Translation:On the Electrodynamics of Moving Bodies (Abraham)

by Max Abraham

On the Electrodynamics of Moving Bodies.

By Max Abraham (Göttingen).

Meeting of January 17, 1909.

§ 1. Introduction.

It's known that the fundamental equations of the electrodynamics of moving bodies as stated by H. Hertz^[1], which could be seen as the obvious generalization of Maxwell's field equations for resting bodies, are insufficient; they contradict the experiments conducted by A. Eichenwald^[2] and H. A. Wilson^[3] concerning the behavior of moving dielectrics.

Those experimental results are in agreement with the electrodynamic theories of H. A. Lorentz^[4] and of E. Cohn^[5] The heuristic ideas by which the two researchers were led, are mutually different throughout; while H. A. Lorentz starts from equations concerning the behavior of electrons and molecules, E. Cohn tries to establish the simplest description of electromagnetic processes in the sense of Kirchhoff.

The inconclusiveness of all attempts up to now, to discover an influence of Earth's motion upon the electromagnetic processes happening at Earth's surface, is satisfactorily explained by the theory of E. Cohn. On the other hand, the electron theory of Lorentz, which is based on the electromagnetic field in the aether, lays the view near at hand, that the motion of a system through the aether should have an influence upon the observations of a co-moving observer. Yet H. A. Lorentz^[6] succeeded – by suitable hypotheses concerning the modifications that the electrical and mechanical properties should experience in their motion through the aether – in adapting his theory to the postulate of relativity. That this is possible, can be explained from the properties of the field equations for the aether, which go over into themselves by certain transformations of coordinates and of the path of light: the so-called^[7] Lorentz transformations.

It is not my intention, to discuss in this paper the whole complex of questions, which are connected to the postulate of relativity; I have taken position to some of these questions at another place^[8]. Here, this postulate is of interest to us, only in so far as it is connected with the electrodynamics of ponderable matter. A paper of H. Minkowski^[9] which appeared recently, has placed just this question at the top; here, such a form is given to the fundamental equations of moving bodies, so that they pass into Maxwell's field equations for moving bodies by the Lorentz transformation.

Minkowski's fundamental equations, as well as the ones of E. Cohn and of H. A. Lorentz, explain all existing experimental results; they and Cohn's fundamental equation – with which they are in agreement (neglecting magnitudes of second order in the ratio of the velocity of matter and that of light) – have the symmetry of electric and magnetic quantities in common. However, Lorentz's fundamental equations in their initial form, in which this symmetry is not present, already deviate in terms of first order from the ones of the two other theories; though this deviation (which was noticed by E. Cohn^[10]) only concerns the para- and diamagnetic isolators, and because of their insignificance they escape any experimental test.

However, it is not hard to modify the relations of electric and magnetic vectors assumed by Lorentz, so that the symmetry is maintained; the paragraphs (8) and (10) of the present investigations are concerned with the form of Lorentz's theory modified in this way. It will be shown, that it is fully in agreement with Minkowski's theory in terms of its actual content. The formal difference lies in the interpretation, which is given to the vectors designated by ${\mathfrak {E}}$ and ${\mathfrak {H}}$; in Lorentz's theory, they represent the electric and magnetic excitation of the aether; while they are lacking an illustrative meaning in Minkowski's theory. In my opinion, exactly in this abandonment of an illustrative interpretation, lies the actuality of Minkowski's theory. After the theory of electrons has borne rich fruit, electrodynamics appears to be entering into a phenomenological phase of its development.

Also the method of the present investigation is a phenomenological one. In view of the perplexity, which is caused by the increasing number of rival theories, it appeared to me as desirable, to possess a system of electrodynamics of moving bodies on a Maxwellian basis, which at first is free from special assumptions of the individual theories. The presuppositions of the system represented here, are contained in the momentum theorems and the energy theorem (§ 3) and also in certain equations, which we denote as "main equations" (§ 4.). Two of them connect – as analogous generalization of the main equations of Maxwell's theory which hold in the case of rest – the line integrals of vectors ${\mathfrak {E'}}$ and ${\mathfrak {H'}}$ (the force upon moving electric and magnetic unit poles) with the temporal changes of the surface integrals of vectors ${\mathfrak {B}}$ and ${\mathfrak {D}}$ (the magnetic and electric excitation). Together with the supplemented three main-equations, which express by those vectors the Joule-heat, the relative ray, and the relative electromagnetic stresses in moving matter, they form a mathematical frame into which the different images of electromagnetic processes can be placed. Any such image is (in the sense of our system) characterized by two relations between the four vectors ${\mathfrak {E',H',D,B}}$; by addition of these relations, the two first main equations pass into the differential equations, which represent (in agreement with the corresponding theory) the temporal change of the electromagnetic field, while the other three main equations determine the energy processes and the ponderomotive forces. However, it is remarkable, as to how far one is able to follow the consequences from the main equations, without using the special connecting equations of the corresponding theory. In particular, the deviations between the different theories in the expression of ponderomotive force (§ 12), are only marginally; in the case of rest, the ponderomotive forces of the theories of Lorentz, Cohn, and Minkowski even become mutually identical.

By placing the different theories of electrodynamics of moving bodies in a general system, I remove those properties of the individual image, which are not determined by the characteristic connecting laws of electromagnetic vectors. That I performed such modifications at some of the mentioned theories, will surely be excused; because in the given mode of representation, the essential points of the concerned images are emphasized even more clearer.

§ 2. Mathematical auxiliary formulas.

The time differentiation for fixed space points, is represented by ${\tfrac {\partial }{\partial t}}$. The temporal change of a surface integral, extended over a surface whose points are moving with velocity ${\mathfrak {w}}$, namely

${\frac {d}{dt}}\int df\ {\mathfrak {A}}_{n}=\int df\left\{{\frac {\partial '{\mathfrak {A}}}{\partial t}}\right\}_{n}$

defines another kind of time differentiation of a vector

(1)	${\frac {\partial '{\mathfrak {A}}}{\partial t}}={\frac {\partial {\mathfrak {A}}}{\partial t}}+{\mathfrak {w}}\ \mathrm {div} {\mathfrak {A}}+\mathrm {curl} [{\mathfrak {Aw}}]$

Furthermore, the derivative (with respect to time) which is related to moving points, is

(2)	${\dot {\mathfrak {A}}}={\frac {\partial {\mathfrak {A}}}{\partial t}}+({\mathfrak {w}}\nabla ){\mathfrak {A}}$

This is connected with the temporal change of the volume integral of a vector, by the relations

(2a)	${\begin{array}{c}{\frac {d}{dt}}\int dv\ {\mathfrak {A}}=\int dv{\frac {\delta {\mathfrak {A}}}{\delta t}}\\\\{\frac {\delta {\mathfrak {A}}}{\delta t}}={\dot {\mathfrak {A}}}+{\mathfrak {A}}\ \mathrm {div} {\mathfrak {w}}\end{array}}$

From (2) and (2a) it follows

(3)	${\frac {\delta {\mathfrak {A}}}{\delta t}}={\frac {\partial {\mathfrak {A}}}{\partial t}}+({\mathfrak {w}}\nabla ){\mathfrak {A}}+{\mathfrak {A}}\ \mathrm {div} {\mathfrak {w}}$

Accordingly it is given for the scalars:

(3a)	${\frac {\delta \psi }{\delta t}}={\frac {\partial \psi }{\partial t}}+\mathrm {div} \psi {\mathfrak {w}}$

From (1) and (3) it finally follows, with respect to the general rule

$\mathrm {curl} [{\mathfrak {Aw}}]=({\mathfrak {w}}\nabla ){\mathfrak {A}}-({\mathfrak {A}}\nabla ){\mathfrak {w}}+{\mathfrak {A}}\ \mathrm {div} {\mathfrak {w}}-{\mathfrak {w}}\ \mathrm {div} {\mathfrak {A}}$,

the relation

(4)	${\frac {\partial '{\mathfrak {A}}}{\partial t}}={\frac {\delta {\mathfrak {A}}}{\delta t}}-({\mathfrak {A}}\nabla ){\mathfrak {w}}$.

Since the time differentiation introduced in (2) follows the ordinary calculation rules, we have with respect to (2a)

$[{\mathfrak {{\dot {A}}B}}]+[{\mathfrak {A{\dot {B}}}}]={\frac {\delta }{\delta t}}[{\mathfrak {AB}}]-[{\mathfrak {AB}}]\mathrm {div} {\mathfrak {w}}$

From this equation, together with the ones following from (4) and (2a)

${\begin{array}{l}{\frac {\partial '{\mathfrak {A}}}{\partial t}}={\mathfrak {\dot {A}}}+{\mathfrak {A}}\ \mathrm {div} {\mathfrak {w}}-({\mathfrak {A}}\nabla ){\mathfrak {w}},\\\\{\frac {\partial '{\mathfrak {B}}}{\partial t}}={\mathfrak {\dot {B}}}+{\mathfrak {B}}\ \mathrm {div} {\mathfrak {w}}-({\mathfrak {B}}\nabla ){\mathfrak {w}},\end{array}}$

one obtains

$\left[{\frac {\partial '{\mathfrak {A}}}{\partial t}}{\mathfrak {B}}\right]+\left[{\mathfrak {A}}{\frac {\partial '{\mathfrak {B}}}{\partial t}}\right]={\frac {\delta }{\delta t}}[{\mathfrak {AB}}]+[{\mathfrak {AB}}]\mathrm {div} {\mathfrak {w}}-\left[{\mathfrak {A}},\ ({\mathfrak {B}}\nabla ){\mathfrak {w}}\right]+\left[{\mathfrak {B}},\ ({\mathfrak {A}}\nabla ){\mathfrak {w}}\right]$

Due to the identity which is easily to be verified

$\left[{\mathfrak {A}},\ ({\mathfrak {B}}\nabla ){\mathfrak {w}}\right]-\left[{\mathfrak {B}},\ ({\mathfrak {A}}\nabla ){\mathfrak {w}}\right]=[{\mathfrak {AB}}]\mathrm {div} {\mathfrak {w}}-([{\mathfrak {AB}}]\nabla ){\mathfrak {w}}-\left[[{\mathfrak {AB}}]\mathrm {curl} {\mathfrak {w}}\right]$

the relation is obtained

(5)

$\left[{\frac {\partial '{\mathfrak {A}}}{\partial t}}{\mathfrak {B}}\right]+\left[{\mathfrak {A}}{\frac {\partial '{\mathfrak {B}}}{\partial t}}\right]={\frac {\delta }{\delta t}}[{\mathfrak {AB}}]+([{\mathfrak {AB}}]\nabla ){\mathfrak {w}}-\left[[{\mathfrak {AB}}]\mathrm {curl} {\mathfrak {w}}\right]$

§ 3. The energy equation and the momentum equations.

We understand under $xyzt$ coordinates and the time, measured in a reference system in which the observer has a fixed location. The ponderomotive force measured by him, which is acting (due to the electromagnetic process) on the unit volume of moving matter, shall have the components:

(6)

${\begin{cases}{\mathfrak {K}}_{x}={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{x}}{\partial t}},\\\\{\mathfrak {K}}_{y}={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{y}}{\partial t}},\\\\{\mathfrak {K}}_{z}={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{z}}{\partial t}}.\end{cases}}$

The vector ${\mathfrak {g}}$ which arises here, is denoted by us as "electromagnetic momentum density" or shortly as "momentum density". The system of "fictitious electromagnetic stresses" consists of six quantities, namely the normal stresses $X_{x},\ Y_{y},\ Z_{z}$, and the pairwise shear-stresses which are mutually equal:

(6a)	$X_{y}=Y_{x},\ Y_{z}=Z_{y},\ Z_{x}=X_{z}$

To the "momentum equations" (6), the energy equation is added:

(7)	${\mathfrak {wK}}+Q=-\mathrm {div} {\mathfrak {S}}-{\frac {\partial \psi }{\partial t}}$

Here, $Q$ means the Joule-head, $\psi$ the electromagnetic energy density, ${\mathfrak {S}}$ the energy current.

While the momentum equations determine the momentum exerted by the electromagnetic field, the energy equation determines which energy-quantity per unit space and time is converted into a non-electromagnetic form (work and heat).

If one introduces into (6) and (7) the temporal derivative defined by (3) and (3a), then one obtains another form of the momentum- and energy theorems

(8)

${\begin{cases}{\mathfrak {K}}_{x}={\frac {\partial X'_{x}}{\partial x}}+{\frac {\partial X'_{y}}{\partial y}}+{\frac {\partial X'_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{x}}{\partial t}},\\\\{\mathfrak {K}}_{y}={\frac {\partial Y'_{x}}{\partial x}}+{\frac {\partial Y'_{y}}{\partial y}}+{\frac {\partial Y'_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{y}}{\partial t}},\\\\{\mathfrak {K}}_{z}={\frac {\partial Z'_{x}}{\partial x}}+{\frac {\partial Z'_{y}}{\partial y}}+{\frac {\partial Z'_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{z}}{\partial t}}.\end{cases}}$

(9)	${\mathfrak {wK}}+Q=-\mathrm {div} \{{\mathfrak {S}}-{\mathfrak {w}}\psi \}-{\frac {\delta \psi }{\delta t}}$

Here, the vector

${\mathfrak {S}}-{\mathfrak {w}}\psi$

represents the "relative energy current". The system of "relative stresses"

(10)

$\left\{{\begin{array}{ccccc}X'_{x}=X_{x}+{\mathfrak {w}}_{x}{\mathfrak {g}}_{x},&&X'_{y}=X_{y}+{\mathfrak {w}}_{y}{\mathfrak {g}}_{x},&&X'_{z}=X_{z}+{\mathfrak {w}}_{z}{\mathfrak {g}}_{x},\\Y'_{x}=Y_{x}+{\mathfrak {w}}_{x}{\mathfrak {g}}_{y},&&Y'_{y}=Y_{y}+{\mathfrak {w}}_{y}{\mathfrak {g}}_{y},&&Y'_{z}=Y_{z}+{\mathfrak {w}}_{z}{\mathfrak {g}}_{y},\\Z'_{x}=Z_{x}+{\mathfrak {w}}_{x}{\mathfrak {g}}_{z},&&Z'_{y}=Z_{y}+{\mathfrak {w}}_{y}{\mathfrak {g}}_{z},&&Z'_{z}=Z_{z}+{\mathfrak {w}}_{z}{\mathfrak {g}}_{z},\end{array}}\right.$

is so defined, that (6) and (8) lead to the equal values of the ponderomotive force.

From (6a) and (10), the relations follow

${\begin{array}{l}Y'_{x}-X'_{y}={\mathfrak {w}}_{x}{\mathfrak {g}}_{y}-{\mathfrak {w}}_{y}{\mathfrak {g}}_{x},\\Z'_{y}-Y'_{z}={\mathfrak {w}}_{y}{\mathfrak {g}}_{z}-{\mathfrak {w}}_{z}{\mathfrak {g}}_{y},\\X'_{z}-Z'_{x}={\mathfrak {w}}_{z}{\mathfrak {g}}_{x}-{\mathfrak {w}}_{x}{\mathfrak {g}}_{z},\end{array}}$

which can be written vectorially

(11)	${\mathfrak {R}}'=[{\mathfrak {wg}}]$

${\mathfrak {R}}'$ is the unit volume related to the torque of relative stresses; it vanishes in ordinary mechanics, since the momentum vector is in agreement with the velocity vector in terms of direction. In electromagnetic mechanics it is not to be neglected in general, but it will be compensated (when related to a fixed point of moment) by that torque which stems from the entrained momentum.

We can imagine the relative energy current as being decomposed into two parts, one of them represents the energy transfer caused by the relative stresses, and the other one the "relative radiation"^[11], which in optics, for example, can be measured by heat production in a black surface:

(12)

${\begin{cases}{\mathfrak {S}}_{x}-{\mathfrak {m}}_{x}\psi ={\mathfrak {S}}'_{x}-\left\{{\mathfrak {w}}_{x}X'_{x}+{\mathfrak {w}}_{y}Y'_{x}+{\mathfrak {w}}_{z}Z'_{x}\right\},\\{\mathfrak {S}}_{y}-{\mathfrak {m}}_{y}\psi ={\mathfrak {S}}'_{y}-\left\{{\mathfrak {w}}_{x}X'_{y}+{\mathfrak {w}}_{y}Y'_{y}+{\mathfrak {w}}_{z}Z'_{y}\right\},\\{\mathfrak {S}}_{z}-{\mathfrak {m}}_{z}\psi ={\mathfrak {S}}'_{z}-\left\{{\mathfrak {w}}_{x}X'_{z}+{\mathfrak {w}}_{y}Y'_{z}+{\mathfrak {w}}_{z}Z'_{z}\right\}.\end{cases}}$

Vector ${\mathfrak {S}}'$ we call the "relative ray".

From the momentum equations (8), we find the expression for the performance of work of the ponderomotive force

${\begin{array}{ll}{\mathfrak {mK}}=-{\mathfrak {w}}{\frac {\delta {\mathfrak {g}}}{\delta t}}&+{\frac {\partial }{\partial x}}\left({\mathfrak {w}}_{x}X'_{x}+{\mathfrak {w}}_{y}Y'_{x}+{\mathfrak {w}}_{z}Z'_{x}\right)\\\\&+{\frac {\partial }{\partial y}}\left({\mathfrak {w}}_{x}X'_{y}+{\mathfrak {w}}_{y}Y'_{y}+{\mathfrak {w}}_{z}Z'_{y}\right)\\\\&+{\frac {\partial }{\partial z}}\left({\mathfrak {w}}_{x}X'_{z}+{\mathfrak {w}}_{y}Y'_{z}+{\mathfrak {w}}_{z}Z'_{z}\right)\end{array}}$ ${\begin{array}{r}-\left\{X'_{x}{\frac {\partial {\mathfrak {w}}_{x}}{\partial x}}+Y'_{x}{\frac {\partial {\mathfrak {w}}_{y}}{\partial x}}+Z'_{x}{\frac {\partial {\mathfrak {w}}_{z}}{\partial x}}+X'_{y}{\frac {\partial {\mathfrak {w}}_{x}}{\partial y}}+Y'_{y}{\frac {\partial {\mathfrak {w}}_{y}}{\partial y}}+Z'_{y}{\frac {\partial {\mathfrak {w}}_{z}}{\partial y}}\right.\\\\\left.+X'_{z}{\frac {\partial {\mathfrak {w}}_{x}}{\partial z}}+Y'_{z}{\frac {\partial {\mathfrak {w}}_{y}}{\partial z}}+Z'_{z}{\frac {\partial {\mathfrak {w}}_{z}}{\partial z}}\right\}\end{array}}$

If we set here for abbreviation

(13)

${\begin{cases}P'&=X'_{x}{\frac {\partial {\mathfrak {w}}_{x}}{\partial x}}+X'_{y}{\frac {\partial {\mathfrak {w}}_{x}}{\partial y}}+X'_{z}{\frac {\partial {\mathfrak {w}}_{x}}{\partial z}}\\\\&+Y'_{x}{\frac {\partial {\mathfrak {w}}_{y}}{\partial x}}+Y'_{y}{\frac {\partial {\mathfrak {w}}_{y}}{\partial y}}+Y'_{y}{\frac {\partial {\mathfrak {w}}_{y}}{\partial z}}\\\\&+Z'_{x}{\frac {\partial {\mathfrak {w}}_{z}}{\partial x}}+Z'_{y}{\frac {\partial {\mathfrak {w}}_{z}}{\partial y}}+Z'_{z}{\frac {\partial {\mathfrak {w}}_{z}}{\partial z}}\end{cases}}$

then the energy equation (9) gives with respect to (12)

(14)	$Q+\mathrm {div} {\mathfrak {S}}'=-{\frac {\delta \psi }{\delta t}}+{\mathfrak {w}}{\frac {\delta {\mathfrak {g}}}{\delta t}}+P'$

This relation gained from the momentum and energy theorem, will prove itself to be still important.

§ 4. The main equations.

Common to all theories of electrodynamics of moving bodies, is the form of the two first main-equations

(I)	$c\ \mathrm {curl{\mathfrak {H}}'={\frac {\partial '{\mathfrak {D}}}{\partial t}}+{\mathfrak {J}},}$

(II)	$c\ \mathrm {curl{\mathfrak {E}}'=-{\frac {\partial '{\mathfrak {B}}}{\partial t}}.}$

They are nothing else than a general scheme, which obtains a physical sense only by addition of two relations between the four arising vectors; and two such relations are necessary, to reduce the number of unknown vectors to two; the temporal change of the field of these two vectors, is then described by the two first main-equations.

We interpret the vectors ${\mathfrak {E}}',\ {\mathfrak {H}}'$ as the forces acting at moving electric and magnetic unit poles. The vectors ${\mathfrak {D,\ B}}$ we call, by using the terminology of the "Enzyklopädie der mathematischen Wissenschaften", the "electric and magnetic excitation".

It corresponds to the importance of vector ${\mathfrak {E}}'$, to make that approach for heat, which is developed for the time- and space-unity of moving matter

(III)

$Q={\mathfrak {JE}}'$

At this third main equations, a equation is added as the fourth one, which connects the relative ray with the vectors $<math>{\mathfrak {E}}'{\mathfrak {H}}'$:

(IV)	${\mathfrak {S}}'=c[{\mathfrak {E}}'{\mathfrak {H}}']$

For the case of rest, this vector passes into the Poynting vector.

Eventually we need an approach, which expresses the quantity $P'$ defined in equation (13) and by that the relative stresses, by the vectors ${\mathfrak {E'H'DB}}$. We put

(V)	$P'={\mathfrak {E}}'({\mathfrak {D}}\nabla ){\mathfrak {w}}+{\mathfrak {H}}'({\mathfrak {B}}\nabla ){\mathfrak {w}}-{\frac {1}{2}}[{\mathfrak {E'D+H'B}}\}\mathrm {div} {\mathfrak {w}}$

and thus we obtain for the relative stresses:

(Va)

${\begin{cases}X'_{x}={\mathfrak {E}}'_{x}{\mathfrak {D}}_{x}+{\mathfrak {H}}'_{x}{\mathfrak {B}}_{x}-{\frac {1}{2}}[{\mathfrak {E'D+H'B}}\}\\X'_{y}={\mathfrak {E}}'_{x}{\mathfrak {D}}_{y}+{\mathfrak {H}}'_{x}{\mathfrak {B}}_{y},\\X'_{z}={\mathfrak {E}}'_{x}{\mathfrak {D}}_{z}+{\mathfrak {H}}'_{x}{\mathfrak {B}}_{z};\\Y'_{x}={\mathfrak {E}}'_{y}{\mathfrak {D}}_{x}+{\mathfrak {H}}'_{y}{\mathfrak {B}}_{x},\\Y'_{y}={\mathfrak {E}}'_{y}{\mathfrak {D}}_{y}+{\mathfrak {H}}'_{y}{\mathfrak {B}}_{y}-{\frac {1}{2}}[{\mathfrak {E'D+H'B}}\},\\Y'_{z}={\mathfrak {E}}'_{y}{\mathfrak {D}}_{z}+{\mathfrak {H}}'_{y}{\mathfrak {B}}_{z};\\Z'_{x}={\mathfrak {E}}'_{z}{\mathfrak {D}}_{x}+{\mathfrak {H}}'_{z}{\mathfrak {B}}_{x},\\Z'_{y}={\mathfrak {E}}'_{z}{\mathfrak {D}}_{y}+{\mathfrak {H}}'_{z}{\mathfrak {B}}_{y},\\Z'_{z}={\mathfrak {E}}'_{z}{\mathfrak {D}}_{z}+{\mathfrak {H}}'_{z}{\mathfrak {B}}_{z}-{\frac {1}{2}}[{\mathfrak {E'D+H'B}}\}.\end{cases}}$

For the case of rest, the known formulas for the fictitious stresses follow from that.

The choice of expressions (IV) and (V) appears to be at first sight as totally arbitrary. Yet it is the simplest generalization of the laws valid in resting bodies, which only uses the vectors occurring in the two first main-equations.

Incidentally, it follows from (Va):

$Y'_{x}-X'_{y}={\mathfrak {D}}_{x}{\mathfrak {E}}'_{y}-{\mathfrak {D}}_{y}{\mathfrak {E}}'_{x}+{\mathfrak {B}}_{x}{\mathfrak {H}}'_{y}-{\mathfrak {B}}_{y}{\mathfrak {H}}'_{x}.$

According to this, the torque of relative stresses is:

(Vb)	${\mathfrak {R}}'=[{\mathfrak {DE}}']+[{\mathfrak {BH}}']$

The mechanical principles laid out in the previous paragraph, and the fife main-equations, are the foundations upon which our system of electrodynamics of moving bodies is resting.

§ 5. Determination of momentum density and energy density.

The various theories of electrodynamics of moving bodies, are differing by the relations assumed between the four vectors ${\mathfrak {E'H'DB}}$ arising in the main equations. However, before we pass to the discussion of special theories, we want to pursue the general developments; there, only a quite general presupposition shall be made about the form of these relations: The vectors ${\mathfrak {E'H'DB}}$ shall be connected by equations, which namely contain the velocity vector ${\mathfrak {w}}$ itself, though not any derivatives of it with respect to time or coordinates.

Main equation (IV) gives:

$\mathrm {div} {\mathfrak {S}}'=c\{{\mathfrak {H}}'\mathrm {curl} {\mathfrak {E}}'-{\mathfrak {E}}'\mathrm {curl} {\mathfrak {H}}'\}$

this becomes with respect to the two first main-equations:

${\mathfrak {JE}}'+\mathrm {div} {\mathfrak {S}}'=-{\mathfrak {E}}'{\frac {\partial '{\mathfrak {D}}}{\partial t}}-{\mathfrak {H}}'{\frac {\partial '{\mathfrak {B}}}{\partial t}}$

From main equation (III) and relation (14) it follows:

(14a)

${\frac {\delta \psi }{\delta t}}-{\mathfrak {w}}{\frac {\delta {\mathfrak {g}}}{\delta t}}-P'={\mathfrak {E}}'{\frac {\partial '{\mathfrak {D}}}{\partial t}}+{\mathfrak {H}}'{\frac {\partial '{\mathfrak {B}}}{\partial t}}$

a condition, which one can also be written in accordance with (4):

(14b)

${\frac {\delta \psi }{\delta t}}-{\mathfrak {w}}{\frac {\delta {\mathfrak {g}}}{\delta t}}-P'={\mathfrak {E}}'{\frac {\delta {\mathfrak {D}}}{\delta t}}+{\mathfrak {H}}'{\frac {\delta {\mathfrak {B}}}{\delta t}}-{\mathfrak {E}}'({\mathfrak {D}}\nabla ){\mathfrak {w}}-{\mathfrak {H}}'({\mathfrak {B}}\nabla ){\mathfrak {w}}$

and which finally, by using main equation (V), passes into:

(15)	${\frac {\delta \psi }{\delta t}}-{\mathfrak {w}}{\frac {\delta {\mathfrak {g}}}{\delta t}}={\mathfrak {E}}'{\frac {\delta {\mathfrak {D}}}{\delta t}}+{\mathfrak {H}}'{\frac {\delta {\mathfrak {B}}}{\delta t}}-{\frac {1}{2}}[{\mathfrak {E'D+H'B}}\}\mathrm {div} {\mathfrak {w}}$

This relation serves to determine the densities of energy and momentum in their dependence from the electromagnetic vectors.

They read with respect to (2a):

(15a)

${\dot {\psi }}={\mathfrak {w{\dot {g}}}}+(\psi -{\mathfrak {wg}})\mathrm {div} {\mathfrak {w}}={\mathfrak {E'{\dot {D}}+H'{\dot {B}}}}+{\frac {1}{2}}[{\mathfrak {E'D+H'B}}\}\mathrm {div} {\mathfrak {w}}$

Since the way of time differentiation now employed, is satisfying the calculation rules, it follows when it is set for brevity's sake:

(16)	$\psi -{\mathfrak {wg}}=\varphi$

(17)	${\dot {\varphi }}+{\mathfrak {g{\dot {w}}}}-{\mathfrak {E'{\dot {D}}-H'{\dot {B}}}}+\left\{\varphi -{\frac {1}{2}}{\mathfrak {E'D}}-{\frac {1}{2}}{\mathfrak {H'B}}\right\}\mathrm {div} {\mathfrak {w}}=0$

As mentioned in the beginning of the paragraph, the relations which connect ${\mathfrak {D,B}}$ with ${\mathfrak {E'H'}}$ shall contain the velocity vector ${\mathfrak {w}}$, though not its derivative with respect to time and space. The same is to be demanded from the expressions, which represent $\psi$ and ${\mathfrak {g}}$ by electromagnetic vectors, and finding them is our next goal. Accordingly, we can separate the terms in (17), which only contain derivatives with respect to time, from those ones, into which the divergence of ${\mathfrak {w}}$ is inserted as a factor; thus the equations are given

(17a)

${\dot {\varphi }}+{\mathfrak {g{\dot {w}}}}={\mathfrak {E'{\dot {D}}+H'{\dot {B}}}}$

(17b)

$\varphi ={\frac {1}{2}}{\mathfrak {E'D}}+{\frac {1}{2}}{\mathfrak {H'B}}$

The elimination of $\varphi$ gives:

(18)	$2{\mathfrak {g{\dot {w}}}}={\mathfrak {E'{\dot {D}}-D{\dot {E}}+H'{\dot {B}}-B{\dot {H}}'}}$

This relation will serve us to determine the components of momentum density, after the right-hand side is expressed as a linear function of the acceleration components, based on the characteristic relations (of this theory) between the electromagnetic vectors.

For the components of ${\mathfrak {g}}$ perpendicular to ${\mathfrak {w}}$, the condition is given from (Vb) and (11)

(18a)

$[{\mathfrak {wg}}]=[{\mathfrak {DE}}']+[{\mathfrak {BH}}']$

This has to be satisfied in any case, since otherwise the system would exhibit an inner contradiction.

From (16) and (17b), the energy density is determined

(19)	$\psi ={\frac {1}{2}}{\mathfrak {E'D}}+{\frac {1}{2}}{\mathfrak {H'B}}+{\mathfrak {wg}}$

According to (Va), the sum of relative normal stresses amounts to

$X'_{x}+Y'_{y}+Z'_{z}=-\left\{{\frac {1}{2}}{\mathfrak {E'D}}+{\frac {1}{2}}{\mathfrak {H'B}}\right\}$

consequently it follows in accordance with (10)

$X{}_{x}+Y{}_{y}+Z{}_{z}=-\left\{{\frac {1}{2}}{\mathfrak {E'D}}+{\frac {1}{2}}{\mathfrak {H'B}}+{\mathfrak {wg}}\right\}$

so that the remarkable relation exists

(19a)

$X{}_{x}+Y{}_{y}+Z{}_{z}+\psi =0$

If one inserts value (19) of $\psi$ as well as expressions (Va) for the relative stresses into (12), then one obtains for the energy current

(20)	${\mathfrak {S}}=c[{\mathfrak {E'H'}}]+{\mathfrak {w}}\{{\mathfrak {E'D}}+{\mathfrak {H'B}}\}-{\mathfrak {D(wE')-B(wH')+w(wg)}}$

an expression, which because of the known calculation rules, passes into

${\frac {\mathfrak {S}}{c}}{\mathfrak {=[E'H']+\left[E'[qD]\right]+\left[H'[qB]\right]+q}}({\mathfrak {q}}c{\mathfrak {g}}),$

when it is put for brevity's sake

${\mathfrak {q=}}{\frac {\mathfrak {w}}{c}}$

Instead of it, it can also be written

(21)	${\frac {\mathfrak {w}}{c}}{\mathfrak {=\left[E'-[qB],\ H'+[qD]\right]-q(qW)}}$

where under ${\mathfrak {W}}$ we have to understand the vector

(22)	${\mathfrak {W=[DB]}}-c{\mathfrak {g}}$

Now, we pass to the discussion of special theories, where we confine ourselves to isotropic bodies throughout.

§ 6. Theory of H. Hertz.

Hertz's electrodynamics of moving bodies sets the vectors ${\mathfrak {D}}$ and ${\mathfrak {B}}$ proportional to ${\mathfrak {E'}}$ and ${\mathfrak {H'}}$

(23)	${\mathfrak {D}}=\epsilon {\mathfrak {E}}',\ {\mathfrak {B}}=\mu {\mathfrak {H}}'$

Accordingly,

${\mathfrak {E'{\dot {D}}-D{\dot {E}}'}}=0,\ {\mathfrak {H'{\dot {B}}-B{\dot {H}}'}}=0$

where $\epsilon$ and $\mu$ are considered as constants for a certain material point of a moving body.

Thus it follows from (18)

(24)	${\mathfrak {g}}=0$

Hertz's theory doesn't know the concept of electromagnetic momentum. It derives the ponderomotive force from the stresses alone, where it is irrelevant according to (10), whether one relates the stresses to stationary or to co-moving surfaces. A torque of the relative stresses doesn't arise, and also both sides of (18a) are equal to zero.

The energy density has a value according to (19)

(25)	$\psi ={\frac {1}{2}}\epsilon {\mathfrak {E}}'^{2}+{\frac {1}{2}}\mu {\mathfrak {H}}'^{2}$

The simple approach, by which Hertz's theory connects the excitations ${\mathfrak {DB}}$ with the electromagnetic forces ${\mathfrak {E'H'}}$, was, however, not confirmed by experiment as mentioned above. Thus only the choice between the theories to be discussed in the following paragraphs, remain.

§ 7. Theory of E. Cohn.

E. Cohn based the electrodynamics of moving bodies on the following connecting equations

(26)	${\begin{cases}{\mathfrak {D}}=\epsilon {\mathfrak {E}}'-[{\mathfrak {qH}}'],\\{\mathfrak {B}}=\mu {\mathfrak {H}}'-[{\mathfrak {qE}}'].\end{cases}}$

From them it follows, that when ${\dot {\epsilon }}$ and ${\dot {\mu }}$ are set equal to zero:

${\begin{array}{l}{\mathfrak {E'{\dot {D}}-D{\dot {E}}'={\dot {q}}[E'H']+q[E'{\dot {H}}']+q[{\dot {E}}'H']}},\\{\mathfrak {H'{\dot {B}}-B{\dot {H}}'={\dot {q}}[E'H']+q[{\dot {E}}'H']+q[E'{\dot {H}}']}},\end{array}}$

Now, since relation (18) requires

$2{\mathfrak {\dot {q}}}c{\mathfrak {g=E'{\dot {D}}-D{\dot {E}}'+H'{\dot {B}}-B{\dot {H}}}}'$

one thus places Cohn's theory in our system, by setting

(27)	$c{\mathfrak {g=[E'H']=}}{\frac {{\mathfrak {S}}'}{c}}$

In Cohn's electrodynamics, the momentum density has to be set equal to the relative ray divided by $c^{2}$.

That relation (18a) is satisfied by (26) and (27), can easily be confirmed by considering, that the identity

${\mathfrak {\left[q[E'H']\right]=\left[E'[qH']\right]-\left[H'[qE']\right]}}$

exists. From (19) it follows now for the electromagnetic energy density.

(28)	$\psi ={\frac {1}{2}}{\mathfrak {E'D}}+{\frac {1}{2}}{\mathfrak {H'B}}+{\mathfrak {q[E'H']}}$,

an expression, which according to (26) can also be written

(28a)

$\psi ={\frac {1}{2}}\epsilon {\mathfrak {E}}'^{2}+{\frac {1}{2}}\mu {\mathfrak {H}}'^{2}+2{\mathfrak {q[E'H']}}$;

it is in agreement with E. Cohns approach.

I later return to the calculation of the ponderomotive force.

§ 8. Theory of H. A. Lorentz.

When we modify the connecting equations of the theory of H. A. Lorentz, so that symmetry exists between the electric and magnetic vectors, then we arrive at the approach:

(29)	${\begin{cases}{\mathfrak {D}}=\epsilon {\mathfrak {E}}'-[{\mathfrak {qH}}],\\{\mathfrak {B}}=\mu {\mathfrak {H}}'+[{\mathfrak {qE}}];\end{cases}}$

(30)	${\begin{cases}{\mathfrak {E'}}={\mathfrak {E}}+[{\mathfrak {qH}}],\\{\mathfrak {H'}}={\mathfrak {H}}-[{\mathfrak {qE}}].\end{cases}}$

Besides four vectors contained in the main equations, two new vectors ${\mathfrak {E,H}}$ occur here. This circumstance makes Lorentz's theory more complicated than Cohn's one. The latter directly connects the components of ${\mathfrak {D,B}}$ with those of ${\mathfrak {E',H'}}$ by equations, which are linear in the velocity components; at this one, however, the connecting equations (§ 10, eq. 37b) given by elimination of ${\mathfrak {EH}}$, are not linear in the velocity components any more.

Though, Lorentz's vectors ${\mathfrak {E}}$ and ${\mathfrak {H}}$ have an illustrative meaning. Namely, the excitations ${\mathfrak {D}}$ and ${\mathfrak {B}}$ can be split into two parts according to eq. (29), (30):

(31)	${\begin{cases}{\mathfrak {D=E+P}},&{\mathfrak {P}}=(\epsilon -1){\mathfrak {E}}';\\{\mathfrak {B=H+M}},&{\mathfrak {M}}=(\mu -1){\mathfrak {H}}'.\end{cases}}$

The first contribution of the electric and magnetic excitation, which is represented by ${\mathfrak {E}}$ and ${\mathfrak {H}}$, is interpreted by Lorentz as electric and magnetic excitation of the aether, and the second contribution, which is represented by the vectors ${\mathfrak {P}}$ and ${\mathfrak {M}}$ (electric and magnetic polarization), is interpreted as the electric and magnetic excitation of matter; the latter is set proportional to the electric and magnetic force ${\mathfrak {E}}'$ and ${\mathfrak {H}}'$, which acts upon the unit charges which is co-moving with matter.

We want to consider $\epsilon$ and $\mu$ in this paragraph as being (for a certain material point) dependent on velocity and time, although we reserve us the right to remove these confinements later.

To find the momentum density on the basis of relation (18), we calculate the quantities

(31a)

${\begin{cases}{\mathfrak {E'{\dot {D}}-D{\dot {E}}'=E'{\dot {E}}-E{\dot {E}}'+E'{\dot {P}}-P{\dot {E}}'}},\\{\mathfrak {H'{\dot {B}}-B{\dot {H}}'=H'{\dot {H}}-H{\dot {H}}'+H'{\dot {M}}-M{\dot {H}}'}}.\end{cases}}$

From (30) it follows

${\begin{array}{l}{\mathfrak {E'{\dot {E}}-E{\dot {E}}'=-q[{\dot {E}}H]+q[E{\dot {H}}]+{\dot {q}}[EH]}},\\{\mathfrak {H'{\dot {H}}-H{\dot {H}}'=-q[E{\dot {H}}]+q[{\dot {E}}H]+{\dot {q}}[EH]}},\end{array}}$

Since the two other terms in (31) are vanishing according to (31), then relation (18) gives

(32)	$c{\mathfrak {g=[EH]}}$

as the value of the electromagnetic momentum density.

Now the question arises, whether this value at the same time satisfies the condition (18a)

$[{\mathfrak {q}}c{\mathfrak {g}}]={\mathfrak {[DE']+[BH']}}$

According to (29), it is

${\mathfrak {[DE']+[BH']=[E'[qH]]-[H'[qE]]}}$

From (30) it also follows

${\mathfrak {[DE']+[BH']=[E[qH]]-[H[qE]]}}$

Due to the known identity

${\mathfrak {[q[EH]]=[E[qH]]-[H[qE]]}}$

it will be seen, that expression (32) for the momentum density really satisfies condition (18a).

Now from (19), the value of the energy density follows

(33)	$\psi ={\frac {1}{2}}{\mathfrak {E'D}}+{\frac {1}{2}}{\mathfrak {H'B}}+{\mathfrak {q[EH]}}$

which one can also write

(33a)

$\psi ={\frac {1}{2}}{\mathfrak {E}}^{2}+{\frac {1}{2}}{\mathfrak {H}}^{2}+{\frac {1}{2}}{\mathfrak {E'P}}+{\frac {1}{2}}{\mathfrak {H'M}}$

The first two terms are contributions of the aether in the sense of Lorentz's theory, the two latter ones are to be seen as contributions of polarized matter to the electromagnetic momentum density.

We begin to calculate the energy current. With respect to (32) and (31), the expression

(34)	${\mathfrak {W=[DB]-[EH]=[EM]+[PH]+[PM]}}$

holds in Lorentz's theory for the vector ${\mathfrak {W}}$ introduced at the end of § 5. According to (31) and (30), we have

${\begin{array}{l}{\mathfrak {E'-[qB]=E-[qM],}}\\{\mathfrak {H'+[qD]=H+[qP],}}\end{array}}$

so that equation (21) assumes the form

${\frac {\mathfrak {S}}{c}}{\mathfrak {=\left[E-[qM],\ H+[wP]\right]-q(qW)}}$

Now, since it is to be set according to (34)

${\mathfrak {q(qW)=\left[[qE][qM]\right]+\left[[qP][qH]\right]+\left[[qP][qM]\right]}}$

then eventually if follows as the value of the energy current

(35)	${\frac {\mathfrak {S}}{c}}{\mathfrak {=[EH]+\left[E'[qP]\right]+\left[H'[qM]\right]}}$

The first term can be interpreted as a contribution of the aether, the second one as the contribution of the electrically polarized matter at the energy current, as G. Nordström^[12] has shown in a recently published paper, which is remarkable also in other respects; the third term which is added at the motion of magnetically polarized matter, corresponds in such a way to the second one, as it is required by the symmetry of electric and magnetic vectors assumed at this place.

§ 9. Theory of H. Minkowski.

In this theory, the following relations between the electromagnetic vectors hold

(36)	${\begin{cases}{\mathfrak {D}}=\epsilon {\mathfrak {E}}'-[{\mathfrak {qH}}],\\{\mathfrak {B}}=\mu {\mathfrak {H}}'+[{\mathfrak {qE}}];\end{cases}}$

(37)	${\begin{cases}{\mathfrak {E'}}={\mathfrak {E}}+[{\mathfrak {qB}}],\\{\mathfrak {H}}'={\mathfrak {H}}-[{\mathfrak {qD}}].\end{cases}}$

Also here, besides the vector pairs contained in the main equations, a new vector pair is added, which mediates the relation between them.

From the standpoint of the system used by us, the task also arises to derive the momentum density from relation (18). It follows from (36)

${\begin{array}{l}{\mathfrak {E'{\dot {D}}-D{\dot {E}}'={\dot {q}}[E'H]+q[E'{\dot {H}}]+q[{\dot {E}}'H]}},\\{\mathfrak {H'{\dot {B}}-B{\dot {H}}'={\dot {q}}[EH']+q[{\dot {E}}H']+q[E{\dot {H}}]}}.\end{array}}$

Thus the right-hand side of (18) becomes

(38)	$\left\{{\begin{array}{c}{\mathfrak {E'{\mathfrak {\dot {D}}}-D{\dot {E}}'+H'{\dot {B}}-B{\dot {H}}'={\dot {q}}\left\{[E'H]+[EH']\right\}}}\\+{\mathfrak {q\left\{[E'{\dot {H}}]+[{\dot {E}}H']-[{\dot {E}}'H]-[E{\dot {H'}}]\right\}}}\end{array}}\right.$

We express, on the basis of (37), ${\mathfrak {EH}}$ and ${\mathfrak {{\dot {E}}{\dot {H}}}}$ by the vectors arising in the main equations, and we find

(38a)

${\mathfrak {[E'H]+[EH']=}}2{\mathfrak {[E'H']+q(E'D)-D(qE')+q(H'B)-B(qH')}}$

(38b)

${\begin{cases}{\mathfrak {\qquad [E'{\dot {H}}]+[{\dot {E}}H']-[{\dot {E}}'H]-[E{\dot {H'}}]}}\\={\mathfrak {{\dot {q}}(E'D)-D({\dot {q}}E')+{\dot {q}}(H'B)-B({\dot {q}}H')}}\\+{\mathfrak {q\{E'{\dot {D}}-D{\dot {E}}'+H'{\dot {B}}-B{\dot {H}}'}}\}\\-\left\{{\mathfrak {{\dot {D}}(qE')-D(q{\dot {E}}')+{\dot {B}}(qH')-B(q{\dot {H}}')}}\right\}.\end{cases}}$

By inserting (38a,b) into (38), we obtain

(38c)

${\begin{cases}\qquad {\mathfrak {E'{\dot {D}}-D{\dot {E}}'+H'{\dot {B}}-B{\dot {H}}'}}\\=2{\mathfrak {\dot {q}}}\left\{{\mathfrak {[E'H']+q(E'D)+q(H'B)-D(qE')-B(qH')}}\right\}\\+{\mathfrak {({\dot {q}}D)(qE')-(qD)({\dot {q}}E')-(q{\dot {D}})(qE')+(qD)(q{\dot {E}}')}}\\+{\mathfrak {({\dot {q}}B)(qH')-(qB)({\dot {q}}H')-(q{\dot {B}})(qH')+(qB)(q{\dot {H}}')}}\\+{\mathfrak {q}}^{2}\{{\mathfrak {E'{\dot {D}}-D{\dot {E}}'+H'{\dot {B}}-B{\dot {H}}'}}\}.\end{cases}}$

However, now it follows from (36)

${\begin{array}{l}{\mathfrak {-(q{\dot {D}})(qE')-(qD)(q{\dot {E}}')=({\dot {q}}D)(qE')+(qD)({\dot {q}}E')}}\\{\mathfrak {-(q{\dot {B}})(qH')-(qB)(q{\dot {H}}')=({\dot {q}}B)(qH')+(qB)({\dot {q}}H')}}\end{array}}$

thus the second and third row of the right-hand side of (28c) assume the values

${\begin{array}{l}2\left\{{\mathfrak {({\dot {q}}D)(qE')-(qD)({\dot {q}}E')}}\right\}=2\left({\mathfrak {[{\dot {q}}q][DE']}}\right),\\2\left\{{\mathfrak {({\dot {q}}B)(qH')-(qB)({\dot {q}}H')}}\right\}=2\left({\mathfrak {[{\dot {q}}q][BH']}}\right).\end{array}}$

If it indeed holds, as required by (18a)

(39)	$[{\mathfrak {q}}c{\mathfrak {g}}]={\mathfrak {[DE']+[BH']}},$

then the second and third row are providing together:

$2\left({\mathfrak {[{\dot {q}}q][{\mathfrak {q}}c{\mathfrak {g}}]}}\right)=2{\mathfrak {[{\dot {q}}q][{\mathfrak {q}}c{\mathfrak {g}}]}}-{\mathfrak {q}}^{2}({\mathfrak {q}}2c{\mathfrak {g}})$

Therefore it eventually follows from (18)

(39a)

$c{\mathfrak {g=[E'H']+q(E'D)+q(H'B)-D(qE')-B(qH')+q(q}}c{\mathfrak {g)}}$

The comparison with (20) gives the important relation

(40)	${\mathfrak {g=}}{\frac {\mathfrak {S}}{c^{2}}}$

If one inserts Minkowski's connecting equations between the electromagnetic vectors into our system, then the momentum density in the moving body becomes equal to the energy current divided by $c^{2}$.

From (40) and (21) it follows, with respect to (37)

(40a)

$c{\mathfrak {g=[EH]-q(qW]}}$

where the vector

(40b)

${\mathfrak {W=[DB]}}-c{\mathfrak {g}}$

is determined from

(40c)

${\mathfrak {W-q(qW)=[DB]-[EH]}}$

Let us direct the $x$-axis into the direction of ${\mathfrak {q}}$, and let us set

(40d)

$k^{2}=1-|{\mathfrak {q}}|^{2}$

then the components of ${\mathfrak {W}}$ become

(41)	${\begin{cases}{\mathfrak {W}}_{x}=k^{-2}\left\{[{\mathfrak {DB}}]_{x}-[{\mathfrak {EH}}]_{x}\right\},\\{\mathfrak {W}}_{y}=[{\mathfrak {DB}}]_{y}-[{\mathfrak {EH}}]_{y},\\{\mathfrak {W}}_{z}=[{\mathfrak {DB}}]_{z}-[{\mathfrak {EH}}]_{z},\end{cases}}$

and it follows from (40a)

(42)

${\begin{cases}c{\mathfrak {g}}_{x}={\frac {{\mathfrak {S}}_{x}}{c}}=k^{-2}[{\mathfrak {EH}}]_{x}-|{\mathfrak {q}}|^{2}k^{-2}[{\mathfrak {DB}}]_{x},\\\\c{\mathfrak {g}}_{y}={\frac {{\mathfrak {S}}_{y}}{c}}=[{\mathfrak {EH}}]_{y},\\\\c{\mathfrak {g}}_{z}={\frac {{\mathfrak {S}}_{z}}{c}}=[{\mathfrak {EH}}]_{z}.\end{cases}}$

The previous derivation has a gap; the proof is missing that equations (39) (assumed as being valid) are really satisfied. In order to prove this, we calculate the vector

${\begin{array}{ll}{\mathfrak {R}}'&={\mathfrak {[DE']+[BH']=\left[E'[qH]\right]-\left[H'[qE]\right]}}\\&={\mathfrak {q(E'H)-q(EH')+E(qH')-H(qE')}}\end{array}}$

Since one has

${\begin{array}{c}{\mathfrak {E'H-EH'=q\left\{[DE']+[BH']\right\}=(qR')}},\\{\mathfrak {E(qH')-H(qE')=E(qH)-H(qE)=\left[q[EH]\right]}},\end{array}}$

then it becomes with respect to (40a)

${\mathfrak {R'-q(qR')}}=[{\mathfrak {q}}c{\mathfrak {g}}]$

One can – because according to the things said, the component of ${\mathfrak {R}}'$ coinciding with the direction of vector ${\mathfrak {q}}$, is equal to zero – also write

(43)	${\mathfrak {R'=[DE']+[BH'}}]=[{\mathfrak {q}}c{\mathfrak {g}}]$

By that, condition (18a) is shown to be valid, and at the same time the gap in the previous derivation of the value of ${\mathfrak {g}}$ is closed.

From (19) the value of the energy density follows:

(44)	$\psi ={\frac {1}{2}}{\mathfrak {E'D}}+{\frac {1}{2}}{\mathfrak {H'B}}+{\mathfrak {q}}c{\mathfrak {g}}$

which according to (37) and (40b) is to be brought into the form

(44a)

$\psi ={\frac {1}{2}}{\mathfrak {ED}}+{\frac {1}{2}}{\mathfrak {HB}}+{\mathfrak {qW}}$

In order to facilitate the comparison of our results with the approaches of Minkowski, we write

${\begin{array}{ccccc}c{\mathfrak {g}}_{x}=X_{t},&&c{\mathfrak {g}}_{y}=Y_{t},&&c{\mathfrak {g}}_{z}=Z_{t}\\{\mathfrak {S}}_{x}=cT_{x},&&{\mathfrak {S}}_{y}=cT_{y},&&{\mathfrak {S}}_{z}=cT_{z},\\ct=l,&&{\mathfrak {wK}}+Q=c{\mathfrak {K}}_{t},&&\psi =T_{t}.\end{array}}$

Then the momentum equations (6) and the energy equation (7) read

${\begin{cases}{\mathfrak {K}}_{x}={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}-{\frac {\partial X_{t}}{\partial l}},\\\\{\mathfrak {K}}_{y}={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}-{\frac {\partial Y_{t}}{\partial l}},\\\\{\mathfrak {K}}_{z}={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}-{\frac {\partial Z_{t}}{\partial l}}.\\\\{\mathfrak {K}}_{t}=-{\frac {\partial T_{x}}{\partial x}}-{\frac {\partial T_{y}}{\partial y}}-{\frac {\partial T_{z}}{\partial z}}-{\frac {\partial T_{t}}{\partial l}}.\end{cases}}$

There, the relation exists according to (19a)

$X_{x}+Y_{y}+Z_{z}+T_{t}=0$

Now, relation (40) means

$X_{t}=T_{x},\ Y_{t}=T_{y},\ Z_{t}=T_{z}.$

Together with (6a), these relations contain a remarkable symmetry property, which cannot be found in Minkowski's approach. Regarding the behavior under Lorentz transformations, the 10 quantities

${\begin{array}{c}X_{x},\ Y_{y},\ Z_{z},\ -T_{t},\ X_{y}=Y_{x},\ Y_{z}=Z_{y},\\Z_{x}=X_{z},\ -X_{t}=-T_{x},\ -Y_{t}=-T_{y},\ -Z_{t}=-T_{z},\end{array}}$

transform as the squares and products of coordinates $xyz$ and of the light-path $l$. Accordingly, this "space-time-tensor" satisfies the "principle of relativity" in the sense of Minkowski; Also the ponderomotive forces, which we are going to calculate in § 12, thus satisfies the relativity principle.

§ 10. The relation between the theories of Lorentz and Minkowski.

We have emphasized the illustrative meaning of vectors ${\mathfrak {E,H}}$ in Lorentz's theory, i.e., as being the contribution of the aether at the electric and magnetic excitation. In the theory of Minkowski, the vectors which connect ${\mathfrak {DB}}$ and ${\mathfrak {E'H'}}$ with each other, have no such illustration. There is also no reason, when one takes the standpoint of the relativity principle, to speak about the aether and its electromagnetic properties. This principle only considers matter in its motion relative to the observer, and the electromagnetic processes in this matter.

However, for our system of electrodynamics of moving bodies, the vectors ${\mathfrak {E}}$ and ${\mathfrak {H}}$ are of lower importance than the vectors ${\mathfrak {DBE'H'}}$. When we, under elimination of ${\mathfrak {E}}$ and ${\mathfrak {H}}$, directly connect those four vectors with each other, then the relatedness of the theories of Minkowski and Lorentz will become clear.

A) Theory of Minkowski.

From equations (36) and (37) of § 9 it follows

(45)	${\begin{cases}{\mathfrak {D+\left[q[qD]\right]}}=\epsilon {\mathfrak {E'-[qH']}},\\{\mathfrak {B+\left[q[qB]\right]}}=\mu {\mathfrak {H'+[qE']}}.\end{cases}}$

If we let the $x$-axis coincide with the direction of ${\mathfrak {q}}$, then it is given for the components taken in this direction

(45a)

${\begin{cases}{\mathfrak {D}}_{x}=&\epsilon {\mathfrak {E}}'_{x},\\{\mathfrak {B}}_{x}=&\mu {\mathfrak {H}}'_{x}.\end{cases}}$

On the other hand, for the components perpendicular to the direction of motion, it is given

(45b)

${\begin{cases}k^{2}{\mathfrak {D}}_{y}=&\epsilon {\mathfrak {E}}'_{y}-[{\mathfrak {qH}}']_{y},\\k^{2}{\mathfrak {B}}_{y}=&\mu {\mathfrak {H}}'_{y}+[{\mathfrak {qE}}']_{y}.\end{cases}}$

B) Theory of Lorentz.

From equations (30) of § 8 it follows

(46)	${\begin{cases}{\mathfrak {E+\left[q[qE]\right]}}={\mathfrak {E'-[qH']}},\\{\mathfrak {H+\left[q[qH]\right]}}={\mathfrak {H'+[qE']}}.\end{cases}}$

Thus the components of ${\mathfrak {E}}$ and ${\mathfrak {H}}$ are parallel and perpendicular to the direction of velocity

(46a)

${\begin{cases}{\mathfrak {E}}_{x}={\mathfrak {E}}'_{x},&k^{2}{\mathfrak {E}}_{y}={\mathfrak {E}}'_{y}-[{\mathfrak {qH}}']_{y};\\{\mathfrak {H}}_{x}={\mathfrak {H}}'_{x},&k^{2}{\mathfrak {H}}_{y}={\mathfrak {H}}'_{y}+[{\mathfrak {qE}}']_{y}.\end{cases}}$

While in Minkowski's theory, $\epsilon$ and $\mu$ in isotropic bodies are independent from the direction, it is allowed in Lorentz's theory, that different values of $\epsilon$ and $\mu$ are possible for excitations parallel and perpendicular to ${\mathfrak {q}}$. Consequently one obtains from (29) and (46a) for the longitudinal components of ${\mathfrak {D}}$ and ${\mathfrak {B}}$

(47a)

${\begin{cases}{\mathfrak {D}}_{x}=&\epsilon _{x}{\mathfrak {E}}'_{x},\\{\mathfrak {B}}_{x}=&\mu _{x}{\mathfrak {H}}'_{x}.\end{cases}}$

and for the transverse components

(47b)

${\begin{cases}k^{2}{\mathfrak {D}}_{y}=&\left(k^{2}\epsilon _{y}+[{\mathfrak {q}}]^{2}\right){\mathfrak {E}}'_{y}-[{\mathfrak {qH}}']_{y},\\k^{2}{\mathfrak {B}}_{y}=&\left(k^{2}\mu _{y}+[{\mathfrak {q}}]^{2}\right){\mathfrak {H}}'_{y}+[{\mathfrak {qE}}']_{y}.\end{cases}}$

If we compare (45a) and (47a) on one hand, and on the other hand (45b) and (47b), then we see that the equations that connect ${\mathfrak {D,B}}$ and ${\mathfrak {E',H'}}$, are in agreement in both theories, when one sets in Lorentz's theory

(48a)

$\epsilon _{x}=\epsilon ,\ \mu _{x}=\mu ;$

(48b)

$\epsilon _{y}-1=k^{-2}(\epsilon -1),\ \mu _{y}-1=k^{-2}(\mu -1)$

Then, the longitudinal and transverse components of the electric and magnetic polarization, become according to (31)

(48c)

${\mathfrak {P}}_{x}=(\epsilon -1){\mathfrak {E}}'_{x},\ {\mathfrak {P}}_{y}=k^{-2}(\epsilon -1){\mathfrak {E}}'_{y},\ {\mathfrak {P}}_{z}=k^{-2}(\epsilon -1){\mathfrak {E}}'_{z};$

(48d)

${\mathfrak {M}}_{x}=(\mu -1){\mathfrak {H}}'_{x},\ {\mathfrak {M}}_{y}=k^{-2}(\mu -1){\mathfrak {H}}'_{y},\ {\mathfrak {M}}_{z}=k^{-2}(\mu -1){\mathfrak {H}}'_{z};$

That the electric polarization of a body being isotropic when at rest, must be influenced by its motion in the way indicated by (48c) – in case the relativity postulate should be compatible with Lorentz's theory – has already been spoken out by Lorentz. If one assumes the symmetry of electric and magnetic vectors, then the corresponding behavior is given for the magnetic polarization.

The presupposition made in § 8, that $\epsilon$ and $\mu$ should be independent from velocity, has become irrelevant now. Thus the values for momentum density, energy density and energy current, which were found there, must be corrected. The quantities included in (31)

(49)	${\begin{cases}{\mathfrak {E'{\dot {P}}-P{\dot {E}}'}}=2{\mathfrak {E'{\dot {P}}}}-{\frac {d}{dt}}({\mathfrak {E'P}}),\\\\{\mathfrak {H'{\dot {M}}-M{\dot {H}}'}}=2{\mathfrak {H'{\dot {M}}}}-{\frac {d}{dt}}({\mathfrak {H'M}}).\end{cases}}$

are not to be neglected any more. It follows from (48c)

(49a)

${\mathfrak {E'P}}=(\epsilon -1)\left\{{\mathfrak {E}}_{x}^{'2}+k^{-2}\left({\mathfrak {E}}_{y}^{'2}+{\mathfrak {E}}_{z}^{'2}\right)\right\}$.

Furthermore one obtains, under consideration of the transverse acceleration and rotation of the polarization ellipsoid caused by it, the expressions for the components of ${\mathfrak {\dot {P}}}$

(49b)

${\begin{cases}{\mathfrak {\dot {P}}}_{x}=(\epsilon -1){\mathfrak {\dot {E}}}'_{x}-{\frac {{\mathfrak {\dot {q}}}_{y}}{|{\mathfrak {q}}|}}{\mathfrak {P}}_{y}-{\frac {{\mathfrak {\dot {q}}}_{z}}{|{\mathfrak {q}}|}}{\mathfrak {P}}_{z},\\\\{\mathfrak {\dot {P}}}_{y}=k^{-2}(\epsilon -1){\mathfrak {\dot {E}}}'_{y}+2{\mathfrak {\dot {q}}}_{x}|{\mathfrak {q}}|k^{-2}{\mathfrak {P}}_{y}-{\frac {{\mathfrak {\dot {q}}}_{y}}{|{\mathfrak {q}}|}}{\mathfrak {P}}_{x},\\\\{\mathfrak {\dot {P}}}_{z}=k^{-2}(\epsilon -1){\mathfrak {\dot {E}}}'_{z}+2{\mathfrak {\dot {q}}}_{x}|{\mathfrak {q}}|k^{-2}{\mathfrak {P}}_{z}+{\frac {{\mathfrak {\dot {q}}}_{z}}{|{\mathfrak {q}}|}}{\mathfrak {P}}_{x},\end{cases}}$.

From that it follows

(49c)

${\begin{cases}2{\mathfrak {E'{\dot {P}}}}&=2(\epsilon -1)\left\{{\mathfrak {E}}'_{x}{\mathfrak {\dot {E}}}'_{x}+k^{-2}{\mathfrak {E}}'_{y}{\mathfrak {\dot {E}}}'_{y}+k^{-2}{\mathfrak {E}}'_{z}{\mathfrak {\dot {E}}}'_{z}\right\}\\\\&+4{\mathfrak {\dot {q}}}_{x}|{\mathfrak {q}}|k^{-2}\left\{{\mathfrak {E}}'_{y}{\mathfrak {P}}_{y}+{\mathfrak {E}}'_{z}{\mathfrak {P}}_{z}\right\}\\\\&-2{\frac {{\mathfrak {\dot {q}}}_{y}}{|{\mathfrak {q}}|}}\left\{{\mathfrak {E}}'_{x}{\mathfrak {P}}_{y}-{\mathfrak {E}}'_{y}{\mathfrak {P}}_{x}\right\}-2{\frac {{\mathfrak {\dot {q}}}_{z}}{|{\mathfrak {q}}|}}\left\{{\mathfrak {E}}'_{x}{\mathfrak {P}}_{z}-{\mathfrak {E}}'_{z}{\mathfrak {P}}_{x}\right\},\end{cases}}$

while (49a) gives

(49d)

${\frac {d}{dt}}({\mathfrak {E'P}})=2(\epsilon -1)\left\{{\mathfrak {E}}'_{x}{\mathfrak {\dot {E}}}'_{x}+k^{-2}{\mathfrak {E}}'_{y}{\mathfrak {\dot {E}}}'_{y}+k^{-2}{\mathfrak {E}}'_{z}{\mathfrak {\dot {E}}}'_{z}\right\}+2{\mathfrak {\dot {q}}}_{x}|{\mathfrak {q}}|k^{-2}\left\{{\mathfrak {E}}'_{y}{\mathfrak {P}}_{y}+{\mathfrak {E}}'_{z}{\mathfrak {P}}_{z}\right\}$.

Since one now has according to (48c)

${\begin{array}{c}{\mathfrak {E}}'_{x}{\mathfrak {P}}_{y}-{\mathfrak {E}}'_{y}{\mathfrak {P}}_{x}=|{\mathfrak {q}}|^{2}{\mathfrak {E}}'_{x}{\mathfrak {P}}_{y},\\{\mathfrak {E}}'_{x}{\mathfrak {P}}_{z}-{\mathfrak {E}}'_{z}{\mathfrak {P}}_{x}=|{\mathfrak {q}}|^{2}{\mathfrak {E}}'_{x}{\mathfrak {P}}_{z},\end{array}}$

thus it is given

(49c)

${\begin{cases}{\mathfrak {E'{\dot {P}}-{\dot {P}}E'}}&=2{\mathfrak {\dot {q}}}_{x}|{\mathfrak {q}}|k^{-2}\left\{{\mathfrak {E}}'_{y}{\mathfrak {P}}_{y}+{\mathfrak {E}}'_{z}{\mathfrak {P}}_{z}\right\}\\&-2{\mathfrak {\dot {q}}}_{y}|{\mathfrak {q}}|{\mathfrak {E}}'_{x}{\mathfrak {P}}_{y}-2{\mathfrak {\dot {q}}}_{z}|{\mathfrak {q}}|{\mathfrak {E}}'_{x}{\mathfrak {P}}_{z}.\end{cases}}$

The insertion of this expression and the corresponding magnetic term into (31), gives (instead of value (32) of momentum density) the corrected value

(50)

${\begin{cases}c{\mathfrak {g}}&={\mathfrak {[EH]+\left[E'[qP]\right]+\left[H'[qM]\right]}}\\&+{\mathfrak {q}}|{\mathfrak {q}}|^{2}k^{-2}\left\{{\mathfrak {E}}'_{y}{\mathfrak {P}}_{y}+{\mathfrak {E}}'_{z}{\mathfrak {P}}_{z}+{\mathfrak {E}}'_{y}{\mathfrak {M}}_{y}+{\mathfrak {H}}'_{z}{\mathfrak {M}}_{z}\right\}\end{cases}}$

That relation (18) is satisfied, can easily be verified.

If the value (50) of $c{\mathfrak {g}}$ is inserted in the general formula (19) for the energy density, then it follows instead of (33)

(51)

${\begin{cases}\psi &={\frac {1}{2}}{\mathfrak {E}}^{2}+{\frac {1}{2}}{\mathfrak {H}}^{2}\\&+{\frac {1}{2}}{\mathfrak {E'P}}+{\frac {1}{2}}{\mathfrak {H'M}}+|{\mathfrak {q}}|^{2}k^{-2}\left\{{\mathfrak {E}}'_{y}{\mathfrak {P}}_{y}+{\mathfrak {E}}'_{z}{\mathfrak {P}}_{z}+{\mathfrak {E}}'_{y}{\mathfrak {M}}_{y}+{\mathfrak {H}}'_{z}{\mathfrak {M}}_{z}\right\}\end{cases}}$

One also obtains, because of (20), the corrected formula for the energy current

(52)

${\begin{cases}{\frac {\mathfrak {S}}{c}}&={\mathfrak {[EH]+\left[E'[qP]\right]+\left[H'[qM]\right]}}\\&+{\mathfrak {q}}|{\mathfrak {q}}|^{2}k^{-2}\left\{{\mathfrak {E}}'_{y}{\mathfrak {P}}_{y}+{\mathfrak {E}}'_{z}{\mathfrak {P}}_{z}+{\mathfrak {E}}'_{y}{\mathfrak {M}}_{y}+{\mathfrak {H}}'_{z}{\mathfrak {M}}_{z}\right\}\end{cases}}$

From (50) and (52) one can see, that also in Lorentz's theory (when modified in the given way) the relation between the energy current and momentum density exists:

(53)	${\frac {\mathfrak {S}}{c}}=c{\mathfrak {g}},$

which we already encountered in Minkowski's theory.

This result was to be expected; after the equations connecting ${\mathfrak {D}}$ and ${\mathfrak {B}}$ with ${\mathfrak {E'}}$ and ${\mathfrak {H'}}$, are brought into agreement, no essential difference exists any more between both theories from the standpoint of our system. Only the meaning of the vectors denoted by ${\mathfrak {E,H}}$, is different. At it can be seen from (50) and (51), Lorentz's definition of these vectors still allows, to separate the contributions of aether and matter to electromagnetic energy and electromagnetic momentum; however, formulas apply to these contributions, which do not allow a simple interpretation any more.

§ 11. Consideration of the temporal change of $\epsilon$ and $\mu$.

Up to now, we have considered the dielectric constant $\epsilon$ and the magnetic permeability $\mu$ as quantities, which have constant values for a given material point, or at least (see § 10) are varying in a specified way with velocity. The case, that these values depend on the state of deformation of the body, and thus on time, we haven't considered yet. Now, as to how are these considerations to be modified, when ${\dot {\epsilon }}$ and ${\dot {\mu }}$ are not equal to zero?

A) Theories of H. Hertz and E. Cohn.

If we employ formulas (23) of Hertz's theory, or formulas (26) of Cohn's theory, then we find in the case that $\epsilon$ and $\mu$ are depending on time, that instead of (18) the following relation takes place

(54)	${\frac {1}{2}}\left\{{\mathfrak {E'{\dot {D}}-D{\dot {E}}'+H'{\dot {B}}-B{\dot {H}}'}}\right\}={\mathfrak {g{\dot {w}}}}+\zeta \epsilon +\eta \mu ,$

where it is set

(54a)

$\zeta ={\frac {1}{2}}{\mathfrak {E}}'^{2},\ \eta ={\frac {1}{2}}{\mathfrak {H}}'^{2};$

there it is assumed, that the earlier expressions (24) and (27) hold for the momentum density.

B) Theories of H. Minkowski and H. A. Lorentz.

The calculation becomes somewhat more complicated, when one employs the connecting equations (36) and (37) of Minkowski’s theory. The terms

${\dot {\epsilon }}{\mathfrak {E}}'^{2}+{\dot {\mu }}{\mathfrak {H}}'^{2}$

are not only to be inserted in the right-hand side of (38), but also – when the terms which contain ${\dot {\mathfrak {D}}}$ and ${\dot {\mathfrak {B}}}$ are calculated – the variability of $\epsilon$ and $\mu$ is to be considered in (38c). Also a relation in the form of (54) is given, when the value of ${\mathfrak {g}}$ is not changed; though the quantities $\zeta ,\ \eta$ have a somewhat different meaning here

(54b)

${\begin{cases}\zeta ={\frac {1}{2}}\left\{{\mathfrak {E}}_{x}^{'2}+k^{-2}\left({\mathfrak {E}}_{y}^{'2}+{\mathfrak {E}}_{z}^{'2}\right)\right\},\\\eta ={\frac {1}{2}}\left\{{\mathfrak {H}}_{x}^{'2}+k^{-2}\left({\mathfrak {H}}_{y}^{'2}+{\mathfrak {H}}_{z}^{'2}\right)\right\},\end{cases}}$

This result also holds for Lorentz's theory in the form which we gave to it in § 10; because all expressions, which only contain the vectors ${\mathfrak {E'H'DB}}$, are in this theory identical with the corresponding expressions of Minkowski's theory.

Now, since equation (54) contradicts relation (18), and since we won't allow a change in the values of momentum density and energy density, we consider it necessary to correct the value of quantity $P'$ given in (V), namely by

$-\zeta {\dot {\epsilon }}-\eta {\dot {\mu }}$

then the considerations of § 5 indeed exactly lead to relation (54) instead of relation (18).

This view finds support in the theory of electrostriction^[13]. In the simplest case present at fluids and gases, where $\epsilon$ and $\mu$ only depend on the density $\sigma$, one has

$-\zeta {\dot {\epsilon }}-\eta {\dot {\mu }}=-{\dot {\sigma }}\left\{\zeta {\frac {d\epsilon }{d\sigma }}+\eta {\frac {d\mu }{d\sigma }}\right\}$

this becomes in consequence of the continuity condition of matter

$-\zeta {\dot {\epsilon }}-\eta {\dot {\mu }}=\mathrm {div} {\mathfrak {w}}\left\{\zeta \sigma {\frac {d\epsilon }{d\sigma }}+\eta \sigma {\frac {d\mu }{d\sigma }}\right\}$

If one considers definition (13) of quantity $P'$, then one sees that this increase corresponds to a growth of relative normal stresses $X'_{x},\ Y'_{y},\ Z'_{z}$ by

(55)	$-p'=\zeta \sigma {\frac {d\epsilon }{d\sigma }}+\eta \sigma {\frac {d\mu }{d\sigma }}$.

In case $\epsilon$ and $\mu$ are increasing with growing density, the additional pressure $p'$ becomes negative, i.e. the fluid tends to contract in the electric and magnetic field. In the case of rest, (55) together with (54a) and (54b) gives the ordinary approach for the theory of electrostriction.

At solid bodies, more general considerations are necessary in order to represent the dependence of electric and magnetic constants from the state of deformation. H. Hertz^[14] has generally calculated the corresponding supplementary stresses from the standpoint of his theory; on the other hand, E. Cohn as well as H. Minkowski didn't introduce such supplementary stresses. This simplification, which is allowed in the light of the insignificance of these supplementary stresses, we will still allow to ourselves too.

§ 12. The ponderomotive force.

Now, since the relative stresses as well as the momentum density in there dependence from the electromagnetic vectors, are determined for every given theory, then the components of the ponderomotive force are given from equations (8). According to (Va) it is

(56)

${\begin{cases}X'_{x}={\frac {1}{2}}\left\{{\mathfrak {E}}'_{x}{\mathfrak {D}}_{x}-{\mathfrak {E}}'_{y}{\mathfrak {D}}_{y}-{\mathfrak {E}}'_{z}{\mathfrak {D}}_{z}\right\}+{\frac {1}{2}}\left\{{\mathfrak {H}}'_{x}{\mathfrak {B}}_{x}-{\mathfrak {H}}'_{y}{\mathfrak {B}}_{y}-{\mathfrak {H}}'_{z}{\mathfrak {B}}_{z}\right\},\\\\X'_{y}={\mathfrak {E}}'_{x}{\mathfrak {D}}_{y}+{\mathfrak {H}}'_{x}{\mathfrak {B}}_{y},\\\\X'_{z}={\mathfrak {E}}'_{x}{\mathfrak {D}}_{z}+{\mathfrak {H}}'_{x}{\mathfrak {B}}_{z}.\end{cases}}$

From that if follows

(57)

${\begin{cases}{\frac {\partial X'_{x}}{\partial x}}+{\frac {\partial X'_{y}}{\partial y}}+{\frac {\partial X'_{z}}{\partial z}}={\mathfrak {E}}'_{x}\mathrm {div} {\mathfrak {D}}+{\mathfrak {H}}'_{x}\mathrm {div} {\mathfrak {B}}\\\\\qquad -{\mathfrak {D}}_{y}\mathrm {curl} _{z}{\mathfrak {E}}'+{\mathfrak {D}}_{z}\mathrm {curl} _{y}{\mathfrak {E}}'-{\mathfrak {B}}_{y}\mathrm {curl} _{z}{\mathfrak {H}}'+{\mathfrak {B}}_{z}\mathrm {curl} _{z}{\mathfrak {H}}'\\\\\qquad -{\frac {1}{2}}\left\{{\mathfrak {E}}'{\frac {\partial {\mathfrak {D}}}{\partial x}}-{\mathfrak {D}}{\frac {\partial {\mathfrak {E'}}}{\partial x}}+{\mathfrak {H}}'{\frac {\partial {\mathfrak {B}}}{\partial x}}-{\mathfrak {B}}{\frac {\partial {\mathfrak {H'}}}{\partial x}}\right\}.\end{cases}}$

If we consider the last row, then the analogy to the left-hand side of (54) leaps out; the expressions only differ by the fact, that it was differentiated with respect to time there, and here with respect to a coordinate.

Since the train of thought that led to relation (54), is not concerned by the meaning of the independent variables, it is given

(57a)

${\frac {1}{2}}\left\{{\mathfrak {E}}'{\frac {\partial {\mathfrak {D}}}{\partial x}}-{\mathfrak {D}}{\frac {\partial {\mathfrak {E'}}}{\partial x}}+{\mathfrak {H}}'{\frac {\partial {\mathfrak {B}}}{\partial x}}-{\mathfrak {B}}{\frac {\partial {\mathfrak {H'}}}{\partial x}}\right\}={\mathfrak {g}}{\frac {\partial {\mathfrak {w}}}{\partial x}}+\zeta {\frac {\partial \epsilon }{\partial x}}+\eta {\frac {\partial \mu }{\partial x}}$

The vectorial generalization of (57) gives as the force contribution stemming from the relative stresses

(58)

${\begin{cases}{\mathfrak {K}}_{1}={\mathfrak {E}}'\mathrm {div} {\mathfrak {D}}+{\mathfrak {H}}'\mathrm {div} {\mathfrak {B}}-[{\mathfrak {D}}\mathrm {curl} {\mathfrak {E'}}]-[{\mathfrak {B}}\mathrm {curl} {\mathfrak {H'}}]\\\quad -({\mathfrak {g}}\nabla ){\mathfrak {w}}-[{\mathfrak {g}}\mathrm {curl} {\mathfrak {w}}]-\zeta \nabla \epsilon -\eta \nabla \mu .\end{cases}}$

The contribution is added to it, which stems from the electromagnetic momentum

(58a)

${\mathfrak {K}}_{2}=-{\frac {\delta {\mathfrak {g}}}{\delta t}}$

By the aid of the two first main equations of § 4, we want to transform the vector products arising in (58), into

${\begin{array}{l}-[{\mathfrak {D}}\mathrm {curl} {\mathfrak {E'}}]={\frac {1}{c}}\left[{\mathfrak {D}}{\frac {\partial '{\mathfrak {B}}}{\partial t}}\right]\\\\-[{\mathfrak {B}}\mathrm {curl} {\mathfrak {H'}}]={\frac {1}{c}}[{\mathfrak {JB}}]+{\frac {1}{c}}\left[{\frac {\partial '{\mathfrak {D}}}{\partial t}}{\mathfrak {B}}\right]\end{array}}$

With respect to rule (5) of § 1, the sum of these two terms is

${\frac {1}{c}}[{\mathfrak {JB}}]+{\frac {1}{c}}\left\{{\frac {\delta }{\delta t}}[{\mathfrak {DB}}]+\left([{\mathfrak {DB}}]\nabla \right){\mathfrak {w}}+\left[[{\mathfrak {DB}}]\mathrm {curl} {\mathfrak {w}}\right]\right\}$

The ponderomotive force is given by addition of the forces ${\mathfrak {K}}_{1}$ and ${\mathfrak {K}}_{2}$; the emerging expression becomes simplified, when one introduces the vector defined in (22)

(59)	${\mathfrak {W=[DB]}}-c{\mathfrak {g}}$,

and if one uses the denotations

(59a)

${\mathfrak {q}}={\frac {\mathfrak {w}}{c}},\ l=ct.$

Furthermore, the density of true electricity may be set to

(59b)

$\mathrm {div} {\mathfrak {D}}=\rho$

and the density of true magnetism shall be assumed as equal to zero

(59c)

$\mathrm {div} {\mathfrak {B}}=0$;

also instead of the electrostatic measure of current strength, the electromagnetic measure shall be introduced

(59d)

${\mathfrak {J}}=ci$.

Then the expression for the ponderomotive force, which acts upon the unit volume of moving matter, reads

(60)	${\mathfrak {K}}={\mathfrak {E}}'\rho +[{\mathfrak {iB}}]-\zeta \nabla \epsilon -\eta \nabla \mu +{\frac {\delta {\mathfrak {W}}}{\delta l}}+({\mathfrak {W}}\nabla ){\mathfrak {q}}+[{\mathfrak {W}}\mathrm {curl} {\mathfrak {q}}]$

The first term represents the force acting upon the moving electricity, the second term represents the force acting upon the electric conduction current; the third and fourth term account for the influence of inhomogeneity of the body. While these four terms already have to be accounted for at static or stationary fields in resting bodies, the latter terms (containing the vector ${\mathfrak {W}}$) only play a role at non-stationary processes, or in moving bodies.

In the obtained expression for the ponderomotive force, the differences between the individual theories only become of importance by the fact – when one neglects the extremely small deviation in the meaning of quantities $\zeta$ and $\eta$ (eq. 54ab) –, that the vector ${\mathfrak {W}}$ assumes different values.

If ${\mathfrak {K}}$ gives the momentum exerted by the electromagnetic field, then the energy converted into non-electromagnetic forms, is given by the sum of Joule-heat and work of the ponderomotive force. For the Joule-heat, according to main equation (III) and (59d), it is given

$Q=cq={\mathfrak {JE}}'=ci{\mathfrak {E}}'$

while the performance of work of force ${\mathfrak {K}}$, is given from (60)

${\mathfrak {qK}}={\mathfrak {E}}'\rho {\mathfrak {q}}-i[{\mathfrak {qB}}]-\zeta ({\mathfrak {q}}\nabla )\epsilon -\eta (q\nabla )\mu +{\mathfrak {q}}\left\{{\frac {\delta {\mathfrak {W}}}{\delta l}}+({\mathfrak {W}}\nabla ){\mathfrak {q}}+[{\mathfrak {W}}\mathrm {curl} {\mathfrak {q}}]\right\}$

If one now considers, that the calculation of the ponderomotive force is based on the presuppositions

${\begin{array}{l}{\frac {1}{c}}{\dot {\epsilon }}={\frac {\partial \epsilon }{\partial l}}+({\mathfrak {q}}\nabla )\epsilon =0,\\\\{\frac {1}{c}}{\dot {\mu }}={\frac {\partial \mu }{\partial l}}+({\mathfrak {q}}\nabla )\mu =0,\end{array}}$

and that from (3) and the known calculation rule

$\nabla ({\mathfrak {qW}})=({\mathfrak {q}}\nabla ){\mathfrak {W}}+[\mathrm {{\mathfrak {q}}curl} {\mathfrak {W}}]+({\mathfrak {W}}\nabla ){\mathfrak {q}}+[{\mathfrak {W}}\mathrm {curl} {\mathfrak {q}}],$

it is given

${\frac {\delta {\mathfrak {W}}}{\delta l}}+({\mathfrak {W}}\nabla ){\mathfrak {q}}+[{\mathfrak {W}}\mathrm {curl} {\mathfrak {q}}]={\frac {\partial {\mathfrak {W}}}{\partial l}}+{\mathfrak {W}}\mathrm {div} {\mathfrak {q}}+\nabla ({\mathfrak {qW}})-[\mathrm {{\mathfrak {q}}curl} {\mathfrak {W}}]$

and that it furthermore follows with respect to (3a)

${\begin{array}{c}{\mathfrak {q}}\left\{{\frac {\delta {\mathfrak {W}}}{\delta l}}+({\mathfrak {W}}\nabla ){\mathfrak {q}}+[{\mathfrak {W}}\mathrm {curl} {\mathfrak {q}}]\right\}\\\\=-{\mathfrak {W}}{\frac {\partial {\mathfrak {q}}}{\partial l}}+{\frac {\partial ({\mathfrak {qW}})}{\partial l}}+\mathrm {div} {\mathfrak {q}}({\mathfrak {qW}})=-{\mathfrak {W}}{\frac {\partial {\mathfrak {q}}}{\partial l}}+{\frac {\delta ({\mathfrak {qW}})}{\delta l}}\end{array}}$

then one finally obtains, for the energy which is emanated in unit time by unit volume, the formula

(60a)

$q+{\mathfrak {qK}}=\{i+\rho {\mathfrak {q}}\}\left\{{\mathfrak {E}}'-[{\mathfrak {qB}}]\right\}+\zeta {\frac {\partial \epsilon }{\partial l}}+\eta {\frac {\partial \mu }{\partial l}}-{\mathfrak {W}}{\frac {\partial {\mathfrak {q}}}{\partial l}}+{\frac {\delta ({\mathfrak {qW}})}{\delta l}}$

Also here, neglecting the meaning (which somewhat deviates for magnitudes of second order) of $\zeta$ and $\eta$, the different theories only differ by the value of the vector ${\mathfrak {W}}$, when considered from the standpoint of our system.

Now one imagines, that the value is set every time for ${\mathfrak {W}}$, which it has in the relevant theory, and then compare our expression (60) of ponderomotive force with the one obtained by other authors.

The value of the ponderomotive force given by E. Cohn is slightly deviating from our value. This partially stems from the fact, that E. Cohn's approach for the relative stresses is not entirely identical with (Va); he namely sets at this place $\epsilon {\mathfrak {E}}'$ instead of ${\mathfrak {D}}$, probably with the intention to remove the torque ${\mathfrak {R'}}$ of the relative stresses. The difference in the value of the force contribution stemming from the relative stresses, which is caused by that, can immediately be found

$({\mathfrak {g}}\nabla ){\mathfrak {w}}+{\mathfrak {w}}\mathrm {div} {\mathfrak {g}}$

Only then we have seen the vanishing of ${\mathfrak {R'}}$ as necessary, when – as in the theory of Hertz – no electromagnetic momentum comes into play. E. Cohn, however, also accounts for the second part of the force which is connected with vector ${\mathfrak {g}}$, namely

${\mathfrak {K}}_{2}=-{\frac {\partial '{\mathfrak {g}}}{\partial t}}$

this expression of the electromagnetic force of inertia deviates from our value (58a), according to (4), by

$({\mathfrak {g}}\nabla ){\mathfrak {w}}$

Thus the difference between the force expression of E. Cohn and the one obtained here, altogether amounts to

$2({\mathfrak {g}}\nabla ){\mathfrak {w}}+{\mathfrak {w}}\mathrm {div} {\mathfrak {g}}$

where ${\mathfrak {g}}$ is determined by (27). It is probably too small, to be accessible to experimental tests.

We now pass to the theory of Minkowski. It was already mentioned in § 9, that the close relationship between momentum density and energy current, which takes place in this theory according to the results of the present investigation, is not assumed in Minkowski's approach. Consequently, also value (60) of the ponderomotive force deviates from Minkowski's approach; especially the term (60) is missing there, which already comes into play in the case of rest. It was already alluded to by A. Einstein and J. Laub^[15], that the force which according to Lorentz shall act in the magnetic field upon the polarization stream, is missing in Minkowski's approach. Now, no experimental confirmation for the existence of this force was provided, however, the conviction of its existence is based on the analogy which exists between conduction current and polarization current according to the concepts of the theory of electrons; this analogy is so useful, that one won't deny that force without weighty reasons. Our force expressions, as it can be see from eq. 63, contains that force; that it doesn't contradict the principle of relativity, was noticed by us at the end of § 9.

In the case of rest, where ${\mathfrak {E,H}}$ has to be written instead of ${\mathfrak {E',H'}}$, the ponderomotive force becomes

(61)	${\mathfrak {K}}={\mathfrak {E}}\rho +[{\mathfrak {iB}}]-{\frac {1}{2}}{\mathfrak {E}}^{2}\nabla \epsilon -{\frac {1}{2}}{\mathfrak {H}}^{2}\nabla \mu +{\frac {\partial {\mathfrak {W}}}{\partial l}}$

Vector ${\mathfrak {W}}$ has the following values in the different theories:

A) Theory of H. Hertz.

Here, it follows from (22) and (24)

(61a)

${\mathfrak {W}}=[{\mathfrak {DB}}]=\epsilon \mu [{\mathfrak {EH}}]$

B) Theories of E. Cohn, H. A. Lorentz and H. Minkowski.

In all three theories, as it follows from (27), (32), (40a), it becomes

(61b)

${\begin{array}{c}cg=[EH]\\{\mathfrak {W=[DB]-[EH]}}=(\epsilon \mu -1)[{\mathfrak {EH}}]\end{array}}$

That all three theories give the same values for the ponderomotive force in resting bodies, is caused (in the sense of our system) by the fact, that the equations connecting ${\mathfrak {D}}$ and ${\mathfrak {B}}$ with ${\mathfrak {E'}}$ and ${\mathfrak {H'}}$, are in agreement (including the terms linear in ${\mathfrak {q}}$). The notation of Lorentz's theory may be employed in the discussion of the force in resting bodies.

If one sets value (61b) for ${\mathfrak {W}}$, then the ponderomotive force (61) can be decomposed into two parts

(62)

${\begin{cases}{\mathfrak {K}}_{e}={\mathfrak {E}}\rho -{\frac {1}{2}}{\mathfrak {E}}^{2}\nabla \epsilon +(\epsilon \mu -1)\left[{\mathfrak {E}}{\frac {\partial {\mathfrak {H}}}{\partial l}}\right],\\\\{\mathfrak {K}}_{m}=[i{\mathfrak {B}}]-{\frac {1}{2}}{\mathfrak {H}}^{2}\nabla \mu +(\epsilon \mu -1)\left[{\frac {\partial {\mathfrak {E}}}{\partial l}}{\mathfrak {H}}\right],\end{cases}}$

which are to be interpreted as the contributions of the electric and magnetic field.

From the main equations for resting bodies

${\begin{array}{l}\mathrm {curl} {\mathit {\mathfrak {H}}}={\frac {\partial {\mathfrak {D}}}{\partial l}}+i,\\\\\mathrm {curl} {\mathit {\mathfrak {E}}}=-{\frac {\partial {\mathfrak {B}}}{\partial l}}\end{array}}$

one derives, by introduction of the electric and magnetic polarization

${\begin{array}{l}{\mathfrak {P=D-E}}=(\epsilon -1){\mathfrak {E}},\\{\mathfrak {M=B-H}}=(\mu -1){\mathfrak {H}}\end{array}}$

the two following relations

${\begin{array}{rl}0=&-[{\mathfrak {P}}\mathrm {curl} {\mathfrak {E}}]-\mu (\epsilon -1)\left[{\mathfrak {E}}{\frac {\partial {\mathfrak {H}}}{\partial l}}\right]\\\\{}[i{\mathfrak {B}}]=&[i{\mathfrak {H}}]-[{\mathfrak {M}}curl{\mathfrak {H}}]-\epsilon (\mu -1)\left[{\frac {\partial {\mathfrak {E}}}{\partial l}}{\mathfrak {H}}\right]\end{array}}$

With respect to them, expressions (62) go over into

(62a)

${\begin{cases}{\mathfrak {K}}_{e}={\mathfrak {E}}\rho -[{\mathfrak {P}}\mathrm {curl} {\mathfrak {E}}]-{\frac {1}{2}}{\mathfrak {E}}^{2}\nabla (\epsilon -1)+\left[{\mathfrak {E}}{\frac {\partial {\mathfrak {M}}}{\partial l}}\right],\\\\{\mathfrak {K}}_{m}=[i{\mathfrak {H}}]-[{\mathfrak {M}}\mathrm {curl} {\mathfrak {H}}]-{\frac {1}{2}}{\mathfrak {H}}^{2}\nabla (\mu -1)+\left[{\frac {\partial {\mathfrak {P}}}{\partial l}}{\mathfrak {H}}\right].\end{cases}}$

Since it furthermore holds

(63)

${\begin{array}{l}{\frac {1}{2}}(\epsilon -1)\nabla {\mathfrak {E}}^{2}+{\frac {1}{2}}{\mathfrak {E}}^{2}\nabla (\epsilon -1)={\frac {1}{2}}\nabla (\epsilon -1){\mathfrak {E}}^{2}={\frac {1}{2}}\nabla ({\mathfrak {PE}}),\\\\{\frac {1}{2}}(\epsilon -1)\nabla {\mathfrak {E}}^{2}=({\mathfrak {P}}\nabla ){\mathfrak {E}}+[{\mathfrak {P}}\mathrm {curl} {\mathfrak {E}}];\\\\{\frac {1}{2}}(\mu -1)\nabla {\mathfrak {H}}^{2}+{\frac {1}{2}}{\mathfrak {H}}^{2}\nabla (\mu -1)={\frac {1}{2}}\nabla (\mu -1){\mathfrak {H}}^{2}={\frac {1}{2}}\nabla ({\mathfrak {MH}}),\\\\{\frac {1}{2}}(\mu -1)\nabla {\mathfrak {H}}^{2}=({\mathfrak {M}}\nabla ){\mathfrak {H}}+[{\mathfrak {M}}\mathrm {curl} {\mathfrak {H}}];\end{array}}$

it eventually becomes

(63)

${\begin{cases}{\mathfrak {K}}_{e}=({\mathfrak {P}}\nabla ){\mathfrak {E}}+{\mathfrak {E}}\rho +\left[{\mathfrak {E}}{\frac {\partial {\mathfrak {M}}}{\partial l}}\right]-{\frac {1}{2}}\nabla ({\mathfrak {PE}}),\\\\{\mathfrak {K}}_{m}=({\mathfrak {M}}\nabla ){\mathfrak {H}}+[i{\mathfrak {H}}]+\left[{\frac {\partial {\mathfrak {P}}}{\partial l}}{\mathfrak {H}}\right]-{\frac {1}{2}}\nabla ({\mathfrak {MH}}).\end{cases}}$

These formulas for the electric and magnetic force contribution which were given here from (61) and (61b), may be compared with the approaches made by A. Einstein and J. Laub^[16] for the ponderomotive force in resting bodies. The first three terms in ${\mathfrak {K}}_{e}$ and ${\mathfrak {K}}_{m}$ can also be found there; the first terms are interpreted as the forces of the field upon electrically and magnetically polarized volume elements, and the vector product of $i$ and ${\mathfrak {H}}$ as the force of the magnetic field upon the electric conduction currents; the previously mentioned force of the magnetic field upon the electric polarization stream is added to it, and the force corresponding to it, which acts upon the magnetic polarization stream in the electric field. Though the two last terms of expressions (63) are missing in the approach of the mentioned authors, which is connected with the fact, that their values of the fictitious normal stresses are somewhat deviating from the ones that are assumed otherwise. If it is about a force upon a section, at whose boundary ${\mathfrak {P}}$ and ${\mathfrak {M}}$ are zero, then those two terms are dropping; because the surface integrals provided by them, vanishes.

In this case which happens not seldom, one may employ the approach of Einstein and Laub. Though the consequence drawn by those authors, namely that vector ${\mathfrak {B}}$ is not decisive for the force upon the conduction currents, seems not correct to me. Because we have seen, that it is just vector ${\mathfrak {B}}$ in (61), that determines the force upon the conductor. However, the force acting in magnetic fields upon a conducting and at the same time magnetized wire, is not at all to be calculated from neither the vector product of $i$ and ${\mathfrak {B}}$, nor from that of $i$ and ${\mathfrak {H}}$; one rather has to add the force $-{\frac {1}{2}}{\mathfrak {H}}^{2}\nabla \mu$ to the vector product, which is located at the transition layer between wire and air, while the force $({\mathfrak {M}}\nabla ){\mathfrak {H}}$ is added, which acts in the magnetized wire, in case the field is not coincidentally homogeneous there. Neglecting this very special case, the force acting at the magnetized volume elements of a homogeneous conducting wire, is to be set equal to the vector product of $i$ and ${\mathfrak {B}}$, though not to that of $i$ and ${\mathfrak {H}}$.

Incidentally, formulas (63) in no way have such a principal importance, as the initial expressions (61) for the ponderomotive force. The latter was given from the system of electrodynamics which also includes moving bodies; yet the former could hardly be so generalized, in order to determine the ponderomotive force in moving bodies as well.

Ospedaletti Ligure, December 1908.

Max Abraham.

---

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