ON THE ELECTRODYNAMICS OF MOVING BODIES


 Clement Wilkins
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1 ON THE ELECTRODYNAMICS OF MOVING BODIES By A. EINSTEIN June 30, 905 It is known that Maxwell s eletrodynamis as usually understood at the present time when applied to moing bodies, leads to asymmetries whih do not appear to be inherent in the phenomena. Take, for example, the reiproal eletrodynami ation of a magnet and a ondutor. The obserable phenomenon here depends only on the relatie motion of the ondutor and the magnet, whereas the ustomary iew draws a sharp distintion between the two ases in whih either the one or the other of these bodies is in motion. For if the magnet is in motion and the ondutor at rest, there arises in the neighbourhood of the magnet an eletri field with a ertain definite energy, produing a urrent at the plaes where parts of the ondutor are situated. But if the magnet is stationary and the ondutor in motion, no eletri field arises in the neighbourhood of the magnet. In the ondutor, howeer, we find an eletromotie fore, to whih in itself there is no orresponding energy, but whih gies rise assuming equality of relatie motion in the two ases disussed to eletri urrents of the same path and intensity as those produed by the eletri fores in the former ase. Examples of this sort, together with the unsuessful attempts to disoer any motion of the earth relatiely to the light medium, suggest that the phenomena of eletrodynamis as well as of mehanis possess no properties orresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of eletrodynamis and optis will be alid for all frames of referene for whih the equations of mehanis hold good. We will raise this onjeture (the purport of whih will hereafter be alled the Priniple of Relatiity ) to the status of a postulate, and also introdue another postulate, whih is only apparently irreonilable with the former, namely, that light is always propagated in empty spae with a definite eloity whih is independent of the state of motion of the emitting body. These two postulates suffie for the attainment of a simple and onsistent theory of the eletrodynamis of moing bodies based on Maxwell s theory for stationary bodies. The introdution of a luminiferous ether will proe to be superfluous inasmuh as the iew here to be deeloped will not require an absolutely stationary spae proided with speial properties, nor The preeding memoir by Lorentz was not at this time known to the author.
2 assign a eloityetor to a point of the empty spae in whih eletromagneti proesses take plae. The theory to be deeloped is based like all eletrodynamis on the kinematis of the rigid body, sine the assertions of any suh theory hae to do with the relationships between rigid bodies (systems of oordinates), loks, and eletromagneti proesses. Insuffiient onsideration of this irumstane lies at the root of the diffiulties whih the eletrodynamis of moing bodies at present enounters. I. KINEMATICAL PART. Definition of Simultaneity Let us take a system of oordinates in whih the equations of Newtonian mehanis hold good. 2 In order to render our presentation more preise and to distinguish this system of oordinates erbally from others whih will be introdued hereafter, we all it the stationary system. If a material point is at rest relatiely to this system of oordinates, its position an be defined relatiely thereto by the employment of rigid standards of measurement and the methods of Eulidean geometry, and an be expressed in Cartesian oordinates. If we wish to desribe the motion of a material point, we gie the alues of its oordinates as funtions of the time. Now we must bear arefully in mind that a mathematial desription of this kind has no physial meaning unless we are quite lear as to what we understand by time. We hae to take into aount that all our judgments in whih time plays a part are always judgments of simultaneous eents. If, for instane, I say, That train arries here at 7 o lok, I mean something like this: The pointing of the small hand of my wath to 7 and the arrial of the train are simultaneous eents. 3 It might appear possible to oerome all the diffiulties attending the definition of time by substituting the position of the small hand of my wath for time. And in fat suh a definition is satisfatory when we are onerned with defining a time exlusiely for the plae where the wath is loated; but it is no longer satisfatory when we hae to onnet in time series of eents ourring at different plaes, or what omes to the same thing to ealuate the times of eents ourring at plaes remote from the wath. We might, of ourse, ontent ourseles with time alues determined by an obserer stationed together with the wath at the origin of the oordinates, and oordinating the orresponding positions of the hands with light signals, gien out by eery eent to be timed, and reahing him through empty spae. But this oordination has the disadantage that it is not independent of the standpoint of the obserer with the wath or lok, as we know from experiene. 2 i.e. to the first approximation. 3 We shall not here disuss the inexatitude whih lurks in the onept of simultaneity of two eents at approximately the same plae, whih an only be remoed by an abstration. 2
3 We arrie at a muh more pratial determination along the following line of thought. If at the point A of spae there is a lok, an obserer at A an determine the time alues of eents in the immediate proximity of A by finding the positions of the hands whih are simultaneous with these eents. If there is at the point B of spae another lok in all respets resembling the one at A, it is possible for an obserer at B to determine the time alues of eents in the immediate neighbourhood of B. But it is not possible without further assumption to ompare, in respet of time, an eent at A with an eent at B. We hae so far defined only an A time and a B time. We hae not defined a ommon time for A and B, for the latter annot be defined at all unless we establish by definition that the time required by light to trael from A to B equals the time it requires to trael from B to A. Let a ray of light start at the A time t A from A towards B, let it at the B time t B be refleted at B in the diretion of A, and arrie again at A at the A time t A. In aordane with definition the two loks synhronize if t B t A = t A t B. We assume that this definition of synhronism is free from ontraditions, and possible for any number of points; and that the following relations are uniersally alid:. If the lok at B synhronizes with the lok at A, the lok at A synhronizes with the lok at B. 2. If the lok at A synhronizes with the lok at B and also with the lok at C, the loks at B and C also synhronize with eah other. Thus with the help of ertain imaginary physial experiments we hae settled what is to be understood by synhronous stationary loks loated at different plaes, and hae eidently obtained a definition of simultaneous, or synhronous, and of time. The time of an eent is that whih is gien simultaneously with the eent by a stationary lok loated at the plae of the eent, this lok being synhronous, and indeed synhronous for all time determinations, with a speified stationary lok. In agreement with experiene we further assume the quantity 2AB t A t =, A to be a uniersal onstant the eloity of light in empty spae. It is essential to hae time defined by means of stationary loks in the stationary system, and the time now defined being appropriate to the stationary system we all it the time of the stationary system. 2. On the Relatiity of Lengths and Times The following reflexions are based on the priniple of relatiity and on the priniple of the onstany of the eloity of light. These two priniples we define as follows: 3
4 . The laws by whih the states of physial systems undergo hange are not affeted, whether these hanges of state be referred to the one or the other of two systems of oordinates in uniform translatory motion. 2. Any ray of light moes in the stationary system of oordinates with the determined eloity, whether the ray be emitted by a stationary or by a moing body. Hene eloity = light path time interal where time interal is to be taken in the sense of the definition in. Let there be gien a stationary rigid rod; and let its length be l as measured by a measuringrod whih is also stationary. We now imagine the axis of the rod lying along the axis of x of the stationary system of oordinates, and that a uniform motion of parallel translation with eloity along the axis of x in the diretion of inreasing x is then imparted to the rod. We now inquire as to the length of the moing rod, and imagine its length to be asertained by the following two operations: (a) The obserer moes together with the gien measuringrod and the rod to be measured, and measures the length of the rod diretly by superposing the measuringrod, in just the same way as if all three were at rest. (b) By means of stationary loks set up in the stationary system and synhronizing in aordane with, the obserer asertains at what points of the stationary system the two ends of the rod to be measured are loated at a definite time. The distane between these two points, measured by the measuringrod already employed, whih in this ase is at rest, is also a length whih may be designated the length of the rod. In aordane with the priniple of relatiity the length to be disoered by the operation (a) we will all it the length of the rod in the moing system must be equal to the length l of the stationary rod. The length to be disoered by the operation (b) we will all the length of the (moing) rod in the stationary system. This we shall determine on the basis of our two priniples, and we shall find that it differs from l. Current kinematis taitly assumes that the lengths determined by these two operations are preisely equal, or in other words, that a moing rigid body at the epoh t may in geometrial respets be perfetly represented by the same body at rest in a definite position. We imagine further that at the two ends A and B of the rod, loks are plaed whih synhronize with the loks of the stationary system, that is to say that their indiations orrespond at any instant to the time of the stationary system at the plaes where they happen to be. These loks are therefore synhronous in the stationary system. We imagine further that with eah lok there is a moing obserer, and that these obserers apply to both loks the riterion established in for the synhronization of two loks. Let a ray of light depart from A at the time 4 t A, 4 Time here denotes time of the stationary system and also position of hands of the moing lok situated at the plae under disussion. 4
5 let it be refleted at B at the time t B, and reah A again at the time t A. Taking into onsideration the priniple of the onstany of the eloity of light we find that t B t A = r AB and t A t B = r AB + where r AB denotes the length of the moing rod measured in the stationary system. Obserers moing with the moing rod would thus find that the two loks were not synhronous, while obserers in the stationary system would delare the loks to be synhronous. So we see that we annot attah any absolute signifiation to the onept of simultaneity, but that two eents whih, iewed from a system of oordinates, are simultaneous, an no longer be looked upon as simultaneous eents when enisaged from a system whih is in motion relatiely to that system. 3. Theory of the Transformation of Coordinates and Times from a Stationary System to another System in Uniform Motion of Translation Relatiely to the Former Let us in stationary spae take two systems of oordinates, i.e. two systems, eah of three rigid material lines, perpendiular to one another, and issuing from a point. Let the axes of X of the two systems oinide, and their axes of Y and Z respetiely be parallel. Let eah system be proided with a rigid measuringrod and a number of loks, and let the two measuringrods, and likewise all the loks of the two systems, be in all respets alike. Now to the origin of one of the two systems (k) let a onstant eloity be imparted in the diretion of the inreasing x of the other stationary system (K), and let this eloity be ommuniated to the axes of the oordinates, the releant measuringrod, and the loks. To any time of the stationary system K there then will orrespond a definite position of the axes of the moing system, and from reasons of symmetry we are entitled to assume that the motion of k may be suh that the axes of the moing system are at the time t (this t always denotes a time of the stationary system) parallel to the axes of the stationary system. We now imagine spae to be measured from the stationary system K by means of the stationary measuringrod, and also from the moing system k by means of the measuringrod moing with it; and that we thus obtain the oordinates x, y, z, and ξ, η, ζ respetiely. Further, let the time t of the stationary system be determined for all points thereof at whih there are loks by means of light signals in the manner indiated in ; similarly let the time τ of the moing system be determined for all points of the moing system at whih there are loks at rest relatiely to that system by applying the method, gien in, of light signals between the points at whih the latter loks are loated. To any system of alues x, y, z, t, whih ompletely defines the plae and time of an eent in the stationary system, there belongs a system of alues ξ, 5
6 η, ζ, τ, determining that eent relatiely to the system k, and our task is now to find the system of equations onneting these quantities. In the first plae it is lear that the equations must be linear on aount of the properties of homogeneity whih we attribute to spae and time. If we plae x = x t, it is lear that a point at rest in the system k must hae a system of alues x, y, z, independent of time. We first define τ as a funtion of x, y, z, and t. To do this we hae to express in equations that τ is nothing else than the summary of the data of loks at rest in system k, whih hae been synhronized aording to the rule gien in. From the origin of system k let a ray be emitted at the time τ 0 along the Xaxis to x, and at the time τ be refleted thene to the origin of the oordinates, arriing there at the time τ 2 ; we then must hae 2 (τ 0 + τ 2 ) = τ, or, by inserting the arguments of the funtion τ and applying the priniple of the onstany of the eloity of light in the stationary system: 2 [ τ(0, 0, 0, t) + τ (0, 0, 0, t + x + x + )] = τ Hene, if x be hosen infinitesimally small, ( 2 + ) τ + t = τ x + τ t, or τ x + τ 2 2 t = 0. (x, 0, 0, t + x It is to be noted that instead of the origin of the oordinates we might hae hosen any other point for the point of origin of the ray, and the equation just obtained is therefore alid for all alues of x, y, z. An analogous onsideration applied to the axes of Y and Z it being borne in mind that light is always propagated along these axes, when iewed from the stationary system, with the eloity 2 2 gies us τ τ = 0, y z = 0. Sine τ is a linear funtion, it follows from these equations that ( ) τ = a t 2 2 x where a is a funtion φ() at present unknown, and where for breity it is assumed that at the origin of k, τ = 0, when t = 0. With the help of this result we easily determine the quantities ξ, η, ζ by expressing in equations that light (as required by the priniple of the onstany of the eloity of light, in ombination with the priniple of relatiity) is also ). 6
7 propagated with eloity when measured in the moing system. For a ray of light emitted at the time τ = 0 in the diretion of the inreasing ξ ( ) ξ = τ or ξ = a t 2 2 x. But the ray moes relatiely to the initial point of k, when measured in the stationary system, with the eloity, so that x = t. If we insert this alue of t in the equation for ξ, we obtain 2 ξ = a 2 2 x. In an analogous manner we find, by onsidering rays moing along the two other axes, that ( ) η = τ = a t 2 2 x when Thus y 2 2 = t, x = 0. η = a 2 y and ζ = a 2 2 z. 2 Substituting for x its alue, we obtain τ = φ()β(t x/ 2 ), ξ = φ()β(x t), η = φ()y, ζ = φ()z, where β = 2 / 2, and φ is an as yet unknown funtion of. If no assumption whateer be made as to the initial position of the moing system and as to the zero point of τ, an additie onstant is to be plaed on the right side of eah of these equations. 7
8 We now hae to proe that any ray of light, measured in the moing system, is propagated with the eloity, if, as we hae assumed, this is the ase in the stationary system; for we hae not as yet furnished the proof that the priniple of the onstany of the eloity of light is ompatible with the priniple of relatiity. At the time t = τ = 0, when the origin of the oordinates is ommon to the two systems, let a spherial wae be emitted therefrom, and be propagated with the eloity in system K. If (x, y, z) be a point just attained by this wae, then x 2 + y 2 + z 2 = 2 t 2. Transforming this equation with the aid of our equations of transformation we obtain after a simple alulation ξ 2 + η 2 + ζ 2 = 2 τ 2. The wae under onsideration is therefore no less a spherial wae with eloity of propagation when iewed in the moing system. This shows that our two fundamental priniples are ompatible. 5 In the equations of transformation whih hae been deeloped there enters an unknown funtion φ of, whih we will now determine. For this purpose we introdue a third system of oordinates K, whih relatiely to the system k is in a state of parallel translatory motion parallel to the axis of Ξ, suh that the origin of oordinates of system K moes with eloity on the axis of Ξ. At the time t = 0 let all three origins oinide, and when t = x = y = z = 0 let the time t of the system K be zero. We all the oordinates, measured in the system K, x, y, z, and by a twofold appliation of our equations of transformation we obtain t = φ( )β( )(τ + ξ/ 2 ) = φ()φ( )t, x = φ( )β( )(ξ + τ) = φ()φ( )x, y = φ( )η = φ()φ( )y, z = φ( )ζ = φ()φ( )z. Sine the relations between x, y, z and x, y, z do not ontain the time t, the systems K and K are at rest with respet to one another, and it is lear that the transformation from K to K must be the idential transformation. Thus φ()φ( ) =. 5 The equations of the Lorentz transformation may be more simply dedued diretly from the ondition that in irtue of those equations the relation x 2 + y 2 + z 2 = 2 t 2 shall hae as its onsequene the seond relation ξ 2 + η 2 + ζ 2 = 2 τ 2. Editor s note: In Einstein s original paper, the symbols (Ξ, H, Z) for the oordinates of the moing system k were introdued without expliitly defining them. In the 923 English translation, (X, Y, Z) were used, reating an ambiguity between X oordinates in the fixed system K and the parallel axis in moing system k. Here and in subsequent referenes we use Ξ when referring to the axis of system k along whih the system is translating with respet to K. In addition, the referene to system K later in this sentene was inorretly gien as k in the 923 English translation. 8
9 We now inquire into the signifiation of φ(). We gie our attention to that part of the axis of Y of system k whih lies between ξ = 0, η = 0, ζ = 0 and ξ = 0, η = l, ζ = 0. This part of the axis of Y is a rod moing perpendiularly to its axis with eloity relatiely to system K. Its ends possess in K the oordinates and x = t, y = l φ(), z = 0 x 2 = t, y 2 = 0, z 2 = 0. The length of the rod measured in K is therefore l/φ(); and this gies us the meaning of the funtion φ(). From reasons of symmetry it is now eident that the length of a gien rod moing perpendiularly to its axis, measured in the stationary system, must depend only on the eloity and not on the diretion and the sense of the motion. The length of the moing rod measured in the stationary system does not hange, therefore, if and are interhanged. Hene follows that l/φ() = l/φ( ), or φ() = φ( ). It follows from this relation and the one preiously found that φ() =, so that the transformation equations whih hae been found beome τ = β(t x/ 2 ), ξ = β(x t), η = y, ζ = z, where β = / 2 / Physial Meaning of the Equations Obtained in Respet to Moing Rigid Bodies and Moing Cloks We enisage a rigid sphere 6 of radius R, at rest relatiely to the moing system k, and with its entre at the origin of oordinates of k. The equation of the surfae of this sphere moing relatiely to the system K with eloity is ξ 2 + η 2 + ζ 2 = R 2. 6 That is, a body possessing spherial form when examined at rest. 9
10 The equation of this surfae expressed in x, y, z at the time t = 0 is x 2 ( 2 / 2 ) 2 + y2 + z 2 = R 2. A rigid body whih, measured in a state of rest, has the form of a sphere, therefore has in a state of motion iewed from the stationary system the form of an ellipsoid of reolution with the axes R 2 / 2, R, R. Thus, whereas the Y and Z dimensions of the sphere (and therefore of eery rigid body of no matter what form) do not appear modified by the motion, the X dimension appears shortened in the ratio : 2 / 2, i.e. the greater the alue of, the greater the shortening. For = all moing objets iewed from the stationary system shriel up into plane figures. For eloities greater than that of light our deliberations beome meaningless; we shall, howeer, find in what follows, that the eloity of light in our theory plays the part, physially, of an infinitely great eloity. It is lear that the same results hold good of bodies at rest in the stationary system, iewed from a system in uniform motion. Further, we imagine one of the loks whih are qualified to mark the time t when at rest relatiely to the stationary system, and the time τ when at rest relatiely to the moing system, to be loated at the origin of the oordinates of k, and so adjusted that it marks the time τ. What is the rate of this lok, when iewed from the stationary system? Between the quantities x, t, and τ, whih refer to the position of the lok, we hae, eidently, x = t and τ = 2 / 2 (t x/2 ). Therefore, τ = t 2 / 2 = t ( 2 / 2 )t whene it follows that the time marked by the lok (iewed in the stationary system) is slow by 2 / 2 seonds per seond, or negleting magnitudes of fourth and higher order by 2 2 / 2. From this there ensues the following peuliar onsequene. If at the points A and B of K there are stationary loks whih, iewed in the stationary system, are synhronous; and if the lok at A is moed with the eloity along the line AB to B, then on its arrial at B the two loks no longer synhronize, but the lok moed from A to B lags behind the other whih has remained at Editor s note: In the 923 English translation, this phrase was erroneously translated as plain figures. I hae used the orret plane figures in this edition. 0
11 B by 2 t2 / 2 (up to magnitudes of fourth and higher order), t being the time oupied in the journey from A to B. It is at one apparent that this result still holds good if the lok moes from A to B in any polygonal line, and also when the points A and B oinide. If we assume that the result proed for a polygonal line is also alid for a ontinuously ured line, we arrie at this result: If one of two synhronous loks at A is moed in a losed ure with onstant eloity until it returns to A, the journey lasting t seonds, then by the lok whih has remained at rest the traelled lok on its arrial at A will be 2 t2 / 2 seond slow. Thene we onlude that a balanelok 7 at the equator must go more slowly, by a ery small amount, than a preisely similar lok situated at one of the poles under otherwise idential onditions. 5. The Composition of Veloities In the system k moing along the axis of X of the system K with eloity, let a point moe in aordane with the equations ξ = w ξ τ, η = w η τ, ζ = 0, where w ξ and w η denote onstants. Required: the motion of the point relatiely to the system K. If with the help of the equations of transformation deeloped in 3 we introdue the quantities x, y, z, t into the equations of motion of the point, we obtain x = y = z = 0. w ξ + + w ξ / 2 t, 2 / 2 + w ξ / 2 w ηt, Thus the law of the parallelogram of eloities is alid aording to our theory only to a first approximation. We set V 2 = ( ) 2 dx + dt w 2 = w 2 ξ + w 2 η, a = tan w η /w ξ, ( ) 2 dy, dt 7 Not a pendulumlok, whih is physially a system to whih the Earth belongs. This ase had to be exluded. Editor s note: This equation was inorretly gien in Einstein s original paper and the 923 English translation as a = tan w y/w x.
12 a is then to be looked upon as the angle between the eloities and w. After a simple alulation we obtain (2 + w V = 2 + 2w os a) (w sin a/) 2 + w os a/ 2. It is worthy of remark that and w enter into the expression for the resultant eloity in a symmetrial manner. If w also has the diretion of the axis of X, we get V = + w + w/ 2. It follows from this equation that from a omposition of two eloities whih are less than, there always results a eloity less than. For if we set = κ, w = λ, κ and λ being positie and less than, then 2 κ λ V = 2 κ λ + κλ/ <. It follows, further, that the eloity of light annot be altered by omposition with a eloity less than that of light. For this ase we obtain V = + w + w/ =. We might also hae obtained the formula for V, for the ase when and w hae the same diretion, by ompounding two transformations in aordane with 3. If in addition to the systems K and k figuring in 3 we introdue still another system of oordinates k moing parallel to k, its initial point moing on the axis of Ξ with the eloity w, we obtain equations between the quantities x, y, z, t and the orresponding quantities of k, whih differ from the equations found in 3 only in that the plae of is taken by the quantity + w + w/ 2 ; from whih we see that suh parallel transformations neessarily form a group. We hae now dedued the requisite laws of the theory of kinematis orresponding to our two priniples, and we proeed to show their appliation to eletrodynamis. II. ELECTRODYNAMICAL PART 6. Transformation of the MaxwellHertz Equations for Empty Spae. On the Nature of the Eletromotie Fores Ourring in a Magneti Field During Motion Let the MaxwellHertz equations for empty spae hold good for the stationary system K, so that we hae Editor s note: X in the 923 English translation. 2
13 X t = N y M Y t = L z N Z t = M x L z, x, y, L t = Y z Z y, M t = Z x X z, N t = X y Y x, where (X, Y, Z) denotes the etor of the eletri fore, and (L, M, N) that of the magneti fore. If we apply to these equations the transformation deeloped in 3, by referring the eletromagneti proesses to the system of oordinates there introdued, moing with the eloity, we obtain the equations τ τ τ τ { β ( Y { ( X = τ η β N Y)} N)} = L ξ { β ( Z + M)} = ξ L = τ ζ { ( β M + Z)} = ξ { ( β N Y)} = X η { ( β M + { ( Z)} β Y { ( N)} β Z + M)} ζ ζ L, η η X, ζ ξ { ( β M + Z)}, { ( β N Y)}, { β ( Z + M)}, { β ( Y N)}, where β = / 2 / 2. Now the priniple of relatiity requires that if the MaxwellHertz equations for empty spae hold good in system K, they also hold good in system k; that is to say that the etors of the eletri and the magneti fore (X, Y, Z ) and (L, M, N ) of the moing system k, whih are defined by their ponderomotie effets on eletri or magneti masses respetiely, satisfy the following equations: X τ = N η M Y τ = L ζ N Z τ = M ξ ζ, ξ, L η, L τ = Y ζ Z η, M τ = Z ξ X ζ, N τ = X η Y ξ. 3
14 Eidently the two systems of equations found for system k must express exatly the same thing, sine both systems of equations are equialent to the MaxwellHertz equations for system K. Sine, further, the equations of the two systems agree, with the exeption of the symbols for the etors, it follows that the funtions ourring in the systems of equations at orresponding plaes must agree, with the exeption of a fator ψ(), whih is ommon for all funtions of the one system of equations, and is independent of ξ, η, ζ and τ but depends upon. Thus we hae the relations X = ψ()x, L = ψ()l, Y = ψ()β ( Y N), M = ψ()β ( M + Z), Z = ψ()β ( Z + M), N = ψ()β ( N Y). If we now form the reiproal of this system of equations, firstly by soling the equations just obtained, and seondly by applying the equations to the inerse transformation (from k to K), whih is haraterized by the eloity, it follows, when we onsider that the two systems of equations thus obtained must be idential, that ψ()ψ( ) =. Further, from reasons of symmetry 8 and therefore and our equations assume the form ψ() =, X = X, L = L, Y = β ( Y N), M = β ( M + Z), Z = β ( Z + M), N = β ( N Y). As to the interpretation of these equations we make the following remarks: Let a point harge of eletriity hae the magnitude one when measured in the stationary system K, i.e. let it when at rest in the stationary system exert a fore of one dyne upon an equal quantity of eletriity at a distane of one m. By the priniple of relatiity this eletri harge is also of the magnitude one when measured in the moing system. If this quantity of eletriity is at rest relatiely to the stationary system, then by definition the etor (X, Y, Z) is equal to the fore ating upon it. If the quantity of eletriity is at rest relatiely to the moing system (at least at the releant instant), then the fore ating upon it, measured in the moing system, is equal to the etor (X, Y, Z ). Consequently the first three equations aboe allow themseles to be lothed in words in the two following ways:. If a unit eletri point harge is in motion in an eletromagneti field, there ats upon it, in addition to the eletri fore, an eletromotie fore whih, if we neglet the terms multiplied by the seond and higher powers of /, is equal to the etorprodut of the eloity of the harge and the magneti fore, diided by the eloity of light. (Old manner of expression.) 8 If, for example, X=Y=Z=L=M=0, and N 0, then from reasons of symmetry it is lear that when hanges sign without hanging its numerial alue, Y must also hange sign without hanging its numerial alue. 4
15 2. If a unit eletri point harge is in motion in an eletromagneti field, the fore ating upon it is equal to the eletri fore whih is present at the loality of the harge, and whih we asertain by transformation of the field to a system of oordinates at rest relatiely to the eletrial harge. (New manner of expression.) The analogy holds with magnetomotie fores. We see that eletromotie fore plays in the deeloped theory merely the part of an auxiliary onept, whih owes its introdution to the irumstane that eletri and magneti fores do not exist independently of the state of motion of the system of oordinates. Furthermore it is lear that the asymmetry mentioned in the introdution as arising when we onsider the urrents produed by the relatie motion of a magnet and a ondutor, now disappears. Moreoer, questions as to the seat of eletrodynami eletromotie fores (unipolar mahines) now hae no point. 7. Theory of Doppler s Priniple and of Aberration In the system K, ery far from the origin of oordinates, let there be a soure of eletrodynami waes, whih in a part of spae ontaining the origin of oordinates may be represented to a suffiient degree of approximation by the equations X = X 0 sin Φ, L = L 0 sin Φ, Y = Y 0 sin Φ, M = M 0 sin Φ, Z = Z 0 sin Φ, N = N 0 sin Φ, where Φ = ω {t } (lx + my + nz). Here (X 0, Y 0, Z 0 ) and (L 0, M 0, N 0 ) are the etors defining the amplitude of the waetrain, and l, m, n the diretionosines of the waenormals. We wish to know the onstitution of these waes, when they are examined by an obserer at rest in the moing system k. Applying the equations of transformation found in 6 for eletri and magneti fores, and those found in 3 for the oordinates and the time, we obtain diretly where X = X 0 sin Φ, L = L 0 sin Φ, Y = β(y 0 N 0 /) sin Φ, M = β(m 0 + Z 0 /) sin Φ, Z = β(z 0 + M 0 /) sin Φ, N = β(n 0 Y 0 /) sin Φ, Φ = ω { τ (l ξ + m η + n ζ) } ω = ωβ( l/), 5
16 l = l / l/, m = m β( l/), n = n β( l/). From the equation for ω it follows that if an obserer is moing with eloity relatiely to an infinitely distant soure of light of frequeny ν, in suh a way that the onneting line soureobserer makes the angle φ with the eloity of the obserer referred to a system of oordinates whih is at rest relatiely to the soure of light, the frequeny ν of the light pereied by the obserer is gien by the equation ν os φ / = ν 2 /. 2 This is Doppler s priniple for any eloities whateer. When φ = 0 the equation assumes the perspiuous form ν / = ν + /. We see that, in ontrast with the ustomary iew, when =, ν =. If we all the angle between the waenormal (diretion of the ray) in the moing system and the onneting line soureobserer φ, the equation for φ assumes the form os φ = os φ / os φ /. This equation expresses the law of aberration in its most general form. If φ = π, the equation beomes simply 2 os φ = /. We still hae to find the amplitude of the waes, as it appears in the moing system. If we all the amplitude of the eletri or magneti fore A or A respetiely, aordingly as it is measured in the stationary system or in the moing system, we obtain whih equation, if φ = 0, simplifies into A 2 = A 2 ( os φ /)2 2 / 2 Editor s note: Erroneously gien as l in the 923 English translation, propagating an error, despite a hange in symbols, from the original 905 paper. 6
17 A 2 = A 2 / + /. It follows from these results that to an obserer approahing a soure of light with the eloity, this soure of light must appear of infinite intensity. 8. Transformation of the Energy of Light Rays. Theory of the Pressure of Radiation Exerted on Perfet Refletors Sine A 2 /8π equals the energy of light per unit of olume, we hae to regard A 2 /8π, by the priniple of relatiity, as the energy of light in the moing system. Thus A 2 /A 2 would be the ratio of the measured in motion to the measured at rest energy of a gien light omplex, if the olume of a light omplex were the same, whether measured in K or in k. But this is not the ase. If l, m, n are the diretionosines of the waenormals of the light in the stationary system, no energy passes through the surfae elements of a spherial surfae moing with the eloity of light: (x lt) 2 + (y mt) 2 + (z nt) 2 = R 2. We may therefore say that this surfae permanently enloses the same light omplex. We inquire as to the quantity of energy enlosed by this surfae, iewed in system k, that is, as to the energy of the light omplex relatiely to the system k. The spherial surfae iewed in the moing system is an ellipsoidal surfae, the equation for whih, at the time τ = 0, is (βξ lβξ/) 2 + (η mβξ/) 2 + (ζ nβξ/) 2 = R 2. If S is the olume of the sphere, and S that of this ellipsoid, then by a simple alulation S S = 2 / 2 os φ /. Thus, if we all the light energy enlosed by this surfae E when it is measured in the stationary system, and E when measured in the moing system, we obtain E E = A 2 S os φ / A 2 = S 2 /, 2 and this formula, when φ = 0, simplifies into E E = / + /. 7
18 It is remarkable that the energy and the frequeny of a light omplex ary with the state of motion of the obserer in aordane with the same law. Now let the oordinate plane ξ = 0 be a perfetly refleting surfae, at whih the plane waes onsidered in 7 are refleted. We seek for the pressure of light exerted on the refleting surfae, and for the diretion, frequeny, and intensity of the light after reflexion. Let the inidental light be defined by the quantities A, os φ, ν (referred to system K). Viewed from k the orresponding quantities are A os φ / = A 2 /, 2 os φ = ν = ν os φ / os φ /, os φ / 2 / 2. For the refleted light, referring the proess to system k, we obtain A = A os φ = os φ ν = ν Finally, by transforming bak to the stationary system K, we obtain for the refleted light A = A + osφ / 2 / 2 = A 2 os φ / + 2 / 2 2 / 2, os φ = os φ + / + os φ / = ( + 2 / 2 ) os φ 2/ 2 os φ / + 2 / 2, ν = ν + os φ / 2 / 2 = ν 2 os φ / + 2 / 2 2 / 2. The energy (measured in the stationary system) whih is inident upon unit area of the mirror in unit time is eidently A 2 ( os φ )/8π. The energy leaing the unit of surfae of the mirror in the unit of time is A 2 ( os φ + )/8π. The differene of these two expressions is, by the priniple of energy, the work done by the pressure of light in the unit of time. If we set down this work as equal to the produt P, where P is the pressure of light, we obtain P = 2 A2 (os φ /) 2 8π 2 / 2. 8
19 In agreement with experiment and with other theories, we obtain to a first approximation P = 2 A2 8π os2 φ. All problems in the optis of moing bodies an be soled by the method here employed. What is essential is, that the eletri and magneti fore of the light whih is influened by a moing body, be transformed into a system of oordinates at rest relatiely to the body. By this means all problems in the optis of moing bodies will be redued to a series of problems in the optis of stationary bodies. 9. Transformation of the MaxwellHertz Equations when ConetionCurrents are Taken into Aount We start from the equations { X { Y t + u xρ } = N y M z, t + u yρ } = L { Z t + u zρ } = M x L z N x, y, L t = Y z Z y, M t = Z x X z, N t = X y Y x, where ρ = X x + Y y + Z z denotes 4π times the density of eletriity, and (u x, u y, u z ) the eloityetor of the harge. If we imagine the eletri harges to be inariably oupled to small rigid bodies (ions, eletrons), these equations are the eletromagneti basis of the Lorentzian eletrodynamis and optis of moing bodies. Let these equations be alid in the system K, and transform them, with the assistane of the equations of transformation gien in 3 and 6, to the system k. We then obtain the equations { X { Y where τ + u ξρ } = N τ + u ηρ } = L { Z τ + u ζρ } = M ξ η M ζ, ζ N ξ, L η, L τ = Y ζ Z η, M τ = Z ξ X ζ, N τ = X η Y ξ, 9
20 u ξ = u η = u ζ = u x u x / 2 u y β( u x / 2 ) u z β( u x / 2 ), and ρ = X ξ + Y η + Z ζ = β( u x / 2 )ρ. Sine as follows from the theorem of addition of eloities ( 5) the etor (u ξ, u η, u ζ ) is nothing else than the eloity of the eletri harge, measured in the system k, we hae the proof that, on the basis of our kinematial priniples, the eletrodynami foundation of Lorentz s theory of the eletrodynamis of moing bodies is in agreement with the priniple of relatiity. In addition I may briefly remark that the following important law may easily be dedued from the deeloped equations: If an eletrially harged body is in motion anywhere in spae without altering its harge when regarded from a system of oordinates moing with the body, its harge also remains when regarded from the stationary system K onstant. 0. Dynamis of the Slowly Aelerated Eletron Let there be in motion in an eletromagneti field an eletrially harged partile (in the sequel alled an eletron ), for the law of motion of whih we assume as follows: If the eletron is at rest at a gien epoh, the motion of the eletron ensues in the next instant of time aording to the equations m d2 x dt 2 = ɛx m d2 y dt 2 = ɛy m d2 z dt 2 = ɛz where x, y, z denote the oordinates of the eletron, and m the mass of the eletron, as long as its motion is slow. 20
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