Translation:The Principle of Relativity and its Application to some Special Physical Phenomena

by Hendrik Lorentz

The Principle of Relativity and its Application to some Special Physical Phenomena.

By H. A. Lorentz.

To discuss Einstein's principle of relativity here in Göttingen, where Minkowski has worked, appears to me as a particular welcomed task.

One can highlight the importance of this principle from different points of views. About the mathematical side of the question, which has found such a gleaming representation by Minkowski and which was further developed by Abraham, Sommerfeld etc., shall not be spoken here. Rather (after some epistemological considerations concerning the concepts of space and time) those physical phenomena shall be discussed, which could contribute to an experimental test of this principle.

The relativity principle asserts the following: When a physical phenomenon is described in reference system $x,y,z,t$ by certain equations, then a phenomenon which can be described in another reference system $x',y',z',t'$ by the same equations, will exist as well. There, both reference systems are connected by relations in which the speed of light occurs, and which express that a system is moving with uniform velocity relative to the other one.

If observer $A$ is in the first, $B$ in the second reference system, and if any of them is equipped with measuring rods and clocks at rest in his respective system, then $A$ will measure the values of $x,y,z,t$, while $B$ measure the values of $x',y',z',t'$, where it is to be remarked, that $A$ and $B$ can use the same measuring rods and the same clocks as well. We have to assume, that when measuring rods and clocks are somehow transfered from the first observer to the second one, then they take over the correct length and the correct rate by themselves, so that $B$ obtains the values of $x',y',z',t'$ form his measurements. Both will now obtain the same value for the speed of light, and generally can make the same observations.

Provided that there is an aether, then under all systems $x,y,z,t$, one is preferred by the fact, that the coordinate axes as well as the clocks are resting in the aether. If one connects with this the idea (which I would abandon only reluctantly) that space and time are completely different things, and that there is a "true time" (simultaneity thus would be independent of the location, in agreement with the circumstance that we can have the idea of infinitely great velocities), then it can be easily seen that this true time should be indicated by clocks at rest in the aether. However, if the relativity principle had general validity in nature, one wouldn't be in the position to determine, whether the reference system just used is the preferred one. Then one comes to the same results, as if one (following Einstein and Minkowski) deny the existence of the aether and of true time, and to see all reference systems as equally valid. Which of these two ways of thinking one is following, can surely be left to the individual.

In order to discuss the physical side of the question, we have to state the transformation formulas first, where we confine ourselves to the special form in which they were already used in the year 1887 by Voigt at investigations concerning Doppler's principle, namely:

$x'=x,\ y'=y,\ z'=az-bct,\ t'=at-{\frac {b}{c}}z;$

there, the constants $a>0,\ b$ satisfy the relation

$a^{2}-b^{2}=1{,}$

which cause the identity

$x'^{2}+y'^{2}+z'^{2}-c^{2}t'^{2}=x^{2}+y^{2}+z^{2}-c^{2}t^{2}\,$

The origin of system $x',y',z'$ moves towards system $x,y,z$ in the $z$-direction with velocity ${\tfrac {b}{a}}c$, which is always smaller than $c$. Generally, any velocity has to be assumed as being smaller than $c$.

All state variables of any phenomenon, measured in one or the other system, are connected by certain transformation formulas. They read, e.g. for the velocity of a point:

${\mathfrak {v}}'_{x}={\frac {{\mathfrak {v}}_{x}}{\omega }},\ {\mathfrak {v}}'_{y}={\frac {{\mathfrak {v}}_{y}}{\omega }},\ {\mathfrak {v}}'_{z}={\frac {a{\mathfrak {v}}_{z}-bc}{\omega }}{,}$

where

$\omega =a-{\frac {b{\mathfrak {v}}_{z}}{c}}$

We furthermore consider a system of pints, whose velocity is a steady function of the coordinates. Let $dS$ be a space element surrounding point $P(x,y,z)$ at time $t$; to this value $t$ and the coordinates of $P$, a moment $t'$ is corresponding in another reference systems according to the transformation equations, and every point lying in $dS$ at time $t'$, has certain $x',y',z'$ for this definite value of $t'$. Points $x',y',z'$ satisfy a space element $dS'$, which is connected with $ds$ as follows:

$dS'={\frac {dS}{\omega }}.$

If we imagine an agent (matter, electricity etc.) as connected with these points, and if we assume that observer $B$ has reason to connect the same amount of that agent with every point as observer $A$, then the space density must be inversely proportional to the volume elements, i.e.,

$\varrho '-\omega \varrho .$

All of these relations are reciprocal, i.e., one can permute the primed and unprimed letters, when one simultaneously replaces $b$ by $-b$.

The fundamental equations of the electromagnetic field retain their form at the transformation, when one introduces the following magnitudes^[1]:

${\begin{array}{ccc}{\mathfrak {d}}'_{x}=a{\mathfrak {d}}_{x}-b{\mathfrak {h}}_{y},&{\mathfrak {d}}'_{y}=a{\mathfrak {d}}_{y}+b{\mathfrak {h}}_{x},&{\mathfrak {d}}'_{z}={\mathfrak {d}}_{z},\\{\mathfrak {h}}'_{x}=a{\mathfrak {h}}_{x}+b{\mathfrak {d}}_{y},&{\mathfrak {h}}'_{y}=a{\mathfrak {h}}_{y}-b{\mathfrak {d}}_{x},&{\mathfrak {h}}'_{z}={\mathfrak {h}}_{z};\end{array}}$

between these ones, and the transformed space density $\varrho '$, and the transformed velocity ${\mathfrak {v}}'$, the following equations hold in system $x',y',z',t'$:

${\begin{array}{l}\operatorname {div} \ {\mathfrak {d}}'=\varrho ',\\\operatorname {div} \ {\mathfrak {h}}'=0,\\\operatorname {rot} \ {\mathfrak {h}}'={\frac {1}{c}}\left({\mathfrak {\dot {d}}}'+\varrho '{\mathfrak {v}}'\right),\\\operatorname {rot} \ {\mathfrak {d}}'=-{\frac {1}{c}}{\mathfrak {\dot {h}}}'.\end{array}}$

In so far, the field equations of the theory of electrons satisfy the relativity principle; though we have to bring the equations of the electron themselves into accordance with this principle.

We will (somewhat more general) consider the motion of an arbitrary material point. At this occasion, the introduction of the concept of "proper time" (a nice invention of Minkowski) is useful. According to this, every point is so to speak connected with its own time which is independent of the reference system chosen; its differential is defined by the equation:

$d\tau ={\sqrt {1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}}}dt.$

The expressions formed by the aid of proper time $\tau$

${\frac {d}{d\tau }}{\frac {dx}{d\tau }},\ {\frac {d}{d\tau }}{\frac {dy}{d\tau }},\ {\frac {d}{d\tau }}{\frac {dz}{d\tau }}{,}$

which are linear homogeneous functions of the ordinary acceleration components, are denoted by us as components of the "Minkowski acceleration". We describe the motion of a point by the equations

$m{\frac {d}{d\tau }}{\frac {dx}{d\tau }}={\mathfrak {K}}_{x}{\text{, etc.,}}$

where $m$ is a constant which we call the "Minkowski mass". Vector ${\mathfrak {K}}$ is denoted by us as "Minkowski force".

The transformation formulas for this acceleration and force can be easily derived; $m$ is left unchanged by us. Then one has

${\mathfrak {K}}'_{x}={\mathfrak {K}}_{x},\ {\mathfrak {K}}'_{y}={\mathfrak {K}}_{y},\ {\mathfrak {K}}'_{z}=a{\mathfrak {K}}_{z}-{\frac {b}{c}}({\mathfrak {v}}\cdot {\mathfrak {K}}).$

The essential thing is now as follows: The relativity principle requires, that (at an actual phenomenon) the Minkowski forces are in a certain way depending on the coordinates, velocities, etc. in one reference system, and the transformed Minkowski forces in the other reference system are depending in the same way on the transformed coordinates, velocities etc. That is a special property, which all forces of nature must have, when the relativity principle shall hold. If we presuppose this, then one can calculate the forces acting on moving bodies, when one know them for the case of rest. If e.g. an electron of charge $e$ is moving, then we imagine a reference system in which it is momentarily at rest. The Minkowski force is acting upon the electron in this system:

${\mathfrak {K}}=e{\mathfrak {d}};$

from that it follows by application of the transformation equations for ${\mathfrak {K}}$ and ${\mathfrak {d}}$, that the Minkowski force acting upon an electron moving with velocity ${\mathfrak {v}}$ in an arbitrary reference system, amounts to

${\mathfrak {K}}=e{\frac {{\mathfrak {d}}+{\frac {1}{c}}[{\mathfrak {v}}\cdot {\mathfrak {h}}]}{\sqrt {1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}}}}$

This formula is not in agreement with the ordinary Ansatz of electron theory, due to the presence of the denominator. The difference stems from the fact, that one usually doesn't operate with our Minkowski force, but with the "Newtonian force" ${\mathfrak {F}}$, and we see, that (for an electron) these two forces are connected as follows:

${\mathfrak {F}}={\mathfrak {K}}{\sqrt {1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}}}.$

One will assume that this relation holds for arbitrary material points.

Thus one can treat the phenomena of motion in two different ways, either with the Minkowski- or with the Newtonian force. In the latter case, the equations of motions read:

${\mathfrak {F}}=m_{1}{\mathfrak {j}}_{1}+m_{2}{\mathfrak {j}}_{2}{,}$

here, ${\mathfrak {j}}_{1}$ means the ordinary acceleration into the direction of motion, and ${\mathfrak {j}}_{2}$ the ordinary normal acceleration, and the factors

${\begin{array}{rl}m_{1}&={\frac {m}{\sqrt {\left(1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}\right)^{3}}}},\\&\\m_{2}&={\frac {m}{\sqrt {1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}}}}\end{array}}$

are called the "longitudinal" and "transverse mass".

In the same way as the Minkowski force, also the Newtonian forces occurring in nature must satisfy certain conditions, when the relativity principle should be satisfied. This is e.g. the case, when (independently from motion) a normal pressure of constant magnitude $p$ is acting per unit area; in the transformed system, a normal pressure of the same magnitude is acting upon the corresponding surface element in motion.

Since we already recognized the invariance of the field equations, then the question, as to whether the motions in a system of electrons are in agreement with the relativity principle, only tantamounts to the experimental test of the formulas for longitudinal and transverse mass $m_{1},\ m_{2}$; although the experiments of Bucherer and Hupka seem to confirm these formulas, we have not arrived at a definite decision.

Regarding the mass of the electron, it is to be considered that they are of electromagnetic nature; thus it will depend on the distribution of the charges in the interior of the electron. Therefore, the formulas for the mass can only then be correct, when the charge distribution and thus also the shape of the electron are variable with velocity in a certain way. One has to assume, that in consequence of the translation of the electron, which is a sphere when at rest, the electron becomes an oblate ellipsoid in the direction of motion; the amount of oblateness is

${\sqrt {1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}}}.$

If we assume, that shape and magnitude of the electron are regulated by inner forces, then they must (to be compatible with the relativity principle) have such properties by which this oblateness is arising by itself. In relation to this, Poincaré has made the following hypothesis. The electron is a charged and expandable shell, and an inner normal stress of invariable magnitude is resisting to the electric repulsions of the individual points. According to the above, such normal stresses indeed satisfy the relativity principle.

In the same way, all molecular forces acting in the interior of ponderable matter, as well as the quasi-elastic forces and resisting forces acting upon the electron, must satisfy certain conditions in order to be in agreement with the relativity principle. Then, every moving body will be invariable for a co-moving observer, yet it will experience a change of dimensions for a stationary observer, which is just a consequence of the change of molecular forces required by those conditions. From that, the contraction of the body – which was already imagined before to explain the negative outcome of Michelson's interference experiment – follows by itself, and also the negative outcome of all similar experiments which should demonstrate an influence of Earth's motion upon optical phenomena.

Regarding the rigid body (with which Born, Herglotz, F. Noether, Levi-Cività were dealing): the difficulties emerging during the consideration of rotations, will probably be solved by ascribing rigidity to the effectiveness of particularly intensive molecular forces.

Eventually we want to turn our attention to gravitation. The relativity principle requires a modification of Newton's laws, above all it requires the propagation of this effect with the speed of light. The possibility of a finite propagation velocity of gravitation was already discussed by Laplace, who imagined a fluid streaming against the sun as the cause of gravity, which pushes the planets towards the sun. He found, that the speed $c$ of this fluid must be assumed to be at least 100 million times greater than the speed of light, so that the calculation remains in agreement with the astronomical observations. The necessity of such a great value of $c$ stems from the fact, that the magnitude ${\tfrac {v}{c}}$ arises in its end formulas in the first power, where $v$ is the planetary velocity. However, if the propagation speed $c$ of gravitation shall have the speed of light, as required by the relativity principle, then a contradiction with observations can only then be avoided, when only magnitudes of second (or higher) order in ${\tfrac {v}{c}}$ arise in the expression for the modified law of gravitation.

If one confines oneself to magnitudes of second order, then a condition can easily be given on the basis of an obvious electron-theoretical analogy, which defines the modified law in a definite manner.

If one namely considers the force acting upon an electron moving with velocity ${\mathfrak {v}}$:

$e\left({\mathfrak {d}}+{\frac {1}{c}}[{\mathfrak {v}}\cdot {\mathfrak {h}}]\right){,}$

then the vectors ${\mathfrak {d}}$ and ${\mathfrak {h}}$ also depend on velocities ${\mathfrak {v}}'$ of the electrons generating the field; therefore, in the vector product $[{\mathfrak {v}}\cdot {\mathfrak {h}}]$ there indeed arise products of the form ${\mathfrak {vv'}}$, yet not the square ${\mathfrak {v}}^{2}$ of the speed of the considered electron. If we accordingly assume, that the square of the velocity ${\mathfrak {v}}_{1}^{2}$ of point 1 doesn't occur in the expression of the attraction acting upon point 1 and exerted by point 2, then this velocity must entirely drop in a reference system in which point 2 is at rest $({\mathfrak {v}}_{2}=0)$; the law in this system thus will be reduced to the ordinary Newtonian one. If one now passes to an arbitrary coordinate system by transformation, then one finds that the force acting upon point 1 is composed of two parts, first an attraction in the direction of the connecting line of amount

$R+{\frac {1}{c^{2}}}\left\{{\frac {1}{2}}{\mathfrak {v}}_{2}^{2}R+{\frac {1}{2}}{\mathfrak {v}}_{2r}^{2}\left(r{\frac {dR}{dr}}-R\right)-\left({\mathfrak {v}}_{1}\cdot {\mathfrak {v}}_{2}\right)R\right\}{,}$

second a force in the direction ${\mathfrak {v}}_{2}$ of amount

${\frac {1}{c^{2}}}{\mathfrak {v}}_{1r}R{\mathfrak {v}}_{2};$

here, $r$ means the distance between two simultaneous points, ${\mathfrak {v}}_{r}$ the component of ${\mathfrak {v}}$ towards the connection line drawn from 1 to 2, and $R$ that function of $r$ which represents the attraction law in the case of rest ($R={\tfrac {k}{r^{2}}}$ at Newtonian attraction, $R=kr$ at quasi-elastic forces). It is to be noticed, that under force we always have to understand the "Newtonian force", not the Minkowski force. Incidentally, Minkowski has given a somewhat different expression than this one. In Poincaré's paper, both this one and also the one written above can be found.

It is to be noticed, that the principle of equality of action and reaction is not satisfied in these laws of gravitation.

Now, the disturbances shall be discussed, which can arise by those supplementary terms of second order. Besides many short-periodic disturbances of no importance, there is a secular motion of the perihelion of the planets. De Sitter calculates, that it amounts to 6,69" per century.^[2] Now, the perihelion anomaly of mercury of amount 44" per century is known since Laplace; although it has the correct sign, it is much too great to be explained by those supplementary terms. It is rather explained by Seeliger as due to a disturbance of the carrier of the zodiacal light, whose mass one can conveniently determine in a plausible way. From that, no decision can be gained as long as the precision of astronomical measurements is not essentially increased. At absolute precision, also the difference between the "proper time" of earth and the time of the solar system has to be considered.

Another method to test the correctness of the modified law of gravitation, can be based upon a procedure proposed by Maxwell for the decision, as to whether the solar system is moving through the aether. If this is the case, then the eclipses of the satellites of Jupiter, depending on the location of these planets with respect to Earth, must suffer earlinesses or delays.

If the distance Jupiter-Earth is $a$ and the velocity component of the solar system in the aether in the direction of the connecting line Jupiter-Earth is $v$, then the time ${\frac {a}{c}}$ which would be required by light (in the case of rest) to traverse distance $a$, is transformed into ${\frac {a}{c\pm v}}$; thus an earliness or a delay occurs due to motion, which amounts to ${\frac {av}{c^{2}}}$ up to terms of second order, and which attains different values, depending on the value of velocity component $v$, which indeed depends on the location of both planets. Now it is clear, that such a dependency of the phenomena from the motion through the aether contradicts the relativity principle.

Fig. 5

To solve this contradiction, we want to simplify the state of facts in a schematic way. We imagine, that sun $S$ shall have a mass which is infinitely great in relation to that of the planets. Let the velocity of the solar system coincide with the $z$-axis, which we let go through the sun. The intersections of the orbit of the planets with the $z$-axis, is denoted by us as the upper and lower transit $A$ and $B$.

We place the observer upon the sun. At every transit of the planet through the $z$-axis, a light signal shall be traveling to the sun. Let $T$ be the orbital period. When the sun is at rest, the time between the upper and lower transit will amount to ${\frac {1}{2}}T$ (at a motion being presupposed as circular); the same is true for the time between the arrival of both light signals. However, if the sun is moving in the $z$-direction, then the light signal of the upper transit must suffer an earliness of $\textstyle {\frac {av}{c^{2}}}$, and the one of the lower transit a delay of the same amount; in case the uniform orbital motion (as presupposed by Maxwell as self-evident) remains conserved in an undisturbed way, then the time interval between the arrival of the light signal of two successive transits would be alternately appear to be increased and diminished by ${\tfrac {2av}{c^{2}}}$. The conservation of uniform circular motion at a translation in the aether which is presupposed here, however, is impossible according to the relativity principle. If we namely describe the process in a coordinate system that doesn't share the motion, then the modified law of gravitation is to be applied, and this gives a non-uniformity of planetary motion, due to which the difference of the time intervals between the arrival of the light signals is exactly canceled.

The demonstration as to whether an earliness or delay of the eclipses actually occurs, can therefore be used for the decision in favor or against the relativity principle. However, the numerical relations are quite unfavorable again. For example, Burton (who has 330 photometric observations at his disposal, which were undertaken at the Harvard-Observatory concerning the eclipses of the 1st Jupiter-satellite) estimates the probable error of the final result for $v$ as being 50 km/s; on the other hand, one has observed star velocities of 70 km/s, and the velocity of the solar system with respect to the fixed stars is estimated to be 20 km/s. The relativity principle is thus hardly supported by Burton's calculations, it can at most be disproved, for example, when eventually a value would be given which exceeds 100 km/s.

Let us leave it undecided, whether or not the new mechanics will experience a confirmation by astronomical observations. Though we won't omit, to learn about its fundamental formulas.

If one defines work as the scalar product of "Newtonian force" and displacement, then the equations of motion give the energy principle in the ordinary form, i.e., that the work performed per unit time is equal to the increase of energy $\varepsilon$:

${\mathfrak {F}}_{x}{\frac {dx}{dt}}+{\mathfrak {F}}_{y}{\frac {dy}{dt}}+{\mathfrak {F}}_{z}{\frac {dz}{dt}}={\frac {d\varepsilon }{dt}}.$

There, the energy has the form:

$\varepsilon =mc^{2}\left({\frac {1}{\sqrt {1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}}}}-1\right)$;

which agrees for small velocities with the value of kinetic energy of ordinary mechanics

$\varepsilon ={\frac {1}{2}}m{\mathfrak {v}}^{2}$

Furthermore, one can derive the Hamiltonian principle

$\int _{t_{1}}^{t_{1}}(\delta L+\delta A)dt=0$

from the equations of motion; here, $\delta A$ is the work of "Newtonian force" at virtual displacement, and $L$ the Lagrangian function which reads as follows:

$L=-mc^{2}\left({\sqrt {1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}}}-1\right)$.

From Hamilton's principle, one conversely can derive the equations of motion again. The quantities

${\frac {\partial L}{\partial {\dot {x}}}},\ {\frac {\partial L}{\partial {\dot {y}}}},\ {\frac {\partial L}{\partial {\dot {z}}}}$

are to be denoted as components of momentum.

All of these formulas can be verified at the electromagnetic laws of motion of an electron; then one has to set the value for the "Minkowski mass" $m$

$m={\frac {e^{2}}{6\pi Rc^{2}}}$,

and to add the energy of those inner stresses to the electric and magnetic energy, which (as we saw) determine the shape of the electron. By specializing on an electron, one can derive them from the general principle of least action for arbitrary electromagnetic systems, which was discussed in the first lecture^[3], though the work of the inner stresses must be considered again.

Now we consider the equations of the electromagnetic field for ponderable bodies. Those are stated by Minkowski in a pure phenomenological way, and then it was shown by M. Born and Ph. Frank that they can be derived from the concepts of the theory of electron; I also have by myself obtained in the latter way the equations, whose shape is formally somewhat different.

In order to obtain the relation between observable magnitudes, one has to blur the details of the phenomena stemming from the electrons, by formation of averages over great quantities of electrons. In this way, one is led to the following equations (which are in agreement with those of ordinary Maxwellian theory):

${\begin{array}{l}\operatorname {div} \ {\mathfrak {D}}=\varrho _{l}{,}\\\operatorname {div} \ {\mathfrak {B}}=0{,}\\\operatorname {rot} \ {\mathfrak {H}}={\frac {1}{c}}({\mathfrak {C}}+{\mathfrak {\dot {D}}}){,}\\\operatorname {rot} \ {\mathfrak {E}}=-{\frac {1}{c}}{\mathfrak {\dot {B}}}.\end{array}}$

Herein, ${\mathfrak {D}}$ is the dielectric displacement, ${\mathfrak {B}}$ the magnetic induction, ${\mathfrak {H}}$ the magnetic force, ${\mathfrak {E}}$ the electric force, ${\mathfrak {C}}$ the electric current, $\varrho _{l}$ the density of the observable electric charges. If one indicates the average formation by overlines, then it is e.g.

${\mathfrak {E}}={\mathfrak {\bar {d}}},{\mathfrak {B}}={\mathfrak {\bar {h}}}{,}$

where ${\mathfrak {d}}$, ${\mathfrak {h}}$ have the earlier meaning; furthermore it is

${\begin{array}{l}{\mathfrak {D}}={\mathfrak {E}}+{\mathfrak {P}}{,}\\{\mathfrak {H}}={\mathfrak {B}}-{\mathfrak {M}}-{\frac {1}{c}}[{\mathfrak {P}}\cdot {\mathfrak {w}}]{,}\end{array}}$

where ${\mathfrak {P}}$ is the electric moment, ${\mathfrak {M}}$ the magnetization per unit volume, and ${\mathfrak {w}}$ the velocity of matter. In the derivation of these formulas, one separates the electrons into three kinds. The first kind, the polarization electrons, produce the electric moment ${\mathfrak {P}}$ by their displacement; the second kind, the magnetization electrons, produce the magnetic moment ${\mathfrak {M}}$ by their orbits; the third kind, the conduction electrons, are freely moving in matter and produce the observable charge density $\varrho _{l}$ and the current ${\mathfrak {C}}$. The latter is still to be separated into two parts; if ${\mathfrak {u}}$ is the relative velocity of the electrons towards matter, then the total velocity of the electrons is ${\mathfrak {v}}={\mathfrak {w}}+{\mathfrak {u}}$, thus the current transported by them

${\mathfrak {C}}={\overline {\varrho {\mathfrak {v}}}}=\varrho {\mathfrak {w}}+{\overline {\varrho {\mathfrak {u}}}}$;

${\bar {\varrho }}$ is the observable charge $\varrho _{l}$, ${\bar {\varrho }}{\mathfrak {w}}$ the convection current, ${\overline {\varrho {\mathfrak {u}}}}$ the actual conduction current ${\mathfrak {C}}_{l}$.

Transformation formulas exist for all these magnitudes, of which some may be given:

${\mathfrak {C}}'_{x}={\mathfrak {C}}_{x},\ {\mathfrak {C}}'_{y}={\mathfrak {C}}_{y},\ {\mathfrak {C}}'_{z}=a{\mathfrak {C}}_{z}-bc\varrho _{l},\ \varrho '_{l}=a\varrho _{l}-{\frac {b}{c}}{\mathfrak {C}}_{z}{,}$ ${\begin{array}{l}{\mathfrak {P}}'_{x}=a{\mathfrak {P}}_{x}-{\frac {b}{c}}({\mathfrak {w}}_{z}{\mathfrak {P}}_{x}-{\mathfrak {w}}_{x}{\mathfrak {P}}_{z})+b{\mathfrak {M}}_{y}{,}\\{\mathfrak {P}}'_{y}=a{\mathfrak {P}}_{y}-{\frac {b}{c}}({\mathfrak {w}}_{z}{\mathfrak {P}}_{y}-{\mathfrak {w}}_{y}{\mathfrak {P}}_{z})-b{\mathfrak {M}}_{x}{,}\\{\mathfrak {P}}'_{z}={\mathfrak {P}}_{z}.\end{array}}$

Furthermore, the following auxiliary vectors are useful:

${\begin{array}{cc}{\mathfrak {H}}_{1}={\mathfrak {H}}-{\frac {1}{c}}[{\mathfrak {w}}\cdot {\mathfrak {D}}],&{\mathfrak {B}}_{1}={\mathfrak {B}}-{\frac {1}{c}}[{\mathfrak {w}}\cdot {\mathfrak {E}}]{,}\\{\mathfrak {E}}_{1}={\mathfrak {E}}+{\frac {1}{c}}[{\mathfrak {w}}\cdot {\mathfrak {B}}],&{\mathfrak {D}}_{1}={\mathfrak {D}}+{\frac {1}{c}}[{\mathfrak {w}}\cdot {\mathfrak {H}}].\end{array}}$

The given field equations now are still to be supplemented by stating the relations, which exist between the vectors ${\mathfrak {E}},{\mathfrak {H}}$ and ${\mathfrak {D}},{\mathfrak {B}}$. One can derive these relations in two ways.

The first phenomenological method follows that procedure: One considers an arbitrarily moving point of matter, and introduces a reference system in which it is at rest; then, in case the volume element surrounding the point is isotropic in the rest system, e.g. the equations valid for resting systems hold between ${\mathfrak {E}}$ and ${\mathfrak {D}}$:

${\mathfrak {D}}=\varepsilon {\mathfrak {E}}{,}$

or also

${\mathfrak {D}}_{1}=\varepsilon {\mathfrak {E}}_{1}{,}$

because the auxiliary vectors ${\mathfrak {D}}_{1}$, ${\mathfrak {E}}_{1}$ are identical with ${\mathfrak {D}}$, ${\mathfrak {E}}$ for ${\mathfrak {w}}=0$. However, ${\mathfrak {D}}_{1}$ and ${\mathfrak {E}}_{1}$ are transformed in the same way, and from that it follows, that the equation

${\mathfrak {D}}_{1}=\varepsilon {\mathfrak {E}}_{1}$

remains valid in the initial reference system as well. Accordingly it is

${\mathfrak {B}}_{1}=\mu {\mathfrak {H}}_{1}.$

As regards the conduction current, we only remark that it depends on ${\mathfrak {E}}_{1}$.

The second method is based upon the mechanics of the electrons. In the same way, as (for resting bodies) equation ${\mathfrak {D}}=\varepsilon {\mathfrak {E}}$ proves to be the consequence of the assumption of quasi-elastic forces, which draw back the electron in their rest states, one will obtain the equation ${\mathfrak {D}}_{1}=\epsilon {\mathfrak {E}}_{1}$ at moving bodies, when one ascribes to those quasi-elastic forces those properties, which are required by the relativity principle. The latter will be satisfied, when one uses the expression of the generalized attraction law for these forces, where $R$ must be taken proportional to $r$.

The similar is valid for the explanation of the conduction resistance. A satisfying electron-theoretical explanation of the magnetic properties of the bodies is not present for the time being.

At last, the importance of the previous equations shall be shown at three remarkable cases.

The first remark is based on the equation

$\varrho '_{l}=a\varrho _{l}-{\frac {b}{c}}{\mathfrak {C}}_{z}$

According to it, $\varrho '_{l}$ can vanish without the need of $\varrho _{l}=0$, as long as only current ${\mathfrak {C}}$ exists; i.e. an observer $A$ will declare the body as charged, while it is uncharged for an observer $B$ moving relative to him. This can be understood when it is considered, that positive and negative electrons of same amount are present in every body, which compensate themselves at uncharged bodies. If the body is moving with velocity ${\mathfrak {w}}$, then (when a conduction current is present) both kinds of electrons will obtain different total velocities, thus also the quantity $\omega =a-b{\tfrac {{\mathfrak {v}}_{z}}{c}}$ will have different values for both kinds. Now, if an observer $B$ which is moving with the body, is calculating the average of charge density ${\overline {\varrho '}}={\overline {\omega \varrho }}$ for both kinds of electrons, then he can obtain the sum zero, even when for an observer $A$ (in whose reference system the body is moving) the averages ${\bar {\varrho }}$ of the positive and negative electrons are not compensating themselves.

This circumstance causes a reminiscence of an old question. Around the year 1880, there was a great discussion among physicists concerning Clausius' fundamental law of electrodynamics. At that time, it was tried to derive a contradiction between this law and the observations by concluding that according to this law, a current-carrying conductor on earth shall exert an action upon a co-moving charge $e$ due to Earth's motion, which could possibly be detected. That the law actually doesn't require this action, was noticed by Budde; it stems from the fact, that the current is acting upon itself due to Earth's motion, and is causing a "compensation charge" upon the traversed conductor, which exactly compensates the first action. The theory of electron leads to similar conclusions, and I find for the density of the compensation charge, when the velocity has the direction of the $z$-axis,

${\frac {1}{c^{2}}}{\mathfrak {w}}_{z}{\mathfrak {C}}_{z}$;

this must be assumed as existent by an observer $A$ who doesn't share the motion with Earth, while it doesn't exist for a co-moving observer $B$. The given value exactly agrees with the formula derived from the relativity principle; if $\varrho '_{l}=0$, then one finds from this formula

$\varrho _{l}={\frac {b}{ac}}{\mathfrak {C}}_{z}{,}$

and since ${\mathfrak {w}}_{z}={\frac {bc}{a}}$ is the mutual velocity of both reference systems according to the things previously said (p. 75), then one indeed finds

$\varrho _{l}={\frac {1}{c^{2}}}{\mathfrak {w}}_{z}{\mathfrak {C}}_{z}.$

Fig. 6.

The second remark is based on the transformation equations for the electric moment ${\mathfrak {P}}$ (p. 84), which shows the impossibility (because the magnetization ${\mathfrak {M}}$ occurs in them) to clearly distinguish between polarization and magnetization electrons. In a magnetized body $({\mathfrak {M}}\neq 0)$, as seen from a reference system, it can rather be ${\mathfrak {P}}=0$, while ${\mathfrak {P}}'$ is different from zero in another reference system. This shall now be applied to a special case, where we confine ourselves to magnitudes of first order. The considered body (e.g. a steel magnet) shall contain only conduction electrons and such ones (when the body is at rest) which produce ${\mathfrak {M}}$, yet not ${\mathfrak {P}}$; it shall have the shape of an infinitely extended even plate, bounded by two planes $a,b$: the middle plane is made by us to the $yz$-plane (Fig. 6). When it is at rest, a constant magnetization ${\mathfrak {M}}_{y}$ may exist in the $y$-direction, while ${\mathfrak {P}}=0$. If the body acquires the velocity $v$ in the $z$-direction, then an observer not participating at the motion, is observing the electric polarization

${\mathfrak {P}}_{x}=-{\frac {v}{c}}{\mathfrak {M}}_{y}$

Now we imagine two conductors $c,d$ at both sides of the body, which together with it are forming two equal condensers, and they shall be short circuited by a wire (from $c$ to $d$). When in motion, $d$ charges will arise upon $c$ now, which can be calculated as follows. Since it is evidently impossible that a current exists in the $x$-direction, it is ${\mathfrak {E}}_{1x}=0$ or ${\mathfrak {E}}_{x}={\frac {v}{c}}{\mathfrak {B}}_{y}$. Since the process is stationary, it becomes ${\mathfrak {\dot {B}}}=0$; then the existence of a potential $\varphi$ follows from $\operatorname {rot} \ {\mathfrak {E}}=0$. If $\Delta$ is the thickness of the plate, then one has

$\varphi _{a}-\varphi _{b}={\frac {v}{c}}\Delta {\mathfrak {B}}_{y}.$

From the symmetry of the arrangement if evidently follows

$\varphi _{d}-\varphi _{a}=\varphi _{b}-\varphi _{c}{,}$

and because the plates $c,d$ are short circuited, it must be

$\varphi _{d}=\varphi _{c}$;

from that if follows

$\varphi _{d}-\varphi _{a}=-{\frac {v}{2c}}\Delta {\mathfrak {B}}_{y}.$

If $\gamma$ is the capacity of one of the two condensers, then the charge of the plate $d$ becomes equal to

$-{\frac {v}{2c}}\gamma \Delta {\mathfrak {B}}_{y}{,}$

and $c$ obtains the oppositely equal amount.

Now we compare this procedure with the inverse case, that the magnet $ab$ is at rests and plates $c,d$ are moving with opposite velocity. Then according to the relativity principle, everything must be quite the same as in the first case. Indeed, one immediately finds from the ordinary law of induction, exactly the mount of charge upon plate $d$ previously given. But now this charge upon $d$ must produce the opposite equal charge upon plane $a$ of the resting magnet by electrostatic induction, and the corresponding must hold for $b$ and $c$. Since no current can flow $({\mathfrak {C}}=0)$, then in both cases (whether the magnet is moving and the plate are at rest, or vice versa) the same charges must be present upon the magnet. Thus we have to consider, as to how it comes that in the first treated case, the opposite charge arises upon plane $a$ of the moving magnet, than upon plate $d$; this is only possible by the polarization $\textstyle {{\mathfrak {P}}_{x}=-{\frac {v}{c}}{\mathfrak {M}}_{y}}$ emerging during the motion. Because one has

${\mathfrak {D}}_{x}={\mathfrak {E}}_{x}+{\mathfrak {P}}_{x}={\frac {v}{c}}{\mathfrak {B}}_{y}-{\frac {v}{c}}{\mathfrak {M}}_{y};$

since ${\mathfrak {P}}$ (thus the term $[{\mathfrak {P}}\cdot {\mathfrak {w}}]$) is to be neglected in the velocity of first order, it becomes

${\mathfrak {B}}-{\mathfrak {M}}={\mathfrak {H}}{,}$

though ${\mathfrak {H}}$ is zero, because the plate is assumed to be infinitely extended. From that if follows

${\mathfrak {D}}_{x}=0{,}$

no dielectric displacement arises in the moving plate, thus the charge upon $a$ agrees with that upon $d$, as required by the relativity principle.

The last remark concerns the circumstance, that the motion of Earth cannot have an influence upon electromagnetic processes according to the relativity principle. However, Liénard alluded to a phenomenon, where such an influence (namely amounting to first order) shall be expected; also Poincaré has discussed this case in his book Electricité et Optique. It is about the ponderomotive force upon an conductor. In order to determine it, one will make the obvious assumption for the force acting upon the conduction electrons per unit force:

${\mathfrak {E}}_{1}={\mathfrak {E}}+{\frac {1}{c}}[{\mathfrak {v}}\cdot {\mathfrak {B}}];$

then the force caused by Earth's motion upon the conductor in the direction of motion, amounts to

${\frac {1}{c^{2}}}({\mathfrak {C}}_{l}\cdot {\mathfrak {E}}){\mathfrak {w}}_{z};$

since $({\mathfrak {C}}_{l}\cdot {\mathfrak {E}})$ is the heat produced by the conduction current ${\mathfrak {C}}_{l}$, this expression is easily to be calculated numerically (however, an unobservable value will be obtained).

If one now asks oneself, as to how this result (which contradicts the relativity principle) can arise, then one sees that one hasn't actually calculated the force acting upon the matter of the conductor, but the force which acts upon the electrons moving in the interior of the conductor. The latter forces are still to be transfered to matter by individually unknown forces, and this only happens (without change of the magnitude) when equality between action and reaction exists for the forces between matter and electrons. However, for moving bodies, the action is not equal to the reaction according to the relativity principle, and this circumstance exactly cancels the force of Liénard.

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In summary one can say, that there is little prospect of testing the relativity principle by experiment; except some astronomic observations, only the measurement of the mass of electrons comes into account. Though one shall not forget, that the negative outcome of different experiments such as Michelson's interference experiment and the experiments to demonstrate double refraction due to Earth's motion, could only be explained by the relativity principle.

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1. Regarding the notations, see Mathematische Encyklopädie Vol. 14.
2. This was a first approximation. By a new calculation, de Sitter found the value 7,15". (Monthly Notices of R. A. Sc. 71 (1911), p. 405).
3. Phys. Zeitschr. 11 (1910) p. 1235.