Translation:On the Theory of Relativity II: Four-dimensional Vector Analysis

by Arnold Sommerfeld

On the Theory of Relativity. II.

Four-dimensional vector analysis^[1];

by A. Sommerfeld.

§ 5. The differential operations of four-dimensional vector analysis.

Instead of Minkowski's general symbol lor (Lorentz operation), we introduce the more specific differential operator

Div, Rot, Grad,

as four-dimensional extensions of the usual operations in ordinary vector-calculus

div, rot, grad.

The summarizing symbol lor will be preferred (similarly to the Hamiltonian $\nabla$ in ordinary vector calculus), when one wants (neglecting the illustrative meaning of the single steps) to symbolically verify the vector formulas. However, since at this place as well as in part I, exactly this geometrical interpretation shall be emphasized, it es recommendable to specialize the symbol lor (depending on its application to six-, four-vectors or scalars) in the given way. There, the divergence operation is employed in a dual meaning, as vector divergence and as scalar divergence, so that four fundamental vector operations actually exist in four dimensions, against the three differential operations of ordinary vector calculus. For distinction, I will write the vector divergence by German letters (${\mathfrak {Div}}$), the scalar divergence by Latin ones (Div).

We will operate at any place, as if the fourth of our world coordinates xyzl were real (in this connection, see the note on p. 752 of part I). This fiction, as far as I see, nowhere encounters difficulties, it is, on the other hand, an essential presupposition for the simplicity of the geometrical way of expression, according to which we will speak in the following, for example, simply about a perpendicularity instead of a non-euclidean perpendicularity, and it makes it possible to supplement the four-dimensional vector expressions in the closest way to the well-known three-dimensional ones.

The following way traversed by us twice in opposite direction, allows us to see that our list might be complete, and that the results obtained in this way, are by their definition independent of the coordinate system.

a) The scalar divergence. Let $\Delta \Sigma$ be an arbitrarily formed, four-dimensional, and infinitely small section of space^[2] in the surrounding of the considered spacetime-point $Q$, $ds$ the element of the (three-dimensional) boundary of $\Delta \Sigma$, $n$ the outer normal to $dS$. Let $P$ be an arbitrary four-vector, $P_{n}$ its normal component formed in the sense of equation (7). From the four-vector $P$ a scalar magnitude Div $P$ emerges, which we define as follows^[3]:

(16)	$\mathrm {Div} \ P={\underset {\Delta \Sigma =0}{\mathsf {Lim}}}{\frac {\int P_{n}d\sigma }{\Delta \Sigma }}$

where the integration with respect to $ds$ is to be extended over the entire boundary of $\Delta \Sigma$.

If we choose $\Delta \Sigma$ especially as a four-dimensional parallelepiped with edge lengths dx dy dz dl, then we have

(16a)

$\mathrm {Div} \ P={\frac {\partial P_{x}}{\partial x}}+{\frac {\partial P_{y}}{\partial y}}+{\frac {\partial P_{z}}{\partial z}}+{\frac {\partial P_{l}}{\partial l}}.$

If $P$ is the four-density defined in (1), then, as it was noticed by Minkowski,

$\mathrm {Div} \ P={\frac {1}{c}}\left({\frac {\partial \varrho {\mathfrak {v}}_{x}}{\partial x}}+{\frac {\partial \varrho {\mathfrak {v}}_{y}}{\partial y}}+{\frac {\partial \varrho {\mathfrak {v}}_{z}}{\partial z}}+{\frac {\partial \varrho }{\partial t}}\right)$

becomes identical with the left-hand side of the continuity equation in ordinary hydrodynamics of compressible fluids (up to the factor $1/c$).

b) The vector divergence. While we started under a) with a vector of first kind (four-vector) where we obtained a "vector of zero kind" (scalar), we now start with a vector of second kind (six-vector) and derive a vector of first kind (four-vector) from it. Its component with respect to an arbitrary direction $s$ is defined by us in the following way. Let $S$ be the three-dimensional space extended through the considered spacetime-point $Q$ perpendicular to $s$, $\Delta S$ an infinitely small area of it in the surrounding of $Q$, $d\sigma$ the element of its (two-dimensional) boundary; the plane perpendicular to it, unequivocally defined by note 2 on p. 753 and containing both the direction $s$ as well as the (outer) normal direction $n$ of $d\sigma$ extended in space $S$, shall be indicated by $sn$, and $f_{sn}$ shall denote the component of six-vector $f$ (formed in the sense of equation (8)) with respect to this plane. Then, let the $s$-component of the vector-divergence of $f$ be:

(17)	${\mathfrak {Div}}\ f_{s}={\underset {\Delta S=0}{\mathsf {Lim}}}{\frac {\int f_{sn}d\sigma }{\Delta S}}$

where the integration with respect to $d\sigma$ is related to the whole boundary of $\Delta S$.

If we especially choose $s$ as $x$-direction, $\Delta S$ as three-dimensional parallelepiped (located in $yzl$-space) of border length dy, dz, dl, then it is given by (17):

${\begin{array}{cc}{\mathfrak {Div}}\ f_{x}=&{\frac {1}{dx\ dz\ dl}}\left\{dz\ dl\cdot {\frac {\partial f_{xy}}{\partial y}}dy+dl\ dy\cdot {\frac {\partial f_{xz}}{\partial z}}dz\right.\\\\&\left.+dy\ dz\cdot {\frac {\partial f_{xl}}{\partial l}}dl\right\}={\frac {\partial f_{xy}}{\partial y}}+{\frac {\partial f_{xz}}{\partial z}}+{\frac {\partial f_{xl}}{\partial l}}\end{array}}$

and somewhat more generally for any of the coordinate directions $j=x,y,z,l$:

(17a)

${\mathfrak {Div}}\ f_{j}={\frac {\partial f_{jx}}{\partial x}}+{\frac {\partial f_{jy}}{\partial y}}+{\frac {\partial f_{jz}}{\partial z}}+{\frac {\partial f_{jl}}{\partial l}}$

in which instant one of the four derivatives, of course, vanishes due to $f_{jj}=0$. The formal agreement of this formation with that in (16a) may motivate us to maintain, despite of the different geometrical meaning, the same name.

If $f$ particularly means the six-vector of the field, then, for example, due to (2a) and (2a) it becomes for $j=x$:

${\mathfrak {Div}}\ f_{x}={\frac {\partial {\mathfrak {H}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {H}}_{y}}{\partial z}}-{\frac {1}{c}}{\frac {\partial {\mathfrak {E}}_{x}}{\partial t}}$

and for $j=l$:

${\mathfrak {Div}}\ f_{l}=i\left({\frac {\partial {\mathfrak {E}}_{x}}{\partial x}}+{\frac {\partial {\mathfrak {E}}_{y}}{\partial y}}+{\frac {\partial {\mathfrak {E}}_{z}}{\partial z}}\right)$

According to the field equations, the first expression is equal to $\varrho {\mathfrak {v}}_{x}/c$ and the latter equal to $i\varrho$, so that the four-density directly emerges from the field vector by operation ${\mathfrak {Div}}$. The first half of the Maxwell-Lorentz equations, including the electric divergence condition, can thus simply be written:

(18)	${\mathfrak {Div}}\ f=P$

c) The supplement of rotation. We start with a vector of third kind ${\mathfrak {A}}$ (three-dimensional space magnitude, see part I, p. 759), with which we simultaneously consider its supplement (a vector of first kind), denote by us as ${\mathfrak {B}}$ for distinction, where ${\mathfrak {B}}_{x}={\mathfrak {A}}_{yzl}$ for example. From that, we derive a vector of second kind, the rotation of ${\mathfrak {A}}$ or the supplement of the rotation of ${\mathfrak {B}}$; its component with respect to a plane $s's''$ is defined by us as follows: Let $\sigma$ be the plane normal to $s's''$ through the considered point $Q$, $\Delta \sigma$ an infinitely small area of $\sigma$ in the surrounding of $Q$, $ds$ a boundary element of $\Delta \sigma$. The normal space with respect to $ds$ contains the directions $s's''$ and the direction of the outer normal $n$ (drawn within $\Delta \sigma$) with respect to $ds$. The component of ${\mathfrak {A}}$ with respect to this normal space is ${\mathfrak {A}}_{s's''n}={\mathfrak {B}}_{s}$. Then, let the component of rotation of ${\mathfrak {A}}$ with respect to plane $s's''$ be:

(19)	$\left\{{\begin{array}{ll}\mathrm {Rot} _{s's''}{\mathfrak {A}}&={\underset {\Delta \sigma =0}{\mathsf {Lim}}}{\frac {\int {\mathfrak {A}}_{s's''n}ds}{\Delta \sigma }}\\\\&={\underset {\Delta \sigma =0}{\mathsf {Lim}}}{\frac {\int {\mathfrak {B}}_{s}ds}{\Delta \sigma }}\end{array}}\right.$

If, for example, $\Delta \sigma$ is a rectangle with side-lengths dn, ds, where the succession of the directions $s's''ns$ is the positive one, i.e. the same as that of the axes xyzl, then:

(19a)

$\left\{{\begin{array}{ll}\mathrm {Rot} _{s's''}{\mathfrak {A}}&={\frac {1}{dn\ ds}}\left\{ds\cdot {\frac {\partial {\mathfrak {B}}_{s}}{\partial n}}dn-dn\cdot {\frac {\partial {\mathfrak {B}}_{n}}{\partial s}}ds\right\}\\\\&={\frac {\partial {\mathfrak {B}}_{s}}{\partial n}}-{\frac {\partial {\mathfrak {B}}_{n}}{\partial s}}\end{array}}\right.$

It is in agreement with the following definition for the rotation of a vector of first kind ${\mathfrak {B}}$ and the earlier equation (4c) for the connection of a six-vector with its supplement, when we also write instead:

(19b)

$\mathrm {Rot} _{ns}{\mathfrak {B}}=\mathrm {Rot} _{s's''}^{*}{\mathfrak {B}}=\mathrm {Rot} _{s's''}{\mathfrak {A}}=\mathrm {Rot} _{ns}^{*}{\mathfrak {A}}.$

By the definition (19) we thus calculate the supplement of rotation with respect to plane $s's''$ for the vector of first kind ${\mathfrak {B}}$, or the rotation with respect to its perpendicular plane $ns$; but simultaneously also the rotation for the plane $s's''$ for the vector of third kind ${\mathfrak {A}}$, or its supplement for its perpendicular plane $ns$.

d) The gradient. It would be in agreement with the things done thus far, to start with a four-dimensional space magnitude $U$, which (as undirected) will have a scalar character, and to derive from it a vector of third kind, which (by its components) shall be taken with respect to an arbitrary space $ss's''$. For that, we would have (in the direction normal to $ss's''$) to separate an infinitely small, linear area $\Delta n$, and the difference $\Delta U$ as replacement for the degenerated integration over the boundary points of $\Delta n$, and eventually to form as component of the emerging vector of third kind:

$\mathrm {Grad} _{ss's''}U={\underset {\Delta n=0}{\mathsf {Lim}}}{\frac {\Delta U}{\Delta n}}={\frac {\partial U}{\partial n}}$

Instead, we will dually reverse our process by starting with a vector of zero kind $V$, i.e. a scalar magnitude too, and derive from it a vector of first kind, the gradient of $V$, by accordingly defining its component with respect to direction $s$:

(20)	$\mathrm {Grad} _{s}V={\frac {\partial V}{\partial s}},$

thus especially its four right-angled components by:

(20a)

${\frac {\partial V}{\partial x}},\ {\frac {\partial V}{\partial y}},\ {\frac {\partial V}{\partial z}},\ {\frac {\partial V}{\partial l}}$

c) Rotation. Now, we start with a vector of first kind ${\mathfrak {B}}$ and derive from it the vector of second kind, its rotation. We obtain its component with respect to any plane, by separating an area $\Delta \sigma$ from this plane, then we extend the line integral of ${\mathfrak {B}}$ around it etc., according to the formula analogous to (19) and (19a,b):

$\mathrm {Rot} _{s's''}{\mathfrak {B}}={\underset {\Delta \sigma =0}{\mathsf {Lim}}}{\frac {\int {\mathfrak {B}}_{s}ds}{\Delta \sigma }}={\frac {\partial {\mathfrak {B}}_{s''}}{\partial s'}}-{\frac {\partial {\mathfrak {B}}_{s'}}{\partial s''}}$

(21)

Under $s's''$ we shall understand two mutually perpendicular directions located in $\Delta \sigma$, so that the rotation from $s'$ to $s''$ has the same orientation, as the rotation of $s$ around $\Delta \sigma$. In the current process we thus directly obtain the rotation of a vector of first kind, instead of its supplement as in c).

A suitable example gives the concept of electrodynamic potential, to whose natural introduction we will resort in the next paragraph at equation (25a), while it is only historically mentioned at this place. Let us combine the vector potential ${\mathfrak {A}}$ and the scalar potential $\varphi$ of the ordinary theory to the "vector potential" $\Phi$, with the components

(21a)

$\Phi _{x}={\mathfrak {A}}_{x},\ \Phi _{y}={\mathfrak {A}}_{y},\ \Phi _{z}={\mathfrak {A}}_{z},\ \Phi _{l}=i\varphi$

From it, the field can be calculated by the uniform formula

(21b)

$f=\mathrm {Rot} \ \Phi$

for example (see (21)):

(21b)

$\left\{{\begin{array}{l}f_{xy}={\frac {\partial \Phi _{y}}{\partial x}}-{\frac {\partial \Phi _{x}}{\partial y}}={\frac {\partial {\mathfrak {A}}_{y}}{\partial x}}-{\frac {\partial {\mathfrak {A}}_{x}}{\partial y}},\\\\f_{xl}={\frac {\partial \Phi _{l}}{\partial x}}-{\frac {\partial \Phi _{x}}{\partial l}}=i\left({\frac {\partial \varphi }{\partial x}}+{\frac {1}{c}}{\frac {\partial {\mathfrak {A}}_{x}}{\partial t}}\right),\end{array}}\right.$

which summarizes the asymmetric formulas of ordinary theory:

${\mathfrak {H}}=\mathrm {rot} \ {\mathfrak {A}},\ {\mathfrak {E}}=-\mathrm {grad} \ \varphi -{\frac {1}{c}}{\frac {\partial {\mathfrak {A}}}{\partial t}}$

Between the scalar and vector potential, in the ordinary theory one has the complicated condition:

$\mathrm {div} \ {\mathfrak {A}}+{\frac {1}{c}}{\frac {\partial \varphi }{\partial t}}=0$

which now simply reads by (16a):

(21c)

$\mathrm {Div} \ \Phi =0$

b') The supplement of vector divergence. Starting from a vector of second kind $f$, we derive the component of a vector of third kind with respect to any space $S$, by separating (within $S$) an infinitely small space section $\Delta S$ with the two-dimensional surface element $d\sigma$ and the mutually perpendicular direction $s's''$ contained in it. While we formed the normal component of $f$ to $d\sigma$ in b), we now consider the surface integral of the tangential component of $f$, namely:

(22)	${\mathfrak {Div}}\ f_{s}^{*}={\underset {\Delta S=0}{\mathsf {Lim}}}{\frac {\int f_{s's''}d\sigma }{\Delta S}}$

If we especially choose $s$ as $x$-direction, and $\Delta S$ as three-dimensional parallelepiped in $yzl$-space, then the three surface pairs $s's''$ become parallel to its boundary or zl, ly, yz respectively, and therefore:

${\mathfrak {Div}}\ f_{x}^{*}={\frac {\partial f_{zl}}{\partial y}}+{\frac {\partial f_{ly}}{\partial z}}+{\frac {\partial f_{yz}}{\partial l}}.$

(22a)

For the right-hand side we can write by (4b):

${\frac {\partial f_{xy}^{*}}{\partial y}}+{\frac {\partial f_{xz}^{*}}{\partial z}}+{\frac {\partial f_{xl}^{*}}{\partial l}}$

by which the chosen denotation ${\mathfrak {Div}}\ f_{x}^{*}$ and more general ${\mathfrak {Div}}\ f_{s}^{*}$ are justified with respect to (17a). If $f$ means the field vector, we have by (22a) and (2):

${\mathfrak {Div}}\ f_{x}^{*}=i\left({\frac {\partial {\mathfrak {E}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {E}}_{y}}{\partial z}}+{\frac {1}{c}}{\frac {\partial {\mathfrak {H}}_{x}}{\partial t}}\right),$

similarly for the $y$- and $z$-direction and for the $l$-axis:

${\mathfrak {Div}}\ f_{l}^{*}={\frac {\partial f_{lx}^{*}}{\partial x}}+{\frac {\partial f_{ly}^{*}}{\partial y}}+{\frac {\partial f_{lz}^{*}}{\partial z}}=-\left({\frac {\partial {\mathfrak {H}}_{x}}{\partial x}}+{\frac {\partial {\mathfrak {H}}_{y}}{\partial y}}+{\frac {\partial {\mathfrak {H}}_{z}}{\partial z}}\right)$

However, these expressions vanish according to the Maxwell-Lorentz field equations; the second half of these equations, including the magnetic divergence condition, can thus be written:

(18*)

${\mathfrak {Div}}\ f^{*}=0$

a') The scalar divergence. Starting from a vector of third kind and its components with respect to the tangential spaces of a four-dimensional space section $\Delta \Sigma$, one could (from the vector of third kind) eventually derive a scalar magnitude - its divergence - by the definition analogous to a). Due to the mutual interchangeability of the vectors of third and first kind (see the end of § 1), nothing new would emerge with respect to a).

The differential operations considered here, are (by their geometric introduction in which coordinate system were not mentioned at all) independent of the choice of reference frame; their coordinate expressions are thus behaving invariant or covariant with respect to Lorentz transformations. This especially applies to the field equations (18) and (18*). The complicated calculations, by which Lorentz (1895 and 1904) and Einstein (1905) proved their applicability independent from the coordinate system, and by which they had to show the meaning of the transformed field vectors, thus become irrelevant in the system of Minkowski's "world".

§ 6. The integral theorems of Gauss, Stokes, Green in four dimensions.

As one directly obtains (in ordinary vector calculus) the theorems of Gauss and Stokes from the concept of div and rot, and Green's theorem is supplemented to that of Gauss by means of the concept of grad, one also will obtain three integral theorems from the concepts of scalar and vectorial divergence and rotation, which we will denote as theorem of Gauss, Gauss-Stokes, and Stokes; there, the "Gauss-Stokes theorem" stands in the middle between the actual theorem of Gauss and Stokes, in the same way as the concept of vector divergence stands between that of scalar divergence and rotation. The theorem of Green thus follows from the connection of the theorem of Gauss with the concept of gradients.

a) The theorem of Gauss. It reads, when $P$ is a four-vector, $\Sigma$ a four-dimensional space area, $S$ its three-dimensional boundary with the outer normal $n$:

(23)	$\int \limits _{\Sigma }\mathrm {Div} \ P\ d\Sigma =\int \limits _{S}P_{n}dS$

Namely, if one separates the space area $\Sigma$ into sufficiently small space-elements $d\Sigma$, and applies to any of them the same equation (16), then over the boundaries all inner ones are canceled from $\int dS$, since they appear twice with opposite sign, and only the parts of the outer boundary $S$ of $\Sigma$ remain.

In particular let be $\mathrm {div} \ P=0$. If one constructs a tube of $P$-lines (lines having everywhere the direction of vector $P$) and if one cuts the tube at an arbitrary place by a ("plane" or curved) space $S$, then according to the theorem of Gauss, one always obtains the same value of $\int P_{n}dS$. Herein lies, when $P$ means the four-density, the independence of charge from the reference system (see I. p. 752).

b) The two forms of the theorem of Gauss-Stokes. Let $f$ be a six-vector, $S$ an arbitrary (not necessarily "plane") three-dimensional space section located within the four-dimensional "world", and $s$ the normal upon an element $dS$ of the same. The boundary of space $S$, which will be a closed two-times extended surface, be $\sigma$; the single element $d\sigma$ we imagine as determined by two mutually perpendicular directions $s's''$, and the surface element normal to $d\sigma$ as determined by the directions $s$ (perpendicular to $S$) and $n$ (within $S$ perpendicular to $d\sigma$). Depending as to whether we project $f$ to the surface element normal to $d\sigma$, or to $d\sigma$ itself, we obtain the components $f_{sn}$ or $f_{s's''}$. Then by equations (17) and (22):

(24)	$\int \limits _{S}{\mathfrak {Div}}\ f_{s}dS=\int \limits _{\sigma }f_{sn}d\sigma$

and

(24*)

$\int \limits _{S}{\mathfrak {Div}}\ f_{s}^{*}dS=\int \limits _{\sigma }f_{s's''}d\sigma$

The proof (decomposition of space $S$ into sufficiently small elements $\Delta S$ etc.) is the same as under a).

One can use these two forms of the theorem of Gauss-Stokes, to rewrite Maxwell's differential equations (18) and (18a) in integral form.

For this purpose, one considers a two-times-extended closed surface $\sigma$ located in the "world", then puts a three-times extended space $S$ through it, and understands $s's''sn$ as direction as it was explained above. Then it applies due to (24) and (18) or due to (24*) and (18*):

(24a)

$\int \limits _{\sigma }f_{sn}d\sigma =\int \limits _{S}P_{s}dS,\ \int \limits _{\sigma }f_{s's''}d\sigma =0$

To emphasize the relation of these formulas to the ordinary integral formulation of Maxwell's equations, we consider two special cases:

1. The surface $\sigma$ lies in $xyz$-space. Thus

$s=l,\ P_{s}=i\varrho ,\ f_{sn}=i{\mathfrak {E}}_{n},\ f_{s's''}={\mathfrak {H}}_{n}$

and we have the known relations:

$\int {\mathfrak {E}}_{n}d\sigma =\int \varrho dS,\ \int {\mathfrak {H}}_{n}d\sigma =0$

2. The surface $\sigma$ be an infinitely flat cylinder, whose basis in $xyz$-space lies, with the generator (length $dl$), parallel to the $l$-axis. Stemming from the mantle of the cylinder, one obtains for the left-hand sides of (24a) (when $s'$ is measured along the contour of the mantle):

$dl\int {\mathfrak {H}}_{s'}ds'$ or $-idl\int {\mathfrak {E}}_{s'}ds'$

On the other hand, if one is taking together the two basis-areas of the cylinder, then for those it is $n=l$, and $s$ means the normal (located in $xyz$-space) upon the basis-area. Its contribution is thus

$-idl\int {\frac {\partial {\mathfrak {E}}_{s}}{\partial l}}d\sigma$ or $-dl\int {\frac {\partial {\mathfrak {H}}_{s}}{\partial l}}d\sigma$

where the first integral measures the displacement current traversing the basis-area, the second measures the temporal change of the magnetic force-line number. At the same time

$\int P_{s}dS=dl\int P_{s}d\sigma ,$

becomes the (up to the factor $dl$) convection current traversing perpendicular to $\sigma$. With respect to the relevant cylindric specialization of the integration area, our surface integrals (24a) therefore go over into the known integral form of Maxwell's equations in ordinary notation:

$\int {\mathfrak {H}}_{s'}ds'={\frac {1}{c}}\int \left({\frac {\partial {\mathfrak {E}}}{\partial t}}+\varrho {\mathfrak {v}}\right),\ \int {\mathfrak {E}}_{s'}ds'=-{\frac {1}{c}}\int {\frac {\partial {\mathfrak {H}}}{\partial t}}d\sigma$

c) The theorem of Stokes. If $s$ means an closed one-dimensional convolution (arbitrarily located in the world), $\sigma$ a two-times extended surface limited by $s$, $\Phi$ a four-vector, then Stokes' theorem is given as a direct consequence of definition equation (21) in the location and order of directions $s's''$ in the form

(25)	$\int \limits _{\sigma }\mathrm {Rot} _{s's''}\Phi d\sigma =\int \limits _{s}\Phi _{s}ds$

One can remark, that one cannot speak (even with respect to the ordinary three-dimensional formulation of Stokes' theorem), as it usually happens, of the normal component, but of the tangential component of rotation, since rotation is also at that place a vector of second kind. If we had started from the supplement of rotation (see the previous paragraph under c), then we would have obtained equation (25) as well.

If it is particularly about a closed surface $\sigma$, then the boundary curve and thus also the right-hand side of (25) vanishes, and thus we have

$\int \limits _{\sigma }\mathrm {Rot} _{s's''}\Phi d\sigma =0$

Accordingly it is given from the second form of Gauss-Stokes' theorem (24*), in which $\sigma$ was to be integrated on the right-hand side over a closed surface $\sigma$, when we include $f$ equal to the rotation of an arbitrary four-vector $\Phi$:

$\int {\mathfrak {Div}}\ \mathrm {Rot} ^{*}\Phi \ dS=0$

Since this equation applies to any section $S$, we conclude the identical relation

(25a)

${\mathfrak {Div}}\ \mathrm {Rot} ^{*}\Phi =0$

"the vector divergence of the supplement of rotation of an arbitrary four-vector vanishes." This can be simply verified using the coordinate expressions (17a) of the vector divergence and (21) of the rotation.

Equation (25a) simultaneously gives the justification for introducing the electrodynamic potential $\Phi$, which was only historically described in the previous paragraph under c'). Namely, since by approach (21b) $f=\mathrm {Rot} \ \Phi$, the second of Maxwell's equations ${\mathfrak {Div}}\ f^{*}=0$ according to (25a) is identically satisfied; it only remains to determine $\Phi$, so that it also satisfies the first of Maxwell's equations ${\mathfrak {Div}}\ f=P$, which now goes over to:

(25b)

${\mathfrak {Div}}\ \mathrm {Rot} \Phi =P$

This four-dimensional vector equation represents the most simple form of Maxwell's theory for vacuum; with its integration, the following paragraph is concerned.

However, by the approach $f=\mathrm {Rot} \Phi$ with given $f$, the vector $\Phi$ is not completely determined. Namely, if $\Phi _{1}$ is such a vector, then we obtain in $\Phi =\Phi _{1}+\Psi$ a more general vector, which also satisfies the condition $f=\mathrm {Rot} \Phi$, in case $\int \Psi ds$ also means a vector of everywhere vanishing rotation. It is, as it simply follows from Stokes' theorem, always representable as gradient of a scalar local function $U$, which itself is given by the line integral $\int \Psi ds$, extended from an arbitrary fixed to the previously considered spacetime point. From that it can be recognized, that one still can impose the constrain to the potential $\Phi$

(25c)

$\mathrm {Div} \ \Phi =0$

Namely, this gives (for the otherwise completely undetermined function $U$) the condition:

$\mathrm {Div} \ \Psi =\mathrm {Div\ Grad} \ U=-\mathrm {Div} \Phi _{1}$

which we can write (following d)) also as $\square U=-\mathrm {Div} \ \Phi _{1}$, and which can be integrated by the method of the following paragraph.

A similar reasoning as the one leading to (25a), we append to the first form of Gauss-Stokes' theorem, equation (24). Namely, if it is about a closed three-times extended space-section $S$, as it is given as boundary of a four-dimensional space-section $\Sigma$, then its boundary surface vanishes and thus also the right-hand side of (24), and we obtain

$\int \limits _{S}{\mathfrak {Div}}\ f_{s}dS=0$

valid for any closed space-section $S$. If we thus include ${\mathfrak {Div}}\ f$ instead of $P$ in Gauss' theorem (23), then the right-hand side of this equation becomes zero as well (here, $n$ was the normal with respect to space-element $dS$, denoted in the previous equation as $s$) and we have

$\int \limits _{\Sigma }\mathrm {Div} \ {\mathfrak {Div}}\ f\ d\Sigma =0$

Since this equation applies to any area $\Sigma$, we conclude the identical relation

$\mathrm {Div} \ {\mathfrak {Div}}\ f=0$

"the scalar divergence of the vector divergence of an arbitrary six-vector vanishes". This can easily be verified with respect to the coordinate expressions of the scalar and vector divergence (equation (16a) and (17a)).

If $f$ in particular means the six-vector of the field again, so that due to the first of Maxwell's equations ${\mathfrak {Div}}\ f=P$, then (24b) expresses the continuity condition $\mathrm {Div} \ P=0$, about which it was spoken in § 5 under a).

d) The theorem of Green. We use a symbol, already introduced by Cauchy and again used by Poincaré,^[4] which has to be applied to a scalar function $U$:

(26)	$\square U=\mathrm {Div\ Grad} \ U={\frac {\partial ^{2}U}{\partial x^{2}}}+{\frac {\partial ^{2}U}{\partial y^{2}}}+{\frac {\partial ^{2}U}{\partial z^{2}}}+{\frac {\partial ^{2}U}{\partial l^{2}}}$

This extends the ordinary Laplacian differential expression $\Delta$ to four dimensions and thus may be denoted as Laplacian expression again. Its geometrical-invariant nature directly follows from the representation $\square =\mathrm {Div\ Grad}$.

If $U$ and V${\displaystyle }$ are now two scalar local functions of the four variables xyzl, then we have by

$P=U\ \mathrm {Grad} \ V-V\ \mathrm {Grad} \ U$

a four-vector of special construction. Its scalar divergence, which one can think of as formed by differentiation with respect to coordinates xyzl, then becomes:

$\mathrm {Div} \ P=U\square V-V\square U$

namely, the two scalar products $\pm (\mathrm {Grad} \ U,\mathrm {Grad} \ V)$ are mutually canceled. Thus if we include this special four-vector into the theorem of Gauss (23), then it is given

(27)	$\int \limits _{\Sigma }(U\square V-V\square U)d\Sigma =\int \limits _{S}\left(U{\frac {\partial V}{\partial n}}-V{\frac {\partial U}{\partial n}}\right)dS$

i.e, the exact analogue to the ordinary theorem of Green. It is related to an arbitrary world-section $\Sigma$ and its three-dimensional boundary $S$. Steadiness of the appearing functions and their first derivations is presupposed as in the other theorems of this paragraph. If it is violated in one world-point, then one would have to exclude it from the integration by a three-dimensional boundary space $S_{0}$, like in the ordinary case, and to supplement the integral over $S_{0}$ of the right-hand side of (27).

This is especially then the case, when $V$ is set equal to the four-dimensional analogue of the Newtonian potential $1/r$:

(27a)

$V={\frac {1}{R^{2}}},\ R^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}+\left(l-l_{0}\right)^{2},$

corresponding to the circumstance, that in four dimensions the mathematical analogue to the Newtonian force would be decreasing by the cube of distance, instead of the square. Here, $R$ means the four-dimensional distance of the fixed world-point $O$ ("reference point") $x_{0}y_{0}z_{0}l_{0}$ with respect to the variable integration point xyzl. The reference point my lie in the integration area and thus may be surrounded by an infinitely small spherical space $S_{0}$ (radius $R_{0}$). If we calculate for it the right-hand side of (27), then it becomes:

${\begin{array}{c}V={\frac {1}{R_{0}^{2}}},\ {\frac {\partial V}{\partial n}}=-{\frac {2}{R_{0}^{3}}}{\frac {\partial R}{\partial n}}=+{\frac {2}{R_{0}^{3}}},\\\\\int U{\frac {\partial V}{\partial n}}dS_{0}=U_{0}\cdot {\frac {2}{R_{0}^{3}}}\int dS_{0}=4\pi ^{2}U_{0},\ \int V{\frac {\partial U}{\partial n}}dS_{0}=0.\end{array}}$

Here, $U_{0}$ means the value of $U$ at the reference point ($R=0$), if the easily verified theorem^[5] is employed, according to which the three-dimensional boundary of a four-dimensional sphere of radius 1 is equal to $2\pi ^{2}$, thus the one of radius $R_{0}$ is equal to $2\pi ^{2}R_{0}^{3}=\int dS_{0}$. From (27) it thus follows

(27b)

$4\pi ^{2}U_{0}=-\int {\frac {\square U}{R^{2}}}d\Sigma -\int \left(U{\frac {\partial }{\partial n}}{\frac {1}{R^{2}}}-{\frac {1}{R^{2}}}{\frac {\partial U}{\partial n}}\right)dS$

On the other hand, it is given from (27) with $V=1$

(27c)

$\int \square U\ d\Sigma =\int {\frac {\partial U}{\partial n}}dS$

If the integration area is extended over the whole infinite space $\Sigma$, then we can choose $S$ as infinitely great sphere (radius $R_{\infty }$). To this it applies, similarly as above:

$\int U{\frac {\partial }{\partial n}}{\frac {1}{R^{2}}}dS=-{\frac {2}{R_{\infty }^{3}}}\int UdS=-4\pi ^{2}U_{m}$

where $U_{m}$ is the average of $U$ on the infinitely distant sphere, and because of (27c)

$\int {\frac {1}{R^{2}}}{\frac {\partial U}{\partial n}}dS={\frac {1}{R_{\infty }^{2}}}\int \square U\ d\Sigma$

If both is included in (27b), then it is given

$4\pi ^{2}\left(U_{0}-U_{m}\right)=-\int \square U\left({\frac {1}{R^{2}}}-{\frac {1}{R_{\infty }^{2}}}\right)d\Sigma$

As $1/R_{\infty }$ is vanishing against $1/R$, it thus becomes

(27d)

$4\pi ^{2}U_{0}=-\int {\frac {\square U}{R^{2}}}d\Sigma +\mathrm {Cons} t.$

By that, we have to calculate for $U$ an arbitrary world-point $O$ except a constant, when $\square U$ in the whole area of real xyzl is given.

To that, however, a remark has to be made concerning the reality relations. As always, we have implicitly presupposed as real the coordinates $l_{0},l$ as well as $x_{0},x\dots$, and assumed for example, that $R$ is only vanishing at one point $O$. This is not the case anymore if it is considered that $\left(l-l_{0}\right)^{2}=-c^{2}\left(t-t_{0}\right)^{2}$, it is rather the case that $R$ becomes zero in the real world-coordinates on a three-times extended cone. Furthermore, with respect to the actually important tasks, $\square U$ is not given for real, but for negative-imaginary values of $l-l_{0}$, namely in the reference system of xyzl, for all times $t<t_{0}$ preceding the time coordinate $t_{0}$ of the origin. Thus one would have (to be able to apply our formulas) to imagine the given values $\square U$ of the negative-imaginary axis of a complex ($l-l_{0}$)-plane (see Fig. 3) as analytically extended with respect to the real axis of that plane, and to extend the integration over these real values of $l-l_{0}$, i.e. over the corresponding values of $l$, in which case $R^{2}$ only vanishes for $l=l_{0}$, when simultaneously $x=x_{0},\ y=y_{0},\ z=z_{0}$. Instead, we will proceed more easily, by deforming the integration path as in Fig. 3 into a slope surrounding the negative-imaginary axis^[6]; the integration in (27d) is then to be understood, so that it is to be led with respect to x y z over all real values, with respect to $l$ over this slope, and (27d) represents the value of $U$ at time $t_{0}$, when for all earlier moments the value of $\square U$ is given. The four-dimensional method proves to be equally fruitful also for these and similar integration tasks, and it allows to solve them quite similar to the calculation of the potential of given masses in ordinary potential theory.

§ 7. Determination of the four-potential and the electrodynamic force.

The differential equation of the four-potential, denoted by us as the most simple formulation of Maxwell's theory, reads:

(25b)

${\mathfrak {Div}}\ \mathrm {Rot} \ \Phi =P$

For one of the four right-angled components $\Phi _{j}$ of $\Phi$ we thus have by (17a) and (21b):

${\begin{array}{r}{\frac {\partial }{\partial x}}\left({\frac {\partial \Phi _{x}}{\partial j}}-{\frac {\partial \Phi _{j}}{\partial x}}\right)+{\frac {\partial }{\partial y}}\left({\frac {\partial \Phi _{y}}{\partial j}}-{\frac {\partial \Phi _{j}}{\partial y}}\right)+{\frac {\partial }{\partial z}}\left({\frac {\partial \Phi _{z}}{\partial j}}-{\frac {\partial \Phi _{j}}{\partial z}}\right)\\\\+{\frac {\partial }{\partial l}}\left({\frac {\partial \Phi _{l}}{\partial j}}-{\frac {\partial \Phi _{j}}{\partial l}}\right)=-\square \Phi _{j}+{\frac {\partial }{\partial j}}\mathrm {Div} \ \Phi =P_{j},\end{array}}$

for which we can write more easily, by constraint (25c)^[7]:

(28)	$\square \Phi =-P$

Thus we have to solve the following problem of four-dimensional potential theory: We seek a solution of equation $\square \Phi =-P$ for an arbitrary spacetime-point $x_{0}y_{0}z_{0}t_{0}$, when the four-density $P$, i.e. the charge and velocity of the considered system, is given for all earlier moments $t<t_{0}$. The solution includes equation (27d) with the slope-path denoted in Fig. 3.

If one includes here for $U$ any of the components of $\Phi$, considers equation (28) and suppresses the constant irrelevant for our potential, then it is given

(29)	$4\pi ^{2}\Phi _{0}=\int {\frac {P}{R^{2}}}d\Sigma$

This most natural representation of electrodynamic potential in the sense of relativity theory, stems from Herglotz^[8]. Factually, this representation of course cannot be distinguished from the older formulas, as long as one remains in the original and and accidentally employed reference system of xyzl.

The integration with respect to $l$ can always carried out in (29) as well as in all analogous later formulas by Cauchy's residue theorem. Namely, within the slope of Fig. 3 lies the place where $R^{2}$ of first order vanishes, thus upon which the integration can be drawn together, namely at the place (see (27a)):

(29a)

$\left\{{\begin{array}{c}l=l_{0}-is,\ t=t_{0}-{\frac {s}{c}},\\\\s={\sqrt {\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}}}\end{array}}\right.$

On the other hand, the principally equally-valid place $l=l_{0}+is$ lies upon the positive-imaginary axis of Fig. 3 and gives no contribution to our slope integration.

Based on the world-line of a certain charge element $de$ (see Fig. 4) we denote the point $L$ of the world-line, which is cut by a cone $R^{2}=0$ constructed at point $O$, with Minkowski as light-point of $O$. Its coordinates are unequivocally determined when the charge element never moves at superluminal velocity, and the fourth coordinate can be determined, as previously shown, by equation $t=t_{0}-s/c$. As it is known, it says that a light signal emanating from world-point $L$, reaches world-point $O$ (i.e. it reaches the space-point $x_{0}y_{0}z_{0}$ at time $t_{0}$).

Thus, if we carry out the integration with respect to $l$ by means of residue-construction, thus the emerging formulas will be related to the light-point $L$ of $O$. For example, in this way the well-known formula of retarded potential directly emerge from (29). We only show this for the case of a point-like charge (of a sufficiently distant reference point).

In this case, one can see $R^{2}$ in (29) as constant during the integration with respect to x y z, and evaluate this integration. However, to avoid from the beginning the introduction of the arbitrary reference system xyzl, we rather use a natural reference system oriented with respect to the world-line of the point charge. Let (see Fig. 4) $dS_{n}$ be the element of normal space of the world-line, $ds$ the curve element of the world line. This is connected with Minkowski's proper time $\tau$, so that

(29b)

$\left\{{\begin{array}{l}ds=icd\tau ,\ \mathrm {by} \\\\ds={\sqrt {dx^{2}+dy^{2}+dz^{2}+dl^{2}}}\\\\d\tau =dt{\sqrt {1-{\frac {1}{c^{2}}}\left(\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}+\left({\frac {dz}{dt}}\right)^{2}\right)}}\end{array}}\right.$

Now one has:

(29c)

$\left\{{\begin{array}{c}d\Sigma =dS_{n}ds=ic\ dS_{n}d\tau ,\\\\ic\int P\ dS_{n}=ie\ {\mathfrak {B}},\ {\mathfrak {B}}=\left({\frac {dx}{d\tau }},\ {\frac {dy}{d\tau }},\ {\frac {dz}{d\tau }},\ {\frac {dl}{d\tau }}\right)\end{array}}\right.$

The first of these formulas directly follows from the fact, that the length element $ds$ and the three-dimensional space-element $dS_{n}$ are mutually normal. In the second formula, $P$ as well as ${\mathfrak {B}}$ denote a four vector directed with respect to the world-line of the charge at the considered place. It only remains to prove, that the vectors on the right-hand and left-hand side of this formula are mutually equal as regards their magnitude. With respect to (29b), $\left|{\mathfrak {B}}\right|=ic$, thus the magnitude of the right-hand side in question is equal to $-ce$. On the left-hand side, one thinks $P$ as decomposed into components with respect to the world-line and perpendicularly to it. The latter ones vanish, the first one becomes equal to $i\varrho _{0}$ by equation (1) part I, where $\varrho _{0}$ is the "rest-density", i.e. the density of charge viewed by a co-moving observer. Accordingly, $\int \varrho _{0}dS_{n}=e$ becomes equal to the total charge. For the magnitude of the left-hand side of (29c), one also has:

$ic\int i\varrho _{0}dS_{n}=-ce$

If one substitutes from (29c) into (29), it follows:

$4\pi ^{2}\Phi =ie\int {\frac {\mathfrak {B}}{R^{2}}}d\tau$

the integration with respect to the new variable $\tau$ is, quite equal as the one by $l$ in Fig. 3, to be extended on an arbitrary complex, clockwise rotation around the light-point $L$, and when calculated by Cauchy's theorem gives^[9]:

(29d)

$4\pi \Phi _{0}={\frac {e{\mathfrak {B}}}{\left({\mathfrak {RB}}\right)}}$

Here,

(29e)

${\mathfrak {R}}=\left(x-x_{0},\ y-y_{0},\ z-z_{0},\ l-l_{0}\right)$

means the four-vector from the reference point with respect to the corresponding light-point of the charge, ${\mathfrak {B}}$ the velocity vector of the charge at the light-point defined in (29c), and $\left({\mathfrak {RB}}\right)$ its scalar product in the sense of § 3 A. Equation (29d) represents the invariant notation (in the sense of relativity theory) of the point-potential law (Liénard-Wiechert); we return to this in the following paragraphs again.

The field of an arbitrarily moving charge at reference point $O$, can be now obtained by formula (21b)

$f=\mathrm {Rot} \ \Phi$, for example $f_{xy}={\frac {\partial \Phi _{y}}{\partial x_{0}}}-{\frac {\partial \Phi _{x}}{\partial y_{0}}}$

If one would like to apply this differentiation in the case of a point-charge upon the calculated formula (29d), then one would be led to complicated considerations,^[10] which stem from the fact, that with a variation of $O$ also a variation of the light-points $L$ is connected. It is much simpler to resort to the original formula (29) and to make the passage to the point charge only at the end. From (29) it is given

$4\pi ^{2}f_{xy}=2\int {\frac {P_{y}\left(x-x_{0}\right)-P_{x}\left(y-y_{0}\right)}{R^{4}}}d\Sigma$

and somewhat more general for $j=x,y,z,l$:

(30)	$2\pi ^{2}f_{xj}=\int {\frac {P_{j}\left(x-x_{0}\right)-P_{x}\left(j-j_{0}\right)}{R^{4}}}d\Sigma$

We immediately pass to the specific electrodynamic force ${\mathfrak {F}}$, by imagining a charge distribution of four-density $P_{0}$ in the surrounding of the reference point $O$, then their $x$-component is specified by equation (11) as $\left(P_{0}f_{x}\right)$; for that, one obtains according to the last formula by using of vector ${\mathfrak {R}}$ explained in (29e), which at first is not yet related with the light-point:

$2\pi ^{2}{\mathfrak {F}}_{x}=2\pi ^{2}\left(P_{0}f_{x}\right)=\int {\frac {\left(P_{0}P\right)\left(x-x_{0}\right)-P_{x}\left(P_{0}{\mathfrak {R}}\right)}{R^{4}}}d\Sigma$

and thus generally:

(30a)

$2\pi ^{2}{\mathfrak {F}}=\int {\frac {\left(P_{0}P\right){\mathfrak {R}}-P\left(P_{0}{\mathfrak {R}}\right)}{R^{4}}}d\Sigma .$

If one immediately goes over to a point charge $e$ again, by means of equations (29c), then its specific force action upon distribution $P_{0}$ is given by:

$2\pi ^{2}{\mathfrak {F}}=ie\int {\frac {\left(P_{0}{\mathfrak {B}}\right){\mathfrak {R}}-{\mathfrak {B}}\left(P_{0}{\mathfrak {R}}\right)}{R^{4}}}d\tau$

If the distribution $P_{0}$ is also point-like of total charge $e_{0}$, then one is able to form the total force ${\mathfrak {K}}$ exerted by $e$ on $e_{0}$. This shall be calculated as co-moving force in the sense of ${\mathfrak {K}}'$ in equation (15). Thus, one shall multiply with the space element $ds'$ normal to the world-line of $O$, and shall form ${\mathfrak {K}}=\int {\mathfrak {F}}\ dS'$. With respect to the second line of (29c) it is given, when ${\mathfrak {B}}_{0}$ means the velocity vector of $O$:

(30b)

$2\pi ^{2}{\mathfrak {K}}={\frac {iee_{0}}{c}}\int {\frac {\left({\mathfrak {B}}_{0}{\mathfrak {B}}\right){\mathfrak {R}}-\left({\mathfrak {B}}_{0}{\mathfrak {R}}\right){\mathfrak {B}}}{R^{4}}}d\tau$

Also here, the integration means a rotation of the complex variable $\tau$ around the light-point of $O$; it can immediately carried out by residue-construction, where now, since the denominator of second order in $R^{2}$ vanishes, the development of numerator and denominator is to be taken up to terms of second order. The obvious calculation is neglected at this place and concerning its result we refer to equation (37) of the next paragraph, where it is derived in a probably more illustrative but essentially less simple way than at this place. Compared with the somewhat composed form of equation (37), the integral representation contained in equation (30b) is in any case remarkable due to its particular clarity.

§ 8. The cyclic or hyperbolic motion and the electrodynamic elementary laws.

As the most simple example of accelerated motion we consider the interesting case of "hyperbolic motion" treated recently by M. Born^[11]. It represents itself (when one again neglects the imaginary character of the time coordinate in terms of expression and drawing) as "cyclic motion", where the reason for its simplicity lies. We namely investigate this motion under the point of view already indicated by Minkowski^[12], that any accelerated motion can always be approximated by "uniformly accelerated" motion, and from that we arrive at an illustrative derivation of the electrodynamic elementary laws.

The electrical system shall be moving, so that for any of its charge elements it applies:

(31)	$x=r\cos \varphi ,\ y=y,\ z=z,\ l=r\sin \varphi .$

At constant $r,y,z$ and variable $\varphi$ these equations give the world-line of the charge element; at constant $\varphi$ and variable $r,y,z$ they determine the "rest-form" of charge, i.e. the simultaneous locations of their elements observed by a co-moving observer. Fig. 5a represents the relations in the $xl$-plane with $l$ and $\varphi$ imagined as real: the world-lines are circles $x^{2}+l^{2}=r^{2}$, the rest-form is projected into the variable radius $r$. Fig. 5b shows, as to how the things are with respect to the imaginary constitution of $l=ict$. If one draws $x$ and $ct$ as real coordinates, and puts $\varphi =i\psi$, where $\psi$ is a real angel, then the world-lines become equally sided hyperbolas $x^{2}-(ct)^{2}=r^{2}$ and the rest-form is given by $\psi =const.$. The asymptotes under 45° are corresponding to a motion with speed of light $c$, which is approximated by hyperbolic motion for $ct=\pm \infty$.

By the cyclic nature of our problem, the four-dimensional polar coordinates $r\ y\ z\ \varphi$ are given instead of the ordinary coordinates xyzl, whose character is mixed of space and time. If we call the corresponding coordinates of the reference point $r_{0}y_{0}z_{0}\varphi _{0}$, then we evidently can choose $\varphi _{0}=0$. This means in the way of expression of Fig. 5a, that we can count the coordinate $\varphi$ of the single charge elements starting from the radius vector extending through the reference point, which becomes the $x$-axis by that. In the way of expression of Fig. 5b we would have to say, that instead of axes $x$ and $ct$, we can introduce new "mutually normal" axes $x'$ and $ct'$, whose first one is going through $O$ and whose last one forms (with the hyperbolic asymptotes) the same angle as $x'$ (harmonic location of axes $x',ct'$ against both asymptotes). Also related to these axes, the world-lines are equally sided hyperbolas and are (non-Euclidean) perpendicular upon them. At the same time, wh have $ct'_{0}=0$ for the reference point, thus also $r_{0}\sin \ i\psi _{0}=0$ or $\psi _{0}=0$. Thus when we would choose $\varphi _{0}=0$ in Fig. 5a, then this means in real terms, that we introduce a new primed $x',y,z,ct'$ instead of $x,y,z,ct$, which is relatively moving with respect to the original one, and which we (starting from the other one) define by equation (31) of our polar coordinates $ryz\varphi$. The introduction of the primed axes is, however, excluding superluminal velocities, only possible when the reference point lies in one of the two space-like quadrants of Fig. 5b (see the note in part I, p. 752), i.e. when in the original coordinates $ct_{0}<x_{0}$ applies, what we want to presuppose. In other cases, i.e. when the reference point lies in one of the time-like quadrants, one only needs to exchange the axes $x'$ and $ct'$, without additionally changing something essential.

Also the vectors $P$ and $\Phi$ are decomposed by us into the components with respect to coordinates $ryz\varphi$, where the four-vector $P$ is drawn in the successive locations of the charge elements, the four-vector $\Phi$ is drawn in the reference point.

Evidently it is:

$P_{x}=P_{y}=P_{z}=0,\ P=P_{\varphi }=i\varrho _{0}$

Vector $P$ namely is directed into the direction of the world-line, thus in the direction of increasing $\varphi$; as well as $i\varrho$ was the fourth component in the $xyzl$-system (see equation (1), $\varrho$ = charge density in $xyz$-space), due to the vector character of $P$, the fourth component in the $ryz\varphi$ system are equal to $i\varrho _{0}$ ($\varrho _{0}$ = charge density in the co-moving $ryz$-space = "rest density" = $\left|P\right|=i\varrho {\sqrt {1-\beta ^{2}}}=i\varrho /\cos \varphi$, see equation (1a) and the explanations to equation (29c) of the previous paragraph). Here, $\varrho _{0}$ is, according to Gauss' theorem, constant along any world-line (independent of $\varphi$), it is possibly variable from world-line to world-line. Due to the vector summation immediately carried out, we also will need the components $P_{x}$ and $P_{l}$ with respect to the axes oriented by $O$ of Fig. 5a (the axes $x'$, $ct'$ of Fig. 5b). For any place of the world line:

(31a)

$\left\{{\begin{array}{lrr}P_{x}=&-P_{\varphi }\cdot \sin \varphi =&-i\varrho _{0}\sin \varphi \\P_{l}=&P_{\varphi }\cdot \cos \varphi =&i\varrho _{0}\cos \varphi \end{array}}\right.$

For the calculation of $\Phi$ we use equation (29) and substitute (for the integration variables $x=r\ \cos \varphi$ there) $l=r\ \sin \varphi$. The slope surrounding the imaginary axis in Fig. 3, is corresponding to an integration in $\varphi$ over a corresponding slope, upon which $\varphi$ goes back from $-i\infty$ over zero to $-i\infty$, and which clockwise envelopes (as earlier) the light-point $\left(R^{2}=0\right)$ belonging to any world-line. The passage to the new integration variables $r,\varphi$ happens according to the scheme of ordinary polar coordinates:

(31b)

$\int \limits _{-\infty }^{+\infty }dx\int dl=\int \limits _{0}^{\infty }r\ dr\int d\varphi$

with the difference, that the integration with respect to $\varphi$ (similar to $l$) is extended over the mentioned slope.

Of the four components of $\Phi$, two are vanishing; namely due to $P_{y}=P_{z}=0$

$\Phi _{y}=0,\ \Phi _{z}=0.\,$

Of the two other components $\Phi _{r}$ and $\Phi _{\varphi }$ it can be said at first, that they are independent of the $\varphi$-coordinate of the reference point, by which the cyclic nature of our problem is expressed. Actually, we could (at any location of the reference point) choose the direction drawn to it as zero-ray; in the expressions of $\Phi _{r}$ (in the direction of the zero-ray) and of $\Phi _{\varphi }$ (perpendicular to it), $\varphi _{0}$ doesn't occur at all. These components become constant for all points or any circle of Fig. 5a (any hyperbola of Fig. 5b), and vector $\Phi$ has a constant magnitude and location against the variable radius $r_{0}$. On the other hand, the components $\Phi _{x},\Phi _{l}$ in a $xl$-system of general location are of course independent of $\varphi _{0}$, namely due to the general formulas for vector transformation:

$\Phi _{x}=\Phi _{r}\cos \varphi _{0}-\Phi _{\varphi }\sin \varphi _{0}$ $\Phi _{l}=\Phi _{r}\sin \varphi _{0}+\Phi _{\varphi }\cos \varphi _{0}$

At the particular location of the $x$-axis as in Fig. 5a (the $x'$-axis as in Fig. 5b), which is convenient for the following, it additionally becomes $\Phi _{r}=\Phi _{x}$, $\Phi _{\varphi }=\Phi _{l}$ due to $\varphi _{0}=0$.

The component $\Phi _{r}=\Phi _{x}$ can easily be executed. At first, it is because of (31a,b) and (29):

(32)

$\left\{{\begin{array}{l}4\pi ^{2}\Phi _{x}=-i\int \limits _{-\infty }^{+\infty }dy\int \limits _{-\infty }^{+\infty }dz\int \limits _{0}^{\infty }r\ dr\int d\varphi {\frac {\varrho _{0}\sin \varphi }{R^{2}}},\\\\R^{2}=r^{2}+r_{0}^{2}-2rr_{0}\cos \varphi +\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\end{array}}\right.$

Since $\varrho _{0}$ is independent (see above) from $\varphi$ and $d\left(dR^{2}\right)=2rr_{0}\sin \varphi \ d\varphi$, then the integral with respect to $\varphi$ is simply:

${\frac {\varrho _{0}}{2rr_{0}}}\int {\frac {d\left(R^{2}\right)}{R^{2}}}=-{\frac {2\pi i\varrho _{0}}{2rr_{0}}}$

since it is to be extended around point $R^{2}=0$ against the positive rotation sense (see Fig. 3). Thus by (32)

(32a)

$4\pi \Phi _{x}=4\pi \Phi _{r}=-{\frac {1}{r_{0}}}\int \limits _{-\infty }^{+\infty }dy\int \limits _{-\infty }^{+\infty }dz\int \limits _{0}^{\infty }dr\ \varrho _{0}={\frac {-e}{r_{0}}};$

here, $e$ means the total charge of the system, obtained by integration of rest density $\varrho _{0}$ over the rest-form of the system. On the other hand, by (31a,b) and (30):

(32b)

$4\pi ^{2}\Phi _{l}=+i\int \limits _{-\infty }^{+\infty }dy\int \limits _{-\infty }^{+\infty }dz\int \limits _{0}^{\infty }r\ dr\int d\varphi {\frac {\varrho _{0}\cos \varphi }{R^{2}}}.$

The integral with respect to $\varphi$ is given, quite similar to above, by residue-construction:

${\frac {\varrho _{0}}{2rr_{0}}}\int {\frac {d\left(R^{2}\right)}{R^{2}}}{\frac {\cos \varphi }{\sin \varphi }}={\frac {-2\pi i\varrho _{0}}{2rr_{0}}}\left({\frac {\cos \varphi }{\sin \varphi }}\right)_{L},$

where for $\cos \varphi$, $\sin \varphi$ the values (following from $R^{2}=0$ and corresponding to the light-point) have to be included:

(32c)

$\left\{{\begin{array}{l}\cos \varphi ={\frac {1}{2rr_{0}}}\left(r^{2}+r_{0}^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right)\\\\\sin \varphi ={\frac {-i}{2rr_{0}}}{\sqrt {\left(\left(r-r_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right)\times \left(\left(r+r_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right)}}\end{array}}\right.$

Thus

(32d)

$4\pi \Phi _{l}=4\pi \Phi _{\varphi }={\frac {1}{r_{0}}}\int \limits _{-\infty }^{+\infty }dy\int \limits _{-\infty }^{+\infty }dz\int \limits _{0}^{\infty }dr\varrho _{0}\left({\frac {\cos \varphi }{\sin \varphi }}\right)_{L}.$

For a far distant point, one can view $\cos \varphi$ and $\sin \varphi$ as approximately constant for all charge elements and execute the three-times integration, where the total charge $e$ of the system emerges. Thus one has for the limiting case:

(33)	$4\pi \Phi _{r}={\frac {-e}{r_{0}}},\ 4\pi \Phi _{\varphi }={\frac {+e}{r_{0}}}\left({\frac {\cos \varphi }{\sin \varphi }}\right)_{L}.$

In consequence of Fig. 6 one can easily convince himself, that direction and magnitude of vector $\Phi$ are only expressed by the state of motion at light-point $L$. For this purpose, we calculate the component of $\Phi$ on the one hand with respect to direction $ML$ perpendicular to the direction of motion at the light-point, and on the other hand with respect to the tangent in $L$, which may be determined by the four-vector ${\mathfrak {B}}$; here it is to be noticed, that $\varphi$ shall mean the angle belonging to $L$ (namely counted from $MO$ as origin); thus:

in the direction $ML$:

$\Phi _{r}\cos \varphi +\Phi _{\varphi }\sin \varphi =-{\frac {e}{r_{0}}}\cos \varphi +{\frac {e}{r_{0}}}\cos \varphi =0$

in the direction ${\mathfrak {B}}$

$-\Phi _{r}\sin \varphi +\Phi _{\varphi }\cos \varphi =+{\frac {e}{r_{0}}}\sin \varphi +{\frac {e}{r_{0}}}{\frac {\cos ^{2}\varphi }{\sin \varphi }}={\frac {e}{r_{0}\sin \varphi }}.$

$r_{0}\sin \varphi$, however, are represented in the figure by the line $ON=PL$, and $PL$ is the projection of vector ${\mathfrak {R}}$ from the reference point with respect to the light-point upon the motion vector ${\mathfrak {B}}$, thus^[13]

(33a)

$r_{0}\sin \varphi =PL=\left|{\mathfrak {R}}\right|\cos \left({\mathfrak {R,B}}\right)={\frac {({\mathfrak {RB)}}}{\left|{\mathfrak {B}}\right|}}$

see part I, § 3 A. Our potential is thus represented in terms of direction and magnitude:

(33b)

$4\pi \Phi ={\frac {e{\mathfrak {B}}}{({\mathfrak {RB)}}}}\dots \left\{{\begin{array}{l}{\mathfrak {R}}=x-x_{0},\ y-y_{0},\ z-z_{0},\ l-l_{0},\\\\{\mathfrak {B}}={\frac {dx}{d\tau }},\ {\frac {dy}{d\tau }},\ {\frac {dz}{d\tau }},\ {\frac {dl}{d\tau }}.\end{array}}\right.$

The special character of hyperbolic motion is vanished from (33b), this representation applies to any motion affecting our hyperbolic motion at the light-point, and was directly taken above (see (29d)) from our general representation (29) by the passage from one point-charge and by residue-construction. As to how the relations are in real terms, is alluded to in Fig. 5b: In the projection of the $x$, $ct$-plane, the light-point $L$ (of a parallel drawn through $O$ with respect to a hyperbolic asymptote) is cut from the world-line, and it is the resultant from the two real components $\Phi _{r}$ and $\Phi _{\varphi }/i$ parallel to the tangent ${\mathfrak {B}}$ at the light-point.

Skipping the calculation of the field, we have to from $f=\mathrm {Rot} \Phi$ by (21b), where we of course choose the required rotations, over which the line integral of $\Phi$ must be extended, in the sense of our polar coordinate system (see Fig. 5a right above). While the $ry,rz,yz$-components of $f$ vanish, since $\Phi _{y}=\Phi _{z}=0$ and $\Phi _{r}$ is independent of $y$ and $z$, it is given^[14] by (21):

(34)	$f_{r\varphi }={\frac {1}{r_{0}}}{\frac {\partial r_{0}\Phi _{\varphi }}{\partial r_{0}}},\ f_{y\varphi }={\frac {\partial \Phi _{\varphi }}{\partial y_{0}}},\ f_{z\varphi }={\frac {\partial \Phi _{\varphi }}{\partial z_{0}}}.$

The field, as well as $\Phi$, is only dependent of the coordinates $r_{0},y_{0},z_{0}$ of the reference point and thus constant on the circles of Fig. 5a (the hyperbolas of 5b). On the other hand, it is of course variable upon the line $x=const.$, since any such line is cut by other hyperbolas for variable $ct$ in Fig. 5b. Thus, while the field is temporally changing in a spatially fixed point, it is constant in a co-moving point. Namely, it has at such a point the character of the electric field throughout. Namely, since the $\varphi$-direction has simultaneously the direction of the time axis in the co-moving ("primed") system, we have to write in consequence of (2):

(34a)

$\left\{{\begin{array}{lll}f_{r\varphi }=-i{\mathfrak {E}}'_{r},&f_{y\varphi }=-i{\mathfrak {E}}'_{y},&f_{z\varphi }=-i{\mathfrak {E}}'_{z},\\f_{yz}={\mathfrak {H}}'_{r}=0,&f_{zr}={\mathfrak {H}}'_{y}=0,&f_{ry}={\mathfrak {H}}'_{z}=0.\end{array}}\right.$

For an observer resting in the $xyzl$-system, on the other hand, the field has, except the electric one, also an magnetic part.

For distant reference points, to which the electric system appears as point-like, it is given from (33) and (34)

$4\pi f_{r\varphi }={\frac {e}{r_{0}}}{\frac {\partial }{\partial r_{0}}}\left({\frac {\cos \varphi }{\sin \varphi }}\right)_{L}={\frac {-e}{r_{0}\sin ^{2}\varphi _{L}}}\left({\frac {\partial \varphi }{\partial r_{0}}}\right)_{L}$

as well as

$4\pi f_{y\varphi }={\frac {-e}{r_{0}\sin ^{2}\varphi _{L}}}\left({\frac {\partial \varphi }{\partial y_{0}}}\right)_{L},\ 4\pi f_{z\varphi }={\frac {-e}{r_{0}\sin ^{2}\varphi _{L}}}\left({\frac {\partial \varphi }{\partial z_{0}}}\right)_{L}.$

From the definition of the light-point:

$R^{2}=r^{2}+r_{0}^{2}-2rr_{0}\cos \varphi +\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}=0$

it follows, however, since r, y, z are constant during hyperbolic motion:

$2\left(r_{0}-r\cos \varphi \right)dr_{0}+2rr_{0}\sin \varphi \ d\varphi =0$

thus

${\frac {\partial \varphi }{\partial r_{0}}}={\frac {r\cos \varphi -r_{0}}{rr_{0}\sin \varphi }}$

as well as

${\frac {\partial \varphi }{\partial y_{0}}}={\frac {y-y_{0}}{rr_{0}\sin \varphi }},\ {\frac {\partial \varphi }{\partial z_{0}}}={\frac {z-z_{0}}{rr_{0}\sin \varphi }};$

therefore, when the index $L$ at $\Phi$ is suppressed:

(34b)

$\left\{{\begin{array}{lll}4\pi f_{r\varphi }={\frac {-e}{rr_{0}^{2}\sin ^{3}\varphi }}\left(r\cos \varphi -r_{0}\right),\\\\4\pi f_{y\varphi }={\frac {-e}{rr_{0}^{2}\sin ^{3}\varphi }}\left(y-y_{0}\right),\\\\4\pi f_{z\varphi }={\frac {-e}{rr_{0}^{2}\sin ^{3}\varphi }}\left(z-z_{0}\right).\end{array}}\right.$

These formulas are the most simple expressions of the field produced by any moving point charge. Namely, by putting the curvature circle (curvature hyperbola) at the world-line of the point-charge in the light-point belonging to our reference point, and by replacing the arbitrary motion by cyclic motion upon the curvature circle, the general case is reduced to our formulas (34b).

These formulas are the most simple expression of the field produced by any moving point charge. Namely, by putting the curvature circle (curvature hyperbola) at the world-line of the point-charge in the light-point belonging to our reference point, and by replacing the arbitrary motion by cyclic motion upon the curvature circle, the general case is reduced to our formulas (34b). The occurrence of the curvature radius $r$, which is connected with the acceleration of motion at the light-point (see below), is characteristic. The difference between longitudinal and transverse acceleration is only a difference in the choice of reference frame.

At first, we want to circumscribe the formulas (34b), so that only four-vectors related to the light-point occur

${\mathfrak {R,\ B,\ {\dot {B}}}}$

${\mathfrak {R}}$ is the vector of $O$ to $L$, ${\mathfrak {B}}$ the velocity vector in $L$ (see equation (33a)) and ${\mathfrak {\dot {B}}}$ the acceleration vector in $L$. In the previous definition (equation (29b)) of $d\tau$ by the world-line element $d\tau =ds/ic$, since in our cyclic coordinates $ds=r\ d\varphi$ applies, it evidently becomes $d\tau =r\ d\varphi /ic$, thus according to equations (31)

${\begin{array}{ll}{\mathfrak {B}}&=\left(-r\ \sin \varphi ,0,0,r\ \cos \varphi \right){\frac {d\varphi }{d\tau }}\\\\&=\left(-\sin \varphi ,0,0,r\ \cos \varphi \right)ic,\ \left|{\mathfrak {B}}\right|=ic,\end{array}}$

consequently is follows:

${\begin{array}{ll}{\mathfrak {\dot {B}}}&={\frac {d{\mathfrak {B}}}{d\tau }}\left(-\cos \varphi ,0,0,-\sin \varphi \right)ic{\frac {d\varphi }{d\tau }}\\\\&=\left(\cos \varphi ,0,0,\ \sin \varphi \right){\frac {c^{2}}{r}},\ \left|{\mathfrak {\dot {B}}}\right|={\frac {c^{2}}{r}}.\end{array}}$

${\dot {\mathfrak {B}}}$ thus has in the light-point the direction of radius $ML$ and the magnitude $c^{2}/r$.

From these four vectors, the following magnitudes independent of reference frame, can be formed:

$({\mathfrak {RB}})$ and $({\mathfrak {R{\dot {B}}}})$

thus by which the formulas (34b) shall be expressed; here, we notice that the other possible invariants have the following simply values according to the above:

(35)	$\left\{{\begin{array}{l}({\mathfrak {RR}})=R^{2}=0,\ ({\mathfrak {BB}})=\left|{\mathfrak {B}}\right|^{2}=-c^{2},\\\\({\mathfrak {{\dot {B}}{\dot {B}}}})=\left|{\mathfrak {\dot {B}}}\right|^{2}={\frac {c^{4}}{r^{2}}},\ ({\mathfrak {B{\dot {B}}}})=0,\end{array}}\right.$

the latter is due to the perpendicular location of ${\mathfrak {B}}$ and ${\mathfrak {\dot {B}}}$. As regards the value of $({\mathfrak {RB}})$, it was given in equation (33a):

(35a)

$({\mathfrak {RB}})=\left|{\mathfrak {B}}\right|r_{0}\sin \varphi =icr_{0}\sin \varphi .$

Similarly, according to Fig. 6 it is given as projection of ${\mathfrak {R}}$ upon ${\mathfrak {\dot {B}}}$

$\left|{\mathfrak {R}}\right|\cos({\mathfrak {R,{\dot {B}}}})={\frac {({\mathfrak {R,{\dot {B}}}})}{\left|{\mathfrak {\dot {B}}}\right|}}=NL=r-r_{0}\cos \varphi ,$

thus

(35b)

$r_{0}\cos \varphi =r\left(1-{\frac {({\mathfrak {R{\dot {B}}}})}{c^{2}}}\right),\ {\frac {r_{0}}{r}}\cos \varphi ={\frac {c^{2}-({\mathfrak {R{\dot {B}}}})}{c^{2}}}$

and by division of (35a) and (35b):

(35c)

${\frac {\cos \varphi }{icr\ \sin \varphi }}={\frac {1}{c^{2}}}{\frac {c^{2}-({\mathfrak {R{\dot {B}}}})}{({\mathfrak {RB}})}}.$

If we now consider the four-vector $f_{\varphi }$ derived from the six-vector $f$ (see part I, equation (6a)):

$4\pi f_{\varphi }={\frac {e}{rr_{0}^{2}\sin ^{3}\varphi }}\left(r\ \cos \varphi -r_{0},\ y-y_{0},\ z-z_{0},\ 0\right)$

Here, the three bracketed magnitudes or the components of ${\mathfrak {R}}$ with respect to the directions $r_{0},y_{0},z_{0}$ drawn through $O$, are namely

${\mathfrak {R}}_{r}=OQ=r\ \cos \varphi -r_{0},\ {\mathfrak {R}}_{y}=y-y_{0},\ {\mathfrak {R}}_{z}=z-z_{0};$

If we thus add the fourth coordinate with respect to the direction (taken through $O$ and perpendicular to $r_{0}$) of the increasing $\varphi _{0}$, namely ${\mathfrak {R}}_{\varphi }=QL$ (see Fig. 6), then

$\left(r\ \cos \varphi -r_{0},\ y-y_{0},\ z-z_{0},\ 0\right)={\mathfrak {R}}-\mathrm {vector} \ QL$

The sum of $QL$ is $r\ \sin \varphi$; the unit vector $\Phi _{1}$ in this direction represents itself by the unit vectors in the directions ${\mathfrak {B}}$ and ${\mathfrak {\dot {B}}}$, which are inclined against that by $\varphi$ and $\pi /2-\varphi$ respectively:

(36)	$\Phi _{1}={\frac {\mathrm {vector} \ QL}{\left|QL\right|}}=\cos \varphi {\frac {\mathfrak {B}}{\left|{\mathfrak {B}}\right|}}+\sin \varphi {\frac {\mathfrak {\dot {B}}}{\left|{\mathfrak {\dot {B}}}\right|}},$

thus

$\mathrm {vector} \ QL=r\ \sin \varphi \ \Phi _{1}=r\sin \varphi \left({\frac {\cos \varphi }{ic}}{\mathfrak {B}}+{\frac {r\ \sin \varphi }{c^{2}}}{\mathfrak {\dot {B}}}\right)$

Therefore, $f_{\varphi }$ is decomposed in two or three four-vectors of directions ${\mathfrak {R}},\Phi _{1}$ or ${\mathfrak {R,B,{\dot {B}}}}$, namely

(36a)

$\left\{{\begin{array}{ll}4\pi f_{\varphi }&={\frac {e}{r\ \sin \varphi r_{0}^{2}\sin ^{2}\varphi }}{\mathfrak {R}}-{\frac {e}{r_{0}^{2}\sin ^{2}\varphi }}\Phi _{1}\\\\&={\frac {e}{r\ \sin \varphi r_{0}^{2}\sin ^{2}\varphi }}{\mathfrak {R}}-{\frac {e}{r_{0}^{2}\sin ^{2}\varphi }}\left({\frac {\cos \varphi }{ic}}{\mathfrak {B}}+{\frac {r\ \cos \varphi }{c^{2}}}{\mathfrak {\dot {B}}}\right).\end{array}}\right.$

We now pass to the specific electrodynamic force ${\mathfrak {F}}$ (see § 4), by imagining a charge at reference point $O$, whose magnitude and motion is given by the four-vector $P_{0}$. The components of ${\mathfrak {F}}$ with respect to the coordinate-directions are according to equation (11)

${\begin{array}{l}{\mathfrak {F}}_{r}=\left(P_{0}f_{r}\right)=P_{0\varphi }f_{r\varphi },\\{\mathfrak {F}}_{y}=\left(P_{0}f_{y}\right)=P_{0\varphi }f_{y\varphi },\\{\mathfrak {F}}_{z}=\left(P_{0}f_{z}\right)=P_{0\varphi }f_{z\varphi },\\{\mathfrak {F}}_{\varphi }=\left(P_{0}f_{\varphi }\right)=P_{0r}f_{\varphi r}+P_{0y}f_{\varphi y}+P_{0z}f_{\varphi z}\end{array}}$

Thus we can vectorially write, when we understand under $\Phi _{1}$ the mentioned unit vector in the $\varphi$-direction extended through $O$

${\mathfrak {F}}=-\left(P_{0}\Phi _{1}\right)f_{\varphi }+\left(P_{0}f_{\varphi }\right)\Phi _{1}$

:

or with respect to (36a):

(36b)

$4\pi {\mathfrak {F}}={\frac {e}{r\ \sin \varphi r_{0}^{2}\sin ^{2}\varphi }}\left(-\left(P_{0}\Phi _{1}\right){\mathfrak {R}}+\left(P_{0}{\mathfrak {R}}\right)\Phi _{1}\right).$

If we now include the value (36) for $\Phi _{1}$, then ${\mathfrak {F}}$ is decomposed into three portions, a location portions of direction ${\mathfrak {R}}$, a velocity- and acceleration portions of direction ${\mathfrak {B}}$ and ${\mathfrak {\dot {B}}}$, namely by using of (35a) and (35c):

(36c)

$\left\{{\begin{array}{l}4\pi {\mathfrak {F_{R}}}={\frac {e}{({\mathfrak {RB}})^{2}}}\left({\frac {c^{2}-({\mathfrak {R{\dot {B}}}})}{({\mathfrak {RB}})}}\left(P_{0}{\mathfrak {B}}\right)+\left(P_{0}{\mathfrak {\dot {B}}}\right)\right){\mathfrak {R}},\\\\4\pi {\mathfrak {F_{B}}}={\frac {-e}{({\mathfrak {RB}})^{2}}}{\frac {c^{2}-({\mathfrak {R{\dot {B}}}})}{({\mathfrak {RB}})}}\left(P_{0}{\mathfrak {R}}\right){\mathfrak {B}}\\\\4\pi {\mathfrak {F_{\dot {B}}}}={\frac {-e}{({\mathfrak {RB}})^{2}}}\left(P_{0}{\mathfrak {R}}\right){\mathfrak {\dot {B}}}.\end{array}}\right.$

By that, the general invariant representation of the specific electrodynamic force ${\mathfrak {F}}$ is achieved. From it, we go over to the total force ${\mathfrak {K}}$, by imagining the total charge $e_{0}$ as point-like in $O$. We will calculate it as a "co-moving force" (in the sense of equation (15) for ${\mathfrak {K}}'$), by considering those values of ${\mathfrak {F}}$, which appear simultaneously to an observer co-moving with $O$, i.e. integrating over a space $dS'$ perpendicular to the world-line of $O$. On the other hand, when integrating over $dS$ (as remarked in §4) a result depending on the reference system would be given. Consequently, since $dS'$ has the same meaning as $dS_{n}$ in equation (29c), it is thus given from this equation

$\int P_{0}dS'={\frac {e_{0}}{c}}{\mathfrak {B}}_{0}$

and from (36c) we obtain the following three portions of the total electrodynamic force:

(37)

$\left\{{\begin{array}{l}4\pi {\mathfrak {K_{R}}}={\frac {ee_{0}}{e({\mathfrak {RB}})^{2}}}\left({\frac {c^{2}-({\mathfrak {R{\dot {B}}}})}{({\mathfrak {RB}})}}\left({\mathfrak {B}}_{0}{\mathfrak {B}}\right)+\left({\mathfrak {B}}_{0}{\mathfrak {\dot {B}}}\right)\right){\mathfrak {R}},\\\\4\pi {\mathfrak {K_{B}}}={\frac {-ee_{0}}{c({\mathfrak {RB}})^{2}}}{\frac {c^{2}-({\mathfrak {R{\dot {B}}}})}{({\mathfrak {RB}})}}\left({\mathfrak {B}}_{0}{\mathfrak {R}}\right){\mathfrak {B}},\\\\4\pi {\mathfrak {K_{\dot {B}}}}={\frac {-ee_{0}}{c({\mathfrak {RB}})^{2}}}\left({\mathfrak {B}}_{0}{\mathfrak {R}}\right){\mathfrak {\dot {B}}}.\end{array}}\right.$

As above in consequence of (33a), one can notice that the special character of hyperbolic motion is vanished from these formulas (hyperbolic motion only served us to conveniently approximate the motion of $e$ at the light-point), and more generally that one obtains the same formulas, when ${\mathfrak {K}}$ is calculated for a quite arbitrary motion of $e$. Indeed, the equations (37) are identical with the result of residue-construction in equation (30b), about which it was spoken on p. 670.

The equations (37) are of course in agreement with the geometric rule given by Minkowski in § V of "Space and Time", and differ from the expressions originally found by Schwarzschild only by supplementing the fourth "energetic component", which by the way is not unimportant for the following, and in the three remaining "dynamic" components only differs by a factor

${\sqrt {1-\beta _{0}^{2}}}$, ($\beta _{0}c$ = velocity of $O$),

which stems from the fact, that in the course of forming the total force, Schwarzschild integrated over a space $dS$, while we integrated over a space $dS'$.

§ 9. Remarks on the laws of Coulomb and Newton.

A. Coulomb's law. Equations (37) are applied to the most simple case of electrostatics, i.e. two mutually resting point charges. In the sense of relativity theory, two charges are at rest whose world-lines are two parallel lines. In this case

${\dot {\mathfrak {B}}}=0,\ \left({\mathfrak {B}}_{0}{\mathfrak {B}}\right)=\left({\mathfrak {B}}{\mathfrak {B}}\right)=-c^{2},\ \left({\mathfrak {B}}_{0}{\mathfrak {R}}\right)=\left({\mathfrak {B}}{\mathfrak {R}}\right)$

and thus

(38)	$4\pi {\mathfrak {K}}=-ee_{0}c{\frac {c^{2}{\mathfrak {K+(RB)B}}}{({\mathfrak {RB}})^{3}}}$

After cancellation of the acceleration-portions, force ${\mathfrak {K}}$ in (38) is thus composed by a location portion and a velocity portion, which at first is related to a light-point, however, it simultaneously has the direction of an arbitrary point of the world-line of $e$ or $e_{0}$ due to the presupposed uniformity. We show, that this velocity portion supplements the location portion exactly to a vector which is directed to the point $P$ (simultaneous with $O$) upon the world-line of $e$. There, simultaneity is evidently to be considered from a reference system moving with velocity ${\mathfrak {B}}$, in our case the only reference system naturally defined, and again it will (in passing) constructed by an ordinary (euclidean) perpendicularity.

As already employed many times (see p. 676 and 679), the projection $PL$ of ${\mathfrak {R}}$ upon ${\mathfrak {B}}$ is equal to ${\mathfrak {(RB)/\left|B\right|}}$, and the unit vector in the direction of ${\mathfrak {B}}$ is equal to ${\mathfrak {B/\left|B\right|}}$, thus with respect to $\left|{\mathfrak {B}}\right|=ic$:

$LP=-PL\cdot {\frac {\mathfrak {B}}{\left|{\mathfrak {B}}\right|}}=-{\frac {({\mathfrak {RB}})}{\left|{\mathfrak {B}}\right|^{2}}}{\mathfrak {B}}=+{\frac {1}{c^{2}}}{\mathfrak {(RB)B}}.$

The nominator in (38) consequently becomes in terms of the meaning of ${\mathfrak {R'}}$ drawn in Fig. 7:

$c^{2}(\mathrm {vector} \ OL+\mathrm {vector} \ LP)=c^{2}\ \mathrm {vector} \ OP=c^{2}{\mathfrak {R'}}.$

If one applies the Pythagoras to the "right-angled" triangle $OLP$, then it is additionally given with $R'=\left|{\mathfrak {R'}}\right|,\ R=\left|{\mathfrak {R}}\right|=0$:

$R^{2}=0=R'^{2}+(PL)^{2},\ (PL)^{2}={\frac {({\mathfrak {RB}})^{2}}{\left|{\mathfrak {B}}\right|^{2}}}=-R'^{2}$

or

(38a)

$({\mathfrak {RB}})^{2}=-\left|{\mathfrak {B}}\right|^{2}R'^{2}=+c^{2}R'^{2}.$

The nominator in (38) thus becomes equal to $c^{3}R'^{3}$, so that we can write instead of (38):

(39)	$4\pi {\mathfrak {K}}=-ee_{0}{\frac {{\mathfrak {R}}'}{R'^{3}}}.$

This is the general invariant expression of Coulomb's law.

If we now introduce the special reference system $xyzl$, whose $l$-axis forms with ${\mathfrak {B}}$ the angle $\varphi$, i.e. in which $e$ and $e_{0}$ are moving. Let the coordinates of $P$ be $x',y',z',l'$, and those of $O$ be $x_{0},y_{0},z_{0},l_{0}$ as earlier. The three-dimensional vector $x'-x_{0},\ y'-y_{0},\ z'-z_{0}$ shall be called ${\mathfrak {r}}',$ $r'$ be its length. The three-dimensional force vector ${\mathfrak {K}}_{x},{\mathfrak {K}}_{y},{\mathfrak {K}}_{z}$ is to be called (according to § 4) as the dynamic portion of the four-vector ${\mathfrak {K}}$. It is of course not directed (in the sense of our arbitrary reference system) to the point simultaneous with $O$, but to that space-point emerging from the spacetime-point $P$ by projection upon the $xyz$-space. This dynamic portion namely becomes

$4\pi \left({\mathfrak {K}}_{x},{\mathfrak {K}}_{y},{\mathfrak {K}}_{z}\right)=-ee_{0}{\frac {{\mathfrak {r}}'}{R'^{3}}};$

its magnitude is:

(39a)

$4\pi {\sqrt {{\mathfrak {K}}_{x}^{2}+{\mathfrak {K}}_{y}^{2}+{\mathfrak {K}}_{z}^{2}}}=ee_{0}{\frac {r'}{R'^{3}}}.$

Thus it is neither inversely proportional to the square of $r'$ nor to that of $R'$. To maintain the simple argument of Coulomb's law, one must rather consider the energetic portion and the form the magnitude of the four-vector

$\left|{\mathfrak {K}}\right|={\sqrt {{\mathfrak {K}}_{x}^{2}+{\mathfrak {K}}_{y}^{2}+{\mathfrak {K}}_{z}^{2}+{\mathfrak {K}}_{l}^{2}}}$

To it, is imply applies

(39b)

$4\pi \left|{\mathfrak {K}}\right|={\frac {ee_{0}}{R'^{2}}},$

thus Coulomb's law in the sense, that the four-dimensional distance of both simultaneous world-points $O$ and $P$ is of importance. We emphasize this to strengthen the remark already given earlier (§4, p. 771), that the energetic component of force has in no way only a formal meaning. It is rather indispensable in our special case, to formulate Coulomb's law (in its simplicity for static conditions) also for moving charges and independently of the choice of a special reference system.

The energetic component of ${\mathfrak {K}}$ of course vanishes, when the reference system is particularly introduced as co-moving, so that both charges are at rest in it. Then $R'=r'$, so that (39a) and (39b) become identical. Since in the general case $r'/R'=\cos \varphi$ (see Fig. 7), one can consider (39a) as a component of the complete vector sum (39b).

B. Newton's law. To this one I only have to make a literary remark. There are two proposals as to the solution of the urgent task, to adapt the law of gravitation to the principles of relativity theory, by Poincaré (1906) and by Minkowski (1908). Minkowski directly transfers^[15] the repulsion law of two electrons with sign-change upon the attraction of two ponderable masses; his proposal thus aims to replace $ee_{0}$ by $-mm_{0}$ in equations (37), where $m_{0}$ and $m$ means the attracted and attracting rest-mass. We want to presuppose the motion of $m$ as uniform (acceleration ${\dot {\mathfrak {B}}}$ evidently comes into play only secondarily, since an influence of ${\mathfrak {B}}$ has never been demonstrated even on gravitation). Then, Minkowski's proposal is simplified to the following expression of Newton's law $\left({\mathfrak {K_{\dot {B}}}}=0,\ {\mathfrak {K=K_{R}+K_{B}}}\right)$, when we also include the inessential factor $f\pi$ in ${\mathfrak {K}}$

(40)	${\mathfrak {K}}=-mm_{0}c{\mathfrak {\frac {\left(B_{0}B\right)R-\left(B_{0}R\right)B}{(RB)^{3}}}}$

On the other hand, Poincaré^[16] raises himself the task, to form (in most general manner) that Lorentz invariant force law, which reduces itself for small velocities (neglecting $\beta ^{2}$ and $\beta _{0}^{2}$) to Newton's law in the ordinary sense. Only in case both velocities (as assumed under A) are equal, the form of the relativistically extended law of gravitation is unequivocally determined; otherwise it remains undetermined in certain limits. Here, Poincaré decidedly operates with four-vectors (the introduction of six-vector, however, was reserved to Minkowski); they were represented by its components in an arbitrary xyzl-system. Namely, the following combination between our vectors (left) and Poincaré's components (right) applies:

${\begin{array}{l}{\mathfrak {K}}=\left(X_{1},Y_{1},Z_{1},iT_{1}\right),\\{\mathfrak {R}}=(x,y,z,-ir),\\{\mathfrak {B}}_{0}=(\xi ,\eta ,\zeta ,\ ic)k_{0},\\{\mathfrak {B}}=(\xi _{1},\eta _{1},\zeta _{1},\ ic)k_{1},\end{array}}$

with the abbreviations

${\begin{array}{l}{\frac {1}{k_{0}^{2}}}=1-{\frac {1}{c^{2}}}\left(\xi ^{2}+\eta ^{2}+\zeta ^{2}\right)=1-\beta _{0}^{2},\\\\{\frac {1}{k_{1}^{2}}}=1-{\frac {1}{c^{2}}}\left(\xi _{1}^{2}+\eta _{1}^{2}+\zeta _{1}^{2}\right)=1-\beta ^{2}.\end{array}}$

Furthermore, it is to be noted that Poincaré has put $c=1$, and also has taken $m=m'=1$ for the attracting and attracted mass; to write his formulas in agreement with dimensions, one has to replace his velocity components $\xi ,\xi _{1},\dots$ by $\xi /c,\ \xi _{1}/c,\dots$ and his force components $X_{1},\dots$ by $X_{1}/mm',\dots$. Consequently, Poincaré's expressions^[17] (11) p. 174 can be written in the first place:

(40a)

${\frac {k_{0}{\mathfrak {K}}}{mm'}}=-{\frac {1}{B^{3}C}}\left(C{\mathfrak {R}}-{\frac {1}{c}}A{\mathfrak {B}}\right).$

The mentioned indetermination of the problems makes it possible, as Poincaré notes himself (9. 175), to replace the factor $1/B^{3}$ by $C/B^{3}$, in which case the following expression is given instead of the preceding one:

(40b)

${\frac {k_{0}{\mathfrak {K}}}{mm'}}=-{\frac {1}{B^{3}}}\left(C{\mathfrak {R}}-{\frac {1}{c}}A{\mathfrak {B}}\right).$

There, A, B, C mean the invariance to be formed from the vectors ${\mathfrak {R,B_{0},B}}$, and are given (by Poincaré) by the second to the fourth of expressions (5) of p. 169, namely:

$A=-{\frac {1}{c}}\left({\mathfrak {RB_{0}}}\right),\ B=-{\frac {1}{c}}({\mathfrak {RB}}),\ C=-{\frac {1}{c^{2}}}\left({\mathfrak {B_{0}B}}\right)$

If one eventually replaces the mass $m'=m_{0}k_{0}$ (as measured in the arbitrary reference system) by the rest-mass $m_{0}$, then it is given from (40a) or (40b):

(40c)

${\mathfrak {K}}=mm_{0}c^{3}{\mathfrak {\frac {\left(B_{0}B\right)R-\left(B_{0}R\right)B}{(RB)^{3}\left(B_{0}B\right)}}}$

or

${\mathfrak {K}}=-mm_{0}c{\mathfrak {\frac {\left(B_{0}B\right)R-\left(B_{0}R\right)B}{(RB)^{3}}}}$

The latter expression perfectly agrees with (40), the first one agrees up to the factor $-c^{2}/\left({\mathfrak {B_{0}B}}\right)$ which differs from unity only by terms of second order in the velocity ratios $\beta ,\beta _{0}$.^[18] Thus we come to the result^[19]: The special Minkowskian formulation of Newton's laws is subsumed under Poincaré's formulation which is naturally undetermined up to a certain grade, when one sets the acceleration of the attracting mass equal to zero; neglecting terms of higher order, the latter is no more general than the first one.

The relation of Minkowski's or Poincaré's version to that of Newton can be explained best by a simple figure, which is different from Fig. 7 only by the fact, that the directions ${\mathfrak {B}}_{0}$ and ${\mathfrak {B}}$ of the world-line of $m_{0}$ and $m$ are not parallel now, but enclose the angle $\psi$. This appears in Fig. 8^[20] as an angle between $QO$ (direction ${\mathfrak {B}}_{0}$) and $QT$ (parallel to ${\mathfrak {B}}$). $P$ is the point simultaneous with $O$ in the sense of reference system moving with ${\mathfrak {B}}$, $S$ is simultaneous with $O$ in the sense of its proper motion ${\mathfrak {B}}_{0}$.

Now we construct the numerator of expressions (40), (40c) and (40d) as follows. It is by § 3 A

(41)	$\left({\mathfrak {B_{0}B}}\right)=\cos \psi {\mathfrak {\left|B_{0}\right|\left|B\right|}}=-c^{2}\cos \psi$

and ${\mathfrak {\left(RB_{0}\right)/\left|B_{0}\right|}}$ is equal to the projection $OQ$ of ${\mathfrak {R}}$ upon ${\mathfrak {B}}_{0}$. From the triangle $OQT$, right-angled at $O$ (perspectively moved in the figure), however, it follows

(41a)

$QO/\cos \psi =QT=LS=-{\mathfrak {\left(RB_{0}\right)/\left|B_{0}\right|}}\cos \psi .$

Therefore by (40) and (41a), the mentioned numerator is equal to

$\left({\mathfrak {RB_{0}}}\right)\left\{{\mathfrak {R}}-{\frac {1}{\cos \psi }}{\mathfrak {\frac {\left(RB_{0}\right)}{\left|B_{0}\right|}}}{\mathfrak {\frac {B}{\left|B\right|}}}\right\}$ $=\left({\mathfrak {B_{0}B}}\right)\left\{{\mathfrak {R}}+LS{\mathfrak {\frac {B}{\left|B\right|}}}\right\}=\left({\mathfrak {B_{0}B}}\right)\left\{{\mathfrak {R}}+\mathrm {vector} \ LS\right\},$ $=\left({\mathfrak {B_{0}B}}\right){\mathfrak {S}},$

since ${\mathfrak {B/\left|B\right|}}$ means the unit vector in the direction of $LS$, and the vectors ${\mathfrak {R}}$ and $LS$ are composing themselves to vector ${\mathfrak {S}}$. The denominator in (40), on the other hand, is equal to $c^{3}R'^{3}$ according to (38a), where ${\mathfrak {R}}'$ means (as in Fig. 7) the vector $OP$ and $R'$ its sum. Consequently, it follows from (40a) with respect to (41) in agreement with Minkowski's proposal:

${\mathfrak {K}}=mm_{0}\cos \psi {\frac {S}{R'^{3}}}$

In Poincaré's proposal (40c), the factor

$-\left({\mathfrak {B_{0}B}}\right)/c^{2}=\cos \psi$

is to be added in the denominator and is thus given:

${\mathfrak {K}}={\frac {mm_{0}}{R'^{3}}}{\mathfrak {S}}.$

It's evident, that one remains in the space which was left open by Poincaré and by the invariance under Lorentz transformations, even when one sets:

(41d)

${\mathfrak {K}}={\frac {mm_{0}}{S^{3}}}{\mathfrak {S}}\ \mathrm {or} \ {\mathfrak {K}}={\frac {mm_{0}}{SR'}}{\frac {\mathfrak {S}}{S}},$

in the latter case, the magnitude $\left|{\mathfrak {K}}\right|=mm_{0}/SR$ would be calculated in symmetrical way from the two distances between $O$ on the one hand, and the two simultaneity-points $P$ and $S$ on the other hand. Also in this symmetrical choice of the force law, the reaction principle wouldn't be satisfied, since the direction of force is determined in an unsymmetrical way by the attracting and attracted mass. The permanence of this principle would rather require a momentum distributed in the surrounding of the center of attraction, as it would be known from electrodynamics, however, with the difference that its localization in the field would be unknown. On the other hand, the ordinary formulation of Newton's law is evidently inadmissible in relativity theory. According to it, the direction and magnitude of the Newtonian force would be given through a point $A$ (see Fig. 8), which is simultaneous with $O$ (in the sense of an arbitrary reference system) and thus physically undetermined. Practically, however, such a formulation might be on equal footing to the preceding one, since it only differs from them by terms of second order in the velocity ratios. Exactly because of this reason, one won't hesitate to leave the ordinary formulation of Newton's law, and to replace it with the preceding formulations which are relative-theoretically possible.

(Received July 13, 1910.

---

1. See part I, Ann. d. Phys. 32. p. 749. 1910, to which the numbering of the equations and paragraphs is supplemented.
2. We throughout denote by $ds,\ d\sigma ,\ dS,\ d\Sigma$ the element of a line, surface, of a three- or four-dimensional space, independent as to whether, for example, $ds$ denotes a curved or "plane" three-dimensional space.
3. So that such a limit exists, $P$ must of course satisfy certain continuity conditions; that the limit is independent of the form of area $\Delta \Sigma$, can be seen for example, when $\Delta \Sigma$ is divided into sufficiently small parallelepipeds. The corresponding remark applies to all following analogous passage of limits.
4. H. Poincaré, Rendiconti Circolo Matematico di Palermo 21, 1906.
5. In general, in space of $\nu$ dimensions, the boundary space of unit sphere becomes ${\tfrac {2\pi ^{\nu /2}}{\Gamma \left({\tfrac {\nu }{2}}\right)}}$, thus in four or three dimensions

   ${\frac {2\pi ^{2}}{\Gamma (2)}}=2\pi ^{2}$ or ${\frac {2\pi ^{3/2}}{{\frac {1}{2}}\Gamma \left({\frac {1}{2}}\right)}}=4\pi$

   See for example Schoute, Mehrdimensionale Geometrie (Sammlung Schubert by Göschen) Nr. 95.

6. In the passage to the negative-, not to the positive-imaginary axis, lies the preference of the waves progressing from its source, over the physical meaningless but mathematically equally-valid converging waves.
7. This equation and the combination of scalar and vector potential to a "world vector" $\Phi$, is already contained in M. Born, Ann. d. Phys. 28. p. 571, 1909
8. G. Herglotz, Göttinger Nachrichten 1904, Heft 6. This short note is especially interesting in the historical sense, because it is dated before relativity theory. Nevertheless, as it was also emphasized by Minkowski in conversation, the four-dimensional symmetry of electrodynamics is latently contained and mathematically applied.
9. At the light-point itself $R^{2}=0$, in its surrounding it is thus

   $R^{2}=\tau {\frac {\partial \left(R^{2}\right)}{\partial \tau }}=2\tau \left\{\left(x-x_{0}\right){\frac {\partial x}{\partial \tau }}+\dots +\left(l-l_{0}\right){\frac {\partial l}{\partial \tau }}\right\}=2\tau \left({\mathfrak {RB}}\right)$

   with the meaning of ${\mathfrak {R}}$ explained in (29e). The integral in question thus becomes

   $\int {\frac {\mathfrak {B}}{R^{2}}}d\tau ={\frac {\mathfrak {B}}{2\left({\mathfrak {RB}}\right)}}\int {\frac {d\tau }{\tau }}=-{\frac {\pi i{\mathfrak {B}}}{\left({\mathfrak {RB}}\right)}}$

10. See for example M. Abraham, Theorie der Elektrizität II. 9. 13
11. M. Born, Ann. d. Phys. 30. p. 1, 1909; corresponding to the previous point of view, only chapter II of this treatise is of relevance for us, whose results will be somewhat simplified by a more free usage of the four-dimensional views.
12. See H. Minkowski, Space and Time § III and V.
13. According to the definition of the light-point, it is $\left|{\mathfrak {R}}\right|=R=0$ and $\cos \left({\mathfrak {R,B}}\right)=\infty$, therefore $\left|{\mathfrak {R}}\right|\cos \left({\mathfrak {R,B}}\right)$ is an unspecified expressions, whose true value is given by $({\mathfrak {RB}})/\left|{\mathfrak {B}}\right|$ according to § 3 A.
14. One obtains, since $r\ dr\ d\varphi$ is the area, $r\ d\varphi$ or $dr$ the sides of the rotation drawn in Fig. 5a:

    $f_{r\varphi }r\ dr\ d\varphi ={\frac {\partial }{\partial r}}\left\{\Phi _{\varphi }r\ d\varphi \right\}dr-{\frac {\partial }{\partial \varphi }}\left\{\Phi _{r}dr\right\}d\varphi$

    This gives the expression (34), since $\Phi _{r}$ is independent of $\varphi$.

15. Space and Time § 5. As regards the formulation given in the "Fundamental equations" see note 2) on p. 686.
16. In the paper cited on 9. 661, from the Rendiconti di Palermo 21. 1906
17. By reversing the inadvertent sign of Poincaré.
18. On the other hand, this expression (40c) exactly matches that form of Newton's law, given by Minkowski in the "Fundamental equations", in case one assumes the motion as acceleration-free also at this place. One can immediately recognize this, when one compares the geometric formulation equivalent with (40c) with the geometric rule (24) in the "Fundamental Equations".
19. The last note in Minkowski's "Fundamental equations" is consequently to be understood, that only Poincaré's method, not the result, is essentially different.
20. To explain this figure the following shall be remarked. Through four points, thus also through two lines ${\mathfrak {B_{0},\ B}}$ of $R_{4}$, there can always be laid a $R_{3}$. All lines of Fig. 8 are therefore following the same three-dimensional space. Namely, the lines $OLSAP$ are lying in a plane, the triangles $OQT$ and $OTS$ as well as the parallelogram $LQTS$ each in one of them, and among each other in a generally different plane. Since $OS$, $OT$ and $LQ$ shall be normal to ${\mathfrak {B}}_{0}$, then also the parallel to $LQ$ (drawn through $T$) is lying in the normal plane (laid through $O$) to ${\mathfrak {B}}_{0}$; since this normal plane cuts line ${\mathfrak {B}}$ only in the single point $S$, then also the mentioned parallel goes through $S$.