Translation:On the Electrodynamics of Moving Systems II

On the Electrodynamics of Moving Systems II  (1904)
by Emil Cohn, translated from German by Wikisource
In German: Zur Elektrodynamik bewegter Systeme II, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften, Zweiter Halbband (43): 1404–1416, Source

(Session of December 15, 1904.)

On the Electrodynamics of Moving Systems II.

By Prof. Emil Cohn

in Straßburg i. E.

Submitted by Mr. Warburg.

Some years ago, I gave an extension of Maxwell's equation for moving bodies, which was in agreement with all then known facts.[1] This approach was found in an inductive way, and has proven itself with respect to subsequent experiences. The decisive test concerns the case of uniform translational velocity; the special case of the equations which corresponds to this case, seems unquestionable to me. Up to now, the general equations were not subject to an equally strict experimental test. In the course of their development, I was led by the principle of "scientific economy";[2] nevertheless, it may be that there can be found a simpler way, which is in agreement with facts as well. Meanwhile, it may be allowed for me, to develop here the characteristic features of that kind of electrodynamics, which follow from my equations. The resulting theorems are in full factual agreement with the content of my older treatise, as far as the properties of the electromagnetic field are concerned; with respect to mechanical forces, they partly deviate from them. That the determination of these forces was arbitrary to some extent, was particularly emphasized by me at that time.

§ 1. The fundamental equations.

 ${\displaystyle -\int \limits _{\circ }{\mathsf {E}}_{s}ds={\frac {d}{dt}}\int {\mathfrak {M}}_{N}dS}$ I
 ${\displaystyle \int \limits _{\circ }{\mathsf {M}}_{s}ds={\frac {d}{dt}}\int {\mathfrak {E}}_{N}dS+\int \Lambda _{N}dS}$ II
 ${\displaystyle {\begin{cases}{\mathfrak {E}}=\epsilon {\mathsf {E}}-[u{\mathsf {M}}]\\\\{\mathfrak {M}}=\mu {\mathsf {M}}+[u{\mathsf {E}}]\\\\\Lambda =\lambda ({\mathsf {E-K}})\end{cases}}}$ III
 ${\displaystyle \Sigma ={\mathsf {EM}}}$ IV
 ${\displaystyle w={\frac {1}{2}}({\mathsf {E}}\cdot {\mathfrak {E}})+{\frac {1}{2}}({\mathsf {M}}\cdot {\mathfrak {M}})+(u\cdot \Sigma )}$ V

Here, E and M denote the two field intensities;

${\displaystyle \epsilon ,\mu ,\lambda }$ scalar constants, K a constant vector;
${\displaystyle u}$ the velocity of matter;
${\displaystyle \Sigma }$ the radiation relative to matter;
${\displaystyle w}$ the electromagnetic energy in unit volume;
${\displaystyle S}$ a surface which continuously goes through the same material particle, ${\displaystyle s}$ its boundary curve, ${\displaystyle N}$ the perpendicular of ${\displaystyle dS}$.

In vacuum, the values apply:

${\displaystyle u=0,\ \lambda =0,\ \epsilon =1,\ \mu =1.}$

By that it is said, that the speed of light in vacuum is chosen as unity. In the preceding, the entirety of our presuppositions is contained. The equations claim applicability for arbitrary velocities ${\displaystyle u}$ as far as Maxwell's equations apply for ${\displaystyle u=0}$.

From our equations it follows, for example, that in case ${\displaystyle u=0}$, the radiation is propagating with the same velocity in all directions. Thus they presuppose a reference system to which this actually applies. That such a references system is existing with respect to the fixed stars, is without question. To what extent it is defined by our equations, shall be examined later.

Equations I and II, related to the unit of area (supposed as infinitely small), shall be written:

 ${\displaystyle -{\mathsf {P(E)}}={\frac {\overline {d{\mathfrak {M}}}}{dt}}}$ I'
 ${\displaystyle {\mathsf {P(M)}}={\frac {\overline {d{\mathfrak {E}}}}{dt}}+\Lambda }$. II'

Consequently, the meaning of the newly introduced symbols is:

 ${\displaystyle {\frac {\overline {dA}}{dt}}={\frac {dA}{dt}}+\Gamma (u)A-(A\cdot \nabla )u}$ (1)
 ${\displaystyle ={\frac {\partial A}{\partial t}}+\Gamma (A)u-{\mathsf {P}}[uA)}$ (2)
where ${\displaystyle {\tfrac {d}{dt}}\left({\tfrac {\partial }{\partial t}}\right)}$ denotes the differentiation with respect to a fixed material point (space point). Furthermore P is rotation, ${\displaystyle \Gamma }$ is divergence, ${\displaystyle \nabla }$ is gradient,

${\displaystyle (A\cdot \nabla )=A_{x}\cdot {\frac {\partial }{\partial x}}+A_{y}\cdot {\frac {\partial }{\partial y}}+A_{z}\cdot {\frac {\partial }{\partial z}}}$.

§ 2. Transformation to a moving coordinate system and local time.

We decompose the velocities ${\displaystyle u}$ into a common translational velocity ${\displaystyle p}$ (which is constant with respect to time) of the whole system, and the "relative" velocity ${\displaystyle v}$:

 ${\displaystyle u=p+v,\ p=const.,}$ (3)

and we denote a differentiation with respect to time, in relation to a relatively stationary point, by ${\displaystyle {\tfrac {\delta }{\delta t}}}$:

 ${\displaystyle {\frac {\delta }{\delta t}}={\frac {\partial }{\partial t}}+(p\cdot \nabla )={\frac {d}{dt}}-(v\cdot \nabla )}$ (4)

Then it is given

${\displaystyle {\frac {\overline {dA}}{dt}}={\frac {\delta A}{\delta t}}+\Gamma (v)A-(A\cdot \nabla )v+(v\cdot \nabla )A}$
 ${\displaystyle ={\frac {\delta A}{\delta t}}+\Gamma (A)v-{\mathsf {P}}[vA)}$. (5)

Simultaneously, instead of the "general time" ${\displaystyle t}$ we introduce the "local time" ${\displaystyle t'}$. It is defined at a point whose radius vector is ${\displaystyle r}$, by:

 ${\displaystyle r'=r-(p\cdot r)\,}$ (6)

Differentiations with respect to relative coordinates, in which local time is assumed as the fourth independent variable, shall be denoted by an upper index prime. Then it is

 ${\displaystyle {\frac {\delta }{\delta t}}={\frac {\delta }{\delta t'}}}$ ${\displaystyle {\mathsf {P}}={\mathsf {P}}'-\left[p\cdot {\frac {\delta }{\delta t'}}\right]}$ ${\displaystyle \Gamma =\Gamma '-\left[p\cdot {\frac {\delta }{\delta t'}}\right]}$ (7)

Eventually, we decompose ${\displaystyle {\mathfrak {E}}}$ and ${\displaystyle {\mathfrak {M}}}$:

 ${\displaystyle {\begin{matrix}{\mathfrak {E}}={\mathfrak {E}}_{0}-[p{\mathsf {M}}]\\\\{\mathfrak {M}}={\mathfrak {M}}_{0}+[p{\mathsf {E}}]\end{matrix}}}$ (8)
By means of (5), (7) and (8), two equations emerge from I' II', which shall be simplified by putting:
 ${\displaystyle p\cdot v=0\,}$ (9)

Then it is given:

 ${\displaystyle -{\mathsf {P'(E)}}={\frac {\delta {\mathfrak {M}}_{0}}{\delta t'}}+\Gamma '({\mathfrak {M}}_{0})v-{\mathsf {P}}'[v{\mathfrak {M}}_{0}]={\frac {\overline {d{\mathfrak {M}}_{0}}}{dt'}}}$ I'a
 ${\displaystyle -{\mathsf {P'(M)}}={\frac {\delta {\mathfrak {E}}_{0}}{\delta t'}}+\Gamma '({\mathfrak {E}}_{0})v-{\mathsf {P}}'[v{\mathfrak {E}}_{0}]+\Lambda ={\frac {\overline {d{\mathfrak {E}}_{0}}}{dt'}}+\Lambda }$ II'a
 ${\displaystyle {\begin{cases}{\mathfrak {E}}_{0}=\epsilon {\mathsf {E}}-[v{\mathsf {M}}]\\\\{\mathfrak {M}}_{0}=\mu {\mathsf {M}}+[v{\mathsf {E}}]\\\\\Lambda =\lambda ({\mathsf {E-K}})\end{cases}}}$ III'a
 ${\displaystyle \Sigma =[{\mathsf {EM}}]}$ IV'a

§ 3. The field in relatively resting media.

Let the only velocity of the system be ${\displaystyle p}$, or which is the same, let it be in relative rest. Then ${\displaystyle v=0}$, and the following equations – in all rigor – shall apply:

 ${\displaystyle -{\mathsf {P'(E)}}={\frac {d{\mathfrak {M}}_{0}}{dt'}}}$ I'b
 ${\displaystyle {\mathsf {P'(M)}}={\frac {d{\mathfrak {E}}_{0}}{dt'}}+\Lambda }$ II'b
 ${\displaystyle {\begin{cases}{\mathfrak {E}}_{0}=\epsilon {\mathsf {E}}\\\\{\mathfrak {M}}_{0}=\mu {\mathsf {M}}\\\\\Lambda =\lambda [{\mathsf {E-K}}]\end{cases}}}$ IIIb
 ${\displaystyle \Sigma =[{\mathsf {EM}}]}$ IVb

These equations have exactly the same form, as Maxwell's equations for a stationary system. Only by those equations, however, the field E, M and thus also the radiation relative to matter ${\displaystyle \Sigma }$ is determined, as far as certain quantities – "electric and magnetic quantities" – which are invariable according to those very equations, are required.[3] Thus for a while we postpone the consideration of these processes in which electromagnetic energy goes over into other forms of energy, especially mechanical work,[4] and then we can say: the electrodynamics of the moving system appears (with respect to a co-moving observer) to be influenced by motion only in so far, as the observer is able to distinguish local time ${\displaystyle t'}$ from general time ${\displaystyle t}$. The difference of both quantities is, according to (6), a fraction of the light propagation-time corresponding to vector ${\displaystyle r}$, which in the worst case (${\displaystyle r}$ parallel to ${\displaystyle p}$) is equal to the ratio of the translational velocity to the speed of light.

Let us apply this to the motion of Earth: Everywhere, where the propagation of radiation is not the object of measurement, we define identical moments of time at different points of Earth's surface, by treating the propagation of light as timeless. In optics, however, we define these identical moments of time by assuming, that the propagation takes place in spherical waves for every relatively resting and isotropic medium.[5] This means: the "time" which actually serves us for the representation of terrestrial process, is the "local time" ${\displaystyle t'}$, for which the equations I'b to IVb hold, – not the "general time" ${\displaystyle t}$.

What is required to experimentally distinguish ${\displaystyle t'}$ from ${\displaystyle t}$, can be shown by a proposal which recently was made by W. Wien "for the decision of the question, as to whether the luminiferous aether is moving with Earth or not."[6] Through the gaps of two gears whose common axis has the direction of Earth's motion, light of the same intensity shall be sent through in both directions. Then both gears shall be set in rotation with equal angular velocity. Wien concludes: If the aether is at rest, then the propagation of light is different for the two paths; – the arriving light hits the gear at the end of the path in different locations upon both stations; – the intensities must have become different.

Now it is clear that for this experiment, not the same angular velocity is required as Wien thinks, but equal collective rotation[7] from the moment of observation at rest until the moment of observation at rotation. If the two collective rotations are equal for equal "general times" t of both stations, then one obtains a difference in brightness in the case of "dragged aether" (but no difference in the case of "stationary aether"). Whether one or the other kind of rotation actually has taken place, there can be (due to logical reasons) no optical, or more general, no electric means of testing. A material (mechanical or acoustical) protection or control is rather required. The scheme would be as follows: the two gears are located at the same shaft, which will be driven in the middle; then we must guarantee the phase equality of both ends up to 110000 of the propagation time of light, which corresponds to the length of the axis.[8]

What we have to understand under "electric and magnetic quantities", still needs a clarification. Those aren't concepts which (besides our equations or independently from them) must be introduced into electrodynamics. The are rather given from these equations as "integration constants". Equation I says, that for every closed surface ${\displaystyle S}$ which goes through invariable material particles, the surface integral of ${\displaystyle {\mathfrak {M}}}$ is a time-independent quantity; we will call this quantity the magnetic quantity within ${\displaystyle S}$. Equation II says the same with respect to the surface integral of ${\displaystyle {\mathfrak {E}}}$ for a surface extended in isolators, and it connects for an arbitrary surface the temporal change of this magnitude with the electric current through ${\displaystyle S}$ in the same way, as the content of a fluid is connected with the current of a fluid. We call this magnitude the quantity of electricity within ${\displaystyle S}$. In the definitions of both magnitudes, it is implicitly presupposed, that we can know the identical moments of time in the various points of the closed surface. From the preceding it follows: When we define identical times at different places, so that the propagation of light becomes uniform with respect to the fixed stars (time ${\displaystyle t}$), then electricity and magnetism express themselves as surface integrals of ${\displaystyle {\mathfrak {E}}}$ and ${\displaystyle {\mathfrak {M}}}$. When we define identical times at different places, so that the propagation of light becomes uniform with respect to Earth (time ${\displaystyle t'}$), then they express themselves as surface integrals of ${\displaystyle {\mathsf {E}}{\mathsf {E}}}$ and ${\displaystyle \mu {\mathsf {M}}}$.

From the equations, into which I', II', III pass for ${\displaystyle u=p=const.}$, it is given by means of (7):

 ${\displaystyle {\begin{matrix}\Gamma ({\mathfrak {M}})=\Gamma '(\mu {\mathsf {M}})\\\\\Gamma ({\mathfrak {E}})=\Gamma '({\mathsf {E}}{\mathsf {E}})+(p\cdot \Lambda )\end{matrix}}}$. (10)
If we at first presuppose, that the field is static, then firstly it is ${\displaystyle {\tfrac {d}{dt}}=0}$ and consequently ${\displaystyle \Gamma =\Gamma '}$, and secondly ${\displaystyle \Lambda =0}$, thus
${\displaystyle \Gamma ({\mathfrak {E}})=\Gamma ({\mathsf {E}}{\boldsymbol {E}})}$

and thus for an arbitrarily closed surface:

${\displaystyle {N}dS=\int \limits _{t'=const.}{\mathsf {E}}{\mathsf {E}}_{N}dS}$

If ${\displaystyle S}$ is extended in isolators, then the first integral is generally of the special value ${\displaystyle t}$, and the last integral is independent of the special value of ${\displaystyle t'}$. The once existing equality of both expressions thus remains during all changes of the field; that is:

 ${\displaystyle \Gamma ({\mathfrak {E}})=\Gamma '({\mathsf {E}}{\mathsf {E}}),}$ in the isolator (11)
 ${\displaystyle \int \limits _{t=const.}{\mathfrak {E}}_{N}dS=\int \limits _{t'=const.}{\mathsf {E}}{\mathsf {E}}_{N}dS}$ for every conductor surface. (12)

(10), (11) and (12) say, that the magnitudes which are to be denoted as magnetic density (${\displaystyle \rho _{0}}$), electric density in the isolator (${\displaystyle \rho _{e}}$) and total quantity of electricity of a conductor (${\displaystyle e}$), have in general the same value in both representations. Thus the result is: identical data ${\displaystyle \rho _{m}}$, ${\displaystyle \rho _{e}}$, ${\displaystyle e}$ determine identical fields E, M, independent of the values of ${\displaystyle p}$.

Everything said in this paragraph applies to media, which are in relative rest with respect to a reference system that itself has a uniform translational velocity. By assuming this references system as being fixed in Earth, we neglect the rotation of axis. Theoretically spoken, the requirement of uniform propagation of light in all directions relative to earth cannot be satisfied by any "local time", because the velocity of diurnal motion has no potential. Namely, this has the consequence that the change which the propagation time of light suffers due to motion, depends on the light path, not on its start- and endpoint. However, if one considers that the diurnal motion (for one meter of distance from the axes each) varies by less than 1100 cmsec, then it becomes clear, that no interference experiment can detect these local velocity differences. (One thinks of an interferometer whose two light paths are the halves of the square of one meter side length; let one side-couple be parallel to the direction of motion; Na-light shall be used. A rotation of the instrument around 180° would cause a replacement of the interference image by a millionth width of a fringe.) Also, that the direction of velocity is changing with time is without observable influence. The proof shall be neglected here. Thus, we practically are allowed to consider the diurnal motion as pure translation as well, which superimposes the motion in the annual path, at every place of Earth's surface in every moment.

§ 4. Relative motions.

Now we consider the general case of relative motions, but we presuppose that the product of common translation velocity and relative velocity is a vanishing magnitude with respect to the square of the speed of light. This condition, which is formulated in (9), has led us to equations from I'a to IVa. They are formally in agreement with I' to IV. The difference only lies in the fact, that in place of the "absolutely resting" spatial reference system, we use a "relatively resting" one, and in place of the "general time" we use "local time". Thus this means when applied to Earth: as far as the product of Earth's velocity assumed by us, and the actually given relative velocity with respect to Earth can be neglected with respect to the square of the speed of light, it is irrelevant as to whether we relate our equations to a coordinate system at rest with respect to Earth and to "terrestrial time" ${\displaystyle t}$, – or to another arbitrary coordinate system, which has the uniform velocity (${\displaystyle -p}$) against Earth and a time ${\displaystyle t}$ defined by (6).

What was spoken out as a condition here, is actually valid for all observations, when we understand under ${\displaystyle p}$ the velocity of Earth against the fixed stars (ca. ${\displaystyle 10^{-4}}$).

We can distinguish two fields of application:

1. Astrophysics. Here, it is either ${\displaystyle v=-p}$ (fixed stars), or it is at most of the order of magnitude ${\displaystyle p}$. Thus, the neglectable magnitudes are at most of order ${\displaystyle 10^{-8}}$, while the measurement of the aberration angle and the corresponding change of wave lengths not nearly reach this precision.

2. Motions of extended bodies at Earth's surface. Here, ${\displaystyle v}$ remains small compared with ${\displaystyle p}$, and ${\displaystyle p\cdot v}$ is neglectable for any observation.

Thus everything strictly derived in §3 for relatively resting systems, applies with practically sufficient precision also to relatively moving systems.

Summarized: the (thus far) known facts of electrodynamics give us the choice for the representation: using the stationary Earth and terrestrial time, or the stationary celestial fixed stars and celestial time.

That our equations, interpreted in the one or the other form, correctly represent the influence of relative motions, was demonstrated partly by me l.c., and partly by others. A summary and a comparison with other theories, is planned by me to be given soon.

§ 5. Energy conservation and mechanical forces.

To obtain the energy equation, we decompose the magnitude ${\displaystyle {\tfrac {\overline {dA}}{dt}}}$ of equation (I) in two parts:

 ${\displaystyle {\frac {\overline {dA}}{dt}}={\frac {d'A}{dt}}+A_{def}}$. (13)

Here, ${\displaystyle {\tfrac {d'A}{dt}}}$ shall denote the change of vector ${\displaystyle A}$ relative to moving matter, or with other words, the change that ${\displaystyle A}$ suffers by the change in a fixed space-point caused by translation and by rotation. This would be the total value of ${\displaystyle {\tfrac {\overline {dA}}{dt}}}$, when matter is not deformed. Therefore, ${\displaystyle A}$ is the contribution that stems from the deformation. In the notation:

 ${\displaystyle {\frac {d'A}{dt}}={\frac {dA}{dt}}+{\frac {1}{2}}[A\cdot {\mathsf {P}}(u)]}$ (14)
 ${\displaystyle (A_{def})_{x}=A_{x}\left({\frac {\partial u_{y}}{\partial y}}+{\frac {\partial u_{z}}{\partial z}}\right)-A_{y}{\frac {1}{2}}\left({\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}\right)-A_{z}{\frac {1}{2}}\left({\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}\right)}$; etc. (15)

or

 ${\displaystyle A_{def}=A\cdot \Gamma (u)+A_{\delta }\,}$ (16)

where

 ${\displaystyle (A_{\delta })_{x}=-A_{x}{\frac {\partial u_{x}}{\partial x}}-A_{y}{\frac {1}{2}}\left({\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}\right)-A_{z}{\frac {1}{2}}\left({\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}\right)}$; etc. (17)

One can easily convince oneself, that ${\displaystyle {\tfrac {\overline {dA}}{dt}}}$, calculated by (13), (14), (15), gives the value in (I). From the defining equations (14) it follows:

 ${\displaystyle \left({\frac {d'A}{dt}}\cdot B\right)+\left(A\cdot {\frac {d'B}{dt}}\right)={\frac {d}{dt}}(A\cdot B)}$ (18)
 ${\displaystyle \left[{\frac {d'A}{dt}}\cdot B\right]+\left[A\cdot {\frac {d'B}{dt}}\right]={\frac {d'}{dt}}[A\cdot B]}$ (19)
Now, we multiply I' by M, II' by E, and sum up; then it follows:
${\displaystyle -\Gamma (\Sigma )-(\Lambda \cdot {\mathsf {E}})=\left({\mathsf {E}}\cdot {\frac {\overline {d{\mathfrak {E}}}}{dt}}\right)+\left({\mathsf {M}}\cdot {\frac {\overline {d{\mathfrak {M}}}}{dt}}\right)}$,

or by (13):

 ${\displaystyle =\left({\mathsf {E}}\cdot {\frac {d'{\mathfrak {E}}}{dt}}\right)+\left({\mathsf {M}}\cdot {\frac {d'{\mathfrak {M}}}{dt}}\right)+({\mathsf {E}}\cdot {\mathfrak {E}}_{def})+({\mathsf {M}}\cdot {\mathfrak {M}}_{def})}$. (20)

We neglect the change that will be suffered by ${\displaystyle \epsilon }$ and ${\displaystyle \mu }$ due to the deformation, thus we put ${\displaystyle {\tfrac {d\epsilon }{dt}}={\tfrac {d\mu }{dt}}=0}$; then by III it becomes:

${\displaystyle \left({\mathsf {E}}\cdot {\frac {d'{\mathfrak {E}}}{dt}}\right)+\left({\mathsf {M}}\cdot {\frac {d'{\mathfrak {M}}}{dt}}\right)={\frac {d}{dt}}\left\{{\frac {1}{2}}(\epsilon {\mathsf {E}}^{2}+u{\mathsf {M}}^{2})\right\}+2\left(\Sigma \cdot {\frac {d'u}{dt}}\right)+\left({\frac {d'\Sigma }{dt}}\cdot u\right)}$,

or by V:

 ${\displaystyle ={\frac {dw}{dt}}-\left(u\cdot {\frac {d'\Sigma }{dt}}\right)}$. (21)

Furthermore it is by (16):

 ${\displaystyle ({\mathsf {E}}\cdot {\mathfrak {E}}_{def})+({\mathsf {M}}\cdot {\mathfrak {M}}_{def})=\left(({\mathsf {E}}\cdot {\mathfrak {E}})+({\mathsf {M}}\cdot {\mathfrak {M}})\right)\Gamma (u)+({\mathsf {E}}\cdot {\mathfrak {E}}_{\delta })+{\mathsf {M}}\cdot {\mathfrak {M}}_{\delta })}$ ${\displaystyle =w\Gamma (u)+{\frac {1}{2}}(\epsilon {\mathsf {E}}^{2}+\mu {\mathsf {M}}^{2})\Gamma (u)+\epsilon ({\mathsf {E}}\cdot {\mathsf {E}}_{\delta })}$ ${\displaystyle +\mu ({\mathsf {M}}\cdot {\mathsf {M}}_{\delta })-({\mathsf {E}}\cdot [u{\mathsf {M}}]_{\delta })+({\mathsf {M}}\cdot [u{\mathsf {E}}]_{\delta })}$. (22)

Eventually, it is given from (17) by arranging with respect to the components of ${\displaystyle u}$:

 ${\displaystyle -({\mathsf {E}}\cdot [u{\mathsf {M}}]_{\delta })+[{\mathsf {M}}\cdot [u{\mathsf {E}}]_{\delta })=-(u\cdot \Sigma _{def})=-\left(u\cdot {\frac {\overline {d\Sigma }}{dt}}\right)+\left(u\cdot {\frac {d'\Sigma }{dt}}\right)}$. (23)

We include (21), (22), (23) in (20), and denote by ${\displaystyle \tau }$ a material element of volume, so that

${\displaystyle {\frac {dw}{dt}}+w\Gamma (u)={\frac {1}{\tau }}{\frac {d}{dt}}(w\cdot \tau )}$

Then it follows:

 ${\displaystyle -{\frac {1}{\tau }}{\frac {d(w\tau )}{dt}}=\Gamma (\Sigma )+(\Lambda \cdot {\mathsf {E}})+{\mathsf {A}}}$, (24)

where

 ${\displaystyle A=-\left(u\cdot {\frac {\overline {d\Sigma }}{dt}}\right)+{\frac {1}{2}}\left(\epsilon {\mathsf {E}}^{2}+\mu {\mathsf {M}}^{2}\right)\Gamma (u)-S_{i,k}\left\{\left(\epsilon {\mathsf {E}}_{i}{\mathsf {E}}_{k}+\mu {\mathsf {M}}_{i}{\mathsf {M}}_{k}\right){\frac {\partial u_{i}}{\partial k}}\right\}\qquad \left.{i \atop k}\right\}=x,y,z.}$. (25)

In (24), the left-hand side is the decrease of electromagnetic energy, the first member of the right-hand side the radiation, the second member the chemical-thermal energy spent, ${\displaystyle A}$ is thus the work spent (always calculated for the unit of time and of the material volume).

The forces which exert this work, consist of the translational force

 ${\displaystyle f_{1}=-{\frac {\overline {d\Sigma }}{dt}}}$ (26)

and of a system of deformation forces, which are entirely in agreement with Maxwell's tensions. They can be decomposed into a universal tension

 ${\displaystyle q=-{\frac {1}{2}}(\epsilon {\mathsf {E}}^{2}+\mu {\mathsf {M}}^{2})}$ (27a)

besides the tensions

 ${\displaystyle q_{ik}=+(\epsilon {\mathsf {E}}_{i}{\mathsf {E}}_{k}+\mu {\mathsf {M}}_{i}{\mathsf {M}}_{k})}$. (27b)

The motions of the material particles are thus determined by the equivalent system of translational forces ${\displaystyle f}$, whose components are:

 ${\displaystyle f_{x}=f_{1x}+{\frac {\partial q}{\partial x}}+{\frac {\partial q_{xx}}{\partial x}}+{\frac {\partial q_{xy}}{\partial y}}+{\frac {\partial q_{xz}}{\partial z}}}$; etc.[9] (28)

If one substitutes the values of (26) and (27), then one obtains

 ${\displaystyle {\begin{matrix}f=-{\frac {d\Sigma }{dt}}-[\epsilon {\mathsf {E}}\cdot {\mathsf {P(E)}}]+\Gamma (\epsilon {\mathsf {E}})\cdot {\mathsf {E}}-{\frac {1}{2}}{\mathsf {E}}^{2}\cdot \nabla \epsilon \\\\-[u{\mathsf {M}}\cdot {\mathsf {P(M)}}+\Gamma (\mu {\mathsf {M}})\cdot {\mathsf {M}}-{\frac {1}{2}}{\mathsf {M}}^{2}\cdot \nabla \mu \end{matrix}}}$ (27)

This is the most general expression for the forces.

We notice at first, that it applies for vacuum:

${\displaystyle u=0}$, ${\displaystyle \lambda }$=0, ${\displaystyle \epsilon =\mu =1}$ and thus ${\displaystyle {\mathfrak {E}}={\mathsf {E}},\ {\mathfrak {M}}={\mathsf {M}},\ \Lambda =0}$;

furthermore ${\displaystyle \Gamma {\mathsf {E}}=\Gamma {\mathsf {M}}=0}$. Thus the last four terms in (29) individually vanish, however, the three first ones give zero according to I' and II'. Force ${\displaystyle f}$ is thus identical with zero at all space locations, where a material substrate of forces is unknown to us. This theorem is a logical postulate, as long as one doesn't ad hoc substitute (into vacuum) a medium with properties of matter. On the other side, this follows from our equations only be means of presupposing ${\displaystyle u=0}$. Thus one can conceptually define the reference system, for which the fundamental equations are valid, so that it is at rest with respect to empty space. By that, however, not the least is gained for the representation of experience.

Since we measure electromagnetic forces with respect to bodies which are at rest with respect to Earth or moving slowly at least, the value which ${\displaystyle f}$ assumes for ${\displaystyle u=p=const.}$ is most important. We obtain it in the most vivid form, by again introducing local time ${\displaystyle t'}$ by means of (6). Then, it follows from (29) by means of (7), or more simple by means of:

${\displaystyle {\frac {\partial }{\partial x}}={\frac {\partial }{\partial x'}}-p_{x}{\frac {d}{dt'}}}$ etc.

and

${\displaystyle {\frac {\bar {d}}{dt}}={\frac {d}{dt}}={\frac {d}{dt'}}}$

directly from (28):

${\displaystyle f_{x}=-{\frac {d\Sigma _{x}}{dt'}}-[\epsilon {\mathsf {E}}\cdot {\mathsf {P'(E)}}]_{x}+\Gamma '(\epsilon {\mathsf {E}})\cdot {\mathsf {E}}_{x}-{\frac {1}{2}}{\mathsf {E}}^{2}{\frac {\partial \epsilon }{\partial x'}}}$
${\displaystyle -[\mu {\mathsf {M\cdot P'(M)}}]_{x}+\Gamma '(\mu {\mathsf {M}})\cdot {\mathsf {M}}_{x}-{\frac {1}{2}}{\mathsf {M}}^{2}{\frac {\partial \mu }{\partial x'}}}$
${\displaystyle -p_{x}{\frac {d}{dt'}}(q+q_{xx})-p_{y}{\frac {d}{dt'}}(q_{xy})-p_{z}{\frac {d}{dt'}}(q_{xz})}$.

Here, it is by I'b to IIIb:

${\displaystyle -{\mathsf {P'(E)}}=\mu {\frac {d{\mathsf {M}}}{dt'}}}$
${\displaystyle {\mathsf {P'(M)}}=\epsilon {\frac {d{\mathsf {E}}}{dt'}}+\Lambda }$;

thus it follows:

 ${\displaystyle {\begin{matrix}f_{x}=f_{0x}+{\frac {d}{dt'}}\left\{(\epsilon u-1)\Sigma _{x}+p_{x}\left({\frac {1}{2}}\left(\epsilon {\mathsf {E}}^{2}+\mu {\mathsf {M}}^{2}\right)-\left(\epsilon {\mathsf {E}}_{x}^{2}+\mu {\mathsf {M}}_{x}^{2}\right)\right)\right.\\\\\left.-p_{y}(\epsilon {\mathsf {E}}_{x}{\mathsf {E}}_{y}+\mu {\mathsf {M}}_{x}{\mathsf {M}}_{y})-p_{z}(\epsilon {\mathsf {E}}_{x}{\mathsf {E}}_{z}+\mu {\mathsf {M}}_{x}{\mathsf {M}}_{z})\right\}\end{matrix}}}$ (30)

where

 ${\displaystyle f_{0}=\Gamma '(\epsilon {\mathsf {E}})\cdot {\mathsf {E}}-{\frac {1}{2}}{\mathsf {E}}^{2}\cdot \nabla '\epsilon +\Gamma '(\mu {\mathsf {M}})\cdot {\mathsf {M}}-{\frac {1}{2}}{\mathsf {M}}^{2}\cdot \nabla '\mu +[\Lambda \cdot \mu {\mathsf {M}}]}$ (31)

The value in (31), considered as a function of relative coordinates and local time, doesn't explicitly depend on ${\displaystyle p}$; but also not implicitly, since according to § 3, also E, M and ${\displaystyle \Lambda }$ are functions (independent from P) of the same four variables. With respect to stationary states it becomes ${\displaystyle f=f_{0}}$. Furthermore, it is irrelevant for the representation of these states, as to whether we use local time or general time. Thus the approach is given: the forces of the stationary field in relatively resting bodies are in all rigor independent from Earth's motion. It additionally gives the amount of these forces in the well-known form, which forms the expression of all certain experiences.

In the case of variable states, an amount of terms are added to ${\displaystyle f_{0}}$, which are all represented as complete derivatives with respect to time. This circumstance excludes the possibility, that momentary actions of periodic processes are arbitrarily summed up. On the other hand, every single term in the {}-brace is a very small quantity: ${\displaystyle \Sigma }$ as well as the quantities in the () are of order ${\displaystyle w}$; however, ${\displaystyle \Sigma }$ is connected with a factor that is very small for all easily movable bodies (gases), and the () have as factors the components of ${\displaystyle p=10^{-4}}$, if we assume that the reference system of our equations is fixed with respect to the fixed stars. What is added to ${\displaystyle f_{0}}$, is thus imperceptible; however, ${\displaystyle f_{0}}$ itself is independent of ${\displaystyle p}$, as soon as one chooses the terrestrial local time as fourth variable.

Thus, also the consideration of mechanical forces verifies our earlier result: no experience hinders us, to arbitrarily relate our fundamental equations to a spatial reference system that is at rest in Earth, or to such a system that has an arbitrary uniform velocity, whose order is the relative velocity of Earth - fixed stars. We only have to adapt the temporal reference system to the freely chosen spatial system.

1. Göttinger Nachrichten 1901, Heft 1; Ann. der Physik 7, 8.29, 1902.
2. See in this respect, below p. 1409 f.
3. See below § 5.
4. The use of this definition presupposes, that bodies exist which can be rotated under all circumstances without changing their dimensions. The presupposition is the basis of our whole geometry. It is nevertheless not superfluous to mention it; since the theory of electrons negates the existence of such bodies.
5. Phys. Zeitschr. 5, p. 585, 1904.
6. "up to whole multiples of the angular distance of the gear", would be a valid but unessential generalization.
7. Also this procedure of course has only a meaning, as soon as we can be sure, that the laws of mechanics for the "general time" is strictly correct.
8. As regards the recent developments, see Lorentz, Math. Enc. V, p. 251 ff. However, it is to be noticed, that Lorentz has neglected terms with ${\displaystyle u^{2}}$ throughout.