Translation:On the spacetime lines of a Minkowski world/Paragraph 5

Ponderomotive force at the reference-point.

As such we take the Template:Sc force

where $𝔴$ is the velocity of the charge at the reference-point, $\rho$ its density, and

It is also

furthermore

in which the $S_{\alpha \beta }=S_{\beta \alpha }$ are the stresses, thus

Spacetime-thread as an image of a body.

The totality of the world lines of the points of a body is called a spacetime-thread by Template:Sc.

In parameter representation:

where ${\xi }^{(4)}$ is a timelike parameter and the ${\xi }^{(1)}{\xi }^{(2)}{\xi }^{(3)}$ are varying within certain limits given by the spatial extension.

We consider a piece of a spacetime-thread, for instance between two values ${\xi }_1^{(4)}$ and ${\xi }_2^{(4)}$, and we form

the latter over the boundary of the spacetime-thread. By application to a rectilinear hyper-cylinder, whose generators are parallel to the $x^{(4)}$-axis, and whose base spaces are plane and parallel to the $x^{(1)}{x}^{(2)}{x}^{(3)}$–space, one obtains from that (at the passage to the limit at infinitesimal height of the cylinder) the known representation of the Newtonian total-force by means of the Maxwell stresses and by the electromagnetic momentum or the energy theorem.

Infinitely thin spacetime-threads as an image of a pointlike charge.

pointlike charges in the mathematical sense of this word do not exist; potentials and fields would indeed become infinitely large within them. The employment of pointlike sources in potential theory is only legitimate in the integral representation of the potential, or otherwise as approximation at distances which are not infinitely small. The same also holds in our case. Therefore we have to imagine the pointlike charge as physically very small. However, Template:Pagenum the individual points of this charge will exert forces upon each other. How can they be computed?

The reaction force of radiation.

Now, it cannot be done without making assumptions on the shape of the infinitely thin spacetime-thread. Since we do not want to make them, we have to confine ourselves with an approximation. So let the world line of the charge $x$ be given, for instance as a function $x^{(\alpha )}(s)$ of the arc. The pointlike charge cannot act on itself; because the distance 0 has to be excluded when the pointlike sources in the potential theory are used, and is excluded here as well. It also cannot act upon a later position of itself; because this would require superluminal speed. Outside of the world line $x$ there should by now charges though. If we therefore extend the integral

over a piece of $S_4$ which does not contain the world line $x$, it will (being extended over a charge-free space) vanish. This is indeed a known property of the ponderomotive force of the electron theory of Lorentz. However, if the piece of $S_4$ contains a part of the world line $x$, then in accordance with the things previously said (and since the integrand would become infinite) it is to be excluded from the integration area by a small hypercylinder which encloses it, whose intersection may be taken as infinitely small. If one executes the integration in this way and then passes to the limit, by letting the hypercylinder contract itself upon the world line, then one finds zero again, as it has to be. Nevertheless, we will be able to use this integral for our purposes; because the charge is pointlike only when seen from infinitely large distances. Thus when we contract the considered piece of $S_4$ as follows: the boundary of the after-cone at point $s=s_1$ or $s=s_2$ of the wordline Template:Pagenum $x$ and if it is cut by a hypercylinder, whose intersection is assumed to be very large, then the field of a non-pointlike charge will not be computable upon the two after-cones by means of the Template:Sc formulas, but it will be computable upon the mantle of the hypercylinder, as long as we assume its intersection to be very large. As it is now

so the third integral can be evaluated by means of the Template:Sc formulas, if the distance

is afterwards increased over all measures.

Introduction of a generalized system of coordinates.

If we anticipate the developments of § 7, then every curve in $S_4$ possesses a comoving tetrad, whose four directions in the respectively considered curve point are given by the tangent $c_1^{(\alpha )}$, the principal normal $c_2^{(\alpha )}$, the binormal $c_3^{(\alpha )}$ and the trinormal $c_4^{(\alpha )}$, which form a system of four mutually pairwise perpendicular unit vectors, changing their directions from curve point to curve point. In the case of a world line, the tangent must have a timelike direction, i.e. by the definition made in § 4: the quantities $c_1^{(1)}{c}_1^{(2)}{c}_1^{(3)}$ are purely imaginary, $c_1^{(4)}$ is real. Now, it is known that

where ${\mathrm{R}}_1$ is the radius of the first curvature, which is computed as follows:

As one can see, ${\mathrm{R}}_1$ is a real quantity and is related to the Template:Sc acceleration $b$ by

Now we want to organize the points $y$ of the integration area with respect to the points $x$ of the world line, which are connected to them as light-points, and we write

Then we have in any case

Template:Pagenum since it is indeed

by assumption.

The $y$ are to be seen as functions of four parameters:

the latter is the timelike coordinate. The integration area is limited:

1. by the after-cone $s=s_1$,

1. by the mantle of the (spherical) hypercylinder

1. by the after-cone $s=s_2$.

Only the integral over 2) can be evaluated by us, if we let $a$ go towards $\mathrm{\infty }$ after finished evaluation.

Evaluation of the integral over the mantle $r=a$.

With the aid of the Template:Sc formulas (see § 7, (19)) one finds the value of

Furthermore

Template:Pagenum If it is now assumed that the determinant is

then it follows from it

For the mantle $r=a$ we have, if $N$ is the normal going to the exterior, i.e. $(N,y-x)>0$:

in which it was set for an instant:

Generally it is therefore:

Template:Pagenum One notices that it is indeed:

$N$ is thus the exterior normal. Now one has also

Thus it follows

thus we have the integral

Template:Pagenum Thus it follows in addition

and at the limit $a=\mathrm{\infty }$

On the other hand, it was

Now, since $F_{\alpha }$ is vanishing everywhere except along the infinitely thin spacetime-thread of the charge, one can write the integral on the right also as follows:

by introducing a vector of total force $K_{\alpha }$, thus

It is known that this result has already been derived by Template:Sc in a different way. Now, guided by dimensional Template:Pagenum considerations, there is still a supplement; to the previous expression we can add

if

i.e. if the motion in the location $s_1$ or $s_2$ is not accelerated.

We remark, that the first solution

gives $K_{\alpha }$ the appearance of a timelike vector; furthermore, the orthogonality condition by which $K\perp V$, is not satisfied in this form, so that also $K_4$ is not representing the mere mechanical power of force $K$. Since we have no reason to allow non-mechanical energy at a single almost pointlike charge, we rather have to assume the second form

since this one satisfies the orthogonality condition. Because it follows from $(V{)}^2\equiv 1$ by two-fold differentiation with respect to $s$:

As the reaction force of radiation we then have:

and as their purely mechanical power

Template:Pagenum The result stated above is obviously the only part of the ponderomotive forces exerted by the electron onto itself, which can be computed without assumptions about its shape. Because on the remaining boundaries, the cones $s=s_1$ or $s=s_2$, the electron cannot be seen as pointlike, because $r$ is decreasing to 0. Since assumptions on the shape should not be made, we have to refrain from further treating the dynamical problem of the forces exerted by the electron onto itself, and by that we have to refrain from treating its motions in terms of dynamics. We will rather consider them as given and then compute the respective potentials and fields at a sufficiently distant reference-point, from which the electron can be seen as pointlike.

The reaction force at Born's hyperbolic motion.

This motion is characterized by circles (equally sided hyperbolas in $M_4:xyzt$) as world lines (see § 6). Therefore it follows for the three radii of the curvatures

and for

and from the Template:Sc formulas § 7 (19):

The reaction force is identical with zero for hyperbolic motion, and only for it, if uniform motion is seen as its special case $(\frac{1}{{\mathrm{R}}_1}\equiv 0)$.

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