Translation:The Theory of the Rigid Electron in the Kinematics of the Principle of Relativity

by Max Born

The Theory of the Rigid Electron in the Kinematics of the Principle of Relativity.

by Max Born.

Dedicated to the memory of Hermann Minkowski.

Introduction.

The great importance of the concepts of rigid body and rigid connection in Newtonian mechanics, is to the closest related with the fundamental views concerning space and time. Because the requirement that lengths shall be mutually comparable at different times, directly leads to the formation of the concept of measuring rods whose length is independent of time and motion, i.e., which are rigid. Later, this concept of the rigid body proves to be fruitful for the development of dynamics itself; because the rigid body as a continuous mass system of only six degrees of freedom, is not only kinematically of the highest simplicity, but also dynamically by allowing the composition of the forces – which are acting at its points – to "resulting" forces and moments of the same magnitude, whose knowledge is sufficient for the description of motion. All these possibilities are principally based on the Galilei-Newtonian connection of space and time into a four-dimensional manifold^{[WS 1]} (which I will call "world" following Minkowski^[1]); a connection essentially contained in the theorem, that the natural laws not only shall be independent from the choice of the origin and the unit of time, as well as from the location of the spatial reference system and the unit of length, but also from a uniform translation given to the reference system under maintenance of the measure of time.

Exactly these foundations of kinematics are the ones to be abandoned, when the electrodynamic relativity principle – as stated by Lorentz, Einstein, Minkowski and others – comes into play. Because here, the connection of space and time into the "world" is different: the independence of the natural laws from the uniform translation of the spatial reference system only then takes place, when also the time parameter experiences a change, which not only tantamounts to a displacement of the origin and the choice of another unit. It is most closely connected to this, that measuring rods that maintain their length at uniform translation in the co-moving coordinate system, suffer a contraction in the direction of their velocity when viewed from a stationary system. By that, the concept of the rigid body fails, at least in its form adapted to Newtonian kinematics.

However, a corresponding concept is by no means to be dispensed with in the new kinematics as well, since otherwise the comparison of lengths of moving bodies at different times becomes illusory. No difficulty arises at the formation of this concept for systems moving relative to each other, and the authors (mentioned above) of the foundational works on this theory, are using this circumstance without giving a particular definition of rigidity.

The difficulty only then arises, when accelerations are present. Only one attempt exists – made by Einstein^[2] –, without completely clarifying the subject. Therefore I have undertaken the elaboration of the kinematics of the rigid body on the basis of the relativity postulate. Its possibility is probable from the outset, because the Newtonian kinematics represents in every relation a limiting case of the new kinematics, namely that one in which the speed of light $c$ is seen as infinitely great. The method used by me, consists in defining rigidity by a differential law instead of an integral law.

Indeed, one arrives in this way at the general rigidity conditions in differential form, which are very analogous to the corresponding conditions of the old kinematics and also pass into them for $c=\infty$. The integration of these conditions, which is very easily executable in the old kinematics in general, and which leads to the constancy of the distance of rigidly connected points, was only executed by me for the case of uniformly accelerated translation; the result is hardly inferior to the old kinematics in terms of simplicity and illustratability, and makes the assumption near at hand, as to what may be the result at arbitrary curvilinear and rotatory motions; though I'm not discussing this. The main result is (at uniform motion), that the motion of a single point of a rigid body co-determines the motion of all other ones by a very simple law, i.e., that the body thus only has one degree of freedom.

Now the question arises, whether (as in the old mechanics) the rigid body has simple properties in its dynamic behavior also in the new kinematics, and of course it will be about electromagnetic forces.

The practical value of the new definition of rigidity must therefore prove itself in the dynamics of the electron; the greater or lesser clarity of the results achieved there, is to be used to a certain degree also in favor or against the acceptance of the relativity principle per se, since experiments have probably given no definite decision and maybe won't give one.

The theory of Abraham, which studies the motion of an electron (being rigid in the ordinary sense) in the force field produced by itself, has not only led to a qualitatively satisfying explanation of the phenomena of inertia of free electrons on a pure electric basis, but has also led to a quantitative law for the dependency of the electrodynamic mass from velocity at very small accelerations, which is probably not to be seen as disproved by the experiments. Though this theory which superimposes the rigid body (which is suited to the old mechanics) upon electrodynamics, doesn't satisfy the relativity principle, and this is the reason why its further development – at which Sommerfeld,^[3] P. Hertz,^[4] Herglotz,^[5] Schwarzschild^[6] and others are participating – leads to extraordinary mathematical complications. Now, already Lorentz tried to adapt Abraham's theory to the relativity principle, and for that purpose he constructed his "deformable" electron. Exactly this electron is to be denoted as rigid according to the definition given by me. That despite of this agreement, Lorentz's theory gives rise to contradictions to which Abraham^[7] has alluded, is due to the fact that the laws of the composition of forces at the rigid body into resulting forces, were taken over without criticism from the old mechanics; as to how these laws are to be modified, will be given by itself in the representation chosen here. Lorentz's formula for the dependency of mass from velocity, which represent the experiments as good as Abraham's formula, proves to be correct also in the more strict theory. Because this law, as it was already noticed by Einstein, and which was discussed by me in the paper^[8] concerning "the inertial mass and the relativity principle" for arbitrary currents, is a direct consequence of kinematics and is not at all essentially connected with the actual electrodynamic mass, the "rest mass".

Yet, my theory strictly provides the dependency of the rest mass on acceleration for a class of motions, which corresponds – being the principally simplest accelerated motions – to the uniformly accelerated ones of the old mechanics, and which I call "hyperbolic motions", namely the rest mass proves to be constant up to enormous accelerations. Equations of motion in the form of the mechanical fundamental equations^[9] which are adapted to the relativity principle, apply to this motion. Yet, since every accelerated motion can be approximated by such hyperbolic motions as long as their acceleration doesn't vary too suddenly, one achieves in this way an electrodynamic foundation of the fundamental equations of mechanics. This theory fails only for very rapidly changing accelerations; then also radiation resistances arise besides the inertial resistances. It is remarkable, that an electron causes no actual radiation, as great as its acceleration may be, but it drags its field along with it, which was up to now only known for uniformly moving electrons. The radiation and the resistance of the radiation only arise at deviations from hyperbolic motion.

My rigidity definition proves to be appropriate for the system of Maxwell's electrodynamics quite in the same way, as the old definition of rigidity for the system of Galilei-Newtonian mechanics. The rigid electron in this sense, represents the dynamically most simple motion of electricity. One can even go so far to assert, that the theory provides clear hints to an atomistic structure of electricity, which is not at all the case in Abraham's theory. Thus my theory is in agreement with the atomistic instinct of so many experimentalists, for which the interesting attempt of Levi-Civita^[10] – to describe the motion of electricity as a freely moving fluid being bound by no kinematic conditions – will hardly find applause.

Since the simplicity of the dynamics is therefore not inferior to the simplicity of kinematics of the new rigid body, then one will ascribe to this concept of rigidity the same fundamental importance in the system of the electromagnetic world-picture, as the ordinary rigid body in the system of the mechanical world-picture.

First chapter. The kinematics of the rigid body.

§ 1. The rigid body of old mechanics.

With respect to the electrodynamic applications of the second and the third chapter, we won't concern ourselves with rigid systems of discrete points, but with continuous rigid bodies. A continuous current of matter can be represented in the way named after Lagrange, by giving the space coordinates $x,y,z$ as functions of time $t$ and of three parameters $\xi ,\eta ,\zeta$ – for example the values of $x,y,z$ at time $t=0$:

(1)	$\left\{{\begin{array}{l}x=x(\xi ,\eta ,\zeta ,t),\\y=y(\xi ,\eta ,\zeta ,t),\\z=z(\xi ,\eta ,\zeta ,t).\end{array}}\right.$

The mass system is rigid, when the distance of any two of its points

(2)	$r={\sqrt {\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}}}$

is independent of time, thus equal to

${\sqrt {\left(\xi _{1}-\xi _{2}\right)^{2}+\left(\eta _{1}-\eta _{2}\right)^{2}+\left(\zeta _{1}-\zeta _{2}\right)^{2}}}$.

Then it follows from that, that equations (1) have the form

(3)	${\begin{cases}x=a_{1}+a_{11}\xi +a_{12}\eta +a_{13}\zeta ,\\y=a_{2}+a_{21}\xi +a_{22}\eta +a_{23}\zeta ,\\z=a_{3}+a_{31}\xi +a_{32}\eta +a_{33}\zeta ,\end{cases}}$

where the quantities $a_{\alpha },\ a_{\alpha \beta }$ are functions of time $t$, and the matrix

$A=\left({\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}}\right)=\left(a_{\alpha \beta }\right)$

is orthogonal^[11]; i.e., when ${\overline {A}}$ denotes the transposed matrix of $A$, and 1 is the unit matrix, then it is

(4)	${\overline {A}}A=1$

Fig. 1.

To oversee the generalization capability of this condition of the kinematics of the relativity principle, it is advantageous to use the interpretation (used by Minkowski in the work just cited) of the variables $x,y,z,t$ as parallel coordinates in a space of four dimensions called "world". In the following, the figures shall always mean the plane cut $y=0,\ z=0$ through a four-dimensional space; within them, we draw the $x$-axis horizontally, and the $t$-axis upwards. The path of a point is represented in the $xyzt$-manifold (world) as a curve, the "world line", and the motion of a body is represented by a bundle of world lines. The previous condition $dr/dt=0$ now means, that the connecting line of the passage points (of two world-lines each) through a three-dimensional structure $t$ = const., has the same length for all those structures. Thus it is related to the three-dimensional points $t$ = const. "parallel" to space $t=0$.

The importance of this rigidity condition for Newtonian mechanics, lies in the fact that it is invariant with respect to transformations, which transfer the Newtonian equations of motion into themselves. These transformations have the form, when the origin is maintained:

(5)	${\begin{cases}x=k_{11}{\overline {x}}+k_{12}{\overline {y}}+k_{13}{\overline {z}}+k_{1}t,\\y=k_{21}{\overline {x}}+k_{22}{\overline {y}}+k_{23}{\overline {z}}+k_{2}t,\\z=k_{31}{\overline {x}}+k_{32}{\overline {y}}+k_{33}{\overline {z}}+k_{3}t,\end{cases}}$

where $k_{\alpha \beta },\ k_{\alpha }$ are constants, and the matrix

$K=\left(k_{\alpha \beta }\right)$

is orthogonal:

(6)	${\overline {K}}K=1$

The orthogonal constituent only denotes the passage from the initial coordinate system to a system rotated around the origin; yet the second part denotes a uniform translation in time. This is represented in our four-dimensional world as passage from the initial $t$-axis to an inclined ${\overline {t}}$-axis. One immediately sees (Fig. 1), that the quantity $r$ indeed remains unchanged at this occasion.

The relativity principle of electrodynamics states an invariance of natural laws with respect to other linear substitutions, and by that the meaning of quantity $r$ becomes irrelevant. These "Lorentz transformations" connect the four magnitudes $x,y,z,t$ with four new ones ${\overline {x}},{\overline {y}},{\overline {z}},{\overline {t}}$ by such linear equations

(7)

${\begin{cases}x=k_{11}{\overline {x}}+k_{12}{\overline {y}}+k_{13}{\overline {z}}+k_{14}{\overline {t}},\\y=k_{21}{\overline {x}}+k_{22}{\overline {y}}+k_{23}{\overline {z}}+k_{24}{\overline {t}},\\z=k_{31}{\overline {x}}+k_{32}{\overline {y}}+k_{33}{\overline {z}}+k_{34}{\overline {t}},\\t=k_{41}{\overline {x}}+k_{42}{\overline {y}}+k_{43}{\overline {z}}+k_{44}{\overline {t}},\end{cases}}$

which transform the expression

(8)	$x^{2}+y^{2}+z^{2}-c^{2}t^{2}$

into itself, where $c$ means the speed of light.

Here, the time (or rather the quantity $ct{\sqrt {-1}}$) is thus transformed with the coordinates in a symmetric way, and not only the $t$-axis

Fig. 2.

becomes inclined at this transformation, but also space $t=0$ obtains another location in the four-dimensional world.^[12] Since spaces $t$ = const. don't go over to spaces ${\overline {t}}$ = const, then neither quantity $r$ nor condition $dr/dt=0$ is invariant.

At first it seems impossible as well, to provide an analogous condition between two world-lines, since there are no three-dimensional spaces with respect to transformations (7), (8), which are so preferred as previously the spaces $t$ = const. with respect to (5).

Therefore for the sake of generalization, one has to look after another definition of rigidity in the old kinematics. For that, one can use the circumstance that one can replace condition $r$ = const. (taking place between two finitely distant world lines) by a differential condition between infinitely adjacent world lines, so that, when the differential condition is satisfied in the whole space, it gives rise to equation $r$ = const.

For that purpose, we consider at time $t$ the distance of two infinitely adjacent world lines, i.e., the line element

$ds={\sqrt {dx^{2}+dy^{2}+dz^{2}}}\,$

If one sets this equal to a constant $\epsilon$, then equation

$ds^{2}=\epsilon ^{2}\,$

represents an infinitely small sphere. This emerges from an infinitely small ellipsoid during the motion represented by (1), which one obtains when one represents the quantity $ds^{2}$ by means of the equations

(9)

${\begin{cases}dx={\frac {\partial x}{\partial \xi }}d\xi +{\frac {\partial x}{\partial \eta }}d\eta +{\frac {\partial x}{\partial \zeta }}d\zeta ,\\\\dy={\frac {\partial y}{\partial \xi }}d\xi +{\frac {\partial y}{\partial \eta }}d\eta +{\frac {\partial y}{\partial \zeta }}d\zeta ,\\\\dz={\frac {\partial z}{\partial \xi }}d\xi +{\frac {\partial z}{\partial \eta }}d\eta +{\frac {\partial z}{\partial \zeta }}d\zeta ,\end{cases}}$

as quadratic form of $d\xi ,d\eta ,d\zeta$; let this form be:

(10)	${\begin{cases}ds^{2}=p_{11}d\xi ^{2}+p_{22}d\eta ^{2}+p_{33}d\zeta ^{2}\\\qquad +2p_{12}d\xi \ d\eta +2p_{13}d\xi \ d\zeta +2p_{23}d\eta \ d\zeta \end{cases}}$

There, the matrix of the "deformation quantities" $p_{\alpha \beta }$

$P=\left(p_{\alpha \beta }\right)$

from the matrix

(11)

$A=\left({\begin{array}{ccccc}{\frac {\partial x}{\partial \xi }}&&{\frac {\partial x}{\partial \eta }}&&{\frac {\partial x}{\partial \zeta }}\\\\{\frac {\partial y}{\partial \xi }}&&{\frac {\partial y}{\partial \eta }}&&{\frac {\partial y}{\partial \zeta }}\\\\{\frac {\partial z}{\partial \xi }}&&{\frac {\partial z}{\partial \eta }}&&{\frac {\partial z}{\partial \zeta }}\end{array}}\right)$

is composed in this way:

(12)	$P={\overline {A}}A$

Now, we will call this motion only in the smallest parts as rigid, when an infinitely small structure is not changed during motion, thus when all $p_{\alpha \beta }$ are independent of time. Thus we have the infinitesimal rigidity conditions:

(13)	${\frac {dp_{\alpha \beta }}{dt}}=0$

When $\xi ,\eta ,\zeta$ are the initial values of $x,y,z$, then matrix $A$ is equal to unit matrix 1 for $t=0$, thus (12) reads:

$P={\overline {A}}A=1$

It is now an elementary theorem of infinitesimal geometry^[13], that when these conditions are satisfied everywhere, it is about the motion of a rigid body.

This infinitesimal rigidity condition (13) can now easily be transfered to the kinematics of the relativity principle.

§ 2. The differential conditions of rigidity.

In the following, only such quantities shall have a meaning, which are invariant with respect to the Lorentz transformations (7), (8).

Now we consider a current, which we represent instead of equations of form (1), by the following equations which better correspond to the symmetry of quantities $x,y,z,t$ required by the relativity principle:

(14)	$\left\{{\begin{array}{l}x=x(\xi ,\eta ,\zeta ,\tau ),\\y=y(\xi ,\eta ,\zeta ,\tau ),\\z=z(\xi ,\eta ,\zeta ,\tau ),\\t=t(\xi ,\eta ,\zeta ,\tau ).\end{array}}\right.$

There, let $\tau$ be the proper time, i.e., the identity exists:

(15)	$\left({\frac {\partial x}{\partial \tau }}\right)^{2}+\left({\frac {\partial y}{\partial \tau }}\right)^{2}+\left({\frac {\partial z}{\partial \tau }}\right)^{2}-c^{2}\left({\frac {\partial t}{\partial \tau }}\right)^{2}=-c^{2}$;

$\tau$ is measured starting at any "cross section" of the currents.

$\xi ,\eta ,\zeta$ shall characterize individual current-filaments, though we leave open their meaning. Now we set for the time being:

(16)	$\left\{{\begin{array}{l}x(0,0,0,\tau )={\mathfrak {x}}(\tau ),\\y(0,0,0,\tau )={\mathfrak {y}}(\tau ),\\z(0,0,0,\tau )={\mathfrak {z}}(\tau ),\\t(0,0,0,\tau )={\mathfrak {t}}(\tau ),\end{array}}\right.$

and consider the filament of world lines, surrounding world line (16) $\xi =\eta =\zeta =0$.

Fig. 3.

This can be represented as follows:

(17)

$\left\{{\begin{array}{l}x={\mathfrak {x}}+x_{\xi }d\xi +x_{\eta }d\eta +x_{\zeta }d\zeta +\dots ,\\y={\mathfrak {y}}+y_{\xi }d\xi +y_{\eta }d\eta +y_{\zeta }d\zeta +\dots ,\\z={\mathfrak {z}}+z_{\xi }d\xi +z_{\eta }d\eta +z{}_{\zeta }d\zeta +\dots ,\\t={\mathfrak {t}}+t_{\xi }d\xi +t_{\eta }d\eta +t_{\zeta }d\zeta +\dots ,\end{array}}\right.$

where we confine ourselves to terms being linear in the increments $d\xi ,\ d\eta ,\ d\zeta$ (which are first to be imagined as small, though finite). Here, ${\mathfrak {x,y,z,t}}$ are the functions defined by (16), and it is set:

$x_{\xi }={\frac {\partial x}{\partial \xi }}(0,0,0,\tau ),\dots$

Two spacetime vectors with components $x_{1},y_{1},z_{1},t_{1}$ and $x_{2},y_{2},z_{2},t_{2}$ are called normal, when their direction are conjugated with respect to the invariant hyperbolic structure

(18)	$x^{2}+y^{2}+z^{2}-c^{2}t^{2}=-1$,

thus when

(19)	$x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2}-c^{2}t_{1}t_{2}=0$

All vectors being normal to a time-like vector^[14] $x_{1},y_{1},z_{1},t_{1}$ are satisfying a three-dimensional linear structure, which can be made to space $t=0$ by a suitable Lorentz transformation; we call it the normal cut of the vector.

The concepts being so defined, are evidently invariant with respect to Lorentz transformations.

Now we consider a certain point $P$ upon the world line $\xi =\eta =\zeta =0$, belonging to the value $\tau _{0}$ of the proper time. Through this point $P$, we lay the normal cut to the velocity vector ${\mathfrak {x'_{0},y'_{0},z'_{0},t'_{0}}}$ in $P$:

{{MathForm1|(20)|${\mathfrak {x}}'_{0}\left(x-{\mathfrak {x}}_{0}\right)+{\mathfrak {y}}'_{0}\left(y-{\mathfrak {y}}_{0}\right)+{\mathfrak {z}}'_{0}\left(z-{\mathfrak {z}}_{0}\right)-c^{2}{\mathfrak {t}}'_{0}\left(t-{\mathfrak {t}}_{0}\right)=0$

where

${\mathfrak {x}}'={\frac {d{\mathfrak {x}}}{d\tau }}=\left[{\frac {\partial x}{\partial \tau }}\right]_{\xi =\eta =\zeta =0},\dots$

and index 0 means, that $\tau =\tau _{0}$ is to be inserted into the functions.

In (20), we replace $x,y,z,t$ by their expressions (17) as functions of $d\xi ,d\eta ,d\zeta$ and $\tau$:

(21)

$\left\{{\begin{array}{l}{\mathfrak {x}}'_{0}\left\{{\mathfrak {x}}-{\mathfrak {x}}_{0}+x_{\xi }d\xi +x_{\eta }d\eta +x_{\zeta }d\zeta +\dots \right\}+\dots \\\qquad \dots -c^{2}{\mathfrak {t}}'_{0}\left\{{\mathfrak {t}}-{\mathfrak {t}}_{0}+t_{\xi }d\xi +t_{\eta }d\eta +t_{\zeta }d\zeta +\dots \right\}=0\end{array}}\right.$

We can see this as an equation for $\tau$, from which one can calculate the values of proper time $\tau$ (belonging to normal cut $\tau _{0}$) upon the neighboring line $d\xi ,d\eta ,d\zeta$. Since the difference $\tau -\tau _{0}=d\tau$ is small, then (21) will be a linear equation in $d\tau$. Namely, if one expands

(22)	$\left\{{\begin{array}{l}{\mathfrak {x}}={\mathfrak {x}}_{0}+{\mathfrak {x}}'_{0}d\tau +\dots ,\\\dots \dots \\x_{\xi }=x_{\xi }^{0}+\left(x'_{\xi }\right)_{0}d\tau +\dots ,\\\dots \dots \end{array}}\right.$

and if one considers, that according to (15) it is identical in $\tau$

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

then it follows from (21), when one neglects all quadratic terms in $d\xi ,d\eta ,d\zeta ,d\tau$:

(24)	$\left\{{\begin{array}{l}c^{2}d\tau ={\mathfrak {x}}'_{0}\left(x_{\xi }^{0}d\xi +x_{\eta }^{0}d\eta +x_{\zeta }^{0}d\zeta \right)+\dots \\\qquad \dots -c^{2}{\mathfrak {t}}'_{0}\left(t_{\xi }^{0}d\xi +t_{\eta }^{0}d\eta +t_{\zeta }^{0}d\zeta \right),\end{array}}\right.$

or when we

(25)

$\left\{{\begin{array}{l}x_{\xi }^{0}d\xi +x_{\eta }^{0}d\eta +x_{\zeta }^{0}d\zeta =\Xi ,\\y_{\xi }^{0}d\xi +y_{\eta }^{0}d\eta +y_{\zeta }^{0}d\zeta =\mathrm {H} ,\\x_{\xi }^{0}d\xi +z_{\eta }^{0}d\eta +z_{\zeta }^{0}d\zeta =\mathrm {Z} ,\\x_{\xi }^{0}d\xi +t_{\eta }^{0}d\eta +t_{\zeta }^{0}d\zeta =\mathrm {T} ,\end{array}}\right.$

set as:

(26)	$c^{2}d\tau ={\mathfrak {x}}'_{0}\Xi +{\mathfrak {y}}'_{0}\mathrm {H} +{\mathfrak {z}}'_{0}\mathrm {Z} -c^{2}{\mathfrak {t}}'_{0}\mathrm {T}$

Now we consider the (one-shell) hyperbolic structure located around the point $\xi =0,\ \eta =0,\ \zeta =0,\ \tau =\tau _{0}$ as center:

(27)	$\left(x-{\mathfrak {x}}_{0}\right)^{2}+\left(y-{\mathfrak {y}}_{0}\right)^{2}+\left(z-{\mathfrak {z}}_{0}\right)^{2}-c^{2}\left(t-{\mathfrak {t}}_{0}\right)^{2}=\epsilon ^{2}$

This cuts the normal cut (20) into a figure, which is to be seen as the "rest shape" of the filament at this place.

If we accordingly replace in (27), $x,y,z,t$ by expressions (17), and then the quantities ${\mathfrak {x,y,z,t}},x_{\xi },\dots$ by the expansions (22), it follows

(28)	$\left\{{\begin{array}{l}\left({\mathfrak {x}}'_{0}d\tau +x_{\xi }^{0}d\xi +x_{\eta }^{0}d\eta +x_{\zeta }^{0}d\zeta \right)^{2}+\dots \\\qquad \dots -c^{2}\left({\mathfrak {t}}'_{0}d\tau +t_{\xi }^{0}d\xi +t_{\eta }^{0}d\eta +t_{\zeta }^{0}d\zeta \right)^{2}=\epsilon ^{2},\end{array}}\right.$

and herein is (upon the normal cut $d\tau$) the function of $d\xi ,d\eta ,d\zeta$ defined by (26); thus (28) goes over into:

(29)

$\left\{{\begin{array}{l}\left\{\left(1+{\frac {{\mathfrak {x}}_{0}'^{2}}{c^{2}}}\right)\Xi +{\frac {{\mathfrak {x}}_{0}'{\mathfrak {y}}_{0}'}{c^{2}}}\mathrm {H} +{\frac {{\mathfrak {x}}_{0}'{\mathfrak {z}}_{0}'}{c^{2}}}\mathrm {Z} -{\mathfrak {x}}_{0}'{\mathfrak {t}}_{0}'\mathrm {T} \right\}^{2}+\dots \\\qquad \dots -c^{2}\left\{{\frac {{\mathfrak {t}}_{0}'{\mathfrak {x}}_{0}'}{c^{2}}}\Xi +{\frac {{\mathfrak {t}}_{0}'{\mathfrak {y}}_{0}'}{c^{2}}}\mathrm {H} +{\frac {{\mathfrak {t}}_{0}'{\mathfrak {z}}_{0}'}{c^{2}}}\mathrm {Z} +\left(1-{\mathfrak {x}}_{0}'^{2}\right)\mathrm {T} \right\}^{2}=\epsilon ^{2},\end{array}}\right.$

By that, the rest shape is given as a quadratic form in $d\xi ,d\eta ,d\zeta$. Since point $\xi =\eta =\zeta =0$, $\tau =\tau _{0}$ was an arbitrary point of the current, one can omit indices 0 and replace ${\mathfrak {x}}'\dots$ by $x_{\tau },\dots$. If we then write (29) in the form

(30)	$\left\{{\begin{array}{l}\left(c_{11}d\xi +c_{12}d\eta +c_{13}d\zeta \right)^{2}+\left(c_{21}d\xi +c_{22}d\eta +c_{23}d\zeta \right)^{2}\\\qquad +\left(c_{31}d\xi +c_{32}d\eta +c_{33}d\zeta \right)^{2}+\left(c_{41}d\xi +c_{42}d\eta +c_{43}d\zeta \right)^{2}=\epsilon ^{2},\end{array}}\right.$

then the rectangular matrix for 4 rows and 3 columns $C=\left(c_{\alpha \beta }\right)$ is equal to the product of two matrices $S$ and $A$, which are formed from the derivatives of functions (14):

(31)

$C=SA\,$

namely it is:

(32)

$S=\left({\begin{array}{ccccccc}1+{\frac {x_{\tau }^{2}}{c^{2}}}&&{\frac {x_{\tau }y_{\tau }}{c^{2}}}&&{\frac {x_{\tau }z_{\tau }}{c^{2}}}&&-{\frac {x_{\tau }t_{\tau }}{ic}}\\\\{\frac {y_{\tau }x_{\tau }}{c^{2}}}&&1+{\frac {y_{\tau }^{2}}{c^{2}}}&&{\frac {y_{\tau }z_{\tau }}{c^{2}}}&&-{\frac {y_{\tau }t_{\tau }}{ic}}\\\\{\frac {z_{\tau }x_{\tau }}{c^{2}}}&&{\frac {z_{\tau }y_{\tau }}{c^{2}}}&&1+{\frac {z_{\tau }^{2}}{c^{2}}}&&-{\frac {z_{\tau }t_{\tau }}{ic}}\\\\-{\frac {t_{\tau }x_{\tau }}{ic}}&&-{\frac {t_{\tau }y_{\tau }}{ic}}&&-{\frac {t_{\tau }z_{\tau }}{ic}}&&1-t_{\tau }^{2}\end{array}}\right)$

(33)	$A=\left({\begin{array}{ccccc}x_{\xi }&&x_{\eta }&&x_{\zeta }\\y_{\xi }&&y_{\eta }&&y_{\zeta }\\\\z_{\xi }&&z_{\eta }&&z_{\zeta }\\ict_{\xi }&&ict_{\eta }&&ict_{\zeta }\end{array}}\right)$

If we now develop the quadratic from (30) to $d\xi ,d\eta ,d\zeta$, then one has:

(34)	$\left\{{\begin{array}{l}p_{11}d\xi ^{2}+p_{22}d\eta ^{2}+p_{33}d\zeta ^{2}\\\qquad +2p_{12}d\xi \ d\eta +2p_{13}d\xi \ d\zeta +2p_{23}d\eta \ d\zeta \end{array}}\right.$

where it becomes

(35)	$P=\left(p_{\alpha \beta }\right)={\overline {C}}C={\overline {A}}{\overline {S}}SA$

With the aid of equation (15) causing the determinant of $S$ to vanish, this relation can still further be simplified; namely it is easily given by computation:

(36)	${\overline {S}}S=S$

and (33) thus goes over into:

(37)	$P={\overline {A}}SA$

This is analogues to equation (12) derived in § 1. The six quantities $p_{\alpha \beta }$ are to be denoted as "deformation quantities", and would be of importance in a theory of elasticity adapted to the relativity principle.

We will call a filament as rigid in the smallest parts, whose rest shape is independent from proper time $\tau$, i.e., for which the six equations

(38)	${\frac {\partial p_{\alpha \beta }}{\partial \tau }}=0$

hold.

When these equations are satisfied in the whole space, then we are dealing with the motion of a rigid body.

By that, we have gained the general differential conditions of rigidity. Since they are solely formed by the aid of such concepts being invariant with respect to Lorentz transformations, they thus have necessarily the same property.

§ 3. The continuity equation and the incompressible current.

If $\varrho$ is the density belonging to the current (1), then we know that it is connected with the velocity components

(39)	$w_{x}={\frac {\partial x}{\partial t}},\ w_{y}={\frac {\partial y}{\partial t}},\ w_{z}={\frac {\partial x}{\partial t}}$

by the continuity equation.

This can be formulated in two ways. According to the one of Euler, one sees $\varrho ,w_{x},w_{y},w_{z}$ as functions of $x,y,z,t$; then the continuity equation reads:

(40)	${\frac {\partial \varrho }{\partial t}}+{\frac {\partial \varrho w_{x}}{\partial x}}+{\frac {\partial \varrho w_{y}}{\partial y}}+{\frac {\partial \varrho w_{z}}{\partial z}}=0$

According to the one of Lagrange, $x,y,z,\varrho$ are seen as functions of $\xi ,\eta ,\zeta ,t$; then the condition reads:

(41)	${\frac {\partial \varrho \Theta }{\partial t}}=0$

where $\Theta$ is the functional determinant

(42)

$\Theta =\left|{\begin{array}{ccccc}{\frac {\partial x}{\partial \xi }}&&{\frac {\partial x}{\partial \eta }}&&{\frac {\partial x}{\partial \zeta }}\\\\{\frac {\partial y}{\partial \xi }}&&{\frac {\partial y}{\partial \eta }}&&{\frac {\partial y}{\partial \zeta }}\\\\{\frac {\partial z}{\partial \xi }}&&{\frac {\partial z}{\partial \eta }}&&{\frac {\partial z}{\partial \zeta }}\end{array}}\right|$

The connection between both formulas is caused by the identity:^[15]

(43)	${\frac {\partial \varrho }{\partial t}}+{\frac {\partial \varrho w_{x}}{\partial x}}+{\frac {\partial \varrho w_{y}}{\partial y}}+{\frac {\partial \varrho w_{z}}{\partial z}}={\frac {1}{\Theta }}{\frac {d\varrho \Theta }{\partial t}}$

Both forms of the continuity equation can be transfered to the representation of the current with the aid of proper time by equations (14). First, it is evidently given:

(44)	$w_{x}={\frac {x_{\tau }}{t_{\tau }}},\ w_{y}={\frac {y_{\tau }}{t_{\tau }}},\ w_{z}={\frac {z_{\tau }}{t_{\tau }}}$

If we furthermore replace $\varrho$ by the "rest density"

(45)	$\varrho ^{*}={\frac {\varrho }{t_{\tau }}}$

then (40) goes over into:

(46)	${\frac {\partial \varrho ^{*}x_{\tau }}{\partial x}}+{\frac {\partial \varrho ^{*}y_{\tau }}{\partial y}}+{\frac {\partial \varrho ^{*}z_{\tau }}{\partial z}}+{\frac {\partial \varrho ^{*}t_{\tau }}{\partial t}}=0$

We get the analogue to formula (41) by showing the correctness of the identity corresponding to (43)

(47)	${\frac {\partial \varrho ^{*}x_{\tau }}{\partial x}}+{\frac {\partial \varrho ^{*}y_{\tau }}{\partial y}}+{\frac {\partial \varrho ^{*}z_{\tau }}{\partial z}}+{\frac {\partial \varrho ^{*}t_{\tau }}{\partial t}}={\frac {1}{D}}{\frac {\partial \varrho ^{*}D}{\partial \tau }}$

where $D$ means the functional determinant

(48)	$D=\left|{\begin{array}{ccccccc}x_{\xi }&&x_{\eta }&&x_{\zeta }&&x_{\tau }\\y_{\xi }&&y_{\eta }&&y_{\zeta }&&y_{\tau }\\z_{\xi }&&z_{\eta }&&z_{\zeta }&&z_{\tau }\\t_{\xi }&&t_{\eta }&&t_{\zeta }&&t_{\tau }\end{array}}\right|$

For this purpose, we momentarily replace for the sake of shortness:

$x,y,z,t$ by $x_{1},x_{2},x_{3},x_{4}$

$\xi ,\eta ,\zeta ,\tau$ by $\xi _{1},\xi _{2},\xi _{3},\xi _{4}$

Then we have for the left-hand side of (47):

$\sum \limits _{\alpha }{\frac {\partial \left(\varrho ^{*}{\frac {\partial x_{\alpha }}{\partial \xi _{4}}}\right)}{\partial x_{\alpha }}}=\sum \limits _{\alpha ,\beta }{\frac {\partial \left(\varrho ^{*}{\frac {\partial x_{\alpha }}{\partial \xi _{4}}}\right)}{\partial \xi _{\beta }}}{\frac {\partial \xi _{\beta }}{\partial x_{\alpha }}}=\sum \limits _{\alpha ,\beta }\left(\varrho ^{*}{\frac {\partial ^{2}x_{\alpha }}{\partial \xi _{\beta }\partial \xi _{4}}}+{\frac {\partial \varrho ^{*}}{\partial \xi _{\beta }}}{\frac {\partial x_{\alpha }}{\partial \xi _{4}}}\right){\frac {\partial \xi _{\beta }}{\partial x_{\alpha }}}$

If we now denote (in the scheme of determinant $D$) by $S\left(\partial x_{\alpha }/\partial \xi _{\beta }\right)$ the sub-determinant belonging to $\partial x_{\alpha }/\partial \xi _{\beta }$, then it is given by successive differentiation of equations (14) with respect to $x_{\alpha }$, and by solving of the linear equations emerging in this way:

(49)	${\frac {\partial \xi _{\beta }}{\partial x_{\alpha }}}={\frac {S\left({\frac {\partial x_{\alpha }}{\partial \xi _{\beta }}}\right)}{D}}$

If this is inserted above, it follows

${\begin{array}{ll}\sum \limits _{\alpha }{\frac {\partial \left(\varrho ^{*}{\frac {\partial x_{\alpha }}{\partial \xi _{4}}}\right)}{\partial x_{\alpha }}}&={\frac {1}{D}}\sum \limits _{\alpha ,\beta }\left\{\varrho ^{*}{\frac {\partial ^{2}x_{\alpha }}{\partial \xi _{\beta }\partial \xi _{4}}}S\left({\frac {\partial x_{\alpha }}{\partial \xi _{\beta }}}\right)+{\frac {\partial \varrho ^{*}}{\partial \xi _{\beta }}}{\frac {\partial x_{\alpha }}{\partial \xi _{4}}}S\left({\frac {\partial x_{\alpha }}{\partial \xi _{\beta }}}\right){\frac {\partial \xi _{\beta }}{\partial \xi _{\alpha }}}\right\}\\\\&={\frac {1}{D}}\left\{\varrho ^{*}{\frac {\partial D}{\partial \xi _{4}}}+{\frac {\partial \varrho ^{*}}{\partial \xi _{4}}}D\right\}\end{array}}$

according to general determinant theorems. Thus it follows

$\sum \limits _{\alpha }{\frac {\partial \left(\varrho ^{*}{\frac {\partial x_{\alpha }}{\partial \xi _{4}}}\right)}{\partial x_{\alpha }}}={\frac {1}{D}}{\frac {\partial \varrho ^{*}D}{\partial \xi _{4}}}$

which is the identity (47) that had to be proven.

Consequently, one can write the continuity condition in the form

(50)	${\frac {\partial \varrho ^{*}D}{\partial \tau }}=0$

Formulas (46), (47), (50) have invariant character with respect to Lorentz transformations

The quantity

$\varrho ^{*}D=\varrho _{0}$

is only dependent from $\xi ,\eta ,\zeta$; when $D$ is equal to 1 for $\tau =0$ (which one can always assume), then $\varrho _{0}$ is the "initial value of the rest density".

A current is called incompressible in the old kinematics, when $\varrho$ is constant and independent from time $t$. In the new kinematics we will define it as follows:

A current is incompressible, when the rest density $\varrho ^{*}$ is constant, i.e., independent from proper time $\tau$.

Two forms of the incompressibility condition are given from (46) and (50).

At first one can namely (46) write:

$\varrho ^{*}\left({\frac {\partial x_{\tau }}{\partial x}}+{\frac {\partial y_{\tau }}{\partial y}}+{\frac {\partial z_{\tau }}{\partial z}}+{\frac {\partial t_{\tau }}{\partial t}}\right)+{\frac {\partial \varrho ^{*}}{\partial x}}x_{\tau }+{\frac {\partial \varrho ^{*}}{\partial y}}y_{\tau }+{\frac {\partial \varrho ^{*}}{\partial z}}z_{\tau }+{\frac {\partial \varrho ^{*}}{\partial t}}t_{\tau }=0$

or

$\varrho ^{*}\left({\frac {\partial x_{\tau }}{\partial x}}+{\frac {\partial y_{\tau }}{\partial y}}+{\frac {\partial z_{\tau }}{\partial z}}+{\frac {\partial t_{\tau }}{\partial t}}\right)+{\frac {\partial \varrho ^{*}}{\partial \tau }}=0$;

now, if $\varrho ^{*}$ shall not depend on $\tau$, then the first form of the incompressibility condition follows:

(53)	${\frac {\partial x_{\tau }}{\partial x}}+{\frac {\partial y_{\tau }}{\partial y}}+{\frac {\partial z_{\tau }}{\partial z}}+{\frac {\partial t_{\tau }}{\partial t}}=0$

The second form is immediately given from (50):

(54)	${\frac {\partial D}{\partial \tau }}=0$

By that, when $D$ is equal to 1 for $\tau =0$, $D$ is identically equal to 1, and by (51):

$\varrho ^{*}=\varrho _{0}(\xi ,\eta ,\zeta )$

§ 4. The uniform translation of the rigid body.

We now want to integrate the differential conditions of rigidity (38) for the simplest case of uniform translation. When we consider, that rigidity must be identical with incompressibility in this case, then we not only achieve by that a criterion as to what our rigidity definition means mutatis mutandis, but simultaneously also a method for integration.

Thus we set

(55)	$y=\eta ,\ z=\zeta$

and assume that $x$ and $t$ only depend on $\xi$ and $\tau$. Then we obtain from (32) and (33):

$S=\left({\begin{array}{ccccccc}1+{\frac {x_{\tau }^{2}}{c^{2}}}&&0&&0&&-{\frac {x_{\tau }t_{\tau }}{ic}}\\\\0&&1&&0&&0\\\\0&&0&&1&&0\\\\-{\frac {t_{\tau }x_{\tau }}{ic}}&&&&&&1-t_{\tau }^{2}\end{array}}\right)$

$A=\left({\begin{array}{ccccc}x_{\xi }&&0&&0\\0&&1&&0\\0&&0&&1\\ict_{\xi }&&0&&0\end{array}}\right)$

If one forms from that the matrix:

$P={\overline {A}}SA$,

then one easily finds

$P=\left({\begin{array}{ccccc}\left(x_{\xi }t_{\tau }-x_{\tau }t_{\xi }\right)^{2}&&0&&0\\0&&0&&0\\0&&0&&0\end{array}}\right)$

The six rigidity conditions thus are reduced to one equation:

(56)	${\frac {d}{d\tau }}\left(x_{\xi }t_{\tau }-x_{\tau }t_{\xi }\right)=0$

On the other hand, determinant (48) becomes:

(57)	$D=\left|{\begin{array}{ccccccc}x_{\xi }&&0&&0&&x_{\tau }\\0&&1&&0&&0\\0&&0&&1&&0\\t_{\xi }&&0&&0&&t_{\tau }\end{array}}\right|=\left|{\begin{array}{ccc}x_{\xi }&&x_{\tau }\\t_{\xi }&&t_{\tau }\end{array}}\right|$

Thus the incompressibility condition

${\frac {dD}{d\tau }}=0$

is identical with the rigidity condition (56).

Consequently we can also replace the latter by the other form (53) of the incompressibility condition, which assumes the form here:

(58)	${\frac {\partial x_{\tau }}{\partial x}}+{\frac {\partial t_{\tau }}{\partial t}}=0$

The integration is now easily to be executed in this form.

If one puts:

(59)	$x_{\tau }=p,\ t_{\tau }=-q$,

then one obtains the two equations for $q,q$:

(60)	${\begin{cases}{\frac {\partial p}{\partial x}}-{\frac {\partial q}{\partial t}}=0,\\\\p^{2}-c^{2}q^{2}=-c^{2}\end{cases}}$

These are equivalent to a partial differential equation for a function of two independent variables. Namely, if one puts

(61)	$p={\frac {\partial \varphi }{\partial t}},\ q={\frac {\partial \varphi }{\partial x}},$

then the first equation (60) is satisfied, and the second one goes over into

(62)	$\varphi _{t}^{2}-c^{2}\varphi _{x}^{2}=-c^{2}$

The simplest solution of these equation is obtained, when one puts $\varphi _{t}$ and $\varphi _{x}$ equal to constants $\gamma$ and $-\delta$, which must satisfy the condition

(63)	$\gamma ^{2}-c^{2}\delta ^{2}=-c^{2}\,$

Then it becomes

$p=x_{\tau }=\gamma ,\ q=-t_{\tau }=-\delta$

from which it follows:

(64)	${\begin{cases}x=W(\xi )+\gamma \tau ,\\t=V(\xi )+\delta \tau ,\end{cases}}$

where $W$ and $V$ are two arbitrary functions of $\xi$. Due to equation (63), the form of equations (64) is indeed conserved, when $x,t$ are subjected to a Lorentz transformation.

Equations (64) together with (55) represent a rectilinear uniform motion. Functions $W(\xi ),\ V(\xi )$ are determined by the value, which $x$ and $t$ shall have for $\tau =0$. It is not convenient here to assume $x=\xi$ for $\tau =0$, but functions $W(\xi ),\ V(\xi )$ are so to be determined, that formulas (64) represent that Lorentz transformation which transforms the body into rest, i.e., it is to be set:

(65)	${\begin{cases}x=\alpha \xi +\gamma \tau ,\\t=\beta \xi +\delta \tau ,\end{cases}}$

where the conditions

(66)	$\alpha ^{2}-c^{2}\beta ^{2}=1,\ \alpha \gamma -c^{2}\beta \delta =0,\ \gamma ^{2}-c^{2}\delta ^{2}=-1$

are satisfied.

As soon as one of the two quantities $\varphi _{t},\ \varphi _{x}$ depends on $t$ in (62), then this must also be the case for the other one. In this case, the integration of (62) can be easily executed by the aid of a Legendre transformation. Then, one can namely introduce the quantity

(67)	$\varphi _{t}=p$

as independent variable besides $x$, and imagine $t$ [from (67)] to be calculated as a function of $x$ and $p$. If one then introduces (instead of $\varphi$) the new unknown function

$\psi (p,x)=\varphi -pt$,

then it is

(68)	${\begin{cases}\psi _{p}=\varphi _{t}t_{p}-pt_{p}-t=-t,\\\psi _{x}=\varphi _{x}+\varphi _{t}t_{x}-pt_{x}=\varphi _{x}.\end{cases}}$

Consequently, (62) goes over into the following equation for $\psi (p,x)$:

$p^{2}-c^{2}\psi _{x}^{2}=-c^{2}\,$

this can be immediately integrated. It is given:

(69)	${\begin{cases}\psi _{x}={\sqrt {1+{\frac {p^{2}}{c^{2}}}}}=q,\\\psi =qx-w(p),\end{cases}}$

where $w$ means an arbitrary function. From that it follows by differentiation with respect to $q$ under consideration of (68):

(70)	${\frac {p}{c^{2}q}}x-w'(p)=-t$

If one consequently imagines $p$ as being calculated as a function of $t$ and inserted into $\varphi =\psi +pt$, then one has the desired most general solution of (62):

(71)	$\varphi =qx-w(p)+pt\,$

According to (59) and (61), it is evidently

${\frac {x_{\tau }}{t_{\tau }}}={\frac {dx}{dt}}=-{\frac {\varphi _{t}}{\varphi _{x}}}$

from which it follows that any equation $\varphi =\mathrm {const} .=-\xi$ represents the world line of a point of the rigid body. From (70) and (71) we consequently find the following representation of the world lines:

(72)	${\begin{cases}{\frac {p}{c^{2}}}x+qt=qw',\\qx+pt=w-\xi ,\end{cases}}$

or, when solved with respect to $x$ and $t$:

(73)	${\begin{cases}x=q(w-\xi )-pqw',\\t=-{\frac {p}{c^{2}}}(w-\xi )+q^{2}w'.\end{cases}}$

Here, the world lines of the rigid body are so described, that $x$ and $t$ are given as functions of the independent variables $\xi ,\ p$. We now want to discuss this representation.

First it is to be noticed, that the rectilinear translatory motion only depends on one arbitrary function of an argument $w(p)$. Thus one can say, that also here only one degree of freedom (as in the old kinematics) is present. There, the usage of the independent variables $p=x_{\tau }$ is essential, which still will be of great importance. Furthermore, equations (73) go over into the corresponding representation of the uniform translation of the old kinematics, when $c=\infty$. Because $q={\sqrt {1+\left(p^{2}/c^{2}\right)}}$ becomes equal to 1 in this case; from the second equation (73) it follows for $c=\infty$, that $p$ only depends on $t$, so that the first one assumes the form

$x=\xi +a(t)$

Finally we direct our attention to the characterization of the world lines in the $xt$-plane. One recognizes, that (72) and (73) have the form of a Lorentz transformation and their inverse ones, which transform the variables $x,y$ into the variables ${\overline {x}}=w-\xi$, ${\overline {t}}=qw'$, and they read:

(74)	${\begin{cases}{\overline {x}}=qx+{\frac {p}{c}}ct,&x=q{\overline {x}}-{\frac {p}{c}}c{\overline {t}},\\\\c{\overline {t}}={\frac {p}{c}}x+qct,&ct=-{\frac {p}{c}}{\overline {x}}+qc{\overline {t}}\end{cases}}$

because the equations (66) between the coefficient are evidently satisfied due to (60).

Thus we are faced with a bundle of Lorentz transformations depending on the parameter $p$. The motion, or rather the corresponding bundle of world lines, can now be described as follows.

If one gives a certain value $\xi _{1}$ to $\xi$, then $x$ and $t$ are given by equations (73) as specific functions of $p$, which represent the world line of point $\xi _{1}$. The components of the velocity world-vector with respect to axes $x$ and $t$, are $p$ and $-q$. By that single curve $\xi _{1}$, all curves of the bundle are co-determined. One has to construct it as follows: To the tangent in a point $p$ of the curve, one lays the line being normal to it in the sense of § 2 (p. 12); this forms (together with the tangent) the $x$- and $t$-axis of a transformed coordinate system. Upon this $x$-axis, one draws the distance $\xi _{1}-\xi$ in the unit of this coordinate system^[16]. If one now moves this coordinate system along curve $\xi _{1}$, then point $\xi$ follows the world line belonging to the parameter value $\xi$. All points of such a normal ($x$-axis) belong to the same value $p$, thus they have the same velocity.

Fig. 4.

The uniform motion of a rigid body is so constituted, that – as soon as one point is transformed to rest – all of its points are transformed to rest by the same transformation. This rest transformation is exactly (74). The lines of same velocity $p$ = const., except at uniform motion, always have an envelope; the regularity of motion stops at this one. At given dimensions of the body, the curvature of world lines thus cannot exceed a certain limit, and vice versa. From that it follows, that a rigid body is necessarily extended into all directions, and has to be the smaller, the bigger the accelerations are that it should experience. Here, we have the first hint at the fundamental importance of atomistics in the new dynamics. If the rigid body carries a substance of rest density $\varrho ^{*}$, then it is independent of $p$, and it is a function of $\xi ,\eta ,\zeta$, which we denote by

$\varrho _{0}(\xi ,\eta ,\zeta )$

§ 5. Hyperbolic motion.

The simplest motion different from uniform translation will be obtained by us, when we set the arbitrary function $w=0$ in (72) and (73). Then it becomes

(75)	${\begin{cases}x=-q\xi ,\\t={\frac {p}{c^{2}}}\xi .\end{cases}}$

If one eliminates $p$ therefrom, then it follows

(76)	$x^{2}-c^{2}t^{2}=\xi ^{2}\,$

Fig. 5.

From that one recognizes, that the corresponding world line in the $xt$-plane and the planes $y=\eta ,\ z=\zeta$ being parallel to it, are hyperbolas having the lines corresponding to the speed of light as asymptotes, and which are cutting the $x$-axis in the distance $\xi$ from the origin. A bundle of such hyperbolas represents a motion, at which the rigid body comes from infinity and is approaching the origin, then it is turning back and is moving away into infinity again, where its velocity decreases from the speed of light to zero first, and after the turning back increases again to $c$. This motion, to some extent analogous to the uniformly accelerated motion of old kinematics, we want to call hyperbolic motion shortly.

Since the origin is a quite arbitrary point, then the hyperbolas

$(x-\alpha )^{2}-c^{2}(t-\beta )^{2}=\xi ^{2}\,$

represent no essentially different motion; only the velocity is then different from zero for $t=0$. Thus we will be able to confine ourselves to formulas (75), (76).

This hyperbolic motion proves to be not only kinematically, but also dynamically as the simplest one. This is closely connected with the circumstance, that any world line is osculated by a hyperbola in any of its points $P$, the "curvature hyperbola", where the vector of magnitude $b=c^{2}/\xi$ directed from its center to point $P$, represents the acceleration vector of the world line in $P$.

Indeed, if we calculate the acceleration components of hyperbolic motion, then we find at first

(78)	${\frac {\partial ^{2}y}{\partial \tau ^{2}}}=0,\ {\frac {\partial ^{2}z}{\partial \tau ^{2}}}=0$

To calculate the $x$- and $t$-components, we consider the equations

(79)	${\begin{cases}\xi _{t}=-p,&p_{t}={\frac {c^{2}q^{2}}{\xi }}\\\xi _{x}=-q,&p_{x}={\frac {pq}{\xi }}.\end{cases}}$

Then it becomes

${\frac {\partial ^{2}x}{\partial \tau ^{2}}}=p_{\tau }=p_{x}x_{\tau }+p_{t}t_{\tau }={\frac {pq}{\xi }}p-{\frac {c^{2}q^{2}}{\xi }}q$

Thus we obtain

(80)	${\frac {\partial ^{2}x}{\partial \tau ^{2}}}=b_{x}=-qb$

as well as

(81)	${\frac {\partial ^{2}t}{\partial \tau ^{2}}}=b_{t}={\frac {p}{c^{2}}}b$

where

(82)	$b={\sqrt {b_{x}^{2}-c^{2}b_{t}^{2}}}={\frac {c^{2}}{\xi }}$

is the magnitude of acceleration. From (81), (81), (82), the previous assertion follows.^[17]

The acceleration is thus constant for every world line of hyperbolic motion in terms of their magnitude; here lies the analogy with the uniformly accelerated motion of old mechanics represented by parabolic world lines. Thus it is the simplest accelerated motion, and every motion can approximated by hyperbolic motions. Based on that, we want to find out more precisely the dynamics of hyperbolic motions, above all we try to determine the force exerted by an electrically charged rigid body upon itself. The result (as approximation) will then also give information about all motions, in which the magnitude of the acceleration vector is only slightly changed.

Second chapter. The field of the rigid electron in hyperbolic motion.

§ 6. Retarded potentials and field strengths.

The forces exerted by moving electric charges, which enter into the equations of motion of these charges, are derived from certain auxiliary quantities, the retarded potentials and field strengths. We want to summarize the expression for these quantities, which are employed in the following.

Let an electric current be represented by equations of form (14); let the initial value of its rest density (see § 3, p. 18, (51)) be:

$\varrho _{0}(\xi ,\eta ,\zeta )$

Then the retarded potentials are given by the following expressions:

(83)

${\begin{cases}&4\pi \Phi _{x}(x,y,z,t)\\=&\iiint \left[{\frac {{\overline {\varrho }}_{0}{\overline {x}}_{\tau }}{(x-{\overline {x}}){\overline {x}}_{\tau }+(y-{\overline {y}}){\overline {y}}_{\tau }+(z-{\overline {z}}){\overline {z}}_{\tau }-c^{2}(t-{\overline {t}}){\overline {t}}_{\tau }}}\right]_{h=0}d{\overline {\xi }}\ d{\overline {\eta }}\ d{\overline {\zeta }}\\&\dots \dots \\&\dots \dots \\&4\pi \Phi (x,y,z,t)\\=&\iiint \left[{\frac {c{\overline {\varrho }}_{0}{\overline {t}}_{\tau }}{(x-{\overline {x}}){\overline {x}}_{\tau }+(y-{\overline {y}}){\overline {y}}_{\tau }+(z-{\overline {z}}){\overline {z}}_{\tau }-c^{2}(t-{\overline {t}}){\overline {t}}_{\tau }}}\right]_{h=0}d{\overline {\xi }}\ d{\overline {\eta }}\ d{\overline {\zeta }}\end{cases}}$

where ${\overline {x}},{\overline {y}},{\overline {z}},{\overline {t}}$ and ${\overline {x}}_{\tau },{\overline {y}}_{\tau },{\overline {z}}_{\tau },{\overline {t}}_{\tau }$ denote the functions (14) or their derivatives with respect to $\tau$, taken from the arguments ${\overline {\xi }},{\overline {\eta }},{\overline {\zeta }}$, and the function of $x,y,z,t,{\overline {\xi }},{\overline {\eta }},{\overline {\zeta }}$ is to be inserted for $\tau$ into the brackets, which is given by solving the equation

(84)	$h=(x-{\overline {x}})^{2}+(y-{\overline {y}})^{2}+(z-{\overline {z}})^{2}-c^{2}(t-{\overline {t}})^{2}=0$

with respect to $\tau$; namely that solution of the equation which is definitely determined is to be taken^[18], for which $t>{\overline {t}}$. As to how the expressions (83), which are surely not used yet for continuous currents in this form, are connected to the ordinary formulas for the retarded potentials, shall be shortly explained in the next paragraphs.

The electric field strength ${\mathfrak {E}}$ and the magnetic one ${\mathfrak {M}}$ are derived from the potentials according to the vector equations:

(85)	${\begin{cases}{\mathfrak {E}}=-{\frac {1}{c}}{\frac {\partial }{\partial t}}\left(\Phi _{x},\Phi _{y},\Phi _{z}\right)-\mathrm {grad} \Phi ,\\{\mathfrak {M}}=\mathrm {curl} \left(\Phi _{x},\Phi _{y},\Phi _{z}\right).\end{cases}}$

The potentials (83) are solutions of equations^[19]

(86)

${\begin{cases}{\frac {\partial }{\partial x}}\mathrm {lor} \Phi +{\frac {1}{c^{2}}}{\frac {\partial ^{2}\Phi _{x}}{\partial t^{2}}}-\Delta \Phi _{x}={\frac {\varrho ^{*}}{c}}x_{\tau },\\\dots \dots \\{\frac {\partial }{\partial t}}\mathrm {lor} \Phi +{\frac {1}{c^{2}}}{\frac {\partial ^{2}\Phi }{\partial t^{2}}}-\Delta \Phi =\varrho ^{*}t_{\tau },\end{cases}}$

namely especially such solutions, for which the quantity

(87)	$\mathrm {lor} \Phi ={\frac {\partial \Phi _{x}}{\partial x}}+{\frac {\partial \Phi _{y}}{\partial y}}+{\frac {\partial \Phi _{z}}{\partial z}}+{\frac {1}{c}}{\frac {\partial \Phi }{\partial t}}$

vanishes for itself.

Equations (86) are the Lagrangian equations regarding the variation problem^[20], to find those functions $\Phi _{x},\Phi _{y},\Phi _{z},\Phi$ for which the integral

(88)	$W=\iiiint \left\{{\frac {1}{2}}\left({\mathfrak {M}}^{2}-{\mathfrak {E}}^{2}\right)-{\frac {\varrho ^{*}}{c}}\left(\Phi _{x}x_{\tau }+\Phi _{y}y_{\tau }+\Phi _{z}z_{\tau }-\Phi t_{\tau }\right)\right\}dx\ dy\ dz\ dt$

extended over an area $G$ of the $xyzt$-manifold, becomes an extremum, where the current of electricity and the values of the potentials are given upon the boundary of $G$.

§ 7. Comparison of the expressions for retarded potentials.

The expressions (83) for the potentials can be seen as the superposition of the elementary potential, stemming from the individually moving points of the current. For the latter, one namely has the expressions according to Liénard and Wiechert^[21]

(89)	${\begin{cases}4\pi \varphi _{x}=\left[{\frac {ew_{x}}{cr\left(1-{\frac {w_{r}}{c}}\right)}}\right]_{t-{\overline {t}}={\frac {r}{c}}}\\\dots \dots \\\dots \dots \\4\pi \varphi =\left[{\frac {e}{r\left(1-{\frac {w_{r}}{c}}\right)}}\right]_{t-{\overline {t}}={\frac {r}{c}}}\end{cases}}$

Where $e$ denotes the charge of the acting point

(90)	$x={\overline {x}}(t),\ y={\overline {y}}(t),\ z={\overline {z}}(t)$;

furthermore

(91)	$r={\sqrt {(x-{\overline {x}})^{2}+(y-{\overline {y}})^{2}+(z-{\overline {z}})^{2}}}$

its distance from reference point $x,y,z$,

(92)	$w_{r}={\frac {1}{r}}\left\{(x-{\overline {x}}){\overline {w}}_{x}+(y-{\overline {y}}){\overline {w}}_{y}+(z-{\overline {z}}){\overline {w}}_{z}\right\}$

is the component of its velocity $w_{x},w_{y},w_{z}$ in the direction of $r$, and ${\overline {t}}$ instead of $t$ is to be set in the brackets, which is given from the equation

(93)	$t-{\overline {t}}={\frac {r}{c}}$

If one now has a continuous current, then world line (90) is to be replaced by a bundle of world lines, by bringing function (90) into the form (1) by inserting three parameters $\xi ,\eta ,\zeta$, and by replacing $e$ by density $\varrho (\xi ,\eta ,\zeta )$. Then, functions $\varphi _{x},\varphi _{y},\varphi _{z},\varphi$ become also dependent on $\xi ,\eta ,\zeta$, and one can integrate them over the entire space. There it is also to be noticed, that it is

$dx\ dy\ dx=\Theta d\xi \ d\eta \ d\zeta$

at the space integration, and that the functional determinant $\Theta$ connects itself with density $\varrho$ to the initial density $\varrho _{0}=\varrho \Theta$ according to § 3, (41).

The emerged expressions can easily be brought into the form (83). For this, one only has to write the equations of motion of the acting point homogeneously in the form:

(94)	$x={\overline {x}}(\tau ),\ y={\overline {y}}(\tau ),\ z={\overline {z}}(\tau ),\ t={\overline {t}}(\tau )$

where $\tau$ denotes the proper time, and to replace $\varrho$ by the rest density $\varrho ^{*}$. Equation (93) then goes over into

(95)	$h=(x-{\overline {x}})^{2}+(y-{\overline {y}})^{2}+(z-{\overline {z}})^{2}-c^{2}(t-{\overline {t}})^{2}=0$

from which $\tau$ is unequivocally given at the additional condition $t>{\overline {t}}$.^[22]

The connection of expressions (83) with the otherwise ordinary expressions for the potentials is also easy to find out. The latter ones read:^[23]

(96)

${\begin{cases}4\pi \Phi _{x}=\iiint {\frac {d{\overline {x}}\ d{\overline {y}}\ d{\overline {z}}}{r}}\left[{\frac {\varrho w_{x}}{c}}\right]_{{\overline {t}}=t-{\frac {r}{c}}}\\\dots \dots \\\dots \dots \\4\pi \Phi =\iiint {\frac {d{\overline {x}}\ d{\overline {y}}\ d{\overline {z}}}{r}}\left[\varrho \right]_{{\overline {t}}=t-{\frac {r}{c}}}\end{cases}}$

There, the current is to be imagined as represented by equations of the form (1) (p. 6), furthermore it is

$r={\sqrt {(x-{\overline {x}})^{2}+(y-{\overline {y}})^{2}+(z-{\overline {z}})^{2}}}$

and ${\overline {x}},{\overline {y}},{\overline {z}},\ {\overline {t}}=t-r/c$ are to be introduced as arguments in the brackets. The integrations in (95) are to be extended over all charges, i.e., over temporally variable boundaries, since they are in motion. The passage from expressions (95) to expressions (83) now exactly consists in this, that one brings the integrals to invariable limits independent of time. This has to be happening in the following way.

In the equations of current (1), we replace $t$ by ${\overline {t}}=t-r/c$, then we obtain equations of the form

(97)

${\begin{cases}{\overline {x}}={\overline {x}}\left\{\xi ,\eta ,\zeta ,{\overline {t}}({\overline {x}},{\overline {y}},{\overline {z}},{\overline {t}})\right\}\\{\overline {y}}={\overline {y}}\left\{\xi ,\eta ,\zeta ,{\overline {t}}({\overline {x}},{\overline {y}},{\overline {z}},{\overline {t}})\right\}\\{\overline {t}}={\overline {t}}\left\{\xi ,\eta ,\zeta ,{\overline {t}}({\overline {x}},{\overline {y}},{\overline {z}},{\overline {t}})\right\}\end{cases}}$

which connect ${\overline {x}},{\overline {y}},{\overline {z}}$ with $\xi ,\eta ,\zeta$ and evidently exactly represent the transformation, which brings the integrals to invariable limits when applied to (96); because this transformation (97) represents ${\overline {x}},{\overline {y}},{\overline {z}}$ as function of their initial values for the relevant instant in the brackets.

To calculate the functional determinant of transformation (97)

(98)	$\Delta =\left[{\frac {\partial ({\overline {x}},{\overline {y}},{\overline {z}})}{\partial (\xi ,\eta ,\zeta )}}\right]_{t=\mathrm {konst.} }$

we want to denote the derivative from ${\overline {x}}$ to $\xi$ at maintained ${\overline {t}}$ with $(\partial {\overline {x}}/\partial \xi )_{\overline {t}}$, and at maintained $t$ with $(\partial {\overline {x}}/\partial \xi )_{t}$. If we differentiate then equations (97) (one after the other) with respect to $\xi ,\eta ,\zeta$, then we obtain three equation systems with identical coefficients; for example at the differentiation with respect to $\xi$:

${\begin{array}{l}\left({\frac {\partial {\overline {x}}}{\partial \xi }}\right)_{t}=\left({\frac {\partial {\overline {x}}}{\partial \xi }}\right)_{\overline {t}}+{\frac {\partial {\overline {x}}}{\partial {\overline {t}}}}\left\{{\frac {\partial {\overline {t}}}{\partial {\overline {x}}}}\left({\frac {\partial {\overline {x}}}{\partial \xi }}\right)_{t}+{\frac {\partial {\overline {t}}}{\partial {\overline {y}}}}\left({\frac {\partial {\overline {y}}}{\partial \xi }}\right)_{t}+{\frac {\partial {\overline {t}}}{\partial {\overline {z}}}}\left({\frac {\partial {\overline {z}}}{\partial \xi }}\right)_{t}\right\},\\\\\left({\frac {\partial {\overline {y}}}{\partial \xi }}\right)_{t}=\left({\frac {\partial {\overline {y}}}{\partial \xi }}\right)_{\overline {t}}+{\frac {\partial {\overline {y}}}{\partial {\overline {t}}}}\left\{{\frac {\partial {\overline {t}}}{\partial {\overline {x}}}}\left({\frac {\partial {\overline {x}}}{\partial \xi }}\right)_{t}+{\frac {\partial {\overline {t}}}{\partial {\overline {y}}}}\left({\frac {\partial {\overline {y}}}{\partial \xi }}\right)_{t}+{\frac {\partial {\overline {t}}}{\partial {\overline {z}}}}\left({\frac {\partial {\overline {z}}}{\partial \xi }}\right)_{t}\right\},\\\\\left({\frac {\partial {\overline {z}}}{\partial \xi }}\right)_{t}=\left({\frac {\partial {\overline {z}}}{\partial \xi }}\right)_{\overline {t}}+{\frac {\partial {\overline {z}}}{\partial {\overline {t}}}}\left\{{\frac {\partial {\overline {t}}}{\partial {\overline {x}}}}\left({\frac {\partial {\overline {x}}}{\partial \xi }}\right)_{t}+{\frac {\partial {\overline {t}}}{\partial {\overline {y}}}}\left({\frac {\partial {\overline {y}}}{\partial \xi }}\right)_{t}+{\frac {\partial {\overline {t}}}{\partial {\overline {z}}}}\left({\frac {\partial {\overline {z}}}{\partial \xi }}\right)_{t}\right\},\end{array}}$

or

${\begin{array}{lll}\left({\frac {\partial {\overline {x}}}{\partial \xi }}\right)_{t}\left(1+{\frac {1}{c}}{\frac {\partial {\overline {x}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {x}}}}\right)&+\left({\frac {\partial {\overline {y}}}{\partial \xi }}\right)_{t}{\frac {1}{c}}{\frac {\partial {\overline {x}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {y}}}}\\\\&+\left({\frac {\partial {\overline {z}}}{\partial \xi }}\right)_{t}{\frac {1}{c}}{\frac {\partial {\overline {x}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {z}}}}&=\left({\frac {\partial {\overline {x}}}{\partial \xi }}\right)_{\overline {t}},\\\\\left({\frac {\partial {\overline {x}}}{\partial \xi }}\right)_{t}{\frac {1}{c}}{\frac {\partial {\overline {y}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {x}}}}&+\left({\frac {\partial {\overline {y}}}{\partial \xi }}\right)_{t}\left(1+{\frac {1}{c}}{\frac {\partial {\overline {y}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {y}}}}\right)\\\\&+\left({\frac {\partial {\overline {z}}}{\partial \xi }}\right)_{t}{\frac {1}{c}}{\frac {\partial {\overline {y}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {z}}}}&=\left({\frac {\partial {\overline {y}}}{\partial \xi }}\right)_{\overline {t}},\\\\\left({\frac {\partial {\overline {x}}}{\partial \xi }}\right)_{t}{\frac {1}{c}}{\frac {\partial {\overline {z}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {x}}}}&+\left({\frac {\partial {\overline {y}}}{\partial \xi }}\right)_{t}{\frac {1}{c}}{\frac {\partial {\overline {z}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {y}}}}\\\\&+\left({\frac {\partial {\overline {z}}}{\partial \xi }}\right)_{t}\left(1+{\frac {1}{c}}{\frac {\partial {\overline {z}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {z}}}}\right)&=\left({\frac {\partial {\overline {z}}}{\partial \xi }}\right)_{\overline {t}},\end{array}}$

Three equation are added twice, at which $\xi$ is replaced by $\eta$ or $\zeta$. If we denote the matrices occurring here as follows:

(99)

$P=\left({\begin{array}{ccccc}\left({\frac {\partial {\overline {x}}}{\partial \xi }}\right)_{t}&&\left({\frac {\partial {\overline {y}}}{\partial \xi }}\right)_{t}&&\left({\frac {\partial {\overline {z}}}{\partial \xi }}\right)_{t}\\\\\left({\frac {\partial {\overline {x}}}{\partial \eta }}\right)_{t}&&\left({\frac {\partial {\overline {y}}}{\partial \eta }}\right)_{t}&&\left({\frac {\partial {\overline {z}}}{\partial \eta }}\right)_{t}\\\\\left({\frac {\partial {\overline {x}}}{\partial \zeta }}\right)_{t}&&\left({\frac {\partial {\overline {y}}}{\partial \zeta }}\right)_{t}&&\left({\frac {\partial {\overline {z}}}{\partial \zeta }}\right)_{t}\end{array}}\right),$

(100)

$Q=\left({\begin{array}{ccccc}1+{\frac {1}{c}}{\frac {\partial {\overline {x}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {x}}}}&&{\frac {1}{c}}{\frac {\partial {\overline {x}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {y}}}}&&{\frac {1}{c}}{\frac {\partial {\overline {x}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {z}}}}\\\\{\frac {1}{c}}{\frac {\partial {\overline {y}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {x}}}}&&1+{\frac {1}{c}}{\frac {\partial {\overline {y}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {y}}}}&&{\frac {1}{c}}{\frac {\partial {\overline {y}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {z}}}}\\\\{\frac {1}{c}}{\frac {\partial {\overline {z}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {x}}}}&&{\frac {1}{c}}{\frac {\partial {\overline {z}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {y}}}}&&1+{\frac {1}{c}}{\frac {\partial {\overline {z}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {z}}}}\end{array}}\right),$

(101)

$R=\left({\begin{array}{ccccc}\left({\frac {\partial {\overline {x}}}{\partial \xi }}\right)_{\overline {t}}&&\left({\frac {\partial {\overline {y}}}{\partial \xi }}\right)_{\overline {t}}&&\left({\frac {\partial {\overline {z}}}{\partial \xi }}\right)_{\overline {t}}\\\\\left({\frac {\partial {\overline {x}}}{\partial \eta }}\right)_{\overline {t}}&&\left({\frac {\partial {\overline {y}}}{\partial \eta }}\right)_{\overline {t}}&&\left({\frac {\partial {\overline {z}}}{\partial \eta }}\right)_{\overline {t}}\\\\\left({\frac {\partial {\overline {x}}}{\partial \zeta }}\right)_{\overline {t}}&&\left({\frac {\partial {\overline {y}}}{\partial \zeta }}\right)_{\overline {t}}&&\left({\frac {\partial {\overline {z}}}{\partial \zeta }}\right)_{\overline {t}}\end{array}}\right),$

then our nine equations can be summarized in the matrix equation:

$P{\overline {Q}}=R$

From that the relation of determinants follows:

(102)

$\left|P\right|\cdot \left|Q\right|=\left|R\right|$

Now it is evidently according to (98)

(103)

$\left|P\right|=\Delta$

furthermore it is according to § 3, (42), p. 16:

(104)

$\left|R\right|=\left|\Theta \right|_{{\overline {t}}=t-{\frac {r}{c}}}$

Finally one finds easily

${\begin{array}{ll}\left|Q\right|&=1+{\frac {1}{c}}\left({\frac {\partial {\overline {x}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {x}}}}+{\frac {\partial {\overline {y}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {y}}}}+{\frac {\partial {\overline {z}}}{\partial {\overline {t}}}}{\frac {\partial r}{\partial {\overline {z}}}}\right),\\\\&=1+{\frac {1}{c}}\left({\overline {w}}_{x}{\frac {\partial r}{\partial {\overline {x}}}}+{\overline {w}}_{y}{\frac {\partial r}{\partial {\overline {y}}}}+{\overline {w}}_{z}{\frac {\partial r}{\partial {\overline {z}}}}\right)\end{array}}$

which is to be written according to (92):

(105)

$Q=\left[1-{\frac {w_{r}}{c}}\right]_{{\overline {t}}=t-{\frac {r}{c}}}$

Consequently it becomes:

(106)

$\Delta =\left[{\frac {\Theta }{1-{\frac {w_{r}}{c}}}}\right]_{{\overline {t}}=t-{\frac {r}{c}}}$

If we insert this into (96) and consider that according to § 3, (41), p. 16, it is to be set:

$\varrho \Theta =\varrho (\xi ,\eta ,\zeta )$,

then it follows:

(107)

${\begin{cases}4\pi \Phi _{x}=\iiint \varrho _{0}(\xi ,\eta ,\zeta )\left[{\frac {w_{x}}{cr\left(1-{\frac {w_{r}}{c}}\right)}}\right]_{t-{\overline {t}}={\frac {r}{c}}}d\xi \ d\eta \ d\zeta ,\\\dots \dots \\4\pi \Phi =\iiint \varrho _{0}(\xi ,\eta ,\zeta )\left[{\frac {1}{r\left(1-{\frac {w_{r}}{c}}\right)}}\right]_{t-{\overline {t}}={\frac {r}{c}}}d\xi \ d\eta \ d\zeta ,\end{cases}}$

If we also replace $w_{x}$ by $x_{\tau }/t_{\tau }$ etc. as on p. 30, then formulas (107) go directly over into expressions (83). Indeed, only the initial density $\varrho _{0}$ occurs in them.

§ 8. Calculation of the potentials at hyperbolic motion.

We now want to evaluate the potentials (83) for hyperbolic motion

(108)

$x=-q\xi ,\ y=\eta ,\ z=\zeta ,\ t={\frac {p}{c^{2}}}\xi$

Since $y_{\tau }=z_{\tau }=0$, then also $\Phi _{y}=\Phi _{z}=0$. Since we have in (108) the quantity $p$ instead of $\tau$ as independent variable besides $\xi ,\eta ,\zeta$, we will see equation (84) $h=0$ as equation for $p$ as well. Then it reads:

(109)

$\left(x+{\overline {q}}{\overline {\xi }}\right)^{2}+\left(y-{\overline {\eta }}\right)^{2}+\left(z-{\overline {\zeta }}\right)^{2}-c^{2}\left(t-{\frac {{\overline {q}}{\overline {\xi }}}{c^{2}}}\right)^{2}=0$;

when one introduces the abbreviations

(110)

${\begin{cases}s=x^{2}-c^{2}t^{2}=\xi ^{2},\\k=-{\frac {1}{2\xi }}\left(s+{\overline {\xi }}^{2}+\left(y-{\overline {\eta }}\right)^{2}+\left(z-{\overline {\zeta }}\right)^{2}\right)\end{cases}}$

one can write (109) as follows:

${\overline {p}}t+{\overline {q}}x=k$;

where it is added:

${\overline {p}}^{2}-c^{2}{\overline {q}}^{2}=-c^{2}$

${\overline {p}}$ is to be calculated from these equations; namely that value of ${\overline {p}}$ is to be chosen, for which $t>{\overline {t}}$. If one inserts the value following from the first equation

${\overline {p}}={\frac {k-{\overline {q}}x}{t}}$

into the second one, then the quadratic equation emerges for ${\overline {q}}$:

${\overline {q}}^{2}-{\overline {q}}{\frac {2kx}{s}}=-{\frac {k^{2}+c^{2}t^{2}}{s}}$

From that if follows:

${\overline {q}}={\frac {1}{s}}\left(kx+ct{\sqrt {k^{2}-s}}\right)$

If we also set (for abbreviation) for the positive square root

(111)

$B={\sqrt {k^{2}-s}}$

and calculate ${\overline {p}}$, then we find

${\begin{array}{lr}{\overline {p}}=&-{\frac {c}{s}}(kct\pm Bx)\\\\{\overline {q}}=&{\frac {1}{s}}(kx\pm Bct)\end{array}}$

Here, that sign is to be chosen, which corresponds to the smaller value of ${\overline {t}}$. Now, since ${\overline {t}}={\overline {q}}{\overline {\xi }}/c^{2}$, then the following is given (presupposed, that the electron is moving on the right side of the origin $x=0$, i.e., ${\overline {\xi }}>0$):

for all reference points, at which $x/s>0$, the positive sign is to be taken,

for all reference points, at which $x/s<0$, the negative sign is to be taken.

The distribution of these reference points can be derived from the figure.

Fig. 6.

We want to presuppose mostly in the following, that $x/s>0$; only such points can be interior points of the electron at hyperbolic motion. Thus the positive square root $B$ is to be taken for them. When we sometimes also consider points for which $x/s<0$, then we have to replace $+B$ by $-B$ everywhere.

Thus we have:

(112)

${\begin{cases}{\overline {p}}=&-{\frac {c}{s}}(kct+Bx)\\{\overline {q}}=&{\frac {1}{s}}(kx+Bct)\end{cases}}$

Now we calculate the denominator of integral (83) for these values of ${\overline {p}},\ {\overline {q}}$.

Due to $y_{\tau }=z_{\tau }=0,\ x_{\tau }=p,\ t_{\tau }=-q$, it becomes:

$(x+{\overline {q}}{\overline {\xi }}){\overline {p}}+c^{2}\left(t-{\frac {\overline {p}}{c^{2}}}{\overline {\xi }}\right){\overline {q}}=x{\overline {p}}+c^{2}t{\overline {q}}=-cB$

This is inserted into the integrals; it follows:

(113)

${\begin{cases}4\pi \Phi _{x}(x,y,z,t)=\iiint {\frac {{\overline {\varrho }}_{0}}{sB}}(kct+Bx)d{\overline {\xi }}\ d{\overline {\eta }}\ d{\overline {\zeta }},\\4\pi \Phi (x,y,z,t)=\iiint {\frac {{\overline {\varrho }}_{0}}{sB}}(kx+Bct)d{\overline {\xi }}\ d{\overline {\eta }}\ d{\overline {\zeta }},\end{cases}}$

If we set for abbreviation:

(114)

${\begin{cases}\psi _{1}(s)={\frac {1}{s}}\iiint {\overline {\varrho }}_{0}d{\overline {\xi }}\ d{\overline {\eta }}\ d{\overline {\zeta }}={\frac {e}{s}},\\\psi _{2}(s)={\frac {1}{s}}\iiint {\overline {\varrho }}_{0}{\frac {k}{B}}d{\overline {\xi }}\ d{\overline {\eta }}\ d{\overline {\zeta }},\end{cases}}$

where $e$ denotes the total charge of the electron, then it simply becomes:

(115)

${\begin{cases}4\pi \Phi _{x}=\psi _{1}(s)\cdot x+\psi _{2}(s)\cdot ct,\\4\pi \Phi =\psi _{2}(s)\cdot x+\psi _{1}(s)\cdot ct.\end{cases}}$

Here, $\psi _{1}$ and $\psi _{2}$ are functions of the connection $s$ from $x$ to $t$ alone.

These potentials particularly satisfy equation (87) $\mathrm {lor} \Phi =0$; because one has, since $\partial s/\partial x=2x$, $\partial s/\partial t=-2c^{2}t$:

${\begin{array}{l}4\pi {\frac {\partial \Phi _{x}}{\partial x}}=\psi _{1}+2\psi _{1}'x^{2}+2\psi _{2}'ctx,\\\\4\pi {\frac {\partial \Phi }{\partial t}}=\psi _{1}c-2\psi _{2}'c^{2}tx-2\psi _{1}'c^{3}t^{2}\end{array}}$

thus