Translation:On the spacetime lines of a Minkowski world/Paragraph 6

If we now go on to consider special motions of a pointlike electron, we have to (by thinking about the metric nature of the relativity principle) study the metrically preferred world lines in advance. These are the curves of constant curvatures. Namely, if any orthogonal transformation of $$S_{4}$$ is carried out, then it can be interpreted as a “motion” of $$S_{4}$$, if the determinant of the transformation is +1. With respect to it, however, the individual points of $$S_{4}$$ follow trajectories of constant curvature, which can be seen by the fact, that these curves have to allow a displacement into themselves. As trajectories of a one-parameter group of motions, these curves have already been considered by during the discussion of the  rigid body.

Types of curves of constant curvatures.
Using suitable coordinate systems, when $$\varphi$$ is the (real) parameter on the curves, one has:

(A) $$\frac{1}{\mathrm{R}_{1}}\ne0,\ \frac{1}{\mathrm{R}_{2}}\ne0,\ \frac{1}{\mathrm{R}_{3}}\ne0$$ (all three constant though):


 * $$\begin{aligned}x^{(1)} & =a\cos\lambda\left(\varphi-\varphi_{0}\right), & x^{(2)} & =a\text{sin }\lambda\left(\varphi-\varphi_{0}\right),\\ x^{(3)} & =b\cos i\varphi, & x^{(4)} & =b\sin i\varphi. \end{aligned} $$

(B) $$\frac{1}{\mathrm{R}_{1}}\ne0,\ \frac{1}{\mathrm{R}_{2}}\ne0$$ (both constant); $$\frac{1}{\mathrm{R}_{3}}=0$$


 * $$\left.\begin{aligned}1.\ x^{(1)} & =a\cos\lambda\left(\varphi-\varphi_{0}\right),x^{(2)}=a\text{sin }\lambda\left(\varphi-\varphi_{0}\right),\\ x^{(3)} & =x_{0}^{(3)},x^{(4)}=i\varphi.\\ 2.\ x^{(1)} & =x_{0}^{(1)}+\alpha\varphi,x^{(2)}=x_{0}^{(2)},\\ x^{(3)} & =b\cos i\varphi,x^{(4)}=b\sin i\varphi. \end{aligned} \right\} $$helical lines of $$S_{3}$$


 * 3. $$\mathrm{R}_{2}=i\mathrm{R}_{1}$$ (Lyon curve of $$S_{3}$$).


 * $$\begin{aligned}x^{(1)} & =x_{0}^{(1)}+\frac{1}{2}\alpha\varphi^{2}, & x^{(2)} & =x_{0}^{(2)}\\ x^{(3)} & =x_{0}^{(3)}+x_{0}^{(1)}\varphi+\frac{1}{6}\alpha\varphi^{3}, & x^{(4)} & =i\left(x_{0}^{(3)}+x_{0}^{(1)}\varphi+\frac{1}{6}\alpha\varphi^{3}\right)+i\alpha\varphi. \end{aligned} $$

(C) $$\frac{1}{\mathrm{R}_{1}}\ne0$$ (constant), $$\frac{1}{\mathrm{R}_{2}}=\frac{1}{\mathrm{R}_{3}}=0$$ (Born's hyperbolic motion, trajectory of a Lorentz transformation, circle of $$S_{3}$$):


 * $$x^{(1)}=x_{0}^{(1)},\ x^{(2)}=x_{0}^{(2)},\ x^{(3)}=b\cos i\varphi,\ x^{(4)}=b\sin i\varphi.$$

(D) $$\frac{1}{\mathrm{R}_{1}}=\frac{1}{\mathrm{R}_{2}}=\frac{1}{\mathrm{R}_{3}}=0$$ (rectilinear uniform translation):


 * $$x^{(1)}=x_{0}^{(1)},\ x^{(2)}=x_{0}^{(2)},\ x^{(3)}=x_{0}^{(3)}+\beta\varphi,\ x^{(4)}=i\varphi.$$

The derivation of these types will be given in § 7. They are quickly given here in the form of a family of $$\infty^{3}$$ trajectories of a one-parameter group of motions. The parameter of the group is $$\varphi$$, the parameters of the family are respectively:


 * (A) $$a,\ b,\ \varphi^{0}$$


 * (B) 1. $$a,\ \varphi^{0},\ x_{0}^{(3)}$$; 2. $$x_{0}^{(1)},\ x_{0}^{(2)},\ b$$; 3. $$x_{0}^{(1)},\ x_{0}^{(2)},\ x_{0}^{(3)}$$;


 * (C) $$x_{0}^{(1)},\ x_{0}^{(2)},\ b$$; (D) $$x_{0}^{(1)},\ x_{0}^{(2)},\ x_{0}^{(3)}$$

On can easily see that the expression for the distance between two points $$x$$ and $$y$$ remain unchanged, if they are displaced along the respective curves of the family, i.e. the two parameter-triple, which characterize the curve of the family passing through $$x$$ or $$y$$, are unchanged, and the parameter $$\varphi_{x}$$ or $$\varphi_{y}$$ are increased by the same increment $$\Delta\varphi$$ (equi-distance)

Newton's trajectories in $$S_{3}\ x^{(4)}=0$$, to which these world lines correspond.
One finds them by insertion of $$x^{(4)}=ict$$ and substitution of $$t$$ instead of $$\varphi$$. For details see, l.c.

To which types of Newtonian mechanics $$(c=\infty)$$ they correspond, can only be discussed in § 8.

Constancy of the field in a reference system varying with the light-point.
This property has already been given by for hyperbolic motion. It generally holds for all curves of constant curvatures, as we now will demonstrate. To that end, we are using generalized coordinates: the three parameters of any family will be used as generalized spacelike coordinates, the parameter $$\varphi$$ as generalized timelike coordinate.

Such a property is geometrically predictable: because an orthogonal transformation of $$S_{4}$$ indeed must transform a mimimal line into a minimal line. Thus $$x$$ and $$y$$ again mean light-point and reference-point, and

$R^{2}=(x-y)^{2}=0$,

then also $$x\left(\varphi_{x}+\Delta\varphi\right)$$ is effectively located with $$y\left(\varphi_{y}+\Delta\varphi\right)$$. In a suitable reference system which participates with the “motion” of $$S_{1}$$, nothing could have changed. The nature of this reference system can only be determined in § 8. Here it is sufficient, that the three family parameter of $$x$$ or $$y$$ remain unchanged during this motion, while $$\varphi_{x}$$ and $$\varphi_{y}$$ experience the same increase $$\Delta\varphi$$. Thus during the transport of the potentials and fields to these four generalized coordinates, it has to be shown that these potentials and fields at the reference-point $$y$$ do not depend on $$\varphi_{y}$$.

In the following, we will only give the formulas for the potentials and the differentiation formulas stated in § 5, by which the (rather complicated) expressions for the field emerge from the potentials.

(A)
$\begin{aligned}\text{Light-point:} & x^{(1)}=a_{x}\cos\lambda\left(\varphi_{x}-\varphi_{x}^{0}\right),\ x^{(2)}=a_{x}\text{sin}\lambda\left(\varphi_{x}-\varphi_{x}^{0}\right),\ x^{(3)}=b_{x}\cos i\varphi_{x},\ x^{(4)}=b_{x}\sin i\varphi_{x}.\\ \text{Reference-point:} & y^{(1)}=a_{y}\cos\lambda\left(\varphi_{y}-\varphi_{y}^{0}\right),\ y^{(2)}=a_{y}\text{sin}\lambda\left(\varphi_{y}-\varphi_{y}^{0}\right),\ y^{(3)}=b_{y}\cos i\varphi_{y},\ y^{(4)}=b_{y}\sin i\varphi_{y}. \end{aligned} $

The transformation matrix reads:

$\begin{array}{

The reciprocal matrix reads:

$\begin{array}{

From that it follows for the arc-element of $$S_{4}$$ (not to be confused with the arc of the spacetime lines):

$d\sigma^{2}=\left(da_{y}\right)^{2}+\left(db_{y}\right)^{2}+a_{y}^{2}\lambda^{2}\left(d\varphi_{y}\right)^{2}-2a_{y}^{2}\lambda^{2}d\varphi_{y}d\varphi_{y}^{0}+\left(\lambda^{2}a_{y}^{2}-b_{y}^{2}\right)\left(d\varphi_{y}\right)^{2}$

The applied coordinates are thus oblique-angled.

For the vectors arising in the formulas

$\begin{aligned}V^{(1)} & =-\frac{a_{x}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}\right)}{i\sqrt{b_{x}^{2}-a_{x}^{2}\lambda^{2}}}, & V^{(2)} & =\frac{a_{x}\lambda\cos\lambda\left(\varphi_{x}-\varphi_{x}^{0}\right)}{i\sqrt{b_{x}^{2}-a_{x}^{2}\lambda^{2}}},\\ V^{(3)} & =-\frac{ib_{x}\sin i\varphi_{x}}{i\sqrt{b_{x}^{2}-a_{x}^{2}\lambda^{2}}}, & V^{(4)} & =\frac{ib_{x}\cos i\varphi_{x}}{i\sqrt{b_{x}^{2}-a_{x}^{2}\lambda^{2}}}. \end{aligned} $ |undefined

$\begin{aligned}R^{(1)} & =a_{x}\cos\lambda\left(\varphi_{x}-\varphi_{x}^{0}\right)-a_{y}\cos\lambda\left(\varphi_{y}-\varphi_{y}^{0}\right),\\ R^{(2)} & =a_{x}\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}\right)-a_{y}\sin\lambda\left(\varphi_{y}-\varphi_{y}^{0}\right),\\ R^{(3)} & =b_{x}\cos i\varphi_{x}-b_{x}\cos i\varphi_{y},\\ R^{(4)} & =b_{x}\sin i\varphi_{x}-b_{y}\sin i\varphi_{y} \end{aligned} $

one finds:

$\begin{aligned}V^{\left(a_{y}\right)} & =\sum_{\alpha=1}^{4}V^{(\alpha)}\frac{\partial a_{y}}{\partial y^{(\alpha)}}=\frac{-a_{x}\lambda}{i\sqrt{b_{x}^{2}-a_{x}^{2}\lambda^{2}}}\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right),\\ V^{\left(b_{y}\right)} & =\sum_{\alpha=1}^{4}V^{(\alpha)}\frac{\partial b_{y}}{\partial y^{(\alpha)}}=\frac{-ib_{x}}{i\sqrt{b_{x}^{2}-a_{x}^{2}\lambda^{2}}}\sin i\left(\varphi_{x}-\varphi_{y}\right),\\ V^{\left(\varphi_{y}^{0}\right)} & =\sum_{\alpha=1}^{4}V^{(\alpha)}\frac{\partial\varphi_{y}^{0}}{\partial y^{(\alpha)}}=\frac{-\frac{a_{x}}{a_{y}}\cos\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)+\frac{b_{x}}{b_{y}}\cos i\left(\varphi_{x}-\varphi_{y}\right)}{i\sqrt{b_{x}^{2}-a_{x}^{2}\lambda^{2}}},\\ V^{\left(\varphi_{y}\right)} & =\sum_{\alpha=1}^{4}V^{(\alpha)}\frac{\partial\varphi_{y}^{0}}{\partial y^{(\alpha)}}=\frac{\frac{b_{x}}{b_{y}}\cos i\left(\varphi_{x}-\varphi_{y}\right)}{i\sqrt{b_{x}^{2}-a_{x}^{2}\lambda^{2}}}, \end{aligned} $|undefined

as well as

$\begin{aligned}R^{\left(a_{y}\right)} & =a_{x}\cos\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)-a_{y},\\ R^{\left(b_{y}\right)} & =b_{x}\cos i\left(\varphi_{x}-\varphi_{y}\right)-b_{y},\\ R^{\left(\varphi_{y}^{0}\right)} & =-\frac{a_{x}}{\lambda a_{y}}\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)+\frac{b_{x}}{ib_{y}}\sin i\left(\varphi_{x}-\varphi_{x}^{0}\right)\\ R^{\left(\varphi_{y}\right)} & =+\frac{b_{x}}{ib_{y}}\sin i\left(\varphi_{x}-\varphi_{x}^{0}\right) \end{aligned} $|undefined

and eventually

$V^{\left(a_{x}\right)}=V^{\left(b_{x}\right)}=V^{\left(\varphi_{x}^{0}\right)},\ V^{\left(\varphi_{x}\right)}\frac{1}{i\sqrt{b_{x}^{2}-a_{x}^{2}\lambda^{2}}}$|undefined

($$V$$ has the parameter line $$\varphi_{x}$$!)

Thus we have by § 4 (15):

I. Potentials in $$y$$:
$\begin{aligned}\frac{4\pi}{e}\Phi_{a_{y}} & =\frac{1}{RV}c_{a_{y}a_{y}}V^{\left(a_{y}\right)}\\ & =-\frac{a_{x}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)}{a_{x}a_{y}\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)+ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)}\\ \frac{4\pi}{e}\Phi_{b_{y}} & =\frac{1}{RV}c_{b_{y}b_{y}}V^{\left(b_{y}\right)}\\ & =-\frac{b_{x}i\sin i\left(\varphi_{x}-\varphi_{y}\right)}{a_{x}a_{y}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)+ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)}\\ \frac{4\pi}{e}\Phi_{\varphi_{y}^{0}} & =\frac{1}{RV}\left\{ c_{\varphi_{y}^{0}\varphi_{y}^{0}}V^{\left(\varphi_{y}^{0}\right)}+c_{\varphi_{y}^{0}\varphi_{y}}V^{\left(\varphi_{y}\right)}\right\} \\ & =-\frac{a_{x}a_{y}\lambda^{2}\cos\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)}{a_{x}a_{y}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)+ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)}\\ \frac{4\pi}{e}\Phi_{\varphi_{y}} & =\frac{1}{RV}\left\{ c_{\varphi_{y}\varphi_{y}^{0}}V^{\left(\varphi_{y}^{0}\right)}+c_{\varphi_{y}\varphi_{y}}V^{\left(\varphi_{y}\right)}\right\} \\ & =\frac{a_{x}a_{y}\lambda^{2}\cos\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)-b_{x}b_{y}\cos i\left(\varphi_{x}-\varphi_{y}\right)}{a_{x}a_{y}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)+ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)} \end{aligned} $ |undefined

Note, that $$\varphi_{y}$$ only appears in the relation $$\varphi_{x}-\varphi_{y}$$, from which one can conclude that the increase of $$\varphi_{x}$$ and $$\varphi_{y}$$ by $$\Delta\varphi$$ each, leaves the potentials unchanged. Furthermore by § 4 (18):

II. Differentiation formulas:
$\begin{aligned}\frac{\partial}{\partial a_{y}} & =\left(\frac{\partial}{\partial a_{y}}\right)+\frac{c_{a_{y}a_{y}}R^{\left(a_{y}\right)}}{RV}\cdot V^{\left(\varphi_{x}\right)}\frac{\partial}{\partial\varphi_{x}}\\ & =\left(\frac{\partial}{\partial a_{y}}\right)_{x}+\frac{a_{x}\cos\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)-a_{y}}{a_{x}a_{y}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)+ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)}\frac{\partial}{\partial\varphi_{x}},\\ \frac{\partial}{\partial b_{y}} & =\left(\frac{\partial}{\partial b_{y}}\right)_{x}+\frac{c_{b_{y}b_{y}}R^{\left(b_{y}\right)}}{RV}\cdot V^{\left(\varphi_{x}\right)}\frac{\partial}{\partial\varphi_{x}}\\ & =\left(\frac{\partial}{\partial b_{y}}\right)_{x}+\frac{b_{x}\cos i\left(\varphi_{x}-\varphi_{y}\right)-b_{y}}{a_{x}a_{y}\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)+ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)}\frac{\partial}{\partial\varphi_{x}},\\ \frac{\partial}{\partial\varphi_{y}^{0}} & =\left(\frac{\partial}{\partial\varphi_{y}^{0}}\right)_{x}+\frac{c_{\varphi_{y}^{0}\varphi_{y}^{0}}R^{\left(\varphi_{y}^{0}\right)}+c_{\varphi_{y}^{0}\varphi_{y}}R^{\left(\varphi_{y}\right)}}{RV}\cdot V^{\left(\varphi_{x}\right)}\frac{\partial}{\partial\varphi_{x}}\\ & =\left(\frac{\partial}{\partial\varphi_{y}^{0}}\right)_{x}-\frac{a_{x}a_{y}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)}{a_{x}a_{y}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)+ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)}\frac{\partial}{\partial\varphi_{x}},\\ \frac{\partial}{\partial\varphi_{y}} & =\left(\frac{\partial}{\partial\varphi_{y}}\right)_{x}+\frac{c_{\varphi_{y}\varphi_{y}^{0}}R^{\left(\varphi_{y}^{0}\right)}+c_{\varphi_{y}\varphi_{y}}R^{\left(\varphi_{y}\right)}}{RV}\cdot V^{\left(\varphi_{x}\right)}\frac{\partial}{\partial\varphi_{x}}\\ & =\left(\frac{\partial}{\partial\varphi_{y}}\right)_{x}+\frac{\partial}{\partial\varphi_{x}}. \end{aligned} $|undefined

By the differentiations, the relation $$\varphi_{x}-\varphi_{y}$$ is therefore not dissolved, and $$\varphi_{y}$$ within that relation will therefore appear in the fields as well. But if $$f\left(\varphi_{x}-\varphi_{y}\right)$$ is an arbitrary function in this relation, it follows

$\frac{\partial}{\partial\varphi_{y}}f\left(\varphi_{x}-\varphi_{y}\right)=\left(\frac{\partial}{\partial\varphi_{y}}f\right)_{x}+\frac{\partial}{\partial\varphi_{x}}f=-f'+f'\equiv0$|undefined

which was to be proven.

(B) 1.
Light-point:

$\begin{aligned}x^{(1)} & =a_{x}\cos\lambda\left(\varphi_{x}-\varphi_{x}^{0}\right), & x^{(2)} & =a_{x}\text{sin }\lambda\left(\varphi_{x}-\varphi_{x}^{0}\right),\\ x^{(3)} & =x_{0}^{(3)}, & x^{(4)} & =i\varphi_{y}. \end{aligned} $

Reference-point:

$\begin{aligned}y^{(1)} & =a_{y}\cos\lambda\left(\varphi_{y}-\varphi_{y}^{0}\right), & y^{(2)} & =a_{y}\text{sin }\lambda\left(\varphi_{y}-\varphi_{y}^{0}\right),\\ y^{(3)} & =y_{0}^{(3)}, & y^{(4)} & =i\varphi_{y}. \end{aligned} $

The details of the computation are analogous as under A.

I. Potentials:
$\begin{aligned}\frac{4\pi}{e}\Phi_{a_{y}} & =-\frac{a_{x}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)}{a_{x}a_{y}\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)-\varphi_{x}+\varphi_{y}}\\ \frac{4\pi}{e}\Phi_{3} & =0\\ \frac{4\pi}{e}\Phi_{\varphi_{y}^{0}} & =-\frac{a_{x}a_{y}\lambda^{2}\cos\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)}{a_{x}a_{y}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)-\varphi_{x}+\varphi_{y}}\\ \frac{4\pi}{e}\Phi_{\varphi_{y}} & =\frac{a_{x}a_{y}\lambda^{2}\cos\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)-1}{a_{x}a_{y}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)-\varphi_{x}+\varphi_{y}} \end{aligned} $|undefined

II. Differentiation formulas:
$\begin{aligned}\frac{\partial}{\partial a_{y}} & =\left(\frac{\partial}{\partial a_{y}}\right)_{x}+\frac{a_{x}\cos\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)-a_{y}}{a_{x}a_{y}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)-\varphi_{x}+\varphi_{y}}\frac{\partial}{\partial\varphi_{x}},\\ \frac{\partial}{\partial y_{0}^{(3)}} & =\left(\frac{\partial}{\partial y_{0}^{(3)}}\right)_{x}+\frac{x_{0}^{(3)}-y_{0}^{(3)}}{a_{x}a_{y}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)-\varphi_{x}+\varphi_{y}}\frac{\partial}{\partial\varphi_{x}},\\ \frac{\partial}{\partial\varphi_{y}^{0}} & =\left(\frac{\partial}{\partial\varphi_{y}^{0}}\right)_{x}+\frac{-a_{x}a_{y}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)}{a_{x}a_{y}\lambda\sin\lambda\left(\varphi_{x}-\varphi_{x}^{0}-\varphi_{y}+\varphi_{y}^{0}\right)-\varphi_{x}+\varphi_{y}}\frac{\partial}{\partial\varphi_{x}},\\ \frac{\partial}{\partial\varphi_{y}} & =\left(\frac{\partial}{\partial\varphi_{y}}\right)_{x}+\frac{\partial}{\partial\varphi_{x}} \end{aligned} $|undefined

The last line again contains the proof.

(B) 2.
Light-point:

$x^{(1)}=x_{0}^{(1)}+\alpha\varphi_{x},\ x^{(2)}=x_{0}^{(2)},\ x^{(3)}=b_{x}\cos i\varphi_{x},\ x^{(4)}=b_{x}\sin i\varphi_{x}.$

Reference-point:

$y^{(1)}=y_{0}^{(1)}+\alpha\varphi_{y},\ y^{(2)}=y_{0}^{(2)},\ y^{(3)}=b_{y}\cos i\varphi_{y},\ y^{(4)}=b_{y}\sin i\varphi_{y}.$

I.
$\begin{aligned}\frac{4\pi}{e}\Phi_{y_{0}^{1}} & =\frac{\alpha}{\alpha\left(x_{0}^{1}-y_{0}^{1}\right)+\alpha^{2}\left(\varphi_{x}-\varphi_{y}\right)+ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)},\\ \frac{4\pi}{e}\Phi_{2} & =0,\\ \frac{4\pi}{e}\Phi_{b_{y}^{0}} & =\frac{-ib_{x}\sin i\left(\varphi_{x}-\varphi_{y}\right)}{\alpha\left(x_{0}^{1}-y_{0}^{1}\right)+\alpha^{2}\left(\varphi_{x}-\varphi_{y}\right)+ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)},\\ \frac{4\pi}{e}\Phi_{\varphi_{y}} & =\frac{\alpha^{2}-b_{x}b_{y}\cos i\left(\varphi_{x}-\varphi_{y}\right)}{\alpha\left(x_{0}^{1}-y_{0}^{1}\right)+\alpha^{2}\left(\varphi_{x}-\varphi_{y}\right)+ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)}, \end{aligned} $|undefined

II.
$\begin{aligned}\frac{\partial}{\partial y_{0}^{1}} & =\left(\frac{\partial}{\partial y_{0}^{1}}\right)_{x}+\frac{x_{0}^{1}-y_{0}^{1}+\alpha\left(\varphi_{x}-\varphi_{y}\right)}{\alpha\left(x_{0}^{1}-y_{0}^{1}\right)+\alpha^{2}\left(\varphi_{x}-\varphi_{y}\right)+ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)}\frac{\partial}{\partial\varphi_{x}},\\ \frac{\partial}{\partial y_{0}^{(2)}} & =\left(\frac{\partial}{\partial y_{0}^{(2)}}\right)_{x}+\frac{x_{0}^{(2)}-y_{0}^{(2)}}{\alpha\left(x_{0}^{1}-y_{0}^{1}\right)+\alpha^{2}\left(\varphi_{x}-\varphi_{y}\right)+ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)}\frac{\partial}{\partial\varphi_{x}},\\ \frac{\partial}{\partial b_{y}} & =\left(\frac{\partial}{\partial b_{y}}\right)_{x}+\frac{b_{x}\cos i\left(\varphi_{x}-\varphi_{y}\right)-b_{y}}{\alpha\left(x_{0}^{1}-y_{0}^{1}\right)+\alpha^{2}\left(\varphi_{x}-\varphi_{y}\right)+ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)}\frac{\partial}{\partial\varphi_{x}},\\ \frac{\partial}{\partial\varphi_{y}} & =\left(\frac{\partial}{\partial\varphi_{y}}\right)_{x}+\frac{\partial}{\partial\varphi_{x}}. \end{aligned} $|undefined

In the last formula the proof is contained again.

(B) 3.
Light-point:

$\begin{aligned}x^{(1)} & =x_{0}^{(1)}+\frac{1}{2}\alpha\varphi_{x}^{2}, & x^{(2)} & =x_{0}^{(2)}\\ x^{(3)} & =x_{0}^{(3)}+x_{0}^{(1)}\varphi_{x}+\frac{1}{6}\alpha\varphi_{x}^{3}, & x^{(4)} & =i\left(x_{0}^{(3)}+x_{0}^{(1)}\varphi_{x}+\frac{1}{6}\alpha\varphi_{x}^{3}\right)+i\alpha\varphi_{x}. \end{aligned} $

Reference-point:

$\begin{aligned}y^{(1)} & =y_{0}^{(1)}+\frac{1}{2}\alpha\varphi_{y}^{2}, & y^{(2)} & =y_{0}^{(2)}\\ y^{(3)} & =y_{0}^{(3)}+y_{0}^{(1)}\varphi_{y}+\frac{1}{6}\alpha\varphi_{y}^{3}, & y^{(4)} & =i\left(y_{0}^{(3)}+y_{0}^{(1)}\varphi_{y}+\frac{1}{6}\alpha\varphi_{y}^{3}\right)+i\alpha\varphi_{y}. \end{aligned} $

$\begin{array}{

$\begin{aligned}V^{(1)} & =\frac{\alpha\varphi_{x}}{i\sqrt{2x_{0}^{(1)}\alpha+\alpha^{2}}}, & V^{(2)} & =0,\\ V^{(3)} & =\frac{x_{0}^{(1)}+\frac{1}{2}\alpha\varphi_{x}^{2}}{i\sqrt{2x_{0}^{(1)}\alpha+\alpha^{2}}}, & V^{(2)} & =\frac{i\left(x_{0}^{(1)}+\frac{1}{2}\alpha\varphi_{x}^{2}\right)+i\alpha}{i\sqrt{2x_{0}^{(1)}\alpha+\alpha^{2}}}, \end{aligned} $|undefined

{{center|$$\left.\begin{aligned}V_{y_{0}^{(1)}} & =\sum_{\alpha=1}^{4}V^{(\alpha)}\frac{\partial y^{(\alpha)}}{\partial y_{0}^{(1)}}=\frac{\alpha\left(\varphi_{x}-\varphi_{y}\right)}{i\sqrt{2x_{0}^{(1)}\alpha+\alpha^{2}}},\\ V_{y_{0}^{(2)}} & =\sum_{\alpha=1}^{4}V^{(\alpha)}\frac{\partial y^{(\alpha)}}{\partial y_{0}^{(2)}}=0,\\ V_{y_{0}^{(3)}} & =\sum_{\alpha=1}^{4}V^{(\alpha)}\frac{\partial y^{(\alpha)}}{\partial y_{0}^{(3)}}=\frac{-\alpha}{i\sqrt{2x_{0}^{(1)}\alpha+\alpha^{2}}},\\ V_{\varphi_{y}} & =\sum_{\alpha=1}^{4}V^{(\alpha)}\frac{\partial\varphi_{y}}{\partial y^{(\alpha)}}=-\frac{\frac{1}{2}\alpha^{2}\left(\varphi_{x}-\varphi_{y}\right)^{2}+\alpha\left(x_{0}^{(1)}-y_{0}^{(1)}\right)+\alpha^{2}}{i\sqrt{2x_{0}^{(1)}\alpha+\alpha^{2}}}, \end{aligned} \right\} $$ }}

$\begin{aligned}R^{(1)} & =x_{0}^{(1)}-y_{0}^{(1)}+\frac{1}{2}\alpha^{2}\left(\varphi_{x}-\varphi_{y}\right)^{2},\\ R^{(2)} & =x_{0}^{(2)}-y_{0}^{(2)},\\ R^{(3)} & =x_{0}^{(3)}-y_{0}^{(3)}+x_{0}^{(1)}\varphi_{x}-y_{0}^{(1)}\varphi_{y}+\frac{1}{6}\alpha\left(\varphi_{x}^{3}-\varphi_{y}^{3}\right),\\ R^{(4)} & =i\left(x_{0}^{(3)}-y_{0}^{(3)}+x_{0}^{(1)}\varphi_{x}-y_{0}^{(1)}\varphi_{y}+\frac{1}{6}\alpha\left[\varphi_{x}^{3}-\varphi_{y}^{3}\right]\right)+i\alpha\left(\varphi_{x}-\varphi_{y}\right) \end{aligned} $

$\begin{aligned}R_{y_{0}^{(1)}} & =x_{0}^{(1)}-y_{0}^{(1)}+\frac{\alpha}{2}\left(\varphi_{x}-\varphi_{y}\right)^{2},\\ R_{y_{0}^{(2)}} & =x_{0}^{(2)}-y_{0}^{(2)},\\ R_{y_{0}^{(3)}} & =-\alpha\left(\varphi_{x}-\varphi_{y}\right),\\ R_{\varphi_{y}} & =-\alpha\left(x_{0}^{(1)}-y_{0}^{(1)}\right)\left(\varphi_{x}-\varphi_{y}\right)-\frac{1}{6}\alpha^{2}\left(\varphi_{x}-\varphi_{y}\right)^{3},\\ & \quad-\alpha\left(x_{0}^{(3)}-y_{0}^{(3)}\right)-\alpha^{2}\left(\varphi_{x}-\varphi_{y}\right) \end{aligned} $|undefined

In relation to that it has to be remarked, that a decomposition with respect to those parameter lines is physically invalid, because the parameter $$y_{0}^{(3)}$$ is always related to minimal directions instead of spacelike ones. Therefore, we can only conclude the constancy of the potentials and the fields from the things stated above, which result is not influenced by the previous circumstance. We have again:

$\frac{\partial}{\partial\varphi_{y}}=\left(\frac{\partial}{\partial\varphi_{y}}\right)_{x}+\frac{\partial}{\partial\varphi_{x}}$|undefined

and because of the appearance of $$\varphi_{y}$$ only in the relation $$\varphi_{x}-\varphi_{y}$$, the proof has been given.

(C) 1. Hyperbolic motion.
Light-point:

$x^{(1)}=x_{0}^{(1)},\ x^{(2)}=x_{0}^{(2)},\ x^{(3)}=b_{x}\cos i\varphi_{x},\ x^{(4)}=b_{x}\sin i\varphi_{x}.$

Reference-point:

$y^{(1)}=y_{0}^{(1)},\ y^{(2)}=y_{0}^{(2)},\ y^{(3)}=b_{y}\cos i\varphi_{y},\ y^{(4)}=b_{y}\sin i\varphi_{y}.$

$\begin{array}{

$V^{(1)}=0,\ V^{(2)}=0,\ V^{(3)}=-\sin i\varphi_{x},\ V^{(4)}=\cos i\varphi_{x}.$

$\frac{dV^{(1)}}{ds}=0,\ \frac{dV^{(2)}}{ds}=0,\ \frac{dV^{(3)}}{ds}=-\frac{\cos i\varphi_{x}}{b_{x}},\ \frac{dV^{(4)}}{ds}=-\frac{\sin i\varphi_{x}}{b_{x}}.$|undefined

$\begin{aligned}R^{(1)} & =x_{0}^{(1)}-y_{0}^{(1)}, & R^{(2)} & =x_{0}^{(2)}-y_{0}^{(2)},\\ R^{(3)} & =b_{x}\cos i\varphi_{x}-b_{y}\cos i\varphi_{y}, & R^{(4)} & =b_{x}\sin i\varphi_{x}-b_{y}\sin i\varphi_{y}. \end{aligned} $

$(RV)=b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right),\ 1+R\frac{dV}{ds}=\frac{b_{y}}{b_{x}}\cos i\left(\varphi_{x}-\varphi_{y}\right)$|undefined

$\begin{aligned}V_{y_{0}^{(1)}} & =V_{y_{0}^{(2)}}=0,\\ V_{b_{y}} & =\sum_{\alpha=1}^{4}V^{(\alpha)}\frac{\partial y^{(\alpha)}}{\partial b_{y}}=-\sin i\left(\varphi_{x}-\varphi_{y}\right),\\ V_{\varphi_{y}} & =\sum_{\alpha=1}^{4}V^{(\alpha)}\frac{\partial y^{(\alpha)}}{\partial\varphi_{y}}=-ib_{y}\cos i\left(\varphi_{x}-\varphi_{y}\right). \end{aligned} $|undefined

$\begin{aligned}\left(\frac{dV}{ds}\right)_{y_{0}^{(1)}} & =\left(\frac{dV}{ds}\right)_{y_{0}^{(2)}}=0,\\ \left(\frac{dV}{ds}\right)_{b_{y}} & =-\frac{\cos i\left(\varphi_{x}-\varphi_{y}\right)}{b_{x}},\\ \left(\frac{dV}{ds}\right)_{\varphi_{y}} & =-i\frac{b_{y}}{b_{x}}\sin i\left(\varphi_{x}-\varphi_{y}\right), \end{aligned} $|undefined

$\begin{aligned}R_{y_{0}^{(1)}} & =x_{0}^{(1)}-y_{0}^{(1)}, & R_{y_{0}^{(2)}} & =x_{0}^{(2)}-y_{0}^{(2)},\\ R_{b_{y}} & =b_{x}\cos i\left(\varphi_{x}-\varphi_{y}\right)-b_{y}, & R_{\varphi_{y}} & =ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right). \end{aligned} $|undefined

Furthermore

$V^{\left(x_{0}^{(1)}\right)}=V^{\left(x_{0}^{(2)}\right)}=V^{\left(b_{x}\right)}=0,\ V^{\left(\varphi_{x}\right)}=\frac{1}{ib_{x}}$|undefined

Thus:

1. Potentials:
$\frac{4\pi}{e}\Phi_{y_{0}^{(1)}}=\frac{V_{y_{0}^{(1)}}}{RV}=0,\ \frac{4\pi}{e}\Phi_{y_{0}^{(2)}}=\frac{V_{y_{0}^{(2)}}}{RV}=0,$|undefined

$\begin{aligned}\frac{4\pi}{e}\Phi_{b_{y}} & =\frac{V_{b_{y}}}{RV}=-\frac{1}{b_{y}}\frac{\sin i\left(\varphi_{x}-\varphi_{y}\right)}{\sin i\left(\varphi_{x}-\varphi_{y}\right)}=-\frac{1}{b_{y}},\\ \frac{4\pi}{e}\Phi_{\varphi_{y}} & =i\frac{\cos i\left(\varphi_{x}-\varphi_{y}\right)}{\sin i\left(\varphi_{x}-\varphi_{y}\right)}. \end{aligned} $|undefined

II. Differentiation formulas:
$\begin{aligned}\frac{\partial}{\partial y_{0}^{(1)}} & =\left(\frac{\partial}{\partial y_{0}^{1}}\right)_{x}+\frac{R_{y_{0}^{(1)}}}{RV}V^{\left(\varphi_{x}\right)}\frac{\partial}{\partial\varphi_{x}}=\left(\frac{\partial}{\partial y_{0}^{1}}\right)_{x}+\frac{x_{0}^{(1)}-y_{0}^{(1)}}{ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)}\frac{\partial}{\partial\varphi_{x}},\\ \frac{\partial}{\partial y_{0}^{(2)}} & =\left(\frac{\partial}{\partial y_{0}^{(2)}}\right)_{x}+\frac{R_{y_{0}^{(2)}}}{RV}V^{\left(\varphi_{x}\right)}\frac{\partial}{\partial\varphi_{x}}=\left(\frac{\partial}{\partial y_{0}^{2}}\right)_{x}+\frac{x_{0}^{(2)}-y_{0}^{(2)}}{ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)}\frac{\partial}{\partial\varphi_{x}},\\ \frac{\partial}{\partial b_{y}} & =\left(\frac{\partial}{\partial b_{y}}\right)_{x}+\frac{R_{b_{y}}}{RV}V^{\left(\varphi_{x}\right)}\frac{\partial}{\partial\varphi_{x}}=\left(\frac{\partial}{\partial b_{y}}\right)_{x}+\frac{b_{x}\cos i\left(\varphi_{x}-\varphi_{y}\right)-b_{y}}{ib_{x}b_{y}\sin i\left(\varphi_{x}-\varphi_{y}\right)}\frac{\partial}{\partial\varphi_{x}},\\ \frac{\partial}{\partial\varphi_{y}} & =\left(\frac{\partial}{\partial\varphi_{y}}\right)_{x}+\frac{R_{\varphi_{y}}}{RV}V^{\left(\varphi_{x}\right)}\frac{\partial}{\partial\varphi_{x}}=\left(\frac{\partial}{\partial\varphi_{y}}\right)_{x}+\frac{\partial}{\partial\varphi_{x}}. \end{aligned} $|undefined

In the last line, the proof for the constancy of the field is contained, since $$\varphi_{y}$$ only appears in the relation $$\varphi_{x}-\varphi_{y}$$.

III. Fields:
By § 5 (16)

$\frac{4\pi}{e}\bar{F}_{\alpha\beta}=\frac{1}{(RV)^{2}}\left[\bar{R}\frac{d\bar{V}}{ds}\right]_{\alpha\beta}-\frac{1+R\frac{dV}{ds}}{(RV)^{3}}[\overline{RV}]_{\alpha\beta}$|undefined

in which

$\left[\bar{R}\frac{d\bar{V}}{ds}\right]_{\alpha\beta}=\bar{R}_{\alpha}\left(\frac{d\bar{V}}{ds}\right)_{\beta}-\bar{R}_{\beta}\left(\frac{d\bar{V}}{ds}\right)_{\alpha}$ etc.|undefined

or, if we set

$\bar{T}_{\alpha}=(RV)\left(\frac{d\bar{V}}{ds}\right)_{\alpha}-\left(1+R\frac{dV}{ds}\right)\bar{V}_{\alpha}$|undefined

it follows

$\frac{4\pi}{e}\bar{F}_{\alpha\beta}=\frac{1}{(RV)^{3}}\left[\overline{RT}\right]_{\alpha\beta}$|undefined

For the computation of $$\bar{T}_{\alpha}$$ we have in the present case

$\begin{aligned}T_{y_{0}^{(1)}} & =0,\quad T_{y_{0}^{(2)}}=0,\\ T_{b_{y}} & =(RV)\left(\frac{dV}{ds}\right)_{b_{y}}-\left(1+R\frac{dV}{ds}\right)_{b}V_{b_{y}}=0,\\ T_{\varphi_{y}} & =(RV)\left(\frac{dV}{ds}\right)_{\varphi_{y}}-\left(1+R\frac{dV}{ds}\right)V_{\varphi_{y}}=-i\frac{b_{y}^{2}}{b_{x}} \end{aligned} $|undefined

Thus the remarkable effect arises in hyperbolic motion, that the vector $$T$$ (of the tangent of the world line of reference point $$W$$) is directed in the parallel direction. We will see, what follows from that. Namely, we have

{{center|$$\frac{4\pi}{e}F_{y_{0}^{(1)}\varphi_{y}^{2}}=\frac{4\pi}{e}F_{y_{0}^{(1)}b_{y}}=\frac{4\pi}{e}F_{y_{0}^{(2)}b_{y}}\equiv0\ \left\{ \begin{matrix}\text{(magnetic}\\ \text{components),} \end{matrix}\right.$$}}

{{center|$$\left.\begin{aligned}\frac{4\pi}{e}F_{y_{0}^{(1)}\varphi_{y}} & =\frac{1}{(RV)^{3}}R_{y_{0}^{(1)}}T_{\varphi_{y}}=\frac{x_{0}^{(1)}-y_{0}^{(1)}}{ib_{x}b_{y}\sin^{3}i\left(\varphi_{x}-\varphi_{y}\right)}\\ \frac{4\pi}{e}F_{y_{0}^{(2)}\varphi_{y}} & =\frac{1}{(RV)^{3}}R_{y_{0}^{(2)}}T_{\varphi_{y}}=\frac{x_{0}^{(2)}-y_{0}^{(2)}}{ib_{x}b_{y}\sin^{3}i\left(\varphi_{x}-\varphi_{y}\right)}\\ \frac{4\pi}{e}F_{b_{y}\varphi_{y}} & =\frac{1}{(RV)^{3}}R_{b_{y}}T_{\varphi_{y}}=\frac{b_{x}\cos i\left(\varphi_{x}-\varphi_{y}\right)-b_{y}}{ib_{x}b_{y}\sin^{3}i\left(\varphi_{x}-\varphi_{y}\right)} \end{aligned} \right\} \ \begin{matrix}\text{(electric}\\ \text{components)} \end{matrix}$$}}

Thus the field ($$F_{\alpha\beta}$$ indeed contains, as predicted, $$\varphi_{y}$$ only in the relation $$\varphi_{x}-\varphi_{y}$$) is not only constant in the varying reference system (this time, it is a system comoving with the reference-point, as can be easily seen), but it is also purely electrical.

Computation by the method of vectorial splitting of § 1.
$\frac{4\pi}{e}\mathfrak{A}_{x}=\frac{4\pi}{e}\Phi_{y_{0}^{(1)}}=0,\ \frac{4\pi}{e}\mathfrak{A}_{y}=\frac{4\pi}{e}\Phi_{y_{0}^{(2)}}=0,$|undefined

$\begin{aligned}\frac{4\pi}{e}\mathfrak{A}_{b} & =-\frac{1}{b_{y}},\\ \frac{4\pi}{e}i\varphi & =\frac{4\pi}{e}\frac{\Phi_{\varphi_{y}}}{\sqrt{c\varphi_{y}\varphi_{y}}}=\frac{1}{\sqrt{-b_{y}^{2}}}\frac{i\cos i\left(\varphi_{x}-\varphi_{y}\right)}{\sin i\left(\varphi_{x}-\varphi_{y}\right)}=\frac{1}{b_{y}}\frac{\cos i\left(\varphi_{x}-\varphi_{y}\right)}{\sin i\left(\varphi_{x}-\varphi_{y}\right)} \end{aligned} $|undefined

furthermore

$\frac{4\pi}{e}\mathfrak{H}_{x}=\frac{4\pi}{e}\mathfrak{H}_{y}=\frac{4\pi}{e}\mathfrak{H}_{b}=0,$

$\begin{aligned}-\frac{4\pi}{e}i\mathfrak{E}_{x} & =\frac{4\pi}{e}\frac{F_{y_{0}^{(1)}\varphi_{y}}}{\sqrt{c\varphi_{y}\varphi_{y}}}=-\frac{x_{0}^{(1)}-y_{0}^{(1)}}{b_{x}b_{y}^{2}\sin^{3}i\left(\varphi_{x}-\varphi_{y}\right)},\\ -\frac{4\pi}{e}i\mathfrak{E}_{y} & =\frac{4\pi}{e}\frac{F_{y_{0}^{(2)}\varphi_{y}}}{\sqrt{c\varphi_{y}\varphi_{y}}}=-\frac{x_{0}^{(2)}-y_{0}^{(2)}}{b_{x}b_{y}^{2}\sin^{3}i\left(\varphi_{x}-\varphi_{y}\right)},\\ -\frac{4\pi}{e}i\mathfrak{E}_{b} & =\frac{4\pi}{e}\frac{F_{b_{y}\varphi_{y}}}{\sqrt{c\varphi_{y}\varphi_{y}}}=-\frac{b_{x}\cos i\left(\varphi_{x}-\varphi_{y}\right)-b_{y}}{b_{x}b_{y}^{2}\sin^{3}i\left(\varphi_{x}-\varphi_{y}\right)}, \end{aligned} $|undefined

which formulas have already been given by.