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From: Paul Stowe
Newsgroups: sci.physics.relativity
Subject: Lorentz' 1904 Paper (Commonly known herein as LET)
Date: Fri, 31 Dec 1999 23:48:00 GMT
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Presented below is a complete rendering of Lorentz' paper. I was quite
an effort to convert this into ASCII format. It is hoped that the
readers will find this helpful, and thus the effort worthwhile.
Paul Stowe

ELECTROMAGNETIC PHENOMENA
IN A SYSTEM MOVING WITH ANY VELOCITY LESS THAN THAT OF LIGHT
By H. A. LORENTZ (1904)
The problem of determining the influence exerted on electric and optical
phenomena by a translation, such as all systems have in virtue of the
Earth's annual motion, admits of a comparatively simple solution, so long
as only those terms need be taken into account, which are proportional to
the first power of the ratio between the velocity of translation v and
the velocity of light c. Cases in which quantities of the second order,
i.e. of the order (v/c)^2, may be perceptible, present more difficulties.
The first example of this kind is Michelson's wellknown interference
experiment, the negative result of which has led Fitzgerald and myself to
the conclusion that the dimensions of solid bodies are slightly altered
by their motion through the ether.
Some new experiments, in which a second order effect was sought for, have
recently been published. Rayleigh[*] and Brace[t] have examined the
question whether the Earth's motion may cause a body to become doubly
refracting. At first sight this might be expected, if the just mentioned
change of dimensions is admitted. Both physicists, however, have
obtained a negative result.
In the second place Trouton and Noble[+] have endeavored to detect a
turning couple acting on a charged condenser, the plates of which make a
certain angle with the direction of translation. The theory of electrons,
unless it be modified by some new hypothesis, would undoubtedly require
the existence of such a couple. In order to see this, it
will suffice to consider a condenser with ether as dielectric. It may be
shown that in every electrostatic system, moving with a velocity v,[*]
there is a certain amount of " electromagnetic momentum." If we represent
this, in direction and magnitude, by a vector G, the couple in question
will be determined by the vector product[i]
[G.v] . . . . . (1)
Now, if the axis of z is chosen perpendicular to the condenser plates,
the velocity v having any direction we like; and if U is the energy of
the condenser, calculated in the ordinary way, the components of G are
given[++] by the following formula, which are exact up to the first
order,
2U 2U
G_x = v_x G_y = v_y G_z = 0
c^2 c^2
Substituting these values in (1), we get for the components of the
couple, up to terms of the second order,
2U 2U
v_yv_z,  v_xv_z, 0
c^2 c^2
These expressions show that the axis of the couple lies in the plane of
the plates, perpendicular to the translation. If a is the angle between
the velocity and the normal to the plates, the moment of the couple will
be U(v/c)^2 sin 2a; it tends to turn the condenser into such a position
that the plates are parallel to the Earth's motion.
In the apparatus of Trouton and Noble the condenser was fixed to the beam
of a torsionbalance, sufficiently delicate to be deflected by a couple
of the above order of magnitude. No effect could however be observed.
2. The experiments of which I have spoken are not the only reason for
which a new examination of the problems connected with the motion of the
Earth is desirable. Poincare[**] has objected to the existing theory of
electric and optical phenomena in moving bodies that, in order to explain
Michelson's negative result, the introduction of a new hypothesis has
been required, and that the same necessity may occur each time new facts
will be brought to light. Surely this course of inventing special
hypotheses for each new experimental result is somewhat artificial. It
would be more satisfactory if it were possible to show by means of
certain fundamental assumptions and without neglecting terms of one order
of magnitude or another, that many electromagnetic actions are entirely
independent of the motion of the system. Some years ago, I already sought
to frame a theory of this kind[tt]. I believe it is now possible to treat
the subject with a better result. The only restriction as regards the
velocity will be that it be less than that of light.
3. 1 shall start from the fundamental equations of the theory of
electrons[23]. Let D be the dielectric displacement in the ether, H the
magnetic force, rho the volume density of the charge of an electron, v
the velocity of a point of such a particle, and F the ponderomotive
force, i.e. the force, reckoned per unit charge, which is exerted by the
ether on a volume element of an electron. Then, if we use a fixed system
of coordinates,
(Note: throughout, the d' in the equations are partial differentials)
Div D = rho 
Div H = 0 
Curl H = 1/c(d'D/d't + rho(v))  . . . (2)
Curl D = 1/c(d'H/d't) 
F = D + 1/c[v.H] 
I shall now suppose that the system as a whole moves in the direction of
x with a constant velocity v, and I shall denote by u any velocity which
a point of an electron may have in addition to this, so that
v_x = v + u_x, v_y = u_y, v_z = u_z
If the equations (2) are at the same time referred to axes moving with
the system, they become
d'H_z d'H_y 1 / d' d' \ 1
   =    v D_x +  rho(v + u_x)
d'y d'dz c \d't d'x / c
d'H_x d'H_z 1 / d' d' \ 1
   =    v D_y +  rho(u_y)
d'z d'dx c \d't d'x / c
d'H_y d'H_x 1 / d' d' \ 1
   =    v D_z +  rho(u_z)
d'x d'dy c \d't d'x / c
d'D_z d'D_y 1 / d' d' \
   =    v H_x
d'y d'dz c \d't d'x /
d'D_x d'D_z 1 / d' d' \
   =    v H_y
d'z d'dx c \d't d'x /
d'D_y d'D_x 1 / d' d' \
   =    v H_z
d'x d'dy c \d't d'x /
1
F_x = D_x + (u_yH_z  u_zH_y)
c
1 1
F_y = D_y + vH_z + (u_zH_x  u_xH_z)
c c
1 1
F_z = D_z + vH_y + (u_xH_y  u_yH_x)
c c
4. We shall further transform these formula by a change of variables.
putting
c^2
 = B^2, . . . (3)
c^2  v^2
and understanding by {L} another numerical quantity, to be determined
further on, I take as new independent variables
x' = BLx, y' = Ly, z' = Lz, . (4)
t' = Lt/B  (BLx/c)(v/c), . . (5)
and I define two new vectors D' and H' by the formula
1
D'_x = D_x
L^2
B / v \
D'_y =  D_y  H_z 
L^2 \ c /
B / v \
D'_z =  D_z  H_y 
L^2 \ c /
1
H'_x = H_x
L^2
B / v \
H'_y =  H_y  D_z 
L^2 \ c /
B / v \
H'_z =  H_z  D_y 
L^2 \ c /
For which, on account of (3), we may also write
D_x = L^2D'x 

D_y = BL^2(D'y + (v/c)H'z) 

D_z = BL^2(D'z + (v/c)H'y) 
 . . (6)
H_x = L^2H'x 

H_y = BL^2(H'y + (v/c)D'z) 

H_z = BL^2(H'z + (v/c)D'y) 
As to the coefficient 1, it is to be considered as a function of v, whose
value is L for v = 0, and which, for small values of v, differs from
unity no more than by a quantity of the second order.
The variable t' maybe called the" local time"; indeed, for B = 1, L = 1
it becomes identical with what I formerly denoted by this name.
If, finally, we put
1
rho = rho' . . . (7)
BL^3
B^2u_x = u'_x, Bu_y = u'_y, Bu_z = u'_z, (8)
These latter quantities being considered as the components of a new
vector u', the equations take the form:
/ vu'x \ 
Div D' = 1   rho' 
\ c^2 / 

Div H' = 0 

1 d'H' . (9)
Curl D' =   
c d't' 

1 /d'D' \
Curl H' =   + rho'(u')
C \d't' /
F_x = L^2{D'_x+(1/c)(u'_yH'_zu'_zH'_y)+(v/c^2)(u'_yD'_y+u'_zD'_z)}

F_y = (L^2/B){D'_y+(1/c)(u'_zH'_xu'_xH'_z)(v/c^2)(u'_xD'_y)} (10)

F_z = (L^2/B){D'_z+(1/c)(u'_xH'_yu'_yH'_x)(v/c^2)(u'_xD'_z)} 
The meaning of the symbols div' and curl' in (9) is similar to that of
div and curl in (2); only, the differentiation with respect to x, y, z
are to be replaced by the corresponding ones with respect to x', y', z'.
5. The equations (9) lead to the conclusion that the vectors D' and H'
may be represented by means of a scalar potential Fee' and a vector
potential A'. These potentials satisfy the equations
1 d'^2Fee'
Del^2(Fee') =   = rho'. . (11)
c^2 d't'^2
1 d'^2A' 1
Del^2(A') =   =  rho'(u'). (12)
c^2 d't'^2 c
and in terms of them D' and H' are given by
1 d'A' v
D' =    grad' Fee' + grad' A'_x . (13)
c d't' c
H' = curl' A' . . . . . . . (14)
the symbol Del^2 is an abbreviation for
d'^2/d'x^2 + d'^2/d'y^2 + d'^2/d'z^2,
and grad' Fee' denotes a vector whose components are
d'Fee'/d'x', d'Fee'/d'y', d'Fee'/d'z',
The expression grad' A'_x has a similar meaning.
In order to obtain the solution of (11) and (12) in a simple form, we may
take x', y', z' as the coordinates of a point Y in a space S', and
ascribe to this point, for each value of t', the values of rho', U',
Fee', A', belonging to the corresponding point P(x, y, z) of the
electromagnetic system. For a definite value t' of the fourth independent
variable, the potentials Fee' and A' at the point P of the ' system or at
the corresponding point P' of the space S', are given by
1 /[rho']
Fee' =   dS'. . . . (15)
4pi / g'
1 /[rho'(u')]
A' =   dS' . . (16)
c4pi / g'
Here dS' is an element of the space S', r' its distance from P', and the
brackets serve to denote the quantity rho' and the vector rho/u' such as
they are in the element dS', for the value t,  g'/c of the fourth
independent variable.
Instead of (15) and (16) we may also write, taking into account (4) and
(7),
1 /[rho]
Fee =   dS . . . . (17)
4pi / g
1 /[rho(u)]
A =   dS . . . (18)
c4pi / g
the integration now extending over the electromagnetic system itself. It
should be kept in mind that in these formula g' does not denote the
distance between the element dS and the point (x, y, z) for which the
calculation is to be performed. If the element lies at the point (x_1,
y_1, z_1,), we must take
g' = Sqrt(B^2(x  x_1)^2 + (y  y_1)^2 + (z  z_1)^2)
It is also to be remembered that, if we wish to determine Fee' and A' for
the instant at which the local time in P is t', we must take rho and
rho(u'), such as they are in the element dS at the instant at which the
local time of that element is t'  g'/c.
6. It will suffice for our purpose to consider two special cases. The
first is that of an electrostatic system, i.e. a system having no other
motion but the translation with the velocity v. In this case u' = 0, and
therefore, by (12), A' = 0. Also, Fee' is independent of t', so that the
equations (11), (13), and (14) reduce to
Del^2 Fee' = p' 
D' = grad' Fee' . . (19)
H' = 0 
After having determined the vector D' by means of these equations, we
know also the ponderomotive force acting on electrons that belong to the
system. For these the formula (10) become, since u' = 0,
F_x = L^2D'_x 
F_y = (L^2/B)D'_y . . (20)
F_z = (L^2/B)D'_z 
The result may be put in a simple form if we compare the moving system
Sigma, with which we are concerned, to another electrostatic system
Sigma' which remains at rest, and into which Sigma is changed if the
dimensions parallel to the axis of x are multiplied by BL, and the
dimensions which have the direction of y or that of z, by La deformation
for which (BL, L, L) is an appropriate symbol. In this new system, which
we may suppose to be placed in the abovementioned space S', we shall
give to the density the value rho', determined by (7), so that the
charges of corresponding elements of volume and of corresponding
electrons are the same in Sigma and Sigma'. Then we shall obtain the
forces acting on the electrons of the moving system Sigma, if we first
determine the corresponding forces in Sigma', and next multiply their
components in the direction of the axis of x by L^2, and their components
perpendicular to that axis by L^2/B. This is conveniently expressed by
the formula
F(Sigma) = (L^2, L^2/B, L^2/B)F(Sigma') (21)
It is further to be remarked that, after having found D' by (19), we can
easily calculate the electromagnetic momentum in the moving system, or
rather its component in the direction of the motion. Indeed, the formula
1/
G = [D.H]dS
c/
shows that
1/
G_x = (D_yH_z  D_zH_y)dS
C/
Therefore, by (6), since H' = 0
B^2L^4v/ BLv/
G_x = (D'_y^2 + D'_z^2)dS = (D'_y^2 + D'_z^2)dS' (22)
c^2 / c^2/
7. Our second special case is that of a particle having an electric
moment, i.e. a small space S, with a total charge (integral rho dS = 0)
but with such a distribution of density that the integrals rho(x) dS,
rho(y) dS, rho(z) dS have values differing from zero.
Let e, f, h be the coordinates, taken relatively to a fixed point A of
the particle, which may be called its center, and let the electric moment
be defined as a vector P whose components are
/ 
P_x = rho(e)dS 
/ 

/ 
P_y = rho(f)dS  . . . (23)
/ 

/ 
P_z = rho(h)dS 
/ 
Then
dP_x / 
 = rho(u_x)dS 
dt / 

dP_y / 
 = rho(u_y)dS 
dt /  . . . (24)

dP_z / 
 = rho(u_z)dS 
dt / 

Of course, if e, f, h are treated as infinitely small, u_x, u_y, u_z must
be so likewise. We shall neglect squares and products of these six
quantities.
We shall now apply the equation (17) to the determination of the scalar
potential Fee' for an exterior point P(x, y, z), at a finite distance
from the polarized particle, and for the instant at which the local time
of this point has some definite value t'. In doing so, we shall give the
symbol [rho], which, in (17), relates to the instant at which the local
time in dS is t'  g'/c, a slightly different meaning. Distinguishing by
g'_o the value of g' for the center A, we shall understand by [rho] the
value of the density existing in the element dS at the point (e, f, h) at
the instant t_o at which the local time of A is t'  g_o/c.
It may be seen from (5) that this instant precedes that for which we have
to take the numerator in (17) by
ve B(g'_o  g') ve B / d'g' d'g' d'g'\
B^2 +  = B^2 + e + f + h
c^2 Lc c^2 Lc\ d'x d'y d'z /
units of time. In this last expression we may put for the differential
coefficients their values at the point A. In (17) we have now to replace
[rho] by
ve/d'P\ B / d'g' d'g' d'g'\/d'P\
[rho] + B^2 + e + f + h . (25)
c^2\d't/ Lc\ d'x d'y d'z /\d't/
where [d'rho/d'tl relates again to the time t_o. Now, the value of t'
for which the calculations are to be performed having been
chosen, this time t_o will be a function of the coordinates x, y, z of
the exterior point P. The value of [rho] will therefore depend on these
coordinates in such a way that
d'{rho] B d'g'd'[rho]
 =   , etc.
d'x Lc d'x d't
by which (25) becomes
ve d'[rho] / d'[rho] d'[rho] d'[rho]\
rho + B^2   e + f + h
c^2 d't \ d'x d'y d'z /
Again, if henceforth we understand by g' what has above been called g'_o,
the factor 1/g' , must be replaced by
1 d'/ 1 \ d'/ 1 \ d'/ 1 \
 = e + f + h
g' d'x\ g'/ d'y\ g'/ d'z\ g'/
so that after all, in the integral (17), the element dS is multiplied by
[rho] ve d'[rho] d' e[rho] d' f[rho] d' h[rho]
 + B^2          
g' c^2g' d't d'x g' d'y g' d'z g'
This is simpler than the primitive form, because neither g', nor the time
for which the quantities enclosed in brackets are to be taken, depend on
x, y, z. Using (23) and remembering that the integral rho dS = 0, we get
B^2v d'P_x 1/ d' P_x d' P_y d' P_z\
Fee' =       +   +  
4pc^2g' d't 4p\ d'x g' d'y g' d'z g'/
a formula in which all the enclosed quantities are to be taken for the
instant at which the local time of the center of the particle is t' 
g'/c.
We shall conclude these calculations by introducing a new vector P',
whose components are
P'_x = BLP_x 
P'_y = LP_y  . . . (26)
P'_z = LP_z 
passing at the same time to x', y', z', t' as independent variables. The
final result is
v d'P'_x 1/ d' P'_x d' P'_y d' P'_z\
Fee' =       +   +  
4pc^2g' d't' 4p\ d'x' g' d'y' g' d'z' g' /
As to the formula (18) for the vector potential, its transformation is
less complicated, because it contains the infinitely small vector u'.
Having regard to (8), (24), (26), and (5), I find
1 d'P'
A' =  
4picg' d't'
The field produced by the polarized particle is now wholly determined.
The formula (13) leads to
1 d'^2 P' 1 / d' P'_x d' P'_y d' P'_z\
D' =     +  grad'  +   +  
4pic^2 d't'^2 g' 4pi \d'x' g' d'y' g' d'z' g' /
(27)
and the vector H' is given by (14). We may further use the equation (20),
instead of the original formula (10), if we wish to consider the forces
exerted by the polarized particle on a similar one placed at some
distance. Indeed, in the second particle, as well as in the first, the
velocities u may be held to be infinitely small.
It is to be remarked that the formulae for a system without translation
are implied in what precedes. For such a system the quantities with
accents become identical to the corresponding ones without accents; also
B = 1 and 1 = 1. The components of (27) are at the same time those of the
electric force which is exerted by one polarized particle on another.
8. Thus far we have used only the fundamental equations without any new
assumptions. I shall now suppose that the electrons, which I take to be
spheres of radius R in the state of rest, have their dimensions changed
by the effect of a translation, the dimensions in the direction of motion
becoming 81 times and those in perpendicular directions 1 times smaller.
In this deformation, which may be represented by (1/BL, 1/L, 1/L), each
element of volume is understood to preserve it's charge.
Our assumption amounts to saying that in an electrostatic system Sigma,
moving with a velocity v, all electrons are flattened ellipsoids with
their smaller axes in the direction of motion. If now, in order to apply
the theorem of section 6, we subject the system to the deformation (BL,
L, L), we shall have again spherical electrons of radius R. Hence, if we
alter the relative position of the centers of the electrons in Sigma by
applying the deformation (BL, L, L), and if, in the points thus obtained,
we place the centers of electrons that remain at rest, we shall get a
system, identical to the imaginary system Sigma', of which we have.
spoken in section 6. The forces in this system and those in Sigma will
bear to each other the relation expressed by (21).
In the second place I shall suppose that the forces between uncharged
particles, as well as those between such particles and electrons, are
influenced by a translation in quite the same way as the electric forces
in an electrostatic system. In other terms, whatever be the nature of the
particles composing a ponderable body, so long as they do not move
relatively to each other, we shall have between the forces acting in a
system (Sigma') without, and the same system (Sigma) with a translation,
the relation specified in (21), if, as regards the relative position of
the particles, Sigma' is got from Sigma by the deformation (BL, L, L), or
Sigma from Sigma' by the deformation ( 1/BL, 1/L, 1/L).
We see by this that, as soon as the resulting force is zero for a
particle in Sigma', the same must be true for the corresponding particle
in Sigma. Consequently, if, neglecting the effects of molecular motion,
we suppose each particle of a solid body to be in equilibrium under the
action of the attraction and repulsion exerted by its neighbors, and if
we take for granted that there is but one configuration of equilibrium,
we may draw the conclusion that the system Sigma', if the velocity v is
imparted to it, will of itself change into the system Sigma. In other
terms, the translation will produce the deformation (1/BL, 1/L, 1/L).
The case of molecular motion will be considered in section 12.
It will easily be seen that the hypothesis which was formerly advanced in
connection with Michelson's experiment, is implied in what has now been
said. However, the present hypothesis is more general, because the only
limitation imposed on the motion is that its velocity be less than that
of light.
9. We are now in a position to calculate the electromagnetic momentum of
a single electron. For simplicity's sake I shall suppose the charge q to
be uniformly distributed over the surface, so long as the electron
remains at rest. Then a distribution of the same kind will exist in the
system Sigma' with which we are concerned in the last integral of (22).
Hence
/ 2/ q^2/ dr q^2
(D'^2_y + D'^2_z)dS' = D'^2 dS' =  = 
/ 3/ 6pi/r^2 6piR
and
q^2
G_x = BLv
c^2R6pi
It must be observed that the product BL is a function of v and that, for
reasons of symmetry, the vector G has the direction of the translation.
In general, representing by v the velocity of this motion, we have the
vector equation
q^2
G = BLv . . . . . (28)
c^2R6pi
Now, every change in the motion of a system will entail a corresponding
change in the electromagnetic momentum and will therefore require a
certain force, which is given in direction and magnitude by
F = dG/dt . . . . . (29)
Strictly speaking, the formula (28) may only be applied in the case of a
uniform rectilinear translation. On account of this circumstancethough
(29) is always truethe theory of rapidly varying motions of an electron
becomes very complicated, the more so, because the hypothesis of § 8
would imply that the direction and amount of the deformation are
continually changing. It is, indeed, hardly probable that the form of the
electron will be determined solely by the velocity existing at the moment
considered.
Nevertheless, provided the changes in the state of motion be sufficiently
slow, we shall get a satisfactory approximation by using (28) at every
instant. The application of (29) to such a quasistationary translation,
as it has been called by Abraham, is a very simple matter. Let, at a
certain instant, a, be the acceleration in the direction of the path, and
a_2 the acceleration perpendicular to it. Then the force F will consist
of two components, having the directions of these accelerations and which
are given by
F_1 = m_1a_1 and F_2 = m_2a_2
if
q^2 d(BLv) q^2BL
m_1 =   and m_2 =  . . (30)
c^2R6pi dv c^2R6pi
Hence, in phenomena in which there is an acceleration in the direction of
motion, the electron behaves as if it had a mass m_l; In those in which
the acceleration is normal to the path, as if the mass were m_2. These
quantities m_1, and m_2 may therefore properly be called the "
longitudinal " and " transverse " electromagnetic masses of the electron.
I shall suppose that there is no other, no " true " or material " mass.
Since B and L differ from unity by quantities of the order (v/c)^2, we
find for very small velocities
q^2
m_1 = m_2 = 
c^2R6pi
This is the mass with which we are concerned, if there are small
vibratory motions of the electrons in a system without translation. If,
on the contrary, motions of this kind are going on in a body moving with
the velocity v in the direction of the axis of x, we shall have to reckon
with the mass m, as given by (30), if we consider the vibrations parallel
to that axis, and with the mass m, if we treat of those that are parallel
to OY or OZ. Therefore, in short terms, referring by the index Sigma to a
moving system and by Sigma' to one that remains at rest,
/dBLv \
m(Sigma) = , BL, BLm(Sigma') . . . (31)
\ dv /
10. We can now proceed to examine the influence of the Earth's motion on
optical phenomena in a system of transparent bodies. In discussing this
problem we shall fix our attention on the variable electric moments in
the particles or "atoms" Of the system. To these moments we may apply
what has been said in § 7. For the sake of simplicity we shall suppose
that, in each particle, the charge is concentrated in a certain number
of separate electrons, and that the " elastic " forces that act on one of
these, and, conjointly with the electric forces, determine its motion,
have their origin within the bounds of the same atom.
I shall show that, if we start from any given state of motion in a system
without translation, we may deduce from it a corresponding state that can
exist in the same system after a' translation has been imparted to it,
the kind of correspondence being as specified in what follows.
(a) Let A_1,, A_2, A_3, etc., be the centers of the particles in
the system without translation (Sigma'); neglecting molecular
motions we shall assume these points to remain at rest. The
system of points A_1, A_2, A_3, etc., formed by the centers
of the particles in the moving system :~, is obtained from
A'_1, A'_2, A'_3, etc., by means of a deformation (1/BL,
1/L, 1/L). According to what has been said in Section 8,
the centers will of themselves take these positions A'_1,
A'_2, A'_3, etc., if originally, before there was a translation,
they occupied the positions A_1, A_2, A_3, etc.
We may conceive any point P' in the space of the system Sigma'
to be displaced by the above deformation, so that a definite
point P of Sigma corresponds to it. For two corresponding
points P' and P we shall define corresponding instants, the
one belonging to P', the other to P, by stating that the true
time at the first instant is equal to the local time, as
determined by (5) for the point P, at the second instant. By
corresponding times for two corresponding particles we shall
understand times that may be said to correspond, if we fix our
attention on the centers A' and A of these particles.
(b) As regards the interior state of the atoms, we shall assume
that the configuration of a particle A in Sigma at a certain
time may be derived by means of the deformation (1/BL, 1/L,
1/L) from the configuration of the corresponding particle in
Sigma, such as it is at the corresponding instant. In so far
as this assumption relates to the form of the electrons
themselves, it is implied in the first hypothesis of section 8.
Obviously, if we start from a state really existing in the
system Sigma', we have now completely defined a state of the
moving system Sigma. The question remains, however, whether
this state will likewise be a possible one.
In order to judge of this, we may remark in the first place that the
electric moments which we have supposed to exist in the moving system and
which we shall denote by P, will be certain definite functions of the
coordinates x, y, z of the centers A of the particles, or, as we shall
say, of the coordinates of the particles themselves, and of the time t.
The equations which express the relations between P on one band and x, y,
z, t on the other, may be replaced by other equations containing the
vectors P' defined by (26) and the quantities x', y', z', t' defined by
(4) and (5). Now, by the above assumptions a and b, if in a particle A of
the moving system, whose coordinates are x, y, z, we find an electric
moment P at the time t, or at the local time t', the vector P' given by
(26) will be the moment which exists in the other system at the true time
t' in a particle whose coordinates are x', y', z'. It appears in this way
that the equations between P', x', y', z', t' are the same for both
systems, the difference being only this, that for the system Sigma'
without translation these symbols indicate the moment, the coordinates,
and the true time, whereas their meaning is different for the moving
system, P', x', y', z', t' being here related to the moment P, the
coordinates x, y, z and the general time t in the manner expressed by
(26), (4), and (5).
It has already been stated that the equation (27) applies to both
systems. The vector D' will therefore be the same in Sigma' and Sigma,
provided we always compare corresponding places and times. However, this
vector has not the same meaning in the two cases. In Sigma' it represents
the electric force, in Sigma it is related to this force in the way
expressed by (20). We may therefore conclude that the ponderomotive
forces acting, in Sigma and in Sigma', on corresponding particles at
corresponding instants, bear to each other the relation determined by
(21). In virtue of our assumption (b), taken in connection with the
second hypothesis of section 8, the same relation will exist between the
" elastic " forces; consequently, the formula (21) may also be regarded
as indicating the relation between the total forces, acting on
corresponding electrons, at corresponding instants.
It is clear that the state we have supposed to exist in the moving system
will really be possible if, in Sigma and Sigma', the products of the mass
m and the acceleration of an electron are to each other in the same
relation as the forces, i.e. if
/ L^2 L^2\
ma(Sigma) = L^2,, ma(Sigma') . . (32)
\ B B /
Now, we have for the accelerations
/ L L L \
a(Sigma) = , , a(Sigma') . . (33)
\B^3 B^2 B^2/
as may be deduced from (4) and (5), and combining this with (32), we find
for the masses
m(Sigma) = (B^3L,BL, BL)m(Sigma')
If this is compared with (31), it appears that, whatever be the value of
1, the condition is always satisfied, as regards the masses with which we
have to reckon when we consider vibrations perpendicular to the
translation. The only condition we have to impose on L is therefore
d(BLv)
 = B^3L
dv
But, on account of (3),
d(Bv)
 = B^3
dv
so that we must put
dL
 = 0, L = const.
dv
The value of the constant must be unity, because we know already that,
for v = 0, L = 1.
We are therefore led to suppose that the influence of a translation on
the dimensions (of the separate electrons and of a ponderable body as a
whole) is confined to those that have the direction of the motion, these
becoming B times smaller than they are in the state of rest. If this
hypothesis is added to those we have already made, we may be sure that
two states, the one in the moving system, the other in the same system
while at rest, corresponding as stated above, may both be possible.
Moreover, this correspondence is not limited to the electric moments of
the particles. In corresponding points that are situated either in the
ether between the particles, or in that surrounding the ponderable
bodies, we shall find at corresponding times the same vector D' and, as
is easily shown, the same vector H'. We may sum up by saying: If, in the
system without translation, there is a state of motion in which, at a
definite place, the components of P, D, and H are certain functions of
the time, then the same system after it has been put in motion (and
thereby deformed) can be the seat of a state of motion in which, at the
corresponding place, the components of V, D', and H' are the same
functions of the local time.
There is one point which requires further consideration. The values of
the masses m_1, and m_2 having been deduced from the theory of quasi
stationary motion, the question arises, whether we are justified in
reckoning with them in the case of the rapid vibrations of light. Now it
is found on closer examination that the motion of an electron may be
treated as quasi stationary if it changes very little during the time a
lightwave takes to travel over a distance equal to the diameter. This
condition is fulfilled in optical phenomena, because the diameter of an
electron is extremely small in comparison with the wavelength.
11. It is easily seen that the proposed theory can account for a large
number of facts.
Let us take in the first place the case of a system without translation,
in some parts of which we have continually P = 0, D = 0, H = 0. Then, in
the corresponding state for the moving system, we shall have in
corresponding parts (or, as we may say, in the same parts of the deformed
system) rho' = 0, D' = 0, H'  0. These equations implying P = 0, D = 0,
H = 0, as is seen by (26) and (6), it appears that those parts which are
dark while the system is at rest, will remain so after it has been put in
motion. It will therefore be impossible to detect an influence of the
Earth's motion on any optical experiment, made with a terrestrial source
of light, in which the geometrical distribution of light and darkness is
observed. Many experiments on interference and diffraction belong to this
class.
In the second place, if, in two points of a system, rays of light of the
same state of polarization are propagated in the same direction, the
ratio between the amplitudes in these points may be shown not to be
altered by a translation. The latter remark applies to those experiments
in which the intensities in adjacent parts of the field of view are
compared.
The above conclusions confirm the results which I formerly obtained by a
similar train of reasoning, in which, however, the terms of the second
order were neglected. They also contain an explanation of Michelson's
negative result, more general than the one previously given, and of a
somewhat different form; and they show why Rayleigh and Brace could find
no signs of double refraction produced by the motion of the Earth.
As to the experiments of Trouton and Noble, their negative result becomes
at once clear, if we admit the hypotheses of section 8. It may be
inferred from these and from our last assumption (section 10) that the
only effect of the translation must have been a contraction of the whole
system of electrons and other particles constituting the charged
condenser and the beam and thread of the torsionbalance. Such a
contraction does not give rise to a sensible change of direction.
It need hardly be said that the present theory is put forward with all
due reserve. Though it seems to me that it can account for all well
established facts, it leads to some consequences that cannot as yet be
put to the test of experiment. One of these is that the result of
Michelson's experiment must remain negative, if the interfering rays of
light are made to travel through some ponderable transparent body.
Our assumption about the contraction of the electrons cannot in itself be
pronounced to be either plausible or inadmissible. What we know about the
nature of electrons is very little, and the only means of pushing our way
farther will be to test such hypotheses as I have here made. Of course,
there will be difficulties, e.g. as soon as we come to consider the
rotation of electrons. Perhaps we shall have to suppose that in those
phenomena in which, if there is no translation, spherical electrons
rotate about a diameter, the points of the electrons in the moving system
will describe elliptic paths, corresponding, in the manner specified in
section 10, to the circular paths described in the other case.
12. There remain to be said a few words about molecular motion. We may
conceive that bodies in which this has a sensible influence or even
predominates, undergo the same deformation as the systems of particles of
constant relative position of which alone we have spoken till now.
Indeed, in two systems of molecules Sigma' and Sigma, the first without
and the second with a translation, we may imagine molecular motions
corresponding to each other in such a way that, if a particle in Sigma'
has a certain position at a definite instant, a particle in Sigma
occupies at the corresponding instant the corresponding position. This
being assumed, we may use the relation (33) between the accelerations in
all those cases in which the velocity of molecular motion is very small
as compared with v. In these cases the molecular forces may be taken to
be determined by the relative positions, independently of the velocities
of molecular motion. If, finally, we suppose these forces to be limited
to such small distances that, for particles acting on each other, the
difference of local times may be neglected, one of the particles,
together with those which lie in its sphere of attraction or repulsion,
will form a system which undergoes the often mentioned deformation. In
virtue of the second hypothesis of section 8 we may therefore apply to
the resulting molecular force acting on a particle, the equation (21).
Consequently, the proper relation between the forces and the
accelerations will exist in the two cases, if we suppose that the masses
of all particles are influenced by a translation to the same degree as
the electromagnetic masses of the electrons.
13. The values (30), which I have found for the longitudinal and
transverse masses of an electron, expressed in terms of its velocity, are
not the same as those that had been previously obtained by Abraham. The
ground for this difference is to be sought solely in the circumstance
that, in his theory, the electrons are treated as spheres of invariable
dimensions. Now, as regards the transverse mass, the results of Abraham
have been confirmed in a most remarkable way by Kaufmann's measurements
of the deflection of radiumrays in electric and magnetic fields.
Therefore, if there is not to be a most serious objection to the theory I
have now proposed, it must be possible to show that those measurements
agree with my values nearly as well as with those of Abraham.
I shall begin by discussing two of the series of measurements published
by Kaufmann in 1902. From each series he has deduced two quantities n and
~, the "reduced" electric and magnetic deflections, which are related as
follows to the ratio g = v/c:
g = k_1(h/f)
. . . . . (34)
Psi(g) = f/k_2h^2
Here Psi(g) is such a function, that the transverse mass is given by
3
m_2 = Sqrt(1  g^2). . . . (35)
4
whereas k_1, and k_2 are constant in each series.
It appears from the second of the formulae (30) that my theory leads
likewise to an equation of the form (35); only Abraham's function Psi(g)
must be replaced by
4 4
B = Sqrt(1  g^2)
3 3
Hence, my theory requires that, if we substitute this value for Psi(g) in
(34), these equations shall still hold. Of course, in seeking to obtain a
good agreement, we shall be justified in giving to k_1, and k_2 other
values than those of Kaufmann, and in taking for every measurement a
proper value of the velocity v, or of the ratio g. Writing
sk_1(3/4)k'_2
and g' for the new values, we may put (34) in the form
g' = sk_1(h/f) . . . . . . (36)
and
Sqrt(1  g'^2) = f/(k'_2h^2). . . . (37)
Kaufmann has tested his equations by choosing for k_1, such a value that,
calculating g and k_2, by means of (34), he obtained values for this
latter number which, as well as might be, remained constant in each
series. This constancy was the proof of a sufficient agreement.
I have followed a similar method, using, however, some of the numbers
calculated by Kaufmann. I have computed for each measurement the value of
the expression
k'_2 = Sqrt(1  g'^2)Psi(g)k_2 . . . (38)
that may be got from (37) combined with the second of the equations (34).
The values of Psi(y) and k_2 have been taken from Kaufmann's tables, and
for g' I have substituted the value he has found for g, multiplied by s,
the latter coefficient being chosen with a view to obtaining a good
constancy of (38). The results are contained in the tables on opposite
page, corresponding to the Tables III and IV in Kaufmann's paper.
The constancy of k'_2 is seen to come out no less satisfactorily than
that of k_2 the more so as in each case the value of s has been
determined by means of only two measurements. The coefficient has been so
chosen that for these two observations, which were in Table III the first
and the last but one, and in Table IV the first and the last, the values
of A'2 should be proportional to those of k_2
I shall next consider two series from a later publication by Kaufmann,
which have been calculated by Runge by means of the method of least
squares, the coefficients k_1, and k_2, having been determined in such a
way that the values of n, calculated, for each observed ~, from
Kaufmann's equations (34), agree as closely as may be with the observed
values of n.
Table III s = 0.933
g Psi(g) k_2 g' K'_2
0.851 2.147 1.721 0.794 2.246
0.766 1.86 1.736 0.715 2.258
0.727 1.78 1.725 0.678 2.256
0.6616 1.66 1.727 0.617 2.256
0.6075 1.595 1.655 0.567 2.175
Table IV s = 0.954
0.963 3.28 8.12 0.919 10.36
0.949 2.86 7.99 0.905 9.70
0.933 2.73 7.46 0.890 9.28
0.683 2.31 8.32 0.842 10.36
0.860 2.195 8.09 0.820 10.16
0.830 2.06 8.13 0.792 10.23
0.801 1.96 8.13 0.764 10.28
0.777 1.89 B.04 0.741 10.20
0.752 1.83 8.02 0.717 10.22
0.732 1.785 7.97 0.698 10.18
I have determined by the same condition, likewise using the method of
least squares, the constants a and b in the formula
f^2 = ah^2 + bh^4
which may be deduced from my equations (36) and (37). Knowing a and b, I
find g for each measurement by means of the relation
g = Sqrt(a)(h/f)
For two plates on which Kaufmann had measured the electric and magnetic
deflections, the results are as follows (p. 34), the deflections being
given in centimeters.
I have not found time for calculating the other tables in Kaufmann's
paper. As they begin, like the table for Plate 15 (next page) with a
rather large negative difference between the values of 17 which have been
deduced from the observations and calculated by Runge, we may expect a
satisfactory agreement with my formula.
Plate No. 15.a  006489, b = 03039.
~ Observed Calc. By E. Diff. Calc. by L. Diff. R. L.
0.1495 0.0388 0.0404 16 0.0400 12 0.997 0.951
0.199 0.0548 0.0550 2 0.0552 4 0.904 0.918
0.2475 0.0716 0.0710 +6 0.0715 +1 0.930 0.881
0296 0.0896 0.0887 +9 0.0895 +1 0.889 0.842
03435 0.1080 0.1081 1 0.1090 10 0.847 0.803
0.391 0.1290 0.1297 7 0.1305 15 0.804 0.763
0.487 0.1524 0.1527 3 0.1532 8 0.763 0.727
0.4025 0.1788 0.1777 +11 0.1777 +11 0.724 0.692
0.5265 0.2033 0.2039 6 0.2033 0 0.688 0.660
0.1495 0.0404 0.0388 +16 0.0379 +25 0.990 0.954
Plate No. 19.a = 005867, b = 02591.
0.199 0.0529 0.0527 +2 0.0522 +7 0.969 0.923
0.247 0.0678 0.0675 +3 0.0674 +4 0.939 0.888
0.296 0.0834 0.0842 8 0.0644 10 0.902 0.849
0.3435 0.1019 0.1022 3 0.1026 7 0.862 0.811
0.391 0.1219 0.1222 3 0.1226 7 0.822 0.773
0.437 0.1429 0.1434 5 0.1487 8 0.782 0.736
0.4825 0.1660 0.1665 5 0.1664 4 0.744 0.702
0.5265 0.1916 0.1906 +10 0.1902 +14 0.709 0.671
* Rayleigh, Phil. Mag. (6), 4, 1902, p. 678.
t Brace, Phil. Mag. (6), 7, 1904, p. 317.
+ Trouton and Noble, Phil. Trans. Roy. Soc. Lond., A 202, 1903, p. 165.
* A vector will be denoted by a Clarendon letter, its magnitude by the
corresponding Latin letter.
i See my article: 11 Weiterbildung der Maxwell'schen Theorie. Electron
entheorie," Mathem. Encyclop4die, V, 14, § 21, a. (This article will be
quoted as 11M.E.")
++ 11 M.E.", § 56, c.
** Poincari, Rapports du Congees de physique de 1900, Paris, 1, pp. 22,
23 Lorentz, Zittingsverslag Akad. v. Wet., 7, 1899, p. 507; Amsterdam
Proc., 189899, p. 427. ++ , M. E.,11 § 2.
From:
Proceedings of the Academy of Sciences of Amsterdam, 6, (1904),
"Electromagnetic Phenomena in a System Moving with any Velocity less
than that of Light", by H. A. Lorentz.
It is the same paper that is reprinted in The Principle of Relativity
published by Dover Publishing. The standard publishing number
is 486600815.
Lorentz also wrote a series of articles in 1903 for the "Enzyklopaedie der
mathematischen Wissenschaften" published in 1904.