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Annalen der Physik (AdP) & Albert Einstein. Special Issue & Supplement: Commemorating Albert Einstein Special Issue, vol. 14, issue 1-3 (2005). (1916) The Foundation of the General Theory of Relativity. Annalen der Physik, 49, 769-822.

By A. Einstein
June 30, 1905

It is known that Maxwell's electrodynamics—as usuallyunderstood at the present time—when applied to moving bodies,leads to asymmetries which do not appear to be inherent in thephenomena. Take, for example, the reciprocal electrodynamic actionof a magnet and a conductor. The observable phenomenon here dependsonly on the relative motion of the conductor and the magnet,whereas the customary view draws a sharp distinction between thetwo cases in which either the one or the other of these bodies isin motion. For if the magnet is in motion and the conductor atrest, there arises in the neighbourhood of the magnet an electricfield with a certain definite energy, producing a current at theplaces where parts of the conductor are situated. But if the magnetis stationary and the conductor in motion, no electric field arisesin the neighbourhood of the magnet. In the conductor, however, wefind an electromotive force, to which in itself there is nocorresponding energy, but which gives rise—assuming equality ofrelative motion in the two cases discussed—to electric currents ofthe same path and intensity as those produced by the electricforces in the former case.

Examples of this sort, together with the unsuccessful attemptsto discover any motion of the earth relatively to the “lightmedium,” suggest that the phenomena of electrodynamics as well asof mechanics possess no properties corresponding to the idea ofabsolute rest. They suggest rather that, as has already been shownto the first order of small quantities, the same laws ofelectrodynamics and optics will be valid for all frames ofreference for which the equations of mechanics hold good.¹ We willraise this conjecture (the purport of which will hereafter becalled the “Principle of Relativity”) to the status of apostulate, and also introduce another postulate, which is onlyapparently irreconcilable with the former, namely, that light isalways propagated in empty space with a definite velocity cwhich is independent of the state of motion of the emitting body.These two postulates suffice for the attainment of a simple andconsistent theory of the electrodynamics of moving bodies based onMaxwell's theory for stationary bodies. The introduction of a“luminiferous ether” will prove to be superfluous inasmuch as theview here to be developed will not require an “absolutelystationary space” provided with special properties, nor assign avelocity-vector to a point of the empty space in whichelectromagnetic processes take place.

The theory to be developed is based—like allelectrodynamics—on the kinematics of the rigid body, since theassertions of any such theory have to do with the relationshipsbetween rigid bodies (systems of co-ordinates), clocks, andelectromagnetic processes. Insufficient consideration of thiscircumstance lies at the root of the difficulties which theelectrodynamics of moving bodies at present encounters.

§ 1. Definition ofSimultaneity

Let us take a system of co-ordinates in which the equations ofNewtonian mechanics hold good.²In order to render our presentationmore precise and to distinguish this system of co-ordinatesverbally from others which will be introduced hereafter, we call itthe “stationary system.”

If a material point is at rest relatively to this system ofco-ordinates, its position can be defined relatively thereto by theemployment of rigid standards of measurement and the methods ofEuclidean geometry, and can be expressed in Cartesianco-ordinates.

If we wish to describe the motion of a material point,we give the values of its co-ordinates as functions of the time.Now we must bear carefully in mind that a mathematical descriptionof this kind has no physical meaning unless we are quite clear asto what we understand by “time.” We have to take into accountthat all our judgments in which time plays a part are alwaysjudgments of simultaneous events. If, for instance, I say,“That train arrives here at 7 o'clock,” I mean something likethis: “The pointing of the small hand of my watch to 7 and thearrival of the train are simultaneous events.”³

It might appear possible to overcome all the difficultiesattending the definition of “time” by substituting “the positionof the small hand of my watch” for “time.” And in fact such adefinition is satisfactory when we are concerned with defining atime exclusively for the place where the watch is located; but itis no longer satisfactory when we have to connect in time series ofevents occurring at different places, or—what comes to the samething—to evaluate the times of events occurring at places remotefrom the watch.

We might, of course, content ourselves with time valuesdetermined by an observer stationed together with the watch at theorigin of the co-ordinates, and co-ordinating the correspondingpositions of the hands with light signals, given out by every eventto be timed, and reaching him through empty space. But thisco-ordination has the disadvantage that it is not independent ofthe standpoint of the observer with the watch or clock, as we knowfrom experience. We arrive at a much more practical determinationalong the following line of thought.

If at the point A of space there is a clock, an observer at Acan determine the time values of events in the immediate proximityof A by finding the positions of the hands which are simultaneouswith these events. If there is at the point B of space anotherclock in all respects resembling the one at A, it is possible foran observer at B to determine the time values of events in theimmediate neighbourhood of B. But it is not possible withoutfurther assumption to compare, in respect of time, an event at Awith an event at B. We have so far defined only an “A time” and a“B time.” We have not defined a common “time” for A and B, forthe latter cannot be defined at all unless we establish bydefinition that the “time” required by light to travel fromA to B equals the “time” it requires to travel from B to A. Let aray of light start at the “A time”from A towards B, let it at the “B time” be reflected at B in thedirection of A, and arrive again at A at the “A time”.

In accordance with definition the two clocks synchronize if

We assume that this definition of synchronism is free fromcontradictions, and possible for any number of points; and that thefollowing relations are universally valid:—

1. If the clock at B synchronizes with the clock at A, the clockat A synchronizes with the clock at B.
2. If the clock at A synchronizes with the clock at B and alsowith the clock at C, the clocks at B and C also synchronize witheach other.

Thus with the help of certain imaginary physical experiments wehave settled what is to be understood by synchronous stationaryclocks located at different places, and have evidently obtained adefinition of “simultaneous,” or “synchronous,” and of“time.” The “time” of an event is that which is givensimultaneously with the event by a stationary clock located at theplace of the event, this clock being synchronous, and indeedsynchronous for all time determinations, with a specifiedstationary clock.

In agreement with experience we further assume the quantity

to be a universal constant—the velocity oflight in empty space.

It is essential to have time defined by means of stationaryclocks in the stationary system, and the time now defined beingappropriate to the stationary system we call it “the time of thestationary system.”

§ 2. On the Relativityof Lengths and Times

The following reflexions are based on the principle of relativityand on the principle of the constancy of the velocity of light.These two principles we define as follows:—

1. The laws by which the states of physical systems undergo changeare not affected, whether these changes of state be referred to theone or the other of two systems of co-ordinates in uniformtranslatory motion.
2. Any ray of light moves in the “stationary” system ofco-ordinates with the determined velocity c, whether the raybe emitted by a stationary or by a moving body. Hence

   where time interval is to be taken in the senseof the definition in § 1.

Let there be given a stationary rigid rod; and let its length bel as measured by a measuring-rod which is also stationary.We now imagine the axis of the rod lying along the axis of xof the stationary system of co-ordinates, and that a uniform motionof parallel translation with velocity v along the axis ofx in the direction of increasing x is then impartedto the rod. We now inquire as to the length of the moving rod, andimagine its length to be ascertained by the following twooperations:—

(a)

The observer moves together with the given measuring-rod andthe rod to be measured, and measures the length of the rod directlyby superposing the measuring-rod, in just the same way as if allthree were at rest.

(b)

By means of stationary clocks set up in the stationary systemand synchronizing in accordance with §1, the observer ascertains at what points of the stationarysystem the two ends of the rod to be measured are located at adefinite time. The distance between these two points, measured bythe measuring-rod already employed, which in this case is at rest,is also a length which may be designated “the length of therod.”

In accordance with the principle of relativity the length to bediscovered by the operation (a)—we will call it “thelength of the rod in the moving system”—must be equal to thelength l of the stationary rod.

The length to be discovered by the operation (b) wewill call “the length of the (moving) rod in the stationarysystem.” This we shall determine on the basis of our twoprinciples, and we shall find that it differs from l.

Current kinematics tacitly assumes that the lengths determinedby these two operations are precisely equal, or in other words,that a moving rigid body at the epoch t may in geometricalrespects be perfectly represented by the same body atrest in a definite position.

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We imagine further that at the two ends A and B of the rod,clocks are placed which synchronize with the clocks of thestationary system, that is to say that their indications correspondat any instant to the “time of the stationary system” at theplaces where they happen to be. These clocks are therefore“synchronous in the stationary system.”

We imagine further that with each clock there is a movingobserver, and that these observers apply to both clocks thecriterion established in § 1 for thesynchronization of two clocks. Let a ray of light depart from A atthe time⁴,let it be reflected at B at the time, and reach A again at the time. Taking into consideration the principle of theconstancy of the velocity of light we find that

wheredenotes the length of the moving rod—measured in the stationarysystem. Observers moving with the moving rod would thus find thatthe two clocks were not synchronous, while observers in thestationary system would declare the clocks to be synchronous.

So we see that we cannot attach any absolutesignification to the concept of simultaneity, but that two eventswhich, viewed from a system of co-ordinates, are simultaneous, canno longer be looked upon as simultaneous events when envisaged froma system which is in motion relatively to that system.

§ 3. Theory of theTransformation of Co-ordinates and Times from a Stationary Systemto another System in Uniform Motion of Translation Relatively tothe Former

Let us in “stationary” space take two systems of co-ordinates,i.e. two systems, each of three rigid material lines, perpendicularto one another, and issuing from a point. Let the axes of X of thetwo systems coincide, and their axes of Y and Z respectively beparallel. Let each system be provided with a rigid measuring-rodand a number of clocks, and let the two measuring-rods, andlikewise all the clocks of the two systems, be in all respectsalike.

Now to the origin of one of the two systems (k) let aconstant velocity v be imparted in the direction of theincreasing x of the other stationary system (K), and letthis velocity be communicated to the axes of the co-ordinates, therelevant measuring-rod, and the clocks. To any time of thestationary system K there then will correspond a definite positionof the axes of the moving system, and from reasons of symmetry weare entitled to assume that the motion of k may be such thatthe axes of the moving 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 space to be measured from the stationary system Kby means of the stationary measuring-rod, and also from the movingsystem k by means of the measuring-rod moving with it; andthat we thus obtain the co-ordinates x, y, z,and ,respectively. Further, let thetime t of the stationary system be determined for all pointsthereof at which there are clocks by means of light signals in themanner indicated in § 1; similarlylet the time of the moving system bedetermined for all points of the moving system at which there areclocks at rest relatively to that system by applying the method,given in § 1, of light signalsbetween the points at which the latter clocks are located.

To any system of values x, y, z, t,which completely defines the place and time of an event in thestationary system, there belongs a system of values, determining that eventrelatively to the system k, and our task is now to find thesystem of equations connecting these quantities.

In the first place it is clear that the equations must belinear on account of the properties of homogeneity whichwe attribute to space and time.

If we place x'=x-vt, it is clear that apoint at rest in the system k must have a system of valuesx', y, z, independent of time. We first define as a function of x',y, z, and t. To do this we have to express inequations thatis nothing else thanthe summary of the data of clocks at rest in system k, whichhave been synchronized according to the rule given in § 1.

From the origin of system k let a ray be emitted at thetime along the X-axis tox', and at the time bereflected thence to the origin of the co-ordinates, arriving thereat the time;we then must have,or, by inserting the arguments of the functionand applying the principle of the constancy of thevelocity of light in the stationary system:—

Hence, if x' be chosen infinitesimallysmall,

or

It is to be noted that instead of the origin of the co-ordinateswe might have chosen any other point for the point of origin of theray, and the equation just obtained is therefore valid for allvalues of x', y, z.

An analogous consideration—applied to the axes of Y and Z—itbeing borne in mind that light is always propagated along theseaxes, when viewed from the stationary system, with the velocitygives us

Since is alinear function, it follows from these equations that

where a is a function at present unknown, and where for brevity it isassumed that at the origin of k, when t=0.

With the help of this result we easily determine the quantities,by expressing in equationsthat light (as required by the principle of the constancy of thevelocity of light, in combination with the principle of relativity)is also propagated with velocity c when measured in themoving system. For a ray of light emitted at the time in the direction of the increasing

But the ray moves relatively to the initialpoint of k, when measured in the stationary system, with thevelocity c-v, so that

If we insert this value of t in theequation for , we obtain

In an analogous manner we find, by consideringrays moving along the two other axes, that

when

Thus

Substituting for x' its value, we obtain

where

and is an as yetunknown function of v. If no assumption whatever be made asto the initial position of the moving system and as to the zeropoint of , an additive constant is to beplaced on the right side of each of these equations.

We now have to prove that any ray of light, measured in themoving system, is propagated with the velocity c, if, as wehave assumed, this is the case in the stationary system; for wehave not as yet furnished the proof that the principle of theconstancy of the velocity of light is compatible with the principleof relativity.

At the time , when the origin ofthe co-ordinates is common to the two systems, let a spherical wavebe emitted therefrom, and be propagated with the velocity cin system K. If (x, y, z) be a point justattained by this wave, then

Transforming this equation with the aid of our equations oftransformation we obtain after a simple calculation

The wave under consideration is therefore no less a sphericalwave with velocity of propagation c when viewed in themoving system. This shows that our two fundamental principles arecompatible.⁵

In the equations of transformation which have been developedthere enters an unknown function ofv, which we will now determine.

For this purpose we introduce a third system of co-ordinates, which relatively to thesystem k is in a state of parallel translatory motionparallel to the axis of ,^*1 suchthat the origin of co-ordinates of system , moves with velocity -v on the axis of. At the time t=0 let all threeorigins coincide, and when t=x=y=z=0let the time t' of the system be zero. We call the co-ordinates, measured in thesystem , x', y',z', and by a twofold application of our equations oftransformation we obtain

Since the relations between x', y', z' andx, y, z do not contain the time t, thesystems K and are at rest withrespect to one another, and it is clear that the transformationfrom K to must be the identicaltransformation. Thus

We now inquire into the signification of. We give our attention tothat part of the axis of Y of system k which lies between and. This part ofthe axis of Y is a rod moving perpendicularly to its axis withvelocity v relatively to system K. Its ends possess in K theco-ordinates

The length of the rod measured in K istherefore ; and this gives us themeaning of the function . From reasonsof symmetry it is now evident that the length of a given rod movingperpendicularly to its axis, measured in the stationary system,must depend only on the velocity and not on the direction and thesense of the motion. The length of the moving rod measured in thestationary system does not change, therefore, if v and-v are interchanged. Hence follows that , or

It follows from this relation and the onepreviously found that , so thatthe transformation equations which have been found become

where

§ 4. Physical Meaningof the Equations Obtained in Respect to Moving Rigid Bodies andMoving Clocks

We envisage a rigid sphere⁶ of radius R, at rest relatively to themoving system k, and with its centre at the origin ofco-ordinates of k. The equation of the surface of thissphere moving relatively to the system K with velocity vis

The equation of this surface expressed inx, y, z at the time t=0 is

A rigid body which, measured in a state ofrest, has the form of a sphere, therefore has in a state ofmotion—viewed from the stationary system—the form of an ellipsoidof revolution with the axes

Thus, whereas the Y and Z dimensions of the sphere (andtherefore of every rigid body of no matter what form) do not appearmodified by the motion, the X dimension appears shortened in theratio , i.e. the greaterthe value of v, the greater the shortening. Forv=c all moving objects—viewed from the“stationary” system—shrivel up into plane figures.^*2 Forvelocities greater than that of light our deliberations becomemeaningless; we shall, however, find in what follows, that thevelocity of light in our theory plays the part, physically, of aninfinitely great velocity.

It is clear that the same results hold good of bodies at rest inthe “stationary” system, viewed from a system in uniformmotion.

Further, we imagine one of the clocks which are qualified tomark the time t when at rest relatively to the stationarysystem, and the time when at restrelatively to the moving system, to be located at the origin of theco-ordinates of k, and so adjusted that it marks the time. What is the rate of thisclock, when viewed from the stationary system?

Between the quantities x, t, and ,which refer to the position of the clock, we have, evidently,x=vt and

Therefore,

whence it follows that the time marked by theclock (viewed in the stationary system) is slow by seconds per second, or—neglectingmagnitudes of fourth and higher order—by .

From this there ensues the following peculiar consequence. If atthe points A and B of K there are stationary clocks which, viewedin the stationary system, are synchronous; and if the clock at A ismoved with the velocity v along the line AB to B, then onits arrival at B the two clocks no longer synchronize, but theclock moved from A to B lags behind the other which has remained atB by (up to magnitudesof fourth and higher order), t being the time occupied inthe journey from A to B.

It is at once apparent that this result still holds good if theclock moves from A to B in any polygonal line, and also when thepoints A and B coincide.

If we assume that the result proved for a polygonal line is alsovalid for a continuously curved line, we arrive at this result: Ifone of two synchronous clocks at A is moved in a closed curve withconstant velocity until it returns to A, the journey lastingt seconds, then by the clock which has remained at rest thetravelled clock on its arrival at A will be second slow. Thence we conclude that abalance-clock⁷ at the equator must go more slowly, bya very small amount, than a precisely similar clock situated at oneof the poles under otherwise identical conditions.

§ 5. The Composition ofVelocities

In the system k moving along the axis of X of the system Kwith velocity v, let a point move in accordance with theequations

where and denote constants.

Required: the motion of the point relatively to the system K. Ifwith the help of the equations of transformation developed in§ 3 we introduce the quantitiesx, y, z, t into the equations of motionof the point, we obtain

Thus the law of the parallelogram of velocities is validaccording to our theory only to a first approximation. We set

a is then to be looked upon as the anglebetween the velocities v and w. After a simplecalculation we obtain^*4

It is worthy of remark that v andw enter into the expression for the resultant velocity in asymmetrical manner. If w also has the direction of the axisof X, we get

It follows from this equation that from acomposition of two velocities which are less than c, therealways results a velocity less than c. For if we set, and beingpositive and less than c, then

It follows, further, that the velocity of light c cannotbe altered by composition with a velocity less than that of light.For this case we obtain

We might also have obtained the formula for V,for the case when v and w have the same direction, bycompounding two transformations in accordance with § 3. If in addition to the systems K andk figuring in § 3 we introducestill another system of co-ordinates k' moving parallel tok, its initial point moving on the axis of ^*5 with the velocity w, we obtainequations between the quantities x, y, z,t and the corresponding quantities of k', whichdiffer from the equations found in §3 only in that the place of “v” is taken by thequantity

from which we see that such paralleltransformations—necessarily—form a group.

We have now deduced the requisite laws of the theory ofkinematics corresponding to our two principles, and we proceed toshow their application to electrodynamics.

§ 6. Transformation ofthe Maxwell-Hertz Equations for Empty Space. On the Nature of theElectromotive Forces Occurring in a Magnetic Field DuringMotion

Let the Maxwell-Hertz equations for empty space hold good for thestationary system K, so that we have

where (X, Y, Z) denotes the vector of theelectric force, and (L, M, N) that of the magnetic force.

If we apply to these equations the transformation developed in§ 3, by referring the electromagneticprocesses to the system of co-ordinates there introduced, movingwith the velocity v, we obtain the equations^*6

where

Now the principle of relativity requires that if theMaxwell-Hertz equations for empty space hold good in system K, theyalso hold good in system k; that is to say that the vectorsof the electric and the magnetic force—(, , ) and (, , )—of themoving system k, which are defined by their ponderomotiveeffects on electric or magnetic masses respectively, satisfy thefollowing equations:—

Evidently the two systems of equations found for system kmust express exactly the same thing, since both systems ofequations are equivalent to the Maxwell-Hertz equations for systemK. Since, further, the equations of the two systems agree, with theexception of the symbols for the vectors, it follows that thefunctions occurring in the systems of equations at correspondingplaces must agree, with the exception of a factor , which is common for all functions of the one systemof equations, and is independent of and butdepends upon v. Thus we have the relations

If we now form the reciprocal of this system of equations,firstly by solving the equations just obtained, and secondly byapplying the equations to the inverse transformation (from kto K), which is characterized by the velocity -v, itfollows, when we consider that the two systems of equations thusobtained must be identical, that . Further, from reasons of symmetry⁸ andtherefore

and our equations assume the form

As to the interpretation of these equations wemake the following remarks: Let a point charge of electricity havethe magnitude “one” when measured in the stationary system K,i.e. let it when at rest in the stationary system exert a force ofone dyne upon an equal quantity of electricity at a distance of onecm. By the principle of relativity this electric charge is also ofthe magnitude “one” when measured in the moving system. If thisquantity of electricity is at rest relatively to the stationarysystem, then by definition the vector (X, Y, Z) is equal to theforce acting upon it. If the quantity of electricity is at restrelatively to the moving system (at least at the relevant instant),then the force acting upon it, measured in the moving system, isequal to the vector (, , ).Consequently the first three equations above allow themselves to beclothed in words in the two following ways:—

1. If a unit electric point charge is in motion in anelectromagnetic field, there acts upon it, in addition to theelectric force, an “electromotive force” which, if we neglect theterms multiplied by the second and higher powers ofv/c, is equal to the vector-product of the velocityof the charge and the magnetic force, divided by the velocity oflight. (Old manner of expression.)
2. If a unit electric point charge is in motion in anelectromagnetic field, the force acting upon it is equal to theelectric force which is present at the locality of the charge, andwhich we ascertain by transformation of the field to a system ofco-ordinates at rest relatively to the electrical charge. (Newmanner of expression.)

The analogy holds with “magnetomotive forces.” We see thatelectromotive force plays in the developed theory merely the partof an auxiliary concept, which owes its introduction to thecircumstance that electric and magnetic forces do not existindependently of the state of motion of the system ofco-ordinates.

Furthermore it is clear that the asymmetry mentioned in theintroduction as arising when we consider the currents produced bythe relative motion of a magnet and a conductor, now disappears.Moreover, questions as to the “seat” of electrodynamicelectromotive forces (unipolar machines) now have no point.

§ 7. Theory ofDoppler's Principle and of Aberration

In the system K, very far from the origin of co-ordinates, letthere be a source of electrodynamic waves, which in a part of spacecontaining the origin of co-ordinates may be represented to asufficient degree of approximation by the equations

where

Here (, ) and (, ) are the vectors defining the amplitude of thewave-train, and l, m, n the direction-cosinesof the wave-normals. We wish to know the constitution of thesewaves, when they are examined by an observer at rest in the movingsystem k.

Applying the equations of transformation found in § 6 for electric and magnetic forces, andthose found in § 3 for theco-ordinates and the time, we obtain directly

where

From the equation for it followsthat if an observer is moving with velocity v relatively toan infinitely distant source of light of frequency , in such a way that the connecting line“source-observer” makes the angle with the velocity of the observer referred to a system ofco-ordinates which is at rest relatively to the source of light,the frequency of the light perceived bythe observer is given by the equation

This is Doppler's principle for any velocitieswhatever. When the equation assumesthe perspicuous form

We see that, in contrast with the customaryview, when .

If we call the angle between the wave-normal (direction of theray) in the moving system and the connecting line“source-observer” , the equationfor ^*7 assumes the form

This equation expresses the law of aberrationin its most general form. If , the equation becomes simply

We still have to find the amplitude of the waves, as it appearsin the moving system. If we call the amplitude of the electric ormagnetic force A or respectively, accordingly as it is measured in the stationarysystem or in the moving system, we obtain

which equation, if , simplifies into

It follows from these results that to an observer approaching asource of light with the velocity c, this source of lightmust appear of infinite intensity.

§ 8. Transformation ofthe Energy of Light Rays. Theory of the Pressure of RadiationExerted on Perfect Reflectors

Since equals the energy oflight per unit of volume, we have to regard , by the principle of relativity, as theenergy of light in the moving system. Thus would be the ratio of the “measured inmotion” to the “measured at rest” energy of a given lightcomplex, if the volume of a light complex were the same, whethermeasured in K or in k. But this is not the case. Ifl, m, n are the direction-cosines of thewave-normals of the light in the stationary system, no energypasses through the surface elements of a spherical surface movingwith the velocity of light:—

We may therefore say that this surfacepermanently encloses the same light complex. We inquire as to thequantity of energy enclosed by this surface, viewed in systemk, that is, as to the energy of the light complex relativelyto the system k.

The spherical surface—viewed in the moving system—is anellipsoidal surface, the equation for which, at the time, is

If S is the volume of the sphere, and that of this ellipsoid,then by a simple calculation

Thus, if we call the light energy enclosed bythis surface E when it is measured in the stationary system, and when measured in themoving system, we obtain

and this formula, when , simplifies into

It is remarkable that the energy and the frequency of a lightcomplex vary with the state of motion of the observer in accordancewith the same law.

Now let the co-ordinate plane be aperfectly reflecting surface, at which the plane waves consideredin § 7 are reflected. We seek for thepressure of light exerted on the reflecting surface, and for thedirection, frequency, and intensity of the light afterreflexion.

Let the incidental light be defined by the quantities A, (referred to system K). Viewed from k thecorresponding quantities are

For the reflected light, referring the processto system k, we obtain

Finally, by transforming back to the stationarysystem K, we obtain for the reflected light

The energy (measured in the stationary system) which is incidentupon unit area of the mirror in unit time is evidently . Theenergy leaving the unit of surface of the mirror in the unit oftime is .The difference of these two expressions is, by the principle ofenergy, the work done by the pressure of light in the unit of time.If we set down this work as equal to the product Pv, where Pis the pressure of light, we obtain

In agreement with experiment and with othertheories, we obtain to a first approximation

All problems in the optics of moving bodies can be solved by themethod here employed. What is essential is, that the electric andmagnetic force of the light which is influenced by a moving body,be transformed into a system of co-ordinates at rest relatively tothe body. By this means all problems in the optics of moving bodieswill be reduced to a series of problems in the optics of stationarybodies.

§ 9. Transformation ofthe Maxwell-Hertz Equations when Convection-Currents are Taken intoAccount

We start from the equations

where

denotes timesthe density of electricity, and(u_x,u_y,u_z)the velocity-vector of the charge. If we imaginethe electric charges to be invariably coupled to small rigid bodies(ions, electrons), these equations are the electromagnetic basis ofthe Lorentzian electrodynamics and optics of moving bodies.

Let these equations be valid in the system K, and transformthem, with the assistance of the equations of transformation givenin §§ 3 and 6, to the system k. We then obtain theequations

where

and

Since—as follows from the theorem of additionof velocities (§ 5)—the vector is nothingelse than the velocity of the electric charge, measured in thesystem k, we have the proof that, on the basis of ourkinematical principles, the electrodynamic foundation of Lorentz'stheory of the electrodynamics of moving bodies is in agreement withthe principle of relativity.

In addition I may briefly remark that the following importantlaw may easily be deduced from the developed equations: If anelectrically charged body is in motion anywhere in space withoutaltering its charge when regarded from a system of co-ordinatesmoving with the body, its charge also remains—when regarded fromthe “stationary” system K—constant.

§ 10. Dynamics of theSlowly Accelerated Electron

Let there be in motion in an electromagnetic field an electricallycharged particle (in the sequel called an “electron”), for thelaw of motion of which we assume as follows:—

If the electron is at rest at a given epoch, the motion of theelectron ensues in the next instant of time according to theequations

where x, y, z denote theco-ordinates of the electron, and m the mass of theelectron, as long as its motion is slow.

Now, secondly, let the velocity of the electron at a given epochbe v. We seek the law of motion of the electron in theimmediately ensuing instants of time.

Without affecting the general character of our considerations,we may and will assume that the electron, at the moment when wegive it our attention, is at the origin of the co-ordinates, andmoves with the velocity v along the axis of X of the systemK. It is then clear that at the given moment (t=0) theelectron is at rest relatively to a system of co-ordinates which isin parallel motion with velocity v along the axis of X.

From the above assumption, in combination with the principle ofrelativity, it is clear that in the immediately ensuing time (forsmall values of t) the electron, viewed from the systemk, moves in accordance with the equations

in which the symbols , , , , , refer to the systemk. If, further, we decide that whent=x=y=z=0 then, the transformation equations of§§ 3 and 6 hold good, so that we have

With the help of these equations we transform the aboveequations of motion from system k to system K, andobtain

Taking the ordinary point of view we now inquire as to the“longitudinal” and the “transverse” mass of the movingelectron. We write the equations (A) in the form

and remark firstly that , ,are the componentsof the ponderomotive force acting upon the electron, and are soindeed as viewed in a system moving at the moment with theelectron, with the same velocity as the electron. (This force mightbe measured, for example, by a spring balance at rest in thelast-mentioned system.) Now if we call this force simply “theforce acting upon the electron,”⁹ and maintain theequation—mass × acceleration = force—and if we also decidethat the accelerations are to be measured in the stationary systemK, we derive from the above equations

With a different definition of force and acceleration we shouldnaturally obtain other values for the masses. This shows us that incomparing different theories of the motion of the electron we mustproceed very cautiously.

We remark that these results as to the mass are also valid forponderable material points, because a ponderable material point canbe made into an electron (in our sense of the word) by the additionof an electric charge, no matter how small.

We will now determine the kinetic energy of the electron. If anelectron moves from rest at the origin of co-ordinates of thesystem K along the axis of X under the action of an electrostaticforce X, it is clear that the energy withdrawn from theelectrostatic field has the value . As the electron is to be slowlyaccelerated, and consequently may not give off any energy in theform of radiation, the energy withdrawn from the electrostaticfield must be put down as equal to the energy of motion W of theelectron. Bearing in mind that during the whole process of motionwhich we are considering, the first of the equations (A)applies, we therefore obtain

Thus, when v=c, W becomes infinite. Velocitiesgreater than that of light have—as in our previous results—nopossibility of existence.

This expression for the kinetic energy must also, by virtue ofthe argument stated above, apply to ponderable masses as well.

We will now enumerate the properties of the motion of theelectron which result from the system of equations (A), andare accessible to experiment.

1. From the second equation of the system (A) it followsthat an electric force Y and a magnetic force N have an equallystrong deflective action on an electron moving with the velocityv, when . Thus wesee that it is possible by our theory to determine the velocity ofthe electron from the ratio of the magnetic power of deflexionto the electric power ofdeflexion , for any velocity, byapplying the law

   This relationship may be tested experimentally, since thevelocity of the electron can be directly measured, e.g. by means ofrapidly oscillating electric and magnetic fields.

2. From the deduction for the kinetic energy of the electron itfollows that between the potential difference, P, traversed and theacquired velocity v of the electron there must be therelationship
3. We calculate the radius of curvature of the path of theelectron when a magnetic force N is present (as the only deflectiveforce), acting perpendicularly to the velocity of the electron.From the second of the equations (A) we obtain

   or

These three relationships are a complete expression for the lawsaccording to which, by the theory here advanced, the electron mustmove.

In conclusion I wish to say that in working at the problem heredealt with I have had the loyal assistance of my friend andcolleague M. Besso, and that I am indebted to him for severalvaluable suggestions.

Footnotes

1.

The preceding memoir by Lorentz was not at this time known tothe author.

2.

i.e. to the first approximation.

3.

We shall not here discuss the inexactitude which lurks in theconcept of simultaneity of two events at approximately the sameplace, which can only be removed by an abstraction.

4.

“Time” here denotes “time of the stationary system” andalso “position of hands of the moving clock situated at the placeunder discussion.”

5.

The equations of the Lorentz transformation may be more simplydeduced directly from the condition that in virtue of thoseequations the relationx²+y²+z²=c²t²shall have as its consequence thesecond relation .

6.

That is, a body possessing spherical form when examined atrest.

7.

Not a pendulum-clock, which is physically a system to which theEarth belongs. This case had to be excluded.

8.

If, for example, X=Y=Z=L=M=0, and N 0,then from reasons of symmetry it is clear that when vchanges sign without changing its numerical value, must also change sign without changing its numericalvalue.

9.

The definition of force here given is not advantageous, as wasfirst shown by M. Planck. It is more to the point to define forcein such a way that the laws of momentum and energy assume thesimplest form.

Editor's Notes

*1

In Einstein's original paper, thesymbols (, H, Z) for the co-ordinates ofthe moving system k were introduced without explicitlydefining them. In the 1923 English translation, (X, Y, Z) wereused, creating an ambiguity between X co-ordinates in the fixedsystem K and the parallel axis in moving system k. Here andin subsequent references we use whenreferring to the axis of system k along which the system istranslating with respect to K. In addition, the reference to system, later in this sentence wasincorrectly given as “k” in the 1923 Englishtranslation.

*2

In the original 1923 Englishedition, this phrase was erroneously translated as “plainfigures”. I have used the correct “plane figures” in thisdocument.

*3

This equation was incorrectlygiven in Einstein's original paper and the 1923 English translationasa=tan^-1w_y/w_x.

*4

The exponent of c in thedenominator of the sine term of this equation was erroneously givenas 2 in the 1923 edition of this paper. It has been corrected tounity here.

*5

“X” in the 1923 Englishtranslation.

*6

In the 1923 Englishtranslation, the quantities “” and “” wereinterchanged in the second equation. They were givencorrectly in the the original 1905 paper.

*7

Erroneously given as l' inthe 1923 English translation, propagating an error, despite achange in symbols, from the original 1905 paper.

In his Annalen der Physik35, 898-908 (1911) article, Einstein explicitly allowed the gravitational field to affect the value of the velocity of light.
German original: “Uber den Einfluβ der Schwerkraft auf die Ausbreitung des Lichtes” (Princeton Univ Press)
English translation: “On the Influence of Gravitation on the Propagation of Light” (Princeton Univ Press)

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From Page 385 in English translation (italic ours):
“If (c_0) denotes the velocity of light at the coordinate origin, then the velocity of light (c) at a point with a gravitational potential (phi) will be given by the relation $$c=c_0left(1+frac{phi}{c^2}right)$$ “The principle of the constancy of the velocity of light does not hold in this theory in the formulation in which it is normally used as the basis of the ordinary theory of relativity [namely, Special Relativity].” Our comment: In this sentence, Einstein was explicit about the limited scope of the principle of the constancy of the velocity of light. To him, that principle is only valid locally.
and, also from Page 385:
“From the proposition just proved, that the velocity of light in the gravitational field is a function of place, once can easily deduce, via Huygens’s principle, that light rays propagated across a gravitational field must undergo deflection.”

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Annalen Der Physik Einstein Pdfcorporationwestern Relativity

Although the deflection of light rays is eventually found in GR to be due to their geodesics on a curved spacetime, the most important point of his ‘1911 Annalen remains: to Einstein, the location-dependent speed of light does NOT contradict the principle of the constancy of light speed (i.e, the Michelson-Morley experiment results and the Lorentz invariance). GR further strengthens his point: all of the latters are meant to hold only locally in the set of reference frames tangent to the manifold at each point.
What if along its geodesic on a curved manifold, the light ray also changes the value of its speed (while the Lorentz symmetry still locally holds in the set of tangent frames at each point, per GR)?
Furthermore, what if the value of light speed is determined by the Riemann tensor itself?
These two questions constitute the body of our work, presented in this blog and in our formal reports.

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Prerequisites:
(star) Introduction: Why a revival of Einstein ‘1911 Idea?

Annalen Der Physik Einstein

Suggested next read:
(star) Ricci scalar as a master ruler and clock: Anisotropic scaling in time direction