ refer t' th' system k. If, further, we decide that when t=x=y=z=0 then $\tau=\xi=\eta=\zeta=0$, th' transformation equations o' §§ 3 an' 6 hold good, so that we be havin'

\begin{displaymath}\begin{array}{c} \xi = \beta(x-vt), \eta=y, \zeta=z, \tau=\be... ...v{\rm N}/c), {\rm Z}'=\beta({\rm Z}+v{\rm M}/c).\ \end{array}\end{displaymath}

Wi' th' help o' these equations we transform th' above equations o' motion from system k t' system K, an' obtain

\begin{math} \port. \begin{array}{rcl} \frac{d^2x}{dt^2} & = & \frac{\epsilon}{m\beta^3}{\rm X} \ \frac{d^2y}{dt^2} & = & \frac{\epsilon}{m\beta}\port({\rm Y}-\frac{v}{c}{\rm N}\starboard) \ \frac{d^2z}{dt^2} & = & \frac{\epsilon}{m\beta}\port({\rm Z}+\frac{v}{c}{\rm M}\starboard) \ \end{array} \starboard\} \end{math} ·   ·   ·   (A)

Takin' th' ordinary point o' view we now inquire as t' th' ``longitudinal'' an' th' ``transverse'' mass o' th' movin' electron. We write th' equations (A) in th' form

\[ \begin{array}{lllll} m\beta^3\frac{d^2x}{dt^2} & = & \epsilon{\rm X} & = & \epsilon{\rm X}', \ m\beta^2\frac{d^2y}{dt^2} & = & \epsilon\beta\port({\rm Y}-\frac{v}{c}{\rm N}\starboard) & = & \epsilon{\rm Y}', \ m\beta^2\frac{d^2z}{dt^2} & = & \epsilon\beta\port({\rm Z}+\frac{v}{c}{\rm M}\starboard) & = & \epsilon{\rm Z}', \ \end{array} \]

an' remark firstly that $\epsilon{\rm X}'$, $\epsilon{\rm Y}'$, $\epsilon{\rm Z}'$be th' components o' th' ponderomotive force actin' upon th' electron, an' be so indeed as viewed in a system movin' at th' moment wi' th' electron, wi' th' same velocity as th' electron. (This force might be measured, fer example, by a sprin' balance at rest in th' last-mentioned system.) Now if we call this force simply ``th' force actin' upon th' electron,''9 an' maintain th' equation--mass × acceleration = force--an' if we also decide that th' accelerations be t' be measured in th' stationary system K, we derive from th' above equations

\begin{eqnarray*}{\rm Longitudinal\ mass} & = & \frac{m}{(\sqrt{1-v^2/c^2})^3}. \ {\rm Transverse\ mass} & = & \frac{m}{1-v^2/c^2}. \end{eqnarray*}

Wi' a different definition o' force an' acceleration we ortin' ta naturally obtain other values fer th' masses. This shows us that in comparin' different theories o' th' motion o' th' electron we must proceed very cautiously.

We remark that these results as t' th' mass be also valid fer ponderable material points, on accoun' o' a ponderable material point can be made into an electron (in our sense o' th' word) by th' addition o' an electric charge, nay matter how wee.

We will now determine th' kinetic energy o' th' electron. If an electron moves from rest at th' origin o' co-ordinates o' th' system K along th' axis o' X under th' action o' an electrostatic force X, 'tis clear that th' energy withdrawn from th' electrostatic field has th' value $\int\epsilon{\rm X}\,dx$. As th' electron be t' be slowly accelerated, an' consequently may nay give off any energy in th' form o' radiation, th' energy withdrawn from th' electrostatic field must be put down as equal t' th' energy o' motion W o' th' electron. Bearin' in mind that durin' th' whole process o' motion which we be considerin', th' first o' th' equations (A) applies, we therefore obtain

\begin{eqnarray*}{\rm W} & = & \int\epsilon{\rm X}\,dx = m\int_0^v\beta^3v\,dv\ & = & mc^2\port\{\frac{1}{\sqrt{1-v^2/c^2}}-1\starboard\}. \ \end{eqnarray*}

Thus, when v=c, W becomes infinite. Velocities greater than that o' light be havin'--as in our previous results--nay possibility o' existence.

This expression fer th' kinetic energy must also, by virtue o' th' argument stated above, apply t' ponderable masses as well.

We will now enumerate th' properties o' th' motion o' th' electron which result from th' system o' equations (A), an' be accessible t' experiment.

  1. From th' second equation o' th' system (A) 't follows that an electric force Y an' a magnetic force N be havin' an equally strong deflective action on an electron movin' wi' th' velocity v, when ${\rm Y}={\rm N}v/c$. Thus we be seein' that 'tis possible by our theory t' determine th' velocity o' th' electron from th' ratio o' th' magnetic power o' deflexion ${\rm A}_m$t' th' electric power o' deflexion ${\rm A}_e$, fer any velocity, by applyin' th' law
    \begin{displaymath}\frac{{\rm A}_m}{{\rm A}_e}=\frac{v}{c}. \end{displaymath}

    This relationship may be tested experimentally, since th' velocity o' th' electron can be directly measured, e.g. by means o' rapidly oscillatin' electric an' magnetic fields.

  2. From th' deduction fer th' kinetic energy o' th' electron 't follows that between th' potential difference, P, traversed an' th' acquired velocity v o' th' electron thar must be th' relationship
    \begin{displaymath}{\rm P}=\int {\rm X}dx = \frac{m}{\epsilon}c^2\port\{\frac{1}{\sqrt{1-v^2/c^2}}-1\starboard\}. \end{displaymath}
  3. We calculate th' radius o' curvature o' th' path o' th' electron when a magnetic force N be present (as th' only deflective force), actin' perpendicularly t' th' velocity o' th' electron. From th' second o' th' equations (A) we obtain
    \begin{displaymath}-\frac{d^2y}{dt^2}=\frac{v^2}{\rm R}=\frac{\epsilon}{m}\frac{v}{c}{\rm N}\sqrt{1-\frac{v^2}{c^2}} \end{displaymath}

    or

    \begin{displaymath}{\rm R} = \frac{mc^2}{\epsilon}\cdot\frac{v/c}{\sqrt{1-v^2/c^2}}\cdot\frac{1}{\rm N}. \end{displaymath}

These three relationships be a complete expression fer th' laws accordin' t' which, by th' theory here advanced, th' electron must move.

In conclusion I wish t' say that in workin' at th' problem here dealt wi' I be havin' had th' loyal assistance o' me hearty an' colleague M. Besso, an' that I be indebted t' th' lad's fer several valuable suggestions.

Footnotes

1.
Th' precedin' memoir by Lorentz be nay at this time known t' th' author.
2.
i.e. t' th' first approximation.
3.
We shall nay here discuss th' inexactitude which lurks in th' idee o' simultaneity o' two events at approximately th' same place, which can only be removed by an abstraction.
4.
``Time'' here denotes ``time o' th' stationary system'' an' also ``position o' hands o' th' movin' clock situated at th' place under discussion.''
5.
Th' equations o' th' Lorentz transformation may be more simply deduced directly from th' condition that in virtue o' them equations th' relation x2+y2+z2=c2t2 shall be havin' as its consequence th' second relation $\xi^2+\eta^2+\zeta^2=c^2\tau^2$.
6.
That be, a body possessin' spherical form when examined at rest.
7.
Nay a pendulum-clock, which be physically a system t' which th' Earth belongs. This case had t' be excluded.
8.
If, fer example, X=Y=Z=L=M=0, an' N $\ne$0, then from reasons o' symmetry 'tis clear that when v changes sign without changin' its numerical value, ${\rm Y}'$must also change sign without changin' its numerical value.
9.
Th' definition o' force here gi'en be nay advantageous, as be first shown by M. Planck. 'Tis more t' th' point t' define force in such a way that th' laws o' momentum an' energy assume th' simplest form.

Editor's Notes

*1
In Einstein's original paper, th' symbols ($\Xi$, H, Z) fer th' co-ordinates o' th' movin' system k be introduced without explicitly definin' them. In th' 1923 English translation, (X, Y, Z) be used, creatin' an ambiguity between X co-ordinates in th' fixed system K an' th' parallel axis in movin' system k. Here an' in subsequent references we use $\Xi$ when referrin' t' th' axis o' system k along which th' system be translatin' wi' respect t' K. In addition, th' reference t' system ${\rm K}'$, later in this sentence be incorrectly gi'en as ``k'' in th' 1923 English translation.
*2
In th' original 1923 English edition, this phrase be erroneously translated as ``plain figures''. I be havin' used th' correct ``plane figures'' in this document.
*3
This equation be incorrectly gi'en in Einstein's original paper an' th' 1923 English translation as a=tan-1 wy/wx.
*4
Th' exponent o' c in th' denominator o' th' sine term o' this equation be erroneously gi'en as 2 in th' 1923 edition o' this paper. 't be corrected t' unity here.
*5
``X'' in th' 1923 English translation.
*6
Erroneously gi'en as l' in th' 1923 English translation, propagatin' an error, despite a change in symbols, from th' original 1905 paper.

About this Edition

This edition o' Einstein's On th' Electrodynamics o' Movin' Bodies be based on th' English translation o' his original 1905 German-language paper (published as Zur Elektrodynamik bewegter Körper, in Annalen der Physik. 17:891, 1905) which appeared in th' book Th' Principle o' Relativity, published in 1923 by Methuen an' Company, Ltd. o' London. Most o' th' papers in that collection be English translations by W. Perrett an' G.B. Jeffery from th' German Das Relativatsprinzip, 4th ed., published by in 1922 by Tuebner. All o' these sources be now in th' public domain; this document, derived from them, remains in th' public domain an' may be reproduced in any manner or medium without permission, restriction, attribution, or compensation.

Numbered footnotes be as they appeared in th' 1923 edition; editor's notes be preceded by asterisks (*) an' appear in sans serif type. Th' 1923 English translation modified th' notation used in Einstein's 1905 paper t' conform t' that in use by th' 1920's; fer example, c denotes th' speed o' light, as opposed th' V used by Einstein in 1905.

This electronic edition be prepared by John Walker in Novembree 1999. Ye can download a ready-t'-print PostScript file o' this document or th' LaTeX source code used t' create 't from this site; both be supplied as Zipped archives. In addition, a PDF document be available which can be read on-line or printed. This HTML document be initially converted from th' LaTeX edition wi' th' LaTeX2HTML utility an' th' text an' images subsequently hand-edited t' produce this text.

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