OF MOVING BODIES

June 30, 1905

It is known that Maxwell's electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.

Examples of this sort, together with the unsuccessful attempts
to discover any motion of the earth relatively to the “light
medium,” suggest that the phenomena of electrodynamics as well as
of mechanics possess no properties corresponding to the idea of
absolute rest. They suggest rather that, as has already been shown
to the first order of small quantities, the same laws of
electrodynamics and optics will be valid for all frames of
reference for which the equations of mechanics hold good . We will
raise this conjecture (the purport of which will hereafter be
called the “Principle of Relativity”) to the status of a
postulate, and also introduce another postulate, which is only
apparently irreconcilable with the former, namely, that light is
always propagated in empty space with a definite velocity *c*
which is independent of the state of motion of the emitting body.
These two postulates suffice for the attainment of a simple and
consistent theory of the electrodynamics of moving bodies based on
Maxwell's theory for stationary bodies. The introduction of a
“luminiferous ether” will prove to be superfluous inasmuch as the
view here to be developed will not require an “absolutely
stationary space” provided with special properties, nor assign a
velocity-vector to a point of the empty space in which
electromagnetic processes take place.

The theory to be developed is based—like all electrodynamics—on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes. Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters.

Let us take a system of co-ordinates in which the equations of
Newtonian mechanics hold good.^{2}
In order to render our presentation
more precise and to distinguish this system of co-ordinates
verbally from others which will be introduced hereafter, we call it
the “stationary system.”

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

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

It might appear possible to overcome all the difficulties attending the definition of “time” by substituting “the position of the small hand of my watch” for “time.” And in fact such a definition is satisfactory when we are concerned with defining a time exclusively for the place where the watch is located; but it is no longer satisfactory when we have to connect in time series of events occurring at different places, or—what comes to the same thing—to evaluate the times of events occurring at places remote from the watch.

We might, of course, content ourselves with time values determined by an observer stationed together with the watch at the origin of the co-ordinates, and co-ordinating the corresponding positions of the hands with light signals, given out by every event to be timed, and reaching him through empty space. But this co-ordination has the disadvantage that it is not independent of the standpoint of the observer with the watch or clock, as we know from experience. We arrive at a much more practical determination along the following line of thought.

If at the point A of space there is a clock, an observer at A
can determine the time values of events in the immediate proximity
of A by finding the positions of the hands which are simultaneous
with these events. If there is at the point B of space another
clock in all respects resembling the one at A, it is possible for
an observer at B to determine the time values of events in the
immediate neighbourhood of B. But it is not possible without
further assumption to compare, in respect of time, an event at A
with an event at B. We have so far defined only an “A time” and a
“B time.” We have not defined a common “time” for A and B, for
the latter cannot be defined at all unless we establish *by
definition* that the “time” required by light to travel from
A to B equals the “time” it requires to travel from B to A. Let a
ray of light start at the “A time”
from A towards B, let it at the “B time”
be reflected at B in the
direction of A, and arrive again at A at the “A time”
.

In accordance with definition the two clocks synchronize if

We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:—

- If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
- If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.

Thus with the help of certain imaginary physical experiments we have settled what is to be understood by synchronous stationary clocks located at different places, and have evidently obtained a definition of “simultaneous,” or “synchronous,” and of “time.” The “time” of an event is that which is given simultaneously with the event by a stationary clock located at the place of the event, this clock being synchronous, and indeed synchronous for all time determinations, with a specified stationary clock.

In agreement with experience we further assume the quantity

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

It is essential to have time defined by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it “the time of the stationary system.”

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

- The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion.
- Any ray of light moves in the “stationary” system of
co-ordinates with the determined velocity
*c*, whether the ray be emitted by a stationary or by a moving body. Hencewhere 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 measuring-rod which is also stationary.
We now imagine the axis of the rod lying along the axis of *x*
of the stationary system of co-ordinates, 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:—

- (
*a*) - The observer moves together with the given measuring-rod and
the rod to be measured, and measures the length of the rod directly
by superposing the measuring-rod, in just the same way as if all
three were at rest.
- (
*b*) - By means of stationary clocks set up in the stationary system and synchronizing in accordance with § 1, the observer ascertains at what points of the stationary system the two ends of the rod to be measured are located at a definite time. The distance between these two points, measured by the measuring-rod already employed, which in this case is at rest, is also a length which may be designated “the length of the rod.”

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
, and reach A again at the time
. Taking into consideration the principle of the
constancy of the velocity of light we find that

where denotes the length of the moving rod—measured 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 co-ordinates, 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 co-ordinates, 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 measuring-rod and a number of clocks, and let the two measuring-rods, 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 co-ordinates, the
relevant measuring-rod, 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 measuring-rod, and also from the moving
system *k* by means of the measuring-rod moving with it; and
that we thus obtain the co-ordinates *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*'=*x*-*vt*, 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 X-axis to
*x*', and at the time
be
reflected thence to the origin of the co-ordinates, 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 co-ordinates
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 consideration—applied to the axes of Y and Z—it 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 *c*-*v*, 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 co-ordinates 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 co-ordinates
, 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 co-ordinates 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 co-ordinates, 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
co-ordinates

and

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
co-ordinates 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 motion—viewed from the stationary system—the 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 objects—viewed from the
“stationary” system—shrivel up into plane figures. 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
co-ordinates 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, or—neglecting magnitudes of fourth and higher order—by .

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
balance-clock^{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

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 co-ordinates *k*' moving parallel to
*k*, its initial point moving on the axis of 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 transformations—necessarily—form 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 Maxwell-Hertz 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 co-ordinates there introduced, moving
with the velocity *v*, we obtain the equations

where

Now the principle of relativity requires that if the
Maxwell-Hertz 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—(, , ) and (, , )—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 Maxwell-Hertz 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 (, , ). Consequently the first three equations above allow themselves to be clothed in words in the two following ways:—

- If a unit electric point charge is in motion in an
electromagnetic field, there acts upon it, in addition to the
electric force, an “electromotive force” which, if we neglect the
terms multiplied by the second and higher powers of
*v*/*c*, is equal to the vector-product of the velocity of the charge and the magnetic force, divided by the velocity of light. (Old manner of expression.) - If a unit electric point charge is in motion in an electromagnetic field, the force acting upon it is equal to the electric force which is present at the locality of the charge, and which we ascertain by transformation of the field to a system of co-ordinates at rest relatively to the electrical charge. (New manner of expression.)

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 co-ordinates.

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 co-ordinates, let there be a source of electrodynamic waves, which in a part of space containing the origin of co-ordinates may be represented to a sufficient degree of approximation by the equations

where

Here (,
, ) and (,
, ) are the vectors defining the amplitude of the
wave-train, and *l*, *m*, *n* the direction-cosines
of the wave-normals. 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 co-ordinates 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
“source-observer” makes the angle
with the velocity of the observer referred to a system of
co-ordinates 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 wave-normal (direction of the ray) in the moving system and the connecting line “source-observer” , the equation for 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 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 direction-cosines of the
wave-normals 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 surface—viewed in the moving system—is an ellipsoidal surface, the equation for which, at the time , is

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

Thus, if we call the light energy enclosed by this surface E when it is measured in the stationary system, and 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 co-ordinate 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 P*v*, 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 co-ordinates 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 velocity-vector 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

Since—as 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 co-ordinates moving with the body, its charge also remains—when regarded from the “stationary” system K—constant.

Let there be in motion in an electromagnetic field an electrically charged particle (in the sequel 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
co-ordinates 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 co-ordinates, 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 co-ordinates 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 , , , , , 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

· · · (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
last-mentioned system.) Now if we call this force simply “the
force acting upon the electron,”^{9} and maintain the
equation—mass × acceleration = force—and if we also decide
that the accelerations are to be measured in the stationary system
K, we derive from the above equations

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 co-ordinates 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 have—as in our previous results—no
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.

- From the second equation of the system
**(A)**it follows that an electric force Y and a magnetic force N have an equally strong deflective action on an electron moving with the velocity*v*, when . Thus we see that it is possible by our theory to determine the velocity of the electron from the ratio of the magnetic power of deflexion to the electric power of deflexion , for any velocity, by applying the lawThis 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.

- From the deduction for the kinetic energy of the electron it
follows that between the potential difference, P, traversed and the
acquired velocity
*v*of the electron there must be the relationship - We calculate the radius of curvature of the path of the
electron when a magnetic force N is present (as the only deflective
force), acting perpendicularly to the velocity of the electron.
From the second of the equations
**(A)**we obtainor

These three relationships are a complete expression for the laws according to which, by the theory here advanced, the electron must 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.