# Translation:On the Electrodynamics of Moving Bodies

On the Electrodynamics of Moving Bodies  (1920)
by Albert Einstein, translated from German by Meghnad Saha and  Wikisource
German original: Einstein, Albert (1905), "Zur Elektrodynamik bewegter Körper", Annalen der Physik 322 (10): 891–921. (Received June 30, 1905; published September 26, 1905). See also the 1923 edition.
• Saha's translation: The Principle of Relativity: Original Papers by A. Einstein and H. Minkowski, University of Calcutta, 1920, pp. 1-34, Online.
• In this Wikisource edition, Saha's notation was replaced by Einstein's original notation. Also many passages were re-written and translated from the German original. (See Saha's original for comparison).

On the electrodynamics of moving bodies;

by A. Einstein.

It is known that the application of Maxwell's electrodynamics, as ordinarily conceived at the present time, to moving bodies, leads to asymmetries which don't seem to be connected with the phenomena. Let us, for example, think of the mutual action between a magnet and a conductor. The observed phenomenon in this case depends only on the relative motion of the conductor and the magnet, while according to the usual conception, a strict distinction must be made between the cases where the one or the other of the bodies is in motion. If, for example, the magnet moves and the conductor is at rest, then an electric field of certain energy-value is produced in the neighbourhood of the magnet, which excites a current in those parts of the field where a conductor exists. But if the magnet be at rest and the conductor be set in motion, no electric field is produced in the neighbourhood of the magnet, but an electromotive force is produced in the conductor which corresponds to no energy per se; however, this causes – equality of the relative motion in both considered cases is assumed – an electric current of the same magnitude and the same course, as the electric force in the first case.

Examples of a similar kind, as well as the unsuccessful attempts to substantiate the motion of the earth relative to the "light-medium", lead us to the supposition that not only in mechanics, but also in electrodynamics, no properties of the phenomena correspond to the concept of absolute rest, but rather that for all coordinate systems for which the mechanical equations hold, the equivalent electrodynamical and optical equations hold also, as has already been shown for magnitudes of the first order. In the following we will elevate this guess to a presupposition (whose content we shall subsequently call the "Principle of Relativity") and introduce the further assumption, — an assumption which is only apparently irreconcilable with the former one — that light in empty space always propagates with a velocity V which is independent of the state of motion of the emitting body. These two assumptions are quite sufficient to give us a simple and consistent theory of electrodynamics of moving bodies on the basis of the Maxwellian theory for bodies at rest. The introduction of a "luminiferous æther" will be proved to be superfluous in so far, as according to the conceptions which will be developed, we shall introduce neither a "space absolutely at rest" endowed with special properties, nor shall we associate a velocity-vector with a point in which electro-magnetic processes take place.

Like every other theory in electrodynamics, the theory to be developed is based on the kinematics of rigid bodies; since in the arguments of every theory, we have to do with relations between rigid bodies (co-ordinate system), clocks, and electromagnetic processes. An insufficient consideration of these circumstances is the cause of difficulties with which the electrodynamics of moving bodies has to fight at present.

## I. Kinematical Part.

### § 1. Definition of Simultaneity.

Let us have a co-ordinate system, in which the Newtonian equations hold. For verbally distinguishing this system from another which will be introduced hereafter, and for clarification of the idea, we shall call it the "stationary system."

If a material point be at rest in this coordinate-system, then its position in this system can be found out by a measuring rod by using the methods of Euclidean Geometry, and can be expressed in Cartesian co-ordinates.

If we wish to describe the motion of a material point, the values of its coordinates must be expressed as functions of time. Now, it is always to be borne in mind that such a mathematical definition has a physical meaning only, when we have a clear notion of what is meant by "time". We have to take into consideration the fact that those of our assessments, in which time plays a role, are always judgements on simultaneous events. For example, we say "that a train arrives here at 7 o'clock"; this means "that the exact pointing of the little hand of my watch to 7, and the arrival of the train are simultaneous events".[1]

It may appear that all difficulties connected with the definition of "time" can be removed when in place of "time", we substitute "the position of the little hand of my watch". Such a definition is in fact sufficient, when it is required to define time exclusively for the place at which the clock is stationed. But the definition is not sufficient when it is required to chronologically connect a series of events taking place at different places — or what amounts to the same thing — to chronologically evaluate the occurrence of events, which take place at places distant from the clock.

However, to chronologically evaluate the events, we can satisfy ourselves by assuming an observer who is stationed at the origin of coordinates with the clock, and who associates a signal of light – giving testimony of the event to be estimated and which comes to him through empty space – with the corresponding position of the hands of the clock. Deficiency of such an association is — as we know by experience — that it depends on the position of the observer provided with the clock. We can attain a far more practical result by the following treatment.

If an observer be stationed at A with a clock, he can estimate the time of events occurring in the immediate neighbourhood of A by looking for the position of the hands of the clock, which are simultaneous with the event. If a clock be stationed also at point B in space, — we should add that "the clock is exactly of the same nature as the one at A", — then the chronological evaluations of the events occurring in the immediate vicinity of B, is possible for an observer located in B. But without further premises, it is not possible to chronologically compare the events at B with the events at A. We have hitherto only an "A-time", and a "B-time", but no "time" common to A and B. This last time (i.e., common time) can now be defined, however, if we establish by definition that the "time" which light requires in travelling from A to B is equivalent to the "time" which light requires in travelling from B to A. For example, a ray of light proceeds from A at "A-time" ${\displaystyle t_{A}}$ towards B, arrives and is reflected from B at B-time ${\displaystyle t_{B}}$, and returns to A at "A-time" ${\displaystyle t'_{A}}$. According to the definition, both clocks are synchronous, if

${\displaystyle t_{B}-t_{A}=t'_{A}-t_{B}\,}$

We assume that this definition of synchronism is possible without involving any inconsistency, for any number of points, therefore the following relations hold:

1. If the clock at B be synchronous with the clock at A, then the clock at A is synchronous with the clock at B.

2. If the clock at A be synchronous with the clock at B as well as with the clock at C, then also the clocks at B and C are synchronous.

Thus with the help of certain (imagined) physical experiences, we have established what we understand when we speak of clocks at rest at different places, and synchronous with one another; and thereby we have arrived at a definition of "synchronism" and "time". The "time" of an event is the simultaneous indication of a stationary clock located at the place of the event, which is synchronous with a certain stationary clock for all time determinations.

In accordance with experience we shall assume that the magnitude

${\displaystyle {\frac {2{\overline {AB}}}{t'_{A}-t_{A}}}=V,}$

is a universal constant (the speed of light in empty space).

We have defined time essentially with a clock at rest in a stationary system. On account of its affiliation to the stationary system, we call the time defined in this way "the time of the stationary system."

### § 2. On the relativity of lengths and times.

The following considerations are based on the Principle of Relativity and on the Principle of Constancy of the velocity of light, both of which we define in the following way.

1. The laws according to which the states of physical systems alter are independent of the choice, to which of two co-ordinate systems (having a uniform translatory motion relative to each other) these state changes are related.

2. Every ray of light moves in the "stationary" co-ordinate system with the definite velocity V, the velocity being independent of the condition, whether this ray of light is emitted by a body at rest or in motion. Here

 velocity = Path of Light Interval of time

where "interval of time," is to be understood as defined in § 1.

Let us have a rigid stationary rod; it has a length l, when measured by a measuring rod also at rest. We shall assume that the axis of the rod is the X-axis of the stationary coordinate-system. Let us now impart to the rod a uniform velocity v, parallel to the axis of X and in the increasing direction of x. What is the length of the moving rod; this can be thought as obtained by either of these operations:

a) The observer provided with the measuring rod moves along with the rod to be measured, and measures by direct superposition the length of the rod, just as if the observer, the measuring rod, and the rod to be measured were at rest.

b) The observer measures, at which points of the stationary system the ends of the rod to be measured are located at a particular time t, by means of clocks placed in the stationary system (the clocks being synchronous as defined in § 1). The distance between these two points, measured by the previously used measuring rod, this time it being at rest, is also a length, which we may call the "length of the rod."

According to the Principle of Relativity, the length found out by the operation a), which we may call "the length of the rod in the moving system", is equal to the length l of the stationary rod.

The length which is found out by operation b), may be called "the length of the (moving) rod in the stationary system". This length is to be estimated on the basis of our two principles, and we shall find it to be different from l.

In the generally employed kinematics, we tacitly assume that the lengths defined by these two operations are equal, or in other words, that at an epoch of time t, a moving rigid body is geometrically replaceable by the same body, when it is at rest in a particular location.

Let us furthermore suppose that the two clocks synchronous with the clocks in the system at rest are brought to the ends A and B of a rod, i.e., the indications of the clocks correspond to the "time of the stationary system" at the places where they happen to arrive; these clocks are therefore "synchronous in the stationary system".

We further imagine that there are two observers at the two clocks, and moving with them, and that these observers apply the criterion for synchronism stated in §1, to the two clocks. At the time[2] tA, a ray of light goes out from A, is reflected from B at the time ${\displaystyle t_{B}}$, and arrives back at A at time t'A. Taking into consideration the principle of constancy of the velocity of light, we have

${\displaystyle t_{B}-t_{A}={\frac {r_{AB}}{V-v}}}$
and
${\displaystyle t'_{A}-t_{B}={\frac {r_{AB}}{V+v}},}$

where ${\displaystyle r_{AB}}$ is the length of the moving rod, measured in the stationary system. Therefore the observers moving with the moving rod, thus would not find the clocks synchronous, though the observers in the stationary system would declare the clocks to be synchronous.

We therefore see that we can attach no absolute significance to the concept of synchronism; two events which are synchronous when viewed from one coordinate-system, will not be synchronous when viewed from a system moving relatively to this system.

### § 3. Theory of co-ordinate and time-transformation from a stationary system to a system which moves relatively to this with uniform translatory velocity.

Let there be given, in "stationary" space, two co-ordinate systems, i.e., two systems of three rigid material lines (mutually perpendicular) issuing from a point. Let the X-axes of each coincide with one another, and the Y and Z-axes be parallel. Let a rigid measuring rod, and a number of clocks be given to each of the systems, and let the two rods and all clocks in both systems be exactly alike each other.

Let the initial point of one of the systems (k) have a (constant) velocity v in the direction of increasing x of the other system, the stationary system (K), and the velocity being also communicated to the coordinate axes, the considered rod, as well as the clocks. Any time t of the stationary system K corresponds to a definite position of the axes of the moving system, and we can assume due to reasons of symmetry, that the motion of k is of such a kind, that the axes of the moving system at time t (by "t", we always mean the time in the stationary system) are parallel to the axes of the stationary system.

We suppose that the space is measured by the stationary measuring rod placed in the stationary system K, as well as by the moving measuring rod placed in the moving system k, and we thus obtain the co-ordinates x, y, z and ξ, η, ζ, respectively. Furthermore, let the time t be determined for each point of the stationary system by means of the clocks which are placed in the stationary system, with the help of light-signals as described in § 1. Let also the time τ of the moving system be determined for each point of the moving system (in which there are clocks which are at rest relative to the moving system), by means of the method of light signals between these points (in which there are clocks) in the manner described in § 1.

To every value of x, y, z, t which fully determines the position and time of an event in the stationary system, there corresponds a system of values ξ, η, ζ, τ determined relative to system k; now the problem is to find out the system of equations connecting these magnitudes.

Primarily it is clear that on account of the property of homogeneity which we ascribe to time and space, the equations must be linear.

If we put ${\displaystyle x'=x-vt}$, then it is clear that at a point at rest in the system k, we have a system of values x', y, z which are independent of time. At first let us find out τ as a function of x', y, z, t. For this purpose we have to express in equations the fact, that τ is nothing other than the time given by the clocks which are at rest in the system k, which must be made synchronous in the manner given by the rule in § 1.

Let a ray of light be sent at time ${\displaystyle \tau _{0}}$ from the origin of the system k along the X-axis towards the increasing ${\displaystyle x'}$, and let it be reflected from that place at time ${\displaystyle \tau _{1}}$ towards the origin of co-ordinates, where it arrives at time ${\displaystyle \tau _{2}}$; then we must have

${\displaystyle {\frac {1}{2}}(\tau _{0}+\tau _{2})=\tau _{1}}$

or, by adding the arguments of function τ, and applying the principle of constancy of the velocity of light in the stationary system, we have

${\displaystyle {\frac {1}{2}}\left[\tau (0,0,0,t)+\tau \left(0,0,0,\left\{t+{\frac {x'}{V-v}}+{\frac {x'}{V+v}}\right\}\right)\right]}$
${\displaystyle =\tau \left(x',0,0,t+{\frac {x'}{V-v}}\right).}$
From this it follows, when we choose ${\displaystyle x'}$ infinitely small:
${\displaystyle {\frac {1}{2}}\left({\frac {1}{V-v}}+{\frac {1}{V+v}}\right){\frac {\partial \tau }{\partial t}}={\frac {\partial \tau }{\partial x'}}+{\frac {1}{V-v}}{\frac {\partial \tau }{\partial t}},}$

or

${\displaystyle {\frac {\partial \tau }{\partial x'}}+{\frac {v}{V^{2}-v^{2}}}{\frac {\partial \tau }{\partial t}}=0.}$

It is to be noticed that instead of the origin of coordinates, we could select any other point as the starting point for rays of light, and therefore the above equation holds for all values of x', y, z.

A similar conception, being applied to the Y- and Z-axis, gives us, when we take into consideration the fact that light when viewed from the stationary system, is always propagated along those axes with the velocity ${\displaystyle {\sqrt {V^{2}-v^{2}}}}$, we have the equations:

 ${\displaystyle {\frac {\partial \tau }{\partial y}}=0,}$ ${\displaystyle {\frac {\partial \tau }{\partial z}}=0.}$

From these equations it follows, because τ is a linear function:

${\displaystyle \tau =a\left(t-{\frac {v}{V^{2}-v^{2}}}x'\right),}$

where a is a yet unknown function of φ(v) and for brevities sake it is assumed that at the origin of k, t = 0 for τ = 0 .

With the help of these results it is easy to obtain the magnitudes ξ, η, ζ, if we express (by means of equations) the fact that light (when measured in the moving system) is always propagated with the constant velocity V (as the principle of constancy of light velocity in conjunction with the principle of relativity requires). For a ray sent in the direction of increasing ξ at time τ = 0, we have

${\displaystyle \xi =V\tau ,\,}$

or

${\displaystyle \xi =aV\left(t-{\frac {v}{V^{2}-v^{2}}}x'\right).}$

However, the ray of light now moves relative to the origin of k with a velocity V - v, measured in the stationary system; therefore we have

${\displaystyle {\frac {x'}{V-v}}=t.}$

Substituting this value of t in the equation for ξ, we obtain

${\displaystyle \xi =a{\frac {V^{2}}{V^{2}-v^{2}}}x'.}$

In an analogous manner, by considering the rays of light moving along the other two axis, we obtain

${\displaystyle \eta =V\tau =aV\left(t-{\frac {v}{V^{2}-v^{2}}}x'\right),}$

where

${\displaystyle {\frac {y}{\sqrt {V^{2}-v^{2}}}}=t,\ x'=0,}$

therefore

${\displaystyle \eta =a{\frac {V}{\sqrt {V^{2}-v^{2}}}}y}$

and

${\displaystyle \zeta =a{\frac {V}{\sqrt {V^{2}-v^{2}}}}z.}$

If for x', we substitute its value, we obtain

${\displaystyle {\begin{array}{lll}\tau &=&\varphi (v)\beta (t-{\frac {v}{V^{2}}}x),\\\\\xi &=&\varphi (v)\beta (x-vt),\\\\\eta &=&\varphi (v)y,\\\\\zeta &=&\varphi (v)z,\end{array}}}$

where

${\displaystyle \beta ={\frac {1}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}}$

and φ is a yet unknown function of v. If we make no assumption about the initial position of the moving system and about the null-point of τ, then an additive constant is to be added to the right-hand side of the equations.

We have now to show, that every ray of light propagates (measured in the moving system) with a velocity V, if, as we have assumed, this is also the case in the stationary system; for we have not as yet provided any proof in support of the assumption that the principle of constant light-velocity is reconcilable with the principle of relativity.

At time τ = t = 0 let a spherical wave be sent out from the common origin of the two systems of co-ordinates, and let it spread with a velocity V in the system K. If (x, y, z) be a point reached by the wave, we have

${\displaystyle x^{2}+y^{2}+z^{2}=V^{2}t^{2}.\,}$

With the aid of our transformation-equations, let us transform this equation, and we obtain by a simple calculation,

${\displaystyle \xi ^{2}+\eta ^{2}+\zeta ^{2}=V^{2}\tau ^{2}.\,}$

Therefore the wave is propagated in the moving system with the same velocity V, and as a spherical wave. Therefore we show that the two fundamental principles are mutually reconcilable.

In the transformations derived, there is still an undetermined function φ of v, and we now proceed to find it out.

Let us introduce for this purpose a third co-ordinate system K', which is set in motion relative to the system k, the motion being parallel to the Ξ-axis, so that the velocity of the origin be -v upon the Ξ-axis. At the time t = 0, all the initial co-ordinate points coincide, and for t = x = y = z = 0, the time t' of the system K' = 0. We shall say that x' y' z' are the co-ordinates measured in the system K', then by a two-fold application of the transformation-equations, we obtain

${\displaystyle {\begin{array}{lllll}t'&=&\varphi (-v)\beta (-v)\left\{\tau +{\frac {v}{V^{2}}}\xi \right\}&=&\varphi (v)\varphi (-v)t,\\\\x'&=&\varphi (-v)\beta (-v)\{\xi +v\tau \}&=&\varphi (v)\varphi (-v)x,\\\\y'&=&\varphi (-v)\eta &=&\varphi (v)\varphi (-v)y,\\\\z'&=&\varphi (-v)\zeta &=&\varphi (v)\varphi (-v)z.\end{array}}}$

Since the relations between x', y', z' and x, y, z do not contain time t, therefore K and K' are relatively at rest. It is clear that the transformation from K to K' has to be identical. Hence:

${\displaystyle \varphi (v)\varphi (-v)=1.\,}$

We now ask after the meaning of φ(v). Let us now turn our attention to the part of the Y-axis of the system k, lying between ξ = 0, η = 0, ζ = 0 and ξ = 0, η = l, ζ = 0. Let this piece of the Y-axis be a rod moving with the velocity v relative to the system K and perpendicular to its axis; the ends of the rod having therefore the co-ordinates in K:

${\displaystyle x_{1}=vt,\ y_{1}={\frac {l}{\varphi (v)}},\ z_{1}=0}$

and

${\displaystyle x_{2}=vt,\ y_{2}=0,\ z_{2}=0.}$

Therefore the length of the rod measured in K, is l/φ(v); thus the meaning of the function φ is given. From grounds of symmetry it is now clear, that the length (measured in the stationary system) of a certain rod, which moves perpendicular to its axis, can only be dependent on the speed, but not from the direction and the sense of motion. Thus, the length of the moving rod as measured in the stationary system does not alter, if v is replaced by -v. From that it follows:

${\displaystyle {\frac {l}{\varphi (v)}}={\frac {l}{\varphi (-v)}},}$

or

${\displaystyle \varphi (v)=\varphi (-v).\,}$

From this and the relation found above, it follows, that it must be φ(v) = 1, so that the obtained transformation equations become:

${\displaystyle {\begin{array}{lll}\tau &=&\beta (t-{\frac {v}{V^{2}}}x),\\\\\xi &=&\beta (x-vt),\\\\\eta &=&y,\\\\\zeta &=&z,\end{array}}}$

where

${\displaystyle \beta ={\frac {1}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}.}$

### § 4. The physical significance of the equations obtained concerning moving rigid bodies and moving clocks.

Let us consider a rigid sphere[3] of radius R which is at rest relative to the moving system k, and whose centre coincides with the origin of k, then the equation of the surface of this sphere, which is moving with a velocity v relative to the system K, is:

${\displaystyle \xi ^{2}+\eta ^{2}+\zeta ^{2}=R^{2}.\,}$

At time t = 0 the equation of this surface is expressed by means of x, y, z, as

${\displaystyle {\frac {x^{2}}{\left({\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}\right)^{2}}}+y^{2}+z^{2}=R^{2}.}$

A rigid body, which has the shape of a sphere when measured in a stationary state, has therefore in the moving condition — when considered from the stationary system – the shape of a rotational ellipsoid with axes

${\displaystyle R{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}},\ R,\ R.}$

While the Y and Z dimensions of the sphere (therefore of any figure also) do not appear to be modified by the motion, the X dimension is shortened in the ratio ${\displaystyle 1\ :\ {\sqrt {1-(v/V)^{2}}}}$; the shortening is the larger, the larger v is. For v = V, all moving bodies, when considered from a "stationary" system, shrink into plane objects. For superluminal velocities, our propositions become meaningless; besides, we will find in the following considerations, that the speed of light physically plays the part of an infinitely large velocity.

It is clear that similar results hold for stationary bodies in a "stationary" system when considered from a uniformly moving system. –

Let us furthermore consider that a clock which is capable of giving the time t when at rest relative to the stationary system, and the time τ when at rest relative to the moving system; suppose it to be placed at the origin of the moving system k, and to be so arranged that it gives the time τ. How fast is the rate of this clock, when viewed from the stationary system?

Between the magnitudes x, t and τ, which refer to the location of that clock, evidently the equations are given

${\displaystyle \tau ={\frac {1}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}(t-{\frac {v}{V^{2}}}x),}$

and

${\displaystyle x=vt.\,}$

Hence

${\displaystyle \tau =t{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}=t-\left(1-{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}\right)t}$.

Therefore the clock loses ${\displaystyle \left(1-{\sqrt {1-(v/V)^{2}}}\right)}$ seconds (as seen in the stationary system) per second, or neglecting magnitudes of fourth and higher order, ${\displaystyle {\tfrac {1}{2}}(v/V)^{2}}$ seconds.

From this, the following peculiar consequence follows. Suppose at two points A and B of K, viewed from the stationary system, two clocks which are synchronous. Suppose the clock at A to be set in motion with velocity v in the line joining it with B, then after the arrival of the clock at B, they will no longer be found synchronous, but the clock which was set in motion from A to B will lag behind the clock which had been all along at B by an amount of ${\displaystyle {\tfrac {1}{2}}tv^{2}/V^{2}}$ seconds (neglecting magnitudes of fourth and higher order), when t is the time required from A to B.

We see forthwith that the result holds also when the clock moves from A to B by a polygonal line, and also when A and B coincide.

If we assume that the result obtained for a polygonal line holds also for a curved line, we obtain the following theorem: If at A, there be two synchronous clocks, and if we set in motion one of them with a constant velocity along a closed curve till it comes back to A, the journey being completed in t-seconds, then after arrival in A, the last mentioned clock will be behind the stationary one by ${\displaystyle {\frac {1}{2}}t(v/V)^{2}}$ seconds. From this, we conclude that a balance wheel clock placed at the equator must be slower by a very small amount than a similarly constructed clock which is placed at the pole, all other conditions being identical.

### § 5. Addition-theorem of velocities.

Let a point move in the system k (which moves with velocity v along the X-axis of the system K) according to the equations

${\displaystyle {\begin{array}{ll}\xi =&w_{\xi }\tau ,\\\\\eta =&w_{\eta }\tau ,\\\\\zeta =&0,\end{array}}}$

where wξ and wη are constants.

It is required to find out the motion of the point relative to the system K. If we now introduce the magnitudes x, y, z, t into the equations of motion by aid of the transformation equations developed in § 3, we obtain

${\displaystyle {\begin{array}{lll}x&=&{\frac {w_{\xi }+v}{1+{\frac {vw_{\xi }}{V^{2}}}}}t,\\\\y&=&{\frac {\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}{1+{\frac {vw_{\xi }}{V^{2}}}}}w_{\eta }t,\\\\z&=&0.\end{array}}}$

The law of parallelogram of velocities only holds up to the first order of approximation according to our theory. We put

${\displaystyle {\begin{array}{lll}U^{2}&=&\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2},\\\\w^{2}&=&w_{\xi }^{2}+w_{\eta }^{2},\end{array}}}$

and

${\displaystyle \alpha =arctg{\frac {w_{\eta }}{w_{\xi }}};}$
i.e., α is put equal to the angle between the velocities v and w. Then we have after simple calculation —
${\displaystyle U={\frac {\sqrt {(v^{2}+w^{2}+2v\ w\ \cos \alpha )-\left({\frac {v\ w\ \sin \alpha }{V}}\right)^{2}}}{1+{\frac {v\ w\ \cos \alpha }{V^{2}}}}}.}$

It should be noticed that v and w symmetrically enter into the expression for the resulting velocity. If w has also the direction of the X-axis (Ξ-axis), we obtain:

${\displaystyle U={\frac {v+w}{1+{\frac {vw}{V^{2}}}}}.}$

From this equation, we see that by combining two velocities, each of which is smaller than V, we obtain a velocity which is always smaller than V. If we put ${\displaystyle v=V-\varkappa ,\ w=V-\lambda }$, where ${\displaystyle \varkappa }$ and λ are positive and smaller than V, then:

${\displaystyle U=V{\frac {2V-\varkappa -\lambda }{2V-\varkappa -\lambda +{\frac {\varkappa \lambda }{V}}}}

It furthermore follows, that the velocity of light V cannot be altered by adding to it a "subliminal velocity". For this case, we obtain

${\displaystyle U={\frac {V+w}{1+{\frac {w}{v}}}}=V.}$

We have obtained the formula for U for the case when v and w have the same direction; it can also be obtained by combining two transformations according § 3. If in addition to the systems K and k used in § 3, we introduce the third system k' (moving parallel to k), whose initial point moves parallel to the Ξ-axis with velocity w, then between the magnitudes x, y, z, t and the corresponding magnitudes of k', we obtain equations which differ from the equations in § 3, only in the respect that in place of "v", we shall have to write,

${\displaystyle {\frac {v+w}{1+{\frac {vw}{V^{2}}}}}.}$
We see that such parallel transformations form (as it has to be) a group.

We have deduced the kinematics corresponding to our two fundamental principles for the laws necessary for us, and we shall now pass over to their application in electrodynamics.

## II. Electrodynamical Part.

### § 6. Transformation of Maxwell-Hertz equations for pure vacuum. On the nature of the electromotive force caused by motion in a magnetic field.

The Maxwell-Hertz equations for pure vacuum may hold for the stationary system K, so that

${\displaystyle {\begin{array}{llllll}{\frac {1}{V}}{\frac {\partial X}{\partial t}}&=&{\frac {\partial N}{\partial y}}-{\frac {\partial M}{\partial z}},&\ {\frac {1}{V}}{\frac {\partial L}{\partial t}}&=&{\frac {\partial Y}{\partial z}}-{\frac {\partial Z}{\partial y}},\\\\{\frac {1}{V}}{\frac {\partial Y}{\partial t}}&=&{\frac {\partial L}{\partial z}}-{\frac {\partial N}{\partial x}},&\ {\frac {1}{V}}{\frac {\partial M}{\partial t}}&=&{\frac {\partial Z}{\partial x}}-{\frac {\partial X}{\partial z}},\\\\{\frac {1}{V}}{\frac {\partial Z}{\partial t}}&=&{\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}},&\ {\frac {1}{V}}{\frac {\partial N}{\partial t}}&=&{\frac {\partial X}{\partial y}}-{\frac {\partial Y}{\partial x}},\end{array}}}$

where (X, Y, Z) are the components of the electric force, (L, M, N) are the components of the magnetic force.

If we apply the transformations developed in § 3 to these equations, and if we refer the electromagnetic processes to the co-ordinate system (introduced at that place) moving with velocity v, we obtain,

 ${\displaystyle {\begin{array}{lllll}{\frac {1}{V}}{\frac {\partial X}{\partial \tau }}&=&{\frac {\partial \beta \left(N-{\frac {v}{V}}Y\right)}{\partial \eta }}&-&{\frac {\partial \beta \left(M+{\frac {v}{V}}Z\right)}{\partial \zeta }},\\\\{\frac {1}{V}}{\frac {\partial \beta \left(Y-{\frac {v}{V}}N\right)}{\partial \tau }}&=&{\frac {\partial L}{\partial \xi }}&-&{\frac {\partial \beta \left(N-{\frac {v}{V}}Y\right)}{\partial \xi }},\\\\{\frac {1}{V}}{\frac {\partial \beta \left(Z+{\frac {v}{V}}M\right)}{\partial \tau }}&=&{\frac {\partial \beta \left(M+{\frac {v}{V}}Z\right)}{\partial \xi }}&-&{\frac {\partial L}{\partial \eta }},\\\\{\frac {1}{V}}{\frac {\partial L}{\partial \tau }}&=&{\frac {\partial \beta \left(Y-{\frac {v}{V}}N\right)}{\partial \zeta }}&-&{\frac {\partial \beta \left(Z+{\frac {v}{V}}M\right)}{\partial \eta }},\end{array}}}$
 ${\displaystyle {\begin{array}{lllll}{\frac {1}{V}}{\frac {\partial \beta \left(M+{\frac {v}{V}}Z\right)}{\partial \tau }}&=&{\frac {\partial \beta \left(Z+{\frac {v}{V}}M\right)}{\partial \xi }}&-&{\frac {\partial X}{\partial \zeta }},\\\\{\frac {1}{V}}{\frac {\partial \beta \left(N-{\frac {v}{V}}Y\right)}{\partial \tau }}&=&{\frac {\partial X}{\partial \eta }}&-&{\frac {\partial \beta \left(Y-{\frac {v}{V}}N\right)}{\partial \xi }},\end{array}}}$

where

${\displaystyle \beta ={\frac {1}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}.}$

The principle of relativity requires that the Maxwell-Hertzian equations for pure vacuum shall hold also for the system k, if they hold for the system K, i.e., for the vectors of the electric and magnetic forces ${\displaystyle \left((X',\ Y',\ Z')\right.}$ and ${\displaystyle \left.(L',\ M',\ N')\right)}$ acting upon electric and magnetic masses in the moving system k, which are defined by their ponderomotive reaction, the equations hold,

${\displaystyle {\begin{array}{llll}{\frac {1}{V}}{\frac {\partial X'}{\partial \tau }}&=&{\frac {\partial N'}{\partial \eta }}-{\frac {\partial M'}{\partial \zeta }},\ {\frac {1}{V}}{\frac {\partial L'}{\partial \tau }}=&{\frac {\partial Y'}{\partial \zeta }}-{\frac {\partial Z'}{\partial \eta }},\\\\{\frac {1}{V}}{\frac {\partial Y'}{\partial \tau }}&=&{\frac {\partial L'}{\partial \zeta }}-{\frac {\partial N'}{\partial \xi }},\ {\frac {1}{V}}{\frac {\partial M'}{\partial \tau }}=&{\frac {\partial Z'}{\partial \xi }}-{\frac {\partial X'}{\partial \zeta }},\\\\{\frac {1}{V}}{\frac {\partial Z'}{\partial \tau }}&=&{\frac {\partial M'}{\partial \xi }}-{\frac {\partial L'}{\partial \eta }},\ {\frac {1}{V}}{\frac {\partial N'}{\partial \tau }}=&{\frac {\partial X'}{\partial \eta }}-{\frac {\partial Y'}{\partial \xi }}.\end{array}}}$

Clearly both the systems of equations (2) and (3) developed for the system k shall express the same things, for both of these systems are equivalent to the Maxwell-Hertzian equations for the system K. Since both systems of equations (2) and (3) agree up to the symbols representing the vectors, it follows that the functions occurring at corresponding places will agree up to a certain factor ψ(v), which possibly depends on v, and is independent of ξ, η, ζ, τ. Hence the relations,

${\displaystyle {\begin{array}{llllll}X'&=&\psi (v)X,&L'&=&\psi (v)L,\\\\Y'&=&\psi (v)\beta \left(Y-{\frac {v}{V}}N\right),&M'&=&\psi (v)\beta \left(M+{\frac {v}{V}}Z\right),\\\\Z'&=&\psi (v)\beta \left(Z+{\frac {v}{V}}M\right),&N'&=&\psi (v)\beta \left(N-{\frac {v}{V}}Y\right).\end{array}}}$
Now, if the reciprocal of this system of equations is formed, firstly by solving the equations just obtained, secondly by applying the equations to the inverse transformation (from k to K), which is characterized by the velocity -v, it follows, by considering that the two systems of equations thus obtained must be identical:
${\displaystyle \varphi (v).\varphi (-v)=1.\,}$

Further, from reasons of symmetry[4]

${\displaystyle \varphi (v)=\varphi (-v);\,}$

hence

${\displaystyle \varphi (v)=1,\,}$

and our equations obtain the form

${\displaystyle {\begin{array}{llllll}X'&=&X,&L'&=&L,\\\\Y'&=&\beta \left(Y-{\frac {v}{V}}N\right),&M'&=&\beta \left(M+{\frac {v}{V}}Z\right),\\\\Z'&=&\beta \left(Z+{\frac {v}{V}}M\right),&N'&=&\beta \left(N-{\frac {v}{V}}Y\right).\end{array}}}$

For the interpretation of these equations, we make the following remarks. Let us have a pointlike quantity of electricity which is of magnitude "unity" measured in the stationary system K, i.e., in the stationary system it exerts the force of 1 dyne upon an equal quantity placed at a distance of 1 cm. According to the principle of relativity, this electrical mass is of magnitude "unity" measured in the moving system as well. If this quantity of electricity be at rest in the stationary system, then the force acting upon it is equivalent to the vector (X, Y, Z) of electric force. But if the quantity of electricity be at rest relative to the moving system (at least for the moment considered), then the force acting upon it, and measured in the moving system is equivalent to the vector (X', Y', Z'). The first three of equations (1), (2), (3), can be expressed in the following way:

1. If a pointlike, electric unit pole moves in an electro-magnetic field, then besides the electric force, an "electromotive force" acts upon it, which, neglecting the numbers involving the second and higher powers of v/V, is equivalent to the vector-product of the velocity of the unit-pole and the magnetic force, divided by the velocity of light. (Old mode of expression).

2. If a point-like, electric unit-pole moves in an electro-magnetic field, then the force acting upon it is equivalent to the electric force existing at the position of the unit-pole, which we obtain by the transformation of the field to a co-ordinate system which is at rest relative to the electric unit-pole. (New mode of expression).

Similar things hold with reference to the "magnetomotive forces". We see that in the theory developed the electromagnetic force only plays the part of an auxiliary concept, which owes its introduction to the circumstance that the electric and magnetic forces possess no existence independent of the state of motion of the co-ordinate system.

It is further clear that the asymmetry mentioned in the introduction which occurs when we treat of the current excited by the relative motion of a magnet and a conductor disappears. Also the question about the "seat" of the electromagnetic electromotive forces (unipolar machines) is seen to be without any meaning.

### § 7. Theory of Doppler's principle and aberration.

In the system K, at a great distance from the origin of co-ordinates, let there be a source of electrodynamic waves, which is represented with sufficient approximation in a part of space containing the origin, by the equations:

${\displaystyle {\begin{array}{lll}X=X_{0}\sin \Phi ,&L=L_{0}\sin \Phi ,\ \\\\Y=Y_{0}\sin \Phi ,&M=M_{0}\sin \Phi ,\ &\Phi =\omega \left(t-{\frac {ax+by+cz}{V}}\right),\\\\Z=Z_{0}\sin \Phi ,&N=N_{0}\sin \Phi .\ \end{array}}}$

Here (${\displaystyle X_{0},Y_{0},Z_{0}}$) and (${\displaystyle L_{0},M_{0},N_{0}}$) are the vectors which determine the amplitude of the train of waves, a, b, c are the direction-cosines of the wave-normal.

Let us now ask after the composition of these waves, when they are investigated by an observer at rest in the moving system k: — By applying the equations of transformation obtained in § 6 for the electric and magnetic forces, and the equations of transformation obtained in § 3 for the co-ordinates and time, we obtain immediately:

 ${\displaystyle {\begin{array}{lrlr}X'=&X_{0}\sin \Phi ',&L'=&L_{0}\sin \Phi ',\\\\Y'=&\beta (Y_{0}-{\frac {v}{V}}N_{0})\sin \Phi ',&M'=&\beta (M_{0}+{\frac {v}{V}}Z_{0})\sin \Phi ',\\\\Z'=&\beta (Z_{0}+{\frac {v}{V}}M_{0})\sin \Phi ',&N'=&\beta (N_{0}-{\frac {v}{V}}Y_{0})\sin \Phi ',\end{array}}}$ ${\displaystyle \Phi '=\omega '\left(\tau -{\frac {a'\xi +b'\eta +c'\zeta }{V}}\right),}$

where it is set

${\displaystyle {\begin{array}{lll}\omega '&=&\omega \beta (1-a{\frac {v}{V}}),\\\\a'&=&{\frac {a-{\frac {v}{V}}}{1-a{\frac {v}{V}}}},\\\\b'&=&{\frac {b}{\beta \left(1-a{\frac {v}{V}}\right)}},\\\\c'&=&{\frac {c}{\beta \left(1-a{\frac {v}{V}}\right)}}.\end{array}}}$

From the equation for ω' it follows: If an observer moves with the velocity v relative to an infinitely distant source of light emitting waves of frequency ν, in such a manner that the connecting line "light source-observer" makes an angle of φ with the velocity of the observer related to a system of co-ordinates which is stationary with regard to the light source, then the frequency ν' which is perceived by the observer is represented by the formula

${\displaystyle \nu '=\nu {\frac {1-\cos \varphi {\frac {v}{V}}}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}.}$

This is Doppler's principle for any velocity. If φ = 0, the equation takes the clear form

${\displaystyle \nu '=\nu {\sqrt {\frac {1-{\frac {v}{V}}}{1+{\frac {v}{V}}}}}.}$

We see that — contrary to the usual conception — v = -∞, for ν = ∞.

If φ' is the angle between the wave-normal (direction of the ray) in the moving system, and the connecting line "light source-observer", the equation for a' takes the form

${\displaystyle \cos \varphi '={\frac {\cos \varphi -{\frac {v}{V}}}{1-{\frac {v}{V}}\cos \varphi }}.}$

This equation expresses the law of aberration in its most general form. If φ = π/2, the equation takes the simple form

${\displaystyle \cos \varphi '=-{\frac {v}{V}}.}$

We have still to search for the amplitude of the waves, which occur in the moving system. If A and A' be the forces (electric or magnetic) measured in the stationary and the moving system, we have

${\displaystyle A'^{2}=A^{2}{\frac {\left(1-{\frac {v}{V}}\cos \varphi \right)^{2}}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}.}$

If φ = 0, this goes over to the simpler form

${\displaystyle A'^{2}=A^{2}{\frac {1-{\frac {v}{V}}}{1+{\frac {v}{V}}}}.}$
From these equations, it appears that for an observer moving with the velocity V towards the source of light, the source should appear infinitely intense.

### § 8. Transformation of the energy of the rays of light. Theory of the radiation-pressure on a perfect mirror.

Since ${\displaystyle A^{2}/8\pi }$ is equal to the energy of light per unit volume, we have (according to the relativity principle) to regard ${\displaystyle A'^{2}/8\pi }$ as the energy of light in the moving system. ${\displaystyle A'^{2}/A^{2}}$ would therefore denote the ratio between the energy of a certain light-complex "measured when moving" and "measured when stationary," the volumes of the light-complex measured in K and k being equal. Yet this is not the case. If a, b, c are the direction-cosines of the wave-normal of light in the stationary system, then no energy passes through the surface elements of the spherical surface

${\displaystyle (x-Vat)^{2}+(y-Vbt)^{2}+(z-Vct)^{2}=R^{2}\,}$

which moves with the velocity of light. We can therefore say, that this surface always encloses the same light-complex. Let us now consider the quantity of energy, which this surface encloses when regarded from the system k, i.e., the energy of the light-complex relative to the system k.

Regarded from the moving system, the spherical surface becomes an ellipsoidal surface, having, at the time τ = 0, the equation:

${\displaystyle \left(\beta \xi -a\beta {\frac {v}{V}}\xi \right)^{2}+\left(\eta -b\beta {\frac {v}{V}}\xi \right)^{2}+\left(\zeta -c\beta {\frac {v}{V}}\xi \right)^{2}=R^{2}.}$

If S denotes the volume of the sphere, S' the volume of this ellipsoid, then a simple calculation shows that:

${\displaystyle {\frac {S'}{S}}={\frac {\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}{1-{\frac {v}{V}}\cos \varphi }}.}$

If E denotes the energy of light measured in the stationary system, E' the quantity measured in the moving system, which are enclosed by the considered surfaces, then one obtains

${\displaystyle {\frac {E'}{E}}={\frac {{\frac {A'^{2}}{8\pi }}S'}{{\frac {A^{2}}{8\pi }}S}}={\frac {1-{\frac {v}{V}}\cos \varphi }{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}.}$
If φ = 0, it goes over to the simpler formula:
${\displaystyle {\frac {E'}{E}}={\sqrt {\frac {1-{\frac {v}{V}}}{1+{\frac {v}{V}}}}}.}$

It is to be noticed that the energy and the frequency of a light-complex vary according to the same law with the state of motion of the observer.

Let there be a perfectly reflecting mirror at the co-ordinate-plane ξ = 0, from which the plane wave considered in the last paragraph is reflected. We ask after the light-pressure exerted on the reflecting surface and the direction, frequency, intensity of the light after reflection.

Let the incident light be defined by the magnitudes A, cos φ, ν (referred to the system K). Regarded from k, we have the corresponding magnitudes:

${\displaystyle {\begin{array}{lll}A'&=&A{\frac {1-{\frac {v}{V}}\cos \varphi }{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}},\\\\\cos \varphi '&=&{\frac {\cos \varphi -{\frac {v}{V}}}{1-{\frac {v}{V}}\cos \varphi }},\\\\\nu '&=&\nu {\frac {1-{\frac {v}{V}}\cos \varphi }{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}.\end{array}}}$

For the reflected light we obtain, when the process is referred to the system k:

${\displaystyle {\begin{array}{rll}A''&=&A',\\\\\cos \varphi ''&=&-\cos \varphi ',\\\\\nu ''&=&\nu '.\end{array}}}$
By means of a back-transformation to the stationary system K, we eventually obtain for the reflected light:
${\displaystyle {\begin{array}{rll}A'''&=&A''{\frac {1+{\frac {v}{V}}cos\varphi ''}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}=A{\frac {1-2{\frac {v}{V}}\cos \varphi +\left({\frac {v}{V}}\right)^{2}}{1-\left({\frac {v}{V}}\right)^{2}}},\\\\\cos \varphi '''&=&{\frac {\cos \varphi ''+{\frac {v}{V}}}{1+{\frac {v}{V}}\cos \varphi ''}}=-{\frac {\left(1+\left({\frac {v}{V}}\right)^{2}\right)\cos \varphi -2{\frac {v}{V}}}{1-2{\frac {v}{V}}\cos \varphi +\left({\frac {v}{V}}\right)^{2}}},\\\\\nu '''&=&\nu ''{\frac {1+{\frac {v}{V}}\cos \varphi ''}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}=\nu {\frac {1-2{\frac {v}{V}}\cos \varphi +\left({\frac {v}{V}}\right)^{2}}{\left(1-{\frac {v}{V}}\right)^{2}}}.\end{array}}}$

The energy falling upon the unit surface of the mirror per unit of time (measured in the stationary system) is evidently ${\displaystyle A^{2}/8\pi (V\ \cos \varphi -v)}$. The energy going away from unit surface of the mirror per unit of time is ${\displaystyle A'''^{2}/8\pi (-V\ \cos \varphi '''+v)}$. The difference of these two expressions is, according to the energy principle, the amount of work exerted by the pressure of light per unit of time. If we put this equal to P.v, where P is the pressure of light, we have:

${\displaystyle P=2{\frac {A^{2}}{8\pi }}{\frac {\left(\cos \varphi -{\frac {v}{V}}\right)^{2}}{1-\left({\frac {v}{V}}\right)^{2}}}.}$

As a first approximation, we obtain

${\displaystyle P=2{\frac {A^{2}}{8\pi }}\cos ^{2}\varphi .}$

which is in accordance with facts, and with other theories.

All problems of optics of moving bodies can be solved after the method used here. The essential point is, that the electric and magnetic forces of light, which are influenced by a moving body, should be transformed to a system of co-ordinates which is stationary relative to the body. In this way, every problem of the optics of moving bodies is reduced to a series of problems of the optics of stationary bodies.

### § 9. Transformation of the Maxwell-Hertz equations with regard to convection currents.

Let us start from the equations:

${\displaystyle {\begin{array}{llllll}{\frac {1}{V}}\left\{u_{x}\varrho +{\frac {\partial X}{\partial t}}\right\}&=&{\frac {\partial N}{\partial y}}-{\frac {\partial M}{\partial z}},&{\frac {1}{V}}{\frac {\partial L}{\partial t}}&=&{\frac {\partial Y}{\partial z}}-{\frac {\partial Z}{\partial y}},\\\\{\frac {1}{V}}\left\{u_{y}\varrho +{\frac {\partial Y}{\partial t}}\right\}&=&{\frac {\partial L}{\partial z}}-{\frac {\partial N}{\partial x}},&{\frac {1}{V}}{\frac {\partial M}{\partial t}}&=&{\frac {\partial Z}{\partial x}}-{\frac {\partial X}{\partial z}},\\\\{\frac {1}{V}}\left\{u_{z}\varrho +{\frac {\partial Z}{\partial t}}\right\}&=&{\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}},&{\frac {1}{V}}{\frac {\partial N}{\partial t}}&=&{\frac {\partial X}{\partial y}}-{\frac {\partial Y}{\partial x}},\end{array}}}$

where

${\displaystyle \varrho ={\frac {\partial X}{\partial x}}+{\frac {\partial Y}{\partial y}}+{\frac {\partial Z}{\partial z}}}$

denotes 4π-times the density of electricity, and (ux, uy, uz) is the velocity-vector of electricity. If we now suppose that the electrical masses are bound unchangeably to small, rigid bodies (ions, electrons), then these equations form the electromagnetic basis of Lorentz's electrodynamics and optics for moving bodies.

If these equations which may hold in the system K, are transformed to the system k with the aid of the transformation-equations given in § 3 and § 6, then we obtain the equations:

${\displaystyle {\begin{array}{llllll}{\frac {1}{V}}\left\{u_{\xi }\varrho '+{\frac {\partial X'}{\partial \tau }}\right\}&=&{\frac {\partial N'}{\partial \eta }}-{\frac {\partial M'}{\partial \zeta }},&{\frac {\partial L'}{\partial \tau }}&=&{\frac {\partial Y'}{\partial \zeta }}-{\frac {\partial Z'}{\partial \eta }},\\\\{\frac {1}{V}}\left\{u_{\eta }\varrho '+{\frac {\partial Y'}{\partial \tau }}\right\}&=&{\frac {\partial L'}{\partial \zeta }}-{\frac {\partial N'}{\partial \xi }},&{\frac {\partial M'}{\partial \tau }}&=&{\frac {\partial Z'}{\partial \xi }}-{\frac {\partial X'}{\partial \zeta }},\\\\{\frac {1}{V}}\left\{u_{\zeta }\varrho '+{\frac {\partial Z'}{\partial \tau }}\right\}&=&{\frac {\partial M'}{\partial \xi }}-{\frac {\partial L'}{\partial \eta }},&{\frac {\partial N'}{\partial \tau }}&=&{\frac {\partial X'}{\partial \eta }}-{\frac {\partial Y'}{\partial \xi }},\end{array}}}$

where

${\displaystyle {\begin{array}{rlll}{\frac {u_{x}-v}{1-{\frac {u_{x}v}{V^{2}}}}}&=&u_{\xi },\\\\{\frac {u_{y}}{\beta \left(1-{\frac {u_{x}v}{V^{2}}}\right)}}&=&u_{\eta },&\varrho '={\frac {\partial X'}{\partial \xi }}+{\frac {\partial Y'}{\partial \eta }}+{\frac {\partial Z'}{\partial \zeta }}=\beta \left(1-{\frac {v\ ux_{x}}{V^{2}}}\right)\varrho .\\\\{\frac {u_{z}}{\beta \left(1-{\frac {u_{x}v}{V^{2}}}\right)}}&=&u_{\zeta }.\end{array}}}$
Since the vector (uξ, uη, uζ) is nothing but the velocity of the electrical mass measured in the system k – as it follows from the addition-theorem of velocities in § 5 — so it is hereby shown, that by taking our kinematical principle as the basis, the electromagnetic basis of Lorentz's theory of electrodynamics of moving bodies correspond to the relativity principle.

It can be briefly remarked that the following important law follows easily from the equations developed: if an electrically charged body moves in any manner in space, and if its charge does not change thereby, when regarded from a system moving along with it, then the charge remains constant even when it is regarded from the "stationary" system K.

### § 10. Dynamics of the electron (slowly accelerated).

Let us suppose that a pointlike particle, having the electrical charge ε (to be called henceforth the "electron") moves in the electromagnetic field; we assume the following about its law of motion:

If the electron be at rest at any definite epoch, then in the next portion of time the motion takes place according to the equations

${\displaystyle {\begin{array}{lll}\mu {\frac {d^{2}x}{dt^{2}}}&=&\epsilon X,\\\\\mu {\frac {d^{2}y}{dt^{2}}}&=&\epsilon Y,\\\\\mu {\frac {d^{2}z}{dt^{2}}}&=&\epsilon Z,\end{array}}}$

where x, y, z are the co-ordinates of the electron, and μ is its mass, as long it is moving slowly.

Let the electron possess the velocity v at a certain epoch of time. Let us now investigate the laws according to which the electron will move in the portion of time immediately following.

Without influencing the generality of treatment, we can and we will assume that, at the moment we are considering, the electron is at the origin of co-ordinates, and moves with the velocity v along the X-axis of the system K. It is clear that at this moment (t = 0) the electron is at rest relative to the system k, which moves parallel to the X-axis with the constant velocity v.

From the suppositions made above, in combination with the principle of relativity, it is clear that regarded from the system k, the electron – in the immediately following time (for small values of t) – moves according to the equations

${\displaystyle {\begin{array}{lll}\mu {\frac {d^{2}\xi }{d\tau ^{2}}}&=&\epsilon X',\\\\\mu {\frac {d^{2}\eta }{d\tau ^{2}}}&=&\epsilon Y',\\\\\mu {\frac {d^{2}\zeta }{d\tau ^{2}}}&=&\epsilon Z',\end{array}}}$

where the symbols ξ, η, ζ, τ, X', Y', Z' refer to the system k. If we now define, that for t = x = y = z = 0, τ = ξ = η = ζ = 0, then the equations of transformation given in §§ 3 and 6 hold, and we have:

${\displaystyle {\begin{array}{llll}\tau =&\beta \left(t-{\frac {v}{V^{2}}}x\right),\\\\\xi =&\beta (x-vt),&X'=&X,\\\\\eta =&y,&Y'=&\beta (Y-{\frac {v}{V}}N),\\\\\zeta =&z,&Z'=&\beta (Z+{\frac {v}{V}}M).\end{array}}}$

With the aid of these equations, we can transform the above equations of motion from the system k to the system K, and obtain:

${\displaystyle (A)\qquad {\begin{cases}{\frac {d^{2}x}{dt^{2}}}&={\frac {\epsilon }{\mu }}{\frac {1}{\beta ^{3}}}X\\\\{\frac {d^{2}y}{dt^{2}}}&={\frac {\epsilon }{\mu }}{\frac {1}{\beta }}\left(Y-{\frac {v}{V}}N\right)\\\\{\frac {d^{2}z}{dt^{2}}}&={\frac {\epsilon }{\mu }}{\frac {1}{\beta }}\left(Z+{\frac {v}{V}}M\right).\end{cases}}}$

We now ask, following the usual method of treatment, after the "longitudinal" and "transverse" mass of a moving electron. We write the equations (A) in the form

${\displaystyle {\begin{array}{lllll}\mu \beta ^{3}{\frac {d^{2}x}{dt^{2}}}&=&\epsilon X&=&\epsilon X',\\\\\mu \beta ^{2}{\frac {d^{2}y}{dt^{2}}}&=&\epsilon \beta \left(Y-{\frac {v}{V}}N\right)&=&\epsilon Y',\\\\\mu \beta ^{2}{\frac {d^{2}z}{dt^{2}}}&=&\epsilon \beta \left(Z+{\frac {v}{V}}M\right)&=&\epsilon Z',\end{array}}}$

and let us first remark, that εX', εY', εZ' are the components of the ponderomotive force acting upon the electron, and are considered in a moving system which, at this moment, moves with a velocity which is equal to that of the electron. (This force can, for example, be measured by means of a spring-balance which is at rest in the latter system.) If we briefly call this force as "the force acting upon the electron," and maintain the equation:

Mass-number × acceleration-number = force-number,

and if we further define that the accelerations are measured in the stationary system K, then from the above equations, we obtain:

Longitudinal mass = ${\displaystyle {\frac {\mu }{\left({\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}\right)^{3}}},}$
Transverse mass = ${\displaystyle {\frac {\mu }{1-\left({\frac {v}{V}}\right)^{2}}}.}$

Naturally, when other definitions are given of the force and the acceleration, other numbers are obtained for the mass; hence we see that we must proceed very carefully in comparing the different theories of the motion of the electron.

We remark that this result about the mass holds also for ponderable material points; because a ponderable material point may be made into an electron (in our sense) by the addition of an electrical charge which may be arbitrarily small.

Let us now determine the kinetic energy of the electron. If the electron moves from the origin of co-ordinates of the system K with the initial velocity 0 steadily along the X-axis under the action of an electrostatic force X, then it is clear that the energy drawn from the electrostatic field has the value ${\displaystyle \int \epsilon X\ dx}$. Since the electron should be only slowly accelerated, and in consequence, no energy is given out in the form of radiation, therefore the energy drawn from the electrostatic field must be put equal to the energy W of motion of the electron. By noticing, that the first of equations (A) holds during the whole process of motion in question, we obtain:

${\displaystyle W=\int \epsilon X\,dx=\int \limits _{0}^{v}\beta ^{3}v\,dv=\mu V^{2}\left\{{\frac {1}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}-1\right\}.}$

For v = V, W becomes infinitely great. As our former result shows, superluminal velocities can have no possibility of existence.

In consequence of the argument mentioned above, this expression for kinetic energy must also hold for ponderable masses.

We can now enumerate the characteristics of the motion of the electrons available for experimental verification, which follow from the system of equations (A).

1. From the second of equations (A), it follows that an electrical force Y and a magnetic force N, produce equal deflections of an electron moving with the velocity v, when Y = N.v/V. Therefore we see that it is possible to obtain the velocity of an electron from the ratio of the magnetic deflection Am and the electric deflection Ae, according to our theory for arbitrary velocities, by applying the law:

${\displaystyle {\frac {A_{m}}{A_{e}}}={\frac {v}{V}}.}$

This relation can be tested by means of experiments, because the velocity of the electron can be directly measured by means, for example, of rapidly oscillating electric and magnetic fields.

2. From the deduction for the kinetic energy of the electron it follows, that between the traversed potential-difference and the velocity v obtained, the relation must hold:

${\displaystyle P=\int X\ dx={\frac {\mu }{\epsilon }}V^{2}\left\{{\frac {1}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}-1\right\}.}$

3. We calculate the radius of curvature R of the path, when the only deflecting force is a magnetic force N acting perpendicular to the velocity of the electron. From the second of equations (A) we obtain:

${\displaystyle -{\frac {d^{2}y}{dt^{2}}}={\frac {v^{2}}{R}}={\frac {\epsilon }{\mu }}{\frac {v}{V}}N\cdot {\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}$

or

${\displaystyle R=V^{2}{\frac {\mu }{\epsilon }}\cdot {\frac {\frac {v}{V}}{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}}\cdot {\frac {1}{N}}.}$

These three relations are complete expressions for the law of motion of the electron, by which the electron has to move according to this theory.

At the end I remark, that my friend and colleague M. Besso loyally stood at my side during the work at the problem discussed here, and that I am indebted to him for some valuable suggestions.

Bern, June 1905.