ON THE ELECTRODYNAMICS
OF MOVING BODIES
By A. Einstein
June 30, 1905
It is known that Maxwell's electrodynamicsas usually understood at the
present timewhen 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 riseassuming equality of relative motion in the
two cases discussedto 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.^{1}
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 velocityvector to a point
of the empty space in which electromagnetic processes take place.
The theory to be developed is basedlike all electrodynamicson the
kinematics of the rigid body, since the assertions of any such theory have
to do with the relationships between rigid bodies (systems of coordinates),
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.
I. KINEMATICAL PART
§ 1. Definition of Simultaneity
Let us take a system of coordinates in which the equations of Newtonian
mechanics hold good.
^{2}
In order to render our presentation more precise and to distinguish this
system of coordinates 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 coordinates,
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 coordinates.
If we wish to describe the motion of a material point, we give
the values of its coordinates 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, orwhat
comes to the same thingto 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 coordinates,
and coordinating 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 coordination 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 constantthe 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.''
§ 2. On the Relativity of Lengths and Times
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 coordinates in uniform translatory motion.

Any ray of light moves in the ``stationary'' system of coordinates with
the determined velocity c, whether the ray be emitted by a stationary
or by a moving body. Hence
where time interval is to be taken in the sense of
the definition in
§ 1.
Let there be given a stationary rigid rod; and let its length be
l
as measured by a measuringrod which is also stationary. We now imagine
the axis of the rod lying along the axis of
x of the stationary
system of coordinates, and that a uniform motion of parallel translation
with velocity
v along the axis of
x in the direction of increasing
x is then imparted to the rod. We now inquire as to the length of
the moving rod, and imagine its length to be ascertained by the following
two operations:

(a)

The observer moves together with the given measuringrod and the rod to
be measured, and measures the length of the rod directly by superposing
the measuringrod, 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 measuringrod 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 rodmeasured in the stationary system.
Observers moving with the moving rod would thus find that the two clocks
were not synchronous, while observers in the stationary system would declare
the clocks to be synchronous.
So we see that we cannot attach any absolute signification to
the concept of simultaneity, but that two events which, viewed from a system
of coordinates, are simultaneous, can no longer be looked upon as simultaneous
events when envisaged from a system which is in motion relatively to that
system.
§ 3. Theory of the Transformation of Coordinates
and Times from a Stationary System to another System in Uniform Motion
of Translation Relatively to the Former
Let us in ``stationary'' space take two systems of coordinates, i.e. two
systems, each of three rigid material lines, perpendicular to one another,
and issuing from a point. Let the axes of X of the two systems coincide,
and their axes of Y and Z respectively be parallel. Let each system be
provided with a rigid measuringrod and a number of clocks, and let the
two measuringrods, and likewise all the clocks of the two systems, be
in all respects alike.
Now to the origin of one of the two systems (k) let a constant
velocity v be imparted in the direction of the increasing x
of the other stationary system (K), and let this velocity be communicated
to the axes of the coordinates, the relevant measuringrod, and the clocks.
To any time of the stationary system K there then will correspond a definite
position of the axes of the moving system, and from reasons of symmetry
we are entitled to assume that the motion of
k may be such that
the axes of the moving system are at the time
t (this ``t''
always denotes a time of the stationary system) parallel to the axes of
the stationary system.
We now imagine space to be measured from the stationary system K by
means of the stationary measuringrod, and also from the moving system
k
by means of the measuringrod moving with it; and that we thus obtain the
coordinates x, y, z, and,,
respectively. Further, let the time t of the stationary system be
determined for all points thereof at which there are clocks by means of
light signals in the manner indicated in
§ 1;
similarly let the time
of the moving system be determined for all points of the moving system
at which there are clocks at rest relatively to that system by applying
the method, given in
§ 1, of light signals
between the points at which the latter clocks are located.
To any system of values x, y, z, t, which
completely defines the place and time of an event in the stationary system,
there belongs a system of values,,,
determining that event relatively to the system k, and our task
is now to find the system of equations connecting these quantities.
In the first place it is clear that the equations must be linear
on account of the properties of homogeneity which we attribute to space
and time.
If we place x'=xvt, it is clear that a point at
rest in the system k must have a system of values x', y,
z, independent of time. We first define
as a function of x', y, z, and t. To do this
we have to express in equations that
is nothing else than the summary of the data of clocks at rest in system
k, which have been synchronized according to the rule given in
§
1.
From the origin of system k let a ray be emitted at the time
along the Xaxis to x', and at the time
be reflected thence to the origin of the coordinates, arriving there at
the time ;
we then must have ,
or, by inserting the arguments of the function
and applying the principle of the constancy of the velocity of light in
the stationary system:
Hence, if x' be chosen infinitesimally small,
or
It is to be noted that instead of the origin of the coordinates we might
have chosen any other point for the point of origin of the ray, and the
equation just obtained is therefore valid for all values of
x',
y,
z.
An analogous considerationapplied to the axes of Y and Zit being
borne in mind that light is always propagated along these axes, when viewed
from the stationary system, with the velocity gives
us
Since
is a
linear function, it follows from these equations that
where
a is a function
at present unknown, and where for brevity it is assumed that at the origin
of
k,
,
when
t=0.
With the help of this result we easily determine the quantities,,
by expressing in equations that light (as required by the principle of
the constancy of the velocity of light, in combination with the principle
of relativity) is also propagated with velocity c when measured
in the moving system. For a ray of light emitted at the time
in the direction of the increasing
But the ray moves relatively to the initial point
of k, when measured in the stationary system, with the velocity
cv, so that
If we insert this value of
t in the equation
for
,
we obtain
In an analogous manner we find, by considering rays
moving along the two other axes, that
when
Thus
Substituting for
x' its value, we obtain
where
and
is an as yet unknown function of
v. If no assumption whatever be
made as to the initial position of the moving system and as to the zero
point of
,
an additive constant is to be placed on the right side of each of these
equations.
We now have to prove that any ray of light, measured in the moving system,
is propagated with the velocity c, if, as we have assumed, this
is the case in the stationary system; for we have not as yet furnished
the proof that the principle of the constancy of the velocity of light
is compatible with the principle of relativity.
At the time ,
when the origin of the coordinates is common to the two systems, let a
spherical wave be emitted therefrom, and be propagated with the velocity
c in system K. If (x, y, z) be a point just
attained by this wave, then
x^{2}+y^{2}+z^{2}=c^{2}t^{2}.
Transforming this equation with the aid of our equations of transformation
we obtain after a simple calculation
The wave under consideration is therefore no less a spherical wave with
velocity of propagation
c when viewed in the moving system. This
shows that our two fundamental principles are compatible.
^{5}
In the equations of transformation which have been developed there enters
an unknown function
of v, which we will now determine.
For this purpose we introduce a third system of coordinates,
which relatively to the system k is in a state of parallel translatory
motion parallel to the axis of X, such that the origin of coordinates
of system k moves with velocity v on the axis of X. At the
time t=0 let all three origins coincide, and when t=x=y=z=0
let the time t' of the system
be zero. We call the coordinates, measured in the system ,
x',
y', z', and by a twofold application of our equations of
transformation we obtain
Since the relations between
x',
y',
z' and
x,
y,
z do not contain the time
t, the systems K and
are at rest with respect to one another, and it is clear that the transformation
from K to
must be the identical transformation. Thus
We now inquire into the signification of
.
We give our attention to that part of the axis of Y of system
k
which lies between
and
.
This part of the axis of Y is a rod moving perpendicularly to its axis
with velocity
v relatively to system K. Its ends possess in K the
coordinates
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
§ 4. Physical Meaning of the Equations Obtained
in Respect to Moving Rigid Bodies and Moving Clocks
We envisage a rigid sphere
^{6}
of radius R, at rest relatively to the moving system
k, and with
its centre at the origin of coordinates of
k. The equation of the
surface of this sphere moving relatively to the system K with velocity
v is
The equation of this surface expressed in x,
y, z at the time t=0 is
A rigid body which, measured in a state of rest,
has the form of a sphere, therefore has in a state of motionviewed from
the stationary systemthe form of an ellipsoid of revolution with the
axes
Thus, whereas the Y and Z dimensions of the sphere (and therefore of every
rigid body of no matter what form) do not appear modified by the motion,
the X dimension appears shortened in the ratio
,
i.e. the greater the value of
v, the greater the shortening. For
v=
c all moving objectsviewed from the ``stationary'' systemshrivel
up into plane figures.
^{*}
For velocities greater than that of light our deliberations become meaningless;
we shall, however, find in what follows, that the velocity of light in
our theory plays the part, physically, of an infinitely great velocity.
It is clear that the same results hold good of bodies at rest in the
``stationary'' system, viewed from a system in uniform motion.
Further, we imagine one of the clocks which are qualified to mark the
time t when at rest relatively to the stationary system, and the
time
when at rest relatively to the moving system, to be located at the origin
of the coordinates of k, and so adjusted that it marks the time .
What is the rate of this clock, when viewed from the stationary system?
Between the quantities x, t, and ,
which refer to the position of the clock, we have, evidently, x=vt
and
Therefore,
whence it follows that the time marked by the clock
(viewed in the stationary system) is slow by
seconds
per second, orneglecting magnitudes of fourth and higher orderby
.
From this there ensues the following peculiar consequence. If at the
points A and B of K there are stationary clocks which, viewed in the stationary
system, are synchronous; and if the clock at A is moved with the velocity
v along the line AB to B, then on its arrival at B the two clocks
no longer synchronize, but the clock moved from A to B lags behind the
other which has remained at B by (up
to magnitudes of fourth and higher order), t being the time occupied
in the journey from A to B.
It is at once apparent that this result still holds good if the clock
moves from A to B in any polygonal line, and also when the points A and
B coincide.
If we assume that the result proved for a polygonal line is also valid
for a continuously curved line, we arrive at this result: If one of two
synchronous clocks at A is moved in a closed curve with constant velocity
until it returns to A, the journey lasting t seconds, then by the
clock which has remained at rest the travelled clock on its arrival at
A will be
second slow. Thence we conclude that a balanceclock^{7}
at the equator must go more slowly, by a very small amount, than a precisely
similar clock situated at one of the poles under otherwise identical conditions.
§ 5. The Composition of Velocities
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 coordinates
k' moving
parallel to
k, its initial point moving on the axis of X with the
velocity
w, we obtain equations between the quantities
x,
y,
z,
t and the corresponding quantities of
k',
which differ from the equations found in
§ 3
only in that the place of ``
v'' is taken by the quantity
from which we see that such parallel transformationsnecessarilyform
a group.
We have now deduced the requisite laws of the theory of kinematics corresponding
to our two principles, and we proceed to show their application to electrodynamics.
II. ELECTRODYNAMICAL PART
§ 6. Transformation of the MaxwellHertz Equations
for Empty Space. On the Nature of the Electromotive Forces Occurring in
a Magnetic Field During Motion
Let the MaxwellHertz equations for empty space hold good for the stationary
system K, so that we have
where (X, Y, Z) denotes the vector of the electric
force, and (L, M, N) that of the magnetic force.
If we apply to these equations the transformation developed in
§
3, by referring the electromagnetic processes to the system of coordinates
there introduced, moving with the velocity v, we obtain the equations
where
Now the principle of relativity requires that if the MaxwellHertz equations
for empty space hold good in system K, they also hold good in system
k;
that is to say that the vectors of the electric and the magnetic force(
,
,
)
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 MaxwellHertz equations for system K. Since, further, the equations
of the two systems agree, with the exception of the symbols for the vectors,
it follows that the functions occurring in the systems of equations at
corresponding places must agree, with the exception of a factor
,
which is common for all functions of the one system of equations, and is
independent of
and
but depends upon
v. Thus we have the relations
If we now form the reciprocal of this system of equations, firstly by solving
the equations just obtained, and secondly by applying the equations to
the inverse transformation (from
k to K), which is characterized
by the velocity 
v, it follows, when we consider that the two systems
of equations thus obtained must be identical, that
.
Further, from reasons of symmetry
^{8}
and therefore
and our equations assume the form
As to the interpretation of these equations we make
the following remarks: Let a point charge of electricity have the magnitude
``one'' when measured in the stationary system K, i.e. let it when at rest
in the stationary system exert a force of one dyne upon an equal quantity
of electricity at a distance of one cm. By the principle of relativity
this electric charge is also of the magnitude ``one'' when measured in
the moving system. If this quantity of electricity is at rest relatively
to the stationary system, then by definition the vector (X, Y, Z) is equal
to the force acting upon it. If the quantity of electricity is at rest
relatively to the moving system (at least at the relevant instant), then
the force acting upon it, measured in the moving system, is equal to the
vector (
,
,
).
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 vectorproduct 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 coordinates 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 coordinates.
Furthermore it is clear that the asymmetry mentioned in the introduction
as arising when we consider the currents produced by the relative motion
of a magnet and a conductor, now disappears. Moreover, questions as to
the ``seat'' of electrodynamic electromotive forces (unipolar machines)
now have no point.
§ 7. Theory of Doppler's Principle and of
Aberration
In the system K, very far from the origin of coordinates, let there be
a source of electrodynamic waves, which in a part of space containing the
origin of coordinates may be represented to a sufficient degree of approximation
by the equations
where
Here (
,
,
)
and (
,
,
)
are the vectors defining the amplitude of the wavetrain, and
l,
m,
n the directioncosines of the wavenormals. We wish to
know the constitution of these waves, when they are examined by an observer
at rest in the moving system
k.
Applying the equations of transformation found in
§
6 for electric and magnetic forces, and those found in
§
3 for the coordinates and the time, we obtain directly
where
From the equation for
it follows that if an observer is moving with velocity
v relatively
to an infinitely distant source of light of frequency
,
in such a way that the connecting line ``sourceobserver'' makes the angle
with the velocity of the observer referred to a system of coordinates
which is at rest relatively to the source of light, the frequency
of the light perceived by the observer is given by the equation
This is Doppler's principle for any velocities whatever.
When
the equation assumes the perspicuous form
We see that, in contrast with the customary view,
when
.
If we call the angle between the wavenormal (direction of the ray)
in the moving system and the connecting line ``sourceobserver'' ,
the equation for l' 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.
§ 8. Transformation of the Energy of Light
Rays. Theory of the Pressure of Radiation Exerted on Perfect Reflectors
Since
equals the energy of light per unit of volume, we have to regard
,
by the principle of relativity, as the energy of light in the moving system.
Thus
would be the ratio of the ``measured in motion'' to the ``measured at rest''
energy of a given light complex, if the volume of a light complex were
the same, whether measured in K or in
k. But this is not the case.
If
l,
m,
n are the directioncosines of the wavenormals
of the light in the stationary system, no energy passes through the surface
elements of a spherical surface moving with the velocity of light:
We may therefore say that this surface permanently
encloses the same light complex. We inquire as to the quantity of energy
enclosed by this surface, viewed in system k, that is, as to the
energy of the light complex relatively to the system k.
The spherical surfaceviewed in the moving systemis an ellipsoidal
surface, the equation for which, at the time,
is
If S is the volume of the sphere, and
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 coordinate plane
be a perfectly reflecting surface, at which the plane waves considered
in
§ 7 are reflected. We seek for the pressure
of light exerted on the reflecting surface, and for the direction, frequency,
and intensity of the light after reflexion.
Let the incidental light be defined by the quantities A, ,
(referred to system K). Viewed from k the corresponding quantities
are
For the reflected light, referring the process to
system k, we obtain
Finally, by transforming back to the stationary system
K, we obtain for the reflected light
The energy (measured in the stationary system) which is incident upon unit
area of the mirror in unit time is evidently
.
The energy leaving the unit of surface of the mirror in the unit of time
is
.
The difference of these two expressions is, by the principle of energy,
the work done by the pressure of light in the unit of time. If we set down
this work as equal to the product 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 coordinates at rest relatively to the body. By this means
all problems in the optics of moving bodies will be reduced to a series
of problems in the optics of stationary bodies.
§ 9. Transformation of the MaxwellHertz Equations
when ConvectionCurrents are Taken into Account
We start from the equations
where
denotes
times the density of electricity, and (
u_{x},
u_{y},
u_{z})
the velocityvector of the charge. If we imagine the electric charges to
be invariably coupled to small rigid bodies (ions, electrons), these equations
are the electromagnetic basis of the Lorentzian electrodynamics and optics
of moving bodies.
Let these equations be valid in the system K, and transform them, with
the assistance of the equations of transformation given in §§
3
and 6, to the system k. We then obtain
the equations
where
and
Sinceas follows from the theorem of addition of
velocities (
§ 5)the vector
is nothing else than the velocity of the electric charge, measured in the
system
k, we have the proof that, on the basis of our kinematical
principles, the electrodynamic foundation of Lorentz's theory of the electrodynamics
of moving bodies is in agreement with the principle of relativity.
In addition I may briefly remark that the following important law may
easily be deduced from the developed equations: If an electrically charged
body is in motion anywhere in space without altering its charge when regarded
from a system of coordinates moving with the body, its charge also remainswhen
regarded from the ``stationary'' system Kconstant.
§ 10. Dynamics of the Slowly Accelerated Electron
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 coordinates
of the electron, and m the mass of the electron, as long as its
motion is slow.
Now, secondly, let the velocity of the electron at a given epoch be
v. We seek the law of motion of the electron in the immediately
ensuing instants of time.
Without affecting the general character of our considerations, we may
and will assume that the electron, at the moment when we give it our attention,
is at the origin of the coordinates, and moves with the velocity v
along the axis of X of the system K. It is then clear that at the given
moment (t=0) the electron is at rest relatively to a system of coordinates
which is in parallel motion with velocity
v along the axis of X.
From the above assumption, in combination with the principle of relativity,
it is clear that in the immediately ensuing time (for small values of t)
the electron, viewed from the system k, moves in accordance with
the equations
in which the symbols
,
,
,
,
,
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
Taking the ordinary point of view we now inquire as to the ``longitudinal''
and the ``transverse'' mass of the moving electron. We write the equations
(A) in the form
and remark firstly that
,
,
are
the components of the ponderomotive force acting upon the electron, and
are so indeed as viewed in a system moving at the moment with the electron,
with the same velocity as the electron. (This force might be measured,
for example, by a spring balance at rest in the lastmentioned system.)
Now if we call this force simply ``the force acting upon the electron,''
^{9}
and maintain the equationmass × acceleration = forceand 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 coordinates of the system K along the
axis of X under the action of an electrostatic force X, it is clear that
the energy withdrawn from the electrostatic field has the value .
As the electron is to be slowly accelerated, and consequently may not give
off any energy in the form of radiation, the energy withdrawn from the
electrostatic field must be put down as equal to the energy of motion W
of the electron. Bearing in mind that during the whole process of motion
which we are considering, the first of the equations (A) applies,
we therefore obtain
Thus, when
v=
c, W becomes infinite. Velocities greater than
that of light haveas in our previous resultsno possibility of existence.
This expression for the kinetic energy must also, by virtue of the argument
stated above, apply to ponderable masses as well.
We will now enumerate the properties of the motion of the electron which
result from the system of equations (A), and are accessible to experiment.

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

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 obtain
or
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.
Footnotes

1.

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

2.

i.e. to the first approximation.

3.

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

4.

``Time'' here denotes ``time of the stationary system'' and also ``position
of hands of the moving clock situated at the place under discussion.''

5.

The equations of the Lorentz transformation may be more simply deduced
directly from the condition that in virtue of those equations the relation
x^{2}+y^{2}+z^{2}=c^{2}t^{2}
shall have as its consequence the second relation .

6.

That is, a body possessing spherical form when examined at rest.

7.

Not a pendulumclock, which is physically a system to which the Earth belongs.
This case had to be excluded.

8.

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

9.

The definition of force here given is not advantageous, as was first shown
by M. Planck. It is more to the point to define force in such a way that
the laws of momentum and energy assume the simplest form.
Editor's Note

*

Editor's note: In the original 1923 English
edition, this phrase was erroneously translated as ``plain figures''. I
have used the correct ``plane figures'' in this document.
About this Edition
This edition of Einstein's
On the Electrodynamics of Moving Bodies is based on the English
translation of his original 1905 Germanlanguage paper (published as
Zur
Elektrodynamik bewegter Körper, in Annalen der Physik.
17:891, 1905) which appeared in the book
The
Principle of Relativity, published in 1923 by Methuen and Company,
Ltd. of London. Most of the papers in that collection are English translations
by W. Perrett and G.B. Jeffery from the German Das Relativatsprinzip,
4th ed., published by in 1922 by Tuebner. All of these sources are now
in the public domain; this document, derived from them, remains in the
public domain and may be reproduced in any manner or medium without permission,
restriction, attribution, or compensation.
Numbered footnotes are as
they appeared in the 1923 edition; an editor's note is marked by an asterisk
(*) and appears in sans serif type. The 1923 English translation modified
the notation used in Einstein's 1905 paper to conform to that in use by
the 1920's; for example, c denotes the speed of light, as opposed
the V used by Einstein in 1905.
This electronic edition
was prepared by
John Walker in November
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