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On the Electrodynamics of Moving Bodies(Einstein's Theory of Special Relativity)
These pages explore Albert Einstein's life, work and philosophy. Albert Einstein was above all a great physicist and mathematician. Because the ideas involved in his professional work were generally obscure, especially in the first part of the 20th century, the public was not aware that many other physicists had been working on the same problems, nor were they in a position to understand the unique contribution of Albert Einstein in each case, which is never quite as it is explained in popular science books. Albert Einstein's work, like all scientific work, reflected advances of others and ideas that were in the air. He did not give proper references in many cases because he worked in a patent office, and had no access to proper libraries at the time he derived these theories, though he was aware of the physical principles involved and the work of others which had led to the problems he was solving. Thes circumstances, together with the great publicity that was suddenly accorded to physics beginning in 1920, to has given rise, among those who understand only popular science accounts, to three different types of misconceptions. First, many attribute to Albert Einstein ideas that really belonged to others, but were popularized when his own work became known. Second, others, having discovered this fact, insist that Einstein was a plagiarist. Third, many attribute to Einstein's scientific theories various philosophical, ethical or moral concepts, which they do not predict or relate to in any way, as Einstein would be the first to acknowledge, and in fact pointed out. Concepts and predictions that preceded Albert Einstein and accusations of plagiarismEinstein's work was based on concepts of physics and mathematics that had been proposed previously, including the problems they had raised. This trivial point is essential to explain why Einstein's work was not regarded as plagiarism by his peers. These concepts include the notion of relativity, first used by Galileo and, in the 19th century by Poincaré, thought experiments about synchronization of clocks, which Poincaré had also used, changes in mass and in size of objects at different velocities, which arise from the equations of Poincaré and Lorentz, and the equivalence of mass and energy, derived in 1902 by Poincaré, and by the Italian Olinto De Pretto, and in the speculations of the 19th century eccentric S. Toliver Preston. The recognition afforded to Einstein was not based on these particular conclusions, but on the ways in which he derived them, the ability of his theories to explain all the different effects together, and the elimination of the need to postulate a " luminiferous Aether," a construct on which all previous derivations depended. Preston, in particular, had expatiated on the properties of this Aether, which he decided was filled with a gas of particles moving at the speed of light, and for which he thought he could even derive the gas pressure. Einstein did not invent the idea that heavy bodies bend light, either. It is a consequence of a Newtonian physics. However, Einstein's General Theory of Relativity predicted twice the value of deflection that Newton's theory did. When this prediction was proven to be correct, it was obvious that Einstein had overturned the foundations of classical Newtonian physics. It was only then, after 1919, that he became world famous. A theory is not worth much if some of the different predictions of that theory are not true. Newton's greatness was not that he observed that bodies fall or that bullets move, but rather in developing a theory and a mathematical methodology for describing all these phenomena. Einstein's greatness, like that of most scientists including Newton, lies in producing theories that tied together the observed phenomena and predictions of others. Poincaré, Lorentz, de Pretto, Newton and even Preston each had different pieces of the puzzle. Einstein put the puzzle together. Those who accuse Einstein of plagiarism must consider that his work was reviewed by his peers and accepted for publication. Notably, Max Planck was on the review board of the Annalen der Physik that published his papers, and was a big enthusiast of Einstein's theory of special relativity. As for Lorentz and Poincaré, had they thought that Einstein plagiarized their work, they would surely have protested! On the contrary, Lorentz was one of those who recommended Einstein for a Nobel prize. Albert Einstein's Theories versus Philosophical and Moral interpretationsIn England, Lord Haldane published a book called "The Reign of Relativity," in which he claimed that Einstein's theory supported his own crusade against dogmatism in society and religion. He warned the archbishop of Canterbury that relativity would have great implications for theology. Haldane invited Albert Einstein to England, and hosted a grand dinner with the greatest British intellectuals. He seated Einstein next to the archbishop, who asked him what implications the theory of relativity might have for religion. Albert Einstein replied, "None. Relativity is a purely scientific matter and has nothing to do with religion." In December of 1939, a rabbi wrote to Albert Einstein about the relation between his theories and religion. Einstein replied as follows:
Nonetheless, the idea of mixing theories of physics with religious, political and ethical arguments persists. For example, one "savant" writes:
Nothing of the sort is necessarily "inherent" in or "akin" to Einstein's theories, and if it is, then it is equally inherent in the formulations of others such as Galileo, Newton and Poincaré. They were all original thinkers and used similar sorts of concepts. Einstein was fortunate to build on the work of others, and to have a bit of extra insight that was not given to his contemporaries. Albert Einstein's On the Electrodynamics of Moving BodiesEinstein's 1905 paper, On the Electrodynamics of Moving Bodies, was the key paper published in his "Annus Mirabilis" (miracle year) of 1905. He considered several problems that had taken the attention of physicists over the last 35 years or so. One of the biggest problems, was that the wave theory of light and the equations of James Clerk Maxwell posited that there must be a medium in which the waves of light (and magnetism and electrical radiation) must vibrate, a "luminiferous Aether." However, nobody was able to demonstrate the existence of this Aether. Because of the conceptual framework of the time, which believed that bodies moved relative to the absolutely motionless Aether, it was also posited that there must be a different explanation for the induction of electric current that occurs when a magnet moves and a coil of wire is at rest, and the induction of current when the magnet is at rest and a coil of wire moves. Einstein solved this problem by assuming that physical laws remain invariant regardless of whether a system is at rest or moving at some constant velocity. The speed of light, according to Einstein, would also remain constant for all observers. He posited that all motion is relative, and therefore it makes no difference whether the coil or the magnet move. Furthermore, in his system, no "luminiferous Aether" was required. The major breakthrough represented by his theory is summed up in the introduction:
In a subsequent paper in 1905, Albert Einstein was able to derive the equivalence of mass and energy from the special relativity theory (the famous E=mc^{2} ). The ideas in this paper became known as the "Special Theory of Relativity," as they applied only to the special case of bodies or systems moving at constant velocity. Einstein summed up the significance of theory as follows:
Einstein went on to develop the General Theory of relativity, which holds for bodies in accelerated motion, and which is based on the equivalence of gravity and acceleration. The General Theory unites gravity and acceleration and explains both. However, having removed the luminiferous ether in 1905, Einstein hinted later, in an address given in Leyden on Ether and Relativity theory, that he was returning to the concept of an "ether"  though one very different from the ether of 19th century physics. Einstein spent much of the latter part of his life working on a Unified Field Theory that would, he hoped, explain gravity, acceleration and electrodynamics based on the same principles. Ami Isseroff Copyright information The above introduction is copyright 2007 by the author. The text below was placed in the public domain and the adaptation is likewise in the public domain, provided that any author copying or using the text below does not claim copyright for themselves. ON THE ELECTRODYNAMICS

velocity= 
light path time interval 
where time interval is to be taken in the sense of the definition in § 1.
Let there be given a stationary rigid rod; and let its length be l as measured by a measuringrod which is also stationary. We now imagine the axis of the rod lying along the axis of x of the stationary system of coordinates, and that a uniform motion of parallel translation with velocity v along the axis of x in the direction of increasing x is then imparted to the rod. We now inquire as to the length of the moving rod, and imagine its length to be ascertained by the following two operations:
In accordance with the principle of relativity the length to be discovered by the operation (a)we will call it ``the length of the rod in the moving system''must be equal to the length l of the stationary rod.
The length to be discovered by the operation (b) we will call ``the length of the (moving) rod in the stationary system.'' This we shall determine on the basis of our two principles, and we shall find that it differs from l.
Current kinematics tacitly assumes that the lengths determined by these two operations are precisely equal, or in other words, that a moving rigid body at the epoch t may in geometrical respects be perfectly 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, clocks are placed which synchronize with the clocks of the stationary system, that is to say that their indications correspond at any instant to the ``time of the stationary system'' at the places where they happen to be. These clocks are therefore ``synchronous in the stationary system.''
We imagine further that with each clock there is a moving observer, and that these observers apply to both clocks the criterion established in § 1 for the synchronization of two clocks. Let a ray of light depart from A at the time^{4} , let it be reflected at B at the time t_{B}, and reach A again at the time t'_{A}. Taking into consideration the principle of the constancy of the velocity of light we find that
where r_{AB} denotes the length of the moving rodmeasured in the stationary system. Observers moving with the moving rod would thus find that the two clocks were not synchronous, while observers in the stationary system would declare the clocks to be synchronous.
So we see that we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of coordinates, are simultaneous, can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system.
Let us in ``stationary'' space take two systems of coordinates, i.e. two systems, each of three rigid material lines, perpendicular to one another, and issuing from a point. Let the axes of X of the two systems coincide, and their axes of Y and Z respectively be parallel. Let each system be provided with a rigid measuringrod and a number of clocks, and let the two measuringrods, and likewise all the clocks of the two systems, be in all respects alike.
Now to the origin of one of the two systems (k) let a constant velocity v be imparted in the direction of the increasing x of the other stationary system (K), and let this velocity be communicated to the axes of the coordinates, the relevant measuringrod, and the clocks. To any time of the stationary system K there then will correspond a definite position of the axes of the moving system, and from reasons of symmetry we are entitled to assume that the motion of k may be such that the 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 K by means of the stationary measuringrod, and also from the moving system k by means of the measuringrod moving with it; and that we thus obtain the coordinates x, y, z, and , , respectively. Further, let the time t of the stationary system be determined for all points thereof at which there are clocks by means of light signals in the manner indicated in § 1; similarly let the time of the moving system be determined for all points of the moving system at which there are clocks at rest relatively to that system by applying the method, given in § 1, of light signals between 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 the stationary system, there belongs a system of values , , , , determining that event relatively to the system k, and our task is now to find the system of equations connecting these quantities.
In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time.
If we place x'=xvt, it is clear that a point at rest in the system k must have a system of values x', y, z, independent of time. We first define as a function of x', y, z, and t. To do this we have to express in equations that is nothing else than the summary of the data of clocks at rest in system k, which have been synchronized according to the rule given in § 1.
From the origin of system k let a ray be emitted at the time along the Xaxis to x', and at the time be reflected thence to the origin of the coordinates, arriving there at the time ; we then must have , or, by inserting the arguments of the function and applying the principle of the constancy of the velocity of light in the stationary system:
Hence, if x' be chosen infinitesimally small,
or
It is to be noted that instead of the origin of the coordinates we might have chosen any other point for the point of origin of the ray, and the equation just obtained is therefore valid for all values of x', y, z.
An analogous considerationapplied to the axes of Y and Zit being borne in mind that light is always propagated along these axes, when viewed from the stationary system, with the velocity gives us
Since is a linear function, it follows from these equations that
where a is a function at present unknown, and where for brevity it is assumed that at the origin of k, , when t=0.
With the help of this result we easily determine the quantities , , by expressing in equations that light (as required by the principle of the constancy of the velocity of light, in combination with the principle of relativity) is also propagated with velocity c when measured in the moving system. For a ray of light emitted at the time in the direction of the increasing
But the ray moves relatively to the initial point of k, when measured in the stationary system, with the velocity cv, so that
If we insert this value of t in the equation for , we obtain
In an analogous manner we find, by considering rays moving along the two other axes, that
when
Thus
Substituting for x' its value, we obtain
where
and is an as yet unknown function of v. If no assumption whatever be made as to the initial position of the moving system and as to the zero point of , an additive constant is to be placed on the right side of each of these equations.
We now have to prove that any ray of light, measured in the moving system, is propagated with the velocity c, if, as we have assumed, this is the case in the stationary system; for we have not as yet furnished the proof that the principle of the constancy of the velocity of light is compatible with the principle of relativity.
At the time , when the origin of the coordinates is common to the two systems, let a spherical wave be emitted therefrom, and be propagated with the velocity c in system K. If (x, y, z) be a point just attained by this wave, then
Transforming this equation with the aid of our equations of transformation we obtain after a simple calculation
The wave under consideration is therefore no less a spherical wave with velocity of propagation c when viewed in the moving system. This shows that our two fundamental principles are compatible.^{5}
In the equations of transformation which have been developed there enters an unknown function of v, which we will now determine.
For this purpose we introduce a third system of coordinates , which relatively to the system k is in a state of parallel translatory motion parallel to the axis of ,^{*1} such that the origin of coordinates of system , moves with velocity v on the axis of . At the time t=0 let all three origins coincide, and when t=x=y=z=0 let the time t' of the system be zero. We call the coordinates, measured in the system , x', y', z', and by a twofold application of our equations of transformation we obtain
Since the relations between x', y', z' and x, y, z do not contain the time t, the systems K and are at rest with respect to one another, and it is clear that the transformation from K to must be the identical transformation. Thus
We now inquire into the signification of . We give our attention to that part of the axis of Y of system k which lies between and . This part of the axis of Y is a rod moving perpendicularly to its axis with velocity v relatively to system K. Its ends possess in K the coordinates
The length of the rod measured in K is therefore ; and this gives us the meaning of the function . From reasons of symmetry it is now evident that the length of a given rod moving perpendicularly to its axis, measured in the stationary system, must depend only on the velocity and not on the direction and the sense of the motion. The length of the moving rod measured in the stationary system does not change, therefore, if v and v are interchanged. Hence follows that , or
It follows from this relation and the one previously found that , so that the transformation equations which have been found become
where
We envisage a rigid sphere^{6} of radius R, at rest relatively to the moving system k, and with its centre at the origin of coordinates of k. The equation of the surface of this sphere moving relatively to the system K with velocity v is
The equation of this surface expressed in x, y, z at the time t=0 is
A rigid body which, measured in a state of rest, has the form of a sphere, therefore has in a state of motionviewed from the stationary systemthe form of an ellipsoid of revolution with the axes
Thus, whereas the Y and Z dimensions of the sphere (and therefore of every rigid body of no matter what form) do not appear modified by the motion, the X dimension appears shortened in the ratio , i.e. the greater the value of v, the greater the shortening. For v=c all moving objectsviewed from the ``stationary'' systemshrivel up into plane figures.^{*2} For velocities greater than that of light our deliberations become meaningless; we shall, however, find in what follows, that the velocity of light in our theory plays the part, physically, of an infinitely great velocity.
It is clear that the same results hold good of bodies at rest in the ``stationary'' system, viewed from a system in uniform motion.
Further, we imagine one of the clocks which are qualified to mark the time t when at rest relatively to the stationary system, and the time when at rest relatively to the moving system, to be located at the origin of the coordinates of k, and so adjusted that it marks the time . What is the rate of this clock, 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 the clock (viewed in the stationary system) is slow by seconds per second, orneglecting magnitudes of fourth and higher orderby .
From this there ensues the following peculiar consequence. If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by (up to magnitudes of fourth and higher order), t being the time occupied in the journey from A to B.
It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, and also when the points A and B coincide.
If we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result: If one of two synchronous clocks at A is moved in a closed curve with constant velocity until it returns to A, the journey lasting t seconds, then by the clock which has remained at rest the travelled clock on its arrival at A will be second slow. Thence we conclude that a balanceclock^{7} at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions.
In the system k moving along the axis of X of the system K with velocity v, let a point move in accordance with the equations
where and denote constants.
Required: the motion of the point relatively to the system K. If with the help of the equations of transformation developed in § 3 we introduce the quantities x, y, z, t into the equations of motion of the point, we obtain
Thus the law of the parallelogram of velocities is valid according to our theory only to a first approximation. We set
a is then to be looked upon as the angle between the velocities v and w. After a simple calculation we obtain^{*4}
It is worthy of remark that v and w enter into the expression for the resultant velocity in a symmetrical manner. If w also has the direction of the axis of X, we get
It follows from this equation that from a composition of two velocities which are less than c, there always results a velocity less than c. For if we set , and being positive and less than c, then
It follows, further, that the velocity of light c cannot be 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, by compounding two transformations in accordance with § 3. If in addition to the systems K and k figuring in § 3 we introduce still another system of coordinates k' moving parallel to k, its initial point moving on the axis of ^{*5} with the velocity w, we obtain equations between the quantities x, y, z, t and the corresponding quantities of k', which differ from the equations found in § 3 only in that the place of ``v'' is taken by the quantity
from which we see that such parallel transformationsnecessarilyform a group.
We have now deduced the requisite laws of the theory of kinematics corresponding to our two principles, and we proceed to show their application to electrodynamics.
Let the MaxwellHertz equations for empty space hold good for the stationary system K, so that we have
where (X, Y, Z) denotes the vector of the electric force, and (L, M, N) that of the magnetic force.
If we apply to these equations the transformation developed in § 3, by referring the electromagnetic processes to the system of coordinates there introduced, moving with the velocity v, we obtain the equations
where
Now the principle of relativity requires that if the MaxwellHertz equations for empty space hold good in system K, they also hold good in system k; that is to say that the vectors of the electric and the magnetic force(X', Y', Z') and (L', M', N')of the moving system k, which are defined by their ponderomotive effects on electric or magnetic masses respectively, satisfy the following equations:
Evidently the two systems of equations found for system k must express exactly the same thing, since both systems of equations are equivalent to the MaxwellHertz equations for system K. Since, further, the equations of the two systems agree, with the exception of the symbols for the vectors, it follows that the functions occurring in the systems of equations at corresponding places must agree, with the exception of a factor , which is common for all functions of the one system of equations, and is independent of and but depends 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 by applying the equations to the inverse transformation (from k to K), which is characterized by the velocity v, it follows, when we consider that the two systems of equations thus obtained must be identical, that . Further, from reasons of symmetry^{8} and therefore
and our equations assume the form
As to the interpretation of these equations we make the following remarks: Let a point charge of electricity have the magnitude ``one'' when measured in the stationary system K, i.e. let it when at rest in the stationary system exert a force of one dyne upon an equal quantity of electricity at a distance of one cm. By the principle of relativity this electric charge is also of the magnitude ``one'' when measured in the moving system. If this quantity of electricity is at rest relatively to the stationary system, then by definition the vector (X, Y, Z) is equal to the force acting upon it. If the quantity of electricity is at rest relatively to the moving system (at least at the relevant instant), then the force acting upon it, measured in the moving system, is equal to the vector (X', Y', Z'). Consequently the first three equations above allow themselves to be clothed in words in the two following ways:
The analogy holds with ``magnetomotive forces.'' We see that electromotive force plays in the developed theory merely the part of an auxiliary concept, which owes its introduction to the circumstance that electric and magnetic forces do not exist independently of the state of motion of the system of coordinates.
Furthermore it is clear that the asymmetry mentioned in the introduction as arising when we consider the currents produced by the relative motion of a magnet and a conductor, now disappears. Moreover, questions as to the ``seat'' of electrodynamic electromotive forces (unipolar machines) now have no point.
In the system K, very far from the origin of coordinates, let there be a source of electrodynamic waves, which in a part of space containing the origin of coordinates may be represented to a sufficient degree of approximation by the equations
where
Here (X_{0}, Y_{0}, Z_{0}) and (L_{0}, M_{0}, N_{0}) are the vectors defining the amplitude of the wavetrain, and l, m, n the directioncosines of the wavenormals. We wish to know the constitution of these waves, when they are examined by an observer at rest in the moving system K.
Applying the equations of transformation found in § 6 for electric and magnetic forces, and those found in § 3 for the coordinates and the time, we obtain directly
where
From the equation for it follows that if an observer is moving with velocity v relatively to an infinitely distant source of light of frequency , in such a way that the connecting line ``sourceobserver'' makes the angle with the velocity of the observer referred to a system of coordinates which is at rest relatively to the source of light, the frequency of the light perceived by the observer is given by the equation
This is Doppler's principle for any velocities whatever. When the equation assumes the perspicuous form
We see that, in contrast with the customary view, when .
If we call the angle between the wavenormal (direction of the ray) in the moving system and the connecting line ``sourceobserver'' , the equation for ^{*6} assumes the form
This equation expresses the law of aberration in its most general form. If , the equation becomes simply
We still have to find the amplitude of the waves, as it appears in the moving system. If we call the amplitude of the electric or magnetic force A or A' respectively, accordingly as it is measured in the stationary system or in the moving system, we obtain
which equation, if , simplifies into
It follows from these results that to an observer approaching a source of light with the velocity c, this source of light must appear of infinite intensity.
Since equals the energy of light per unit of volume, we have to regard , by the principle of relativity, as the energy of light in the moving system. Thus would be the ratio of the ``measured in motion'' to the ``measured at rest'' energy of a given light complex, if the volume of a light complex were the same, whether measured in K or in k. But this is not the case. If l, m, n are the directioncosines of the wavenormals of the light in the stationary system, no energy passes through the surface elements of a spherical surface moving with the velocity of light:
We may therefore say that this surface permanently encloses the same light complex. We inquire as to the quantity of energy enclosed by this surface, viewed in system k, that is, as to the energy of the light complex relatively to the system k.
The spherical surfaceviewed in the moving systemis an ellipsoidal surface, the equation for which, at the time , is
If S is the volume of the sphere, and S' that of this ellipsoid, then by a simple calculation
Thus, if we call the light energy enclosed by this surface E when it is measured in the stationary system, and E' when measured in the moving system, we obtain
and this formula, when , simplifies into
It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law.
Now let the coordinate plane be a perfectly reflecting surface, at which the plane waves considered in § 7 are reflected. We seek for the pressure of light exerted on the reflecting surface, and for the direction, frequency, and intensity of the light after reflexion.
Let the incidental light be defined by the quantities A, , (referred to system K). Viewed from k the corresponding quantities are
For the reflected light, referring the process to system k, we obtain
Finally, by transforming back to the stationary system K, we obtain for the reflected light
The energy (measured in the stationary system) which is incident upon unit area of the mirror in unit time is evidently . The energy leaving the unit of surface of the mirror in the unit of time is . The difference of these two expressions is, by the principle 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 product Pv, where P is the pressure of light, we obtain
In agreement with experiment and with other theories, we obtain to a first approximation
All problems in the optics of moving bodies can be solved by the method here employed. What is essential is, that the electric and magnetic force of the light which is influenced by a moving body, be transformed into a system of coordinates at rest relatively to the body. By this means all problems in the optics of moving bodies will be reduced to a series of problems in the optics of stationary bodies.
We start from the equations
where
denotes times the density of electricity, and (u_{x},u_{y},u_{z}) the velocityvector of the charge. If we imagine the electric charges to be invariably coupled to small rigid bodies (ions, electrons), these equations are the electromagnetic basis of the Lorentzian electrodynamics and optics of moving bodies.
Let these equations be valid in the system K, and transform them, with the assistance of the equations of transformation given in §§ 3 and 6, to the system k. We then obtain the equations
where
and
Sinceas follows from the theorem of addition of velocities (§ 5)the vector is nothing else than the velocity of the electric charge, measured in the system k, we have the proof that, on the basis of our kinematical principles, the electrodynamic foundation of Lorentz's theory of the electrodynamics of moving bodies is in agreement with the principle of relativity.
In addition I may briefly remark that the following important law may easily be deduced from the developed equations: If an electrically charged body is in motion anywhere in space without altering its charge when regarded from a system of coordinates moving with the body, its charge also remainswhen regarded from the ``stationary'' system Kconstant.
Let there be in motion in an electromagnetic field an electrically charged particle (in the sequel [following] called an ``electron''), for the law of motion of which we assume as follows:
If the electron is at rest at a given epoch, the motion of the electron ensues in the next instant of time according to the equations
where x, y, z denote the coordinates of the electron, and m the mass of the electron, as long as its motion is slow.
Now, secondly, let the velocity of the electron at a given epoch be v. We seek the law of motion of the electron in the immediately ensuing instants of time.
Without affecting the general character of our considerations, we may and will assume that the electron, at the moment when we give it our attention, is at the origin of the coordinates, and moves with the velocity v along the axis of X of the system K. It is then clear that at the given moment (t=0) the electron is at rest relatively to a system of coordinates which is in parallel motion with velocity v along the axis of X.
From the above assumption, in combination with the principle of relativity, it is clear that in the immediately ensuing time (for small values of t) the electron, viewed from the system k, moves in accordance with the equations
in which the symbols , , , X', Y', Z' refer to the system k. If, further, we decide that when t=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 above equations of motion from system k to system K, and obtain
· · · bsp;(A) 
Taking the ordinary point of view we now inquire as to the ``longitudinal'' and the ``transverse'' mass of the moving electron. We write the equations (A) in the form
and remark firstly that , , are the components of the ponderomotive force acting upon the electron, and are so indeed as viewed in a system moving at the moment with the electron, with the same velocity as the electron. (This force might be measured, for example, by a spring balance at rest in the lastmentioned system.) Now if we call this force simply ``the force acting upon the electron,''^{9}
With a different definition of force and acceleration we should naturally obtain other values for the masses. This shows us that in comparing different theories of the motion of the electron we must proceed very cautiously.
We remark that these results as to the mass are also valid for ponderable material points, because a ponderable material point can be made into an electron (in our sense of the word) by the addition of an electric charge, no matter how small.
We will now determine the kinetic energy of the electron. If an electron moves from rest at the origin of coordinates of the system K along the axis of X under the action of an electrostatic force X, it is clear that the energy withdrawn from the electrostatic field has the value . As the electron is to be slowly accelerated, and consequently may not give off any energy in the form of radiation, the energy withdrawn from the electrostatic field must be put down as equal to the energy of motion W of the electron. Bearing in mind that during the whole process of motion which we are considering, the first of the equations (A) applies, we therefore obtain
Thus, when v=c, W becomes infinite. Velocities greater than that of light haveas in our previous resultsno possibility of existence.
This expression for the kinetic energy must also, by virtue of the argument stated above, apply to ponderable masses as well.
We will now enumerate the properties of the motion of the electron which result from the system of equations (A), and are accessible to experiment.
This relationship may be tested experimentally, since the velocity of the electron can be directly measured, e.g. by means of rapidly oscillating electric and magnetic fields..
or
These three relationships are a complete expression for the laws according to which, by the theory here advanced, the electron must move. move.
In conclusion I wish to say that in working at the problem here dealt with I have had the loyal assistance of my friend and colleague M. Besso, and that I am indebted to him for several valuable suggestions.
About this EditionThis edition of Einstein's On the Electrodynamics of Moving Bodies is based on the English translation of his original 1905 Germanlanguage paper (published as Zur Elektrodynamik bewegter Körper, in Annalen der Physik. 17:891, 1905) which appeared in the book The Principle of Relativity, published in 1923 by Methuen and Company, Ltd. of London. Most of the papers in that collection are English translations by W. Perrett and G.B. Jeffery from the German Das Relativatsprinzip, 4th ed., published by in 1922 by Tuebner. All of these sources are now in the public domain; this document, derived from them, remains in the public domain and may be reproduced in any manner or medium without permission, restriction, attribution, or compensation. Numbered footnotes are as they appeared in the 1923 edition; editor's notes are preceded by asterisks (*) and appear in sans serif type. The 1923 English translation modified the notation used in Einstein's 1905 paper to conform to that in use by the 1920's; for example, c denotes the speed of light, as opposed the V used by Einstein in 1905. This electronic edition was prepared by John Walker in November 1999 and adapted by zionismisrael.com in 2007. 
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