Translation:On the spacetime lines of a Minkowski world/Paragraph 8

Now we have to make the announced applications of the differential geometric theories upon the world lines of $S_4$.

Curves of constant curvatures.

For the reference-point $y$, a trajectory of the family (which was definitely determined by the world line of $x$) was required in § 6. We found as the typical form of such a trajectory

in which the $c_k$ are the direction cosines of the $k^{\mathrm{t}\mathrm{h}}$ axis of the comoving tetrad in $x$, and the $\mathrm{\Lambda }$ are constants, which also have to satisfy the condition

Of course, if for instance the curve $x$ lies in a plane $S_3$ with $(\frac{1}{{\mathrm{R}}_3}=0)$, then $c_4$ must be replaced by the normal of $S_3$ etc. For the directions of the generalized axis-cross used in § 6, we have the typical representation as $\sum _{k=1}^4{A}^{(k)}{c}_k$ with constant $A^{(k)}$. The underlying reference system is essentially nothing other as the comoving tetrad $c_{1,x}{c}_{2,x}{c}_{3,x}{c}_{4,x}$ of $x$, or if one likes it better, the comoving tetrad $c_{1,y}{c}_{2,y}{c}_{3,y}{c}_{4,y}$ of $y$, since also for $c_{h,y}$ the representation as $\sum _{k=1}^4{A}^{(\varkappa )}{c}_{k,x}$ with constant $A^{(\varkappa )}$ holds. $y$ rests in the latter (time axis $c_{1,y}\equiv W$), while $x$ rests in the former ($c_{1,x}\equiv V$), but they are both simultanesouly at rest in none of them. But one surely finds by (19)

Template:Pagenum thus for

the representation as $\sum _k^4{A}^{(k)}{c}_{k,x}$, thus for the scalar product

The relative velocity $𝔲$ of $y$ against $x$ and of $x$ against $y$ is not only constant in terms of direction (of course in system $c_{1,x}{c}_{2,x}{c}_{3,x}{c}_{4,x}$ or $c_{1,y}{c}_{2,y}{c}_{3,y}{c}_{4,y}$), but also in terms of magnitude; thus in the respective relative system, $y$ is uniformly and rectilinearly in motion with respect to $x$, or $x$ with respect to $y$.

Constancy of the field in system $c_{1,x}{c}_{2,x}{c}_{3,x}{c}_{4,x}$.

The Template:Sc formulas (10) read:

Now we set

then one finds:

Template:Pagenum Thus when the directions $c_{1,x}{c}_{2,x}{c}_{3,x}{c}_{4,x}$ are used as axis-directions 4, 3, 2, 1:

$\begin{array}{cc}\frac{4\pi }{e}{F}_{23}& =\frac{4\pi }{e}{F}_{c_4{c}_2}=-\frac{{\mathrm{\Lambda }}^{(4)}/{\mathrm{R}}_1}{{\left({\mathrm{\Lambda }}^{(1)}\right)}^2},\\ \frac{4\pi }{e}{F}_{31}& =\frac{4\pi }{e}{F}_{c_2{c}_3}=+\frac{{\mathrm{\Lambda }}^{(3)}/{\mathrm{R}}_1}{{\left({\mathrm{\Lambda }}^{(1)}\right)}^2},\\ \frac{4\pi }{e}{F}_{12}& =\frac{4\pi }{e}{F}_{c_3{c}_4}=0;\\ \frac{4\pi }{e}{F}_{14}& =\frac{4\pi }{e}{F}_{c_3{c}_1}=-\frac{(1-{\mathrm{\Lambda }}^{(2)}/{\mathrm{R}}_1){\mathrm{\Lambda }}^{(3)}}{{\left({\mathrm{\Lambda }}^{(1)}\right)}^3},\\ \frac{4\pi }{e}{F}_{24}& =\frac{4\pi }{e}{F}_{c_4{c}_1}=-\frac{(1-{\mathrm{\Lambda }}^{(2)}/{\mathrm{R}}_1){\mathrm{\Lambda }}^{(4)}}{{\left({\mathrm{\Lambda }}^{(1)}\right)}^3},\\ \frac{4\pi }{e}{F}_{34}& =\frac{4\pi }{e}{F}_{c_2{c}_1}=\frac{1/{\mathrm{R}}_1}{{\mathrm{\Lambda }}^{(1)}}-\frac{(1-{\mathrm{\Lambda }}^{(2)}/{\mathrm{R}}_1){\mathrm{\Lambda }}^{(2)}}{{\left({\mathrm{\Lambda }}^{(1)}\right)}^3},\end{array}$

Template:Optional style|(29)

by which the constancy of the field is proven in reference system $c_{1,x}{c}_{2,x}{c}_{3,x}{c}_{4,x}$, and therefore in any system whose axes appear to be given by expressions $\sum A^{(k)}{c}_{k,x}$. In this respect, ${\mathrm{\Lambda }}^{(1)}$ is of course purely imaginary, ${\mathrm{\Lambda }}^{(2)}{\mathrm{\Lambda }}^{(3)}{\mathrm{\Lambda }}^{(4)}$ are real.

Since also $x$ behaves with respect to $y$ as well as $y$ behaves with respect to $x$, one can also write for the wordline of $x$

where, of course, another value as the previous $x$ must be inserted when $x$ shall be the reference point for $y$, then the things said above are also valid for the field in $x$ of a charge located in $y$. The mutual influence is therefore constant in a system $c_{1,x}{c}_{2,x}{c}_{3,x}{c}_{4,x}$ or $c_{1,y}{c}_{2,y}{c}_{3,y}{c}_{4,y}$; the electron, whose points describe such world lines as $x$ and $y$ (thus curves of constant curvatures belonging to one and the same family), retains its shape in such a continuously varying reference system . Template:Pagenum

Generalizations to curves, in which only ${\mathrm{R}}_1$ is constant.

One could say: “The passage from one point $x$ to a consecutive $x+dx$ causes an orthogonal (infinitesimal) transformation of the comoving tetrad; for such one, however, the fundamental equations are covariant according to the relativity principle, thus invariant in a suitable reference system, namely the tetrad in $x$; now, if I define a reference-point fixed in the tetrad $c_{1,x}{c}_{2,x}{c}_{3,x}{c}_{4,x}$, thus

where the $\mathrm{\Lambda }$ are constants satisfying the condition $\sum {\left({\mathrm{\Lambda }}^{(\varkappa )}\right)}^2=0$, do the fields in $c_{1,x}{c}_{2,x}{c}_{3,x}{c}_{4,x}$ have to be constant in the general case of an arbitrary world line $x$?” However, as one can see in the previous formulas (29), this is only the case for ${\mathrm{R}}_1=\text{const}$. The reason for this is, that the Template:Sc formulas go to second order, thus they require the consideration not only of $x$ and one consecutive point, but of $x$ and two consecutive points; both consecutive orthogonal transformations that have to be carried out, will in general not be identical; this is only the case when ${\left(\frac{d{\mathrm{R}}_1}{ds}\right)}_x=0$, since then the trajectory of the orthogonal transformation considered in § 7, which goes through $x$, is in contact in terms of the third order, thus it also goes through the three consecutive points. For the Template:Sc formulas which only go to second order, of course this constancy holds in system $c_{1,x}{c}_{2,x}{c}_{3,x}{c}_{4,x}$ for arbitrary wordlines $x$, as it can be seen from

But also in the case ${\mathrm{R}}_1=\text{const}$ (unlike the curves ${\mathrm{R}}_1=\text{const}$ ${\mathrm{R}}_2=\text{const}$ ${\mathrm{R}}_3=\text{const}$), the reciprocity between $x$ and $y$ does not hold here any more, in particular the $c_{k,y}$ are not representable by $\sum A^{(k)}{c}_{k,x}$ with constant $A^{(k)}$; thus the Template:Pagenum conclusions drawn there regarding the shape of the electron are here not in place anymore.

What corresponds in Newtonian mechanics to the curves of § 6?

To that end, let the world line be given

where the real parameter $t$ means the time $t$, and compute from that the matrix $D$ of § 7; one finds

Thus for

If one now wants to pass to the limit $c=\mathrm{\infty }$, one has to consider that the values ${\mathrm{R}}_1$${\mathrm{R}}_2$${\mathrm{R}}_3$ are represented by means of the arc $s=ic\tau$, which itself contains $c$, but that in the equations of motions of Template:Scan mechanics Template:Pagenum

the variable $\tau$ arises. We therefore have to transform our formulas to a $M_4$ with the arc-theorem $d{x}^2+d{y}^2+d{z}^2-c^{2}d{t}^2$ and then would obtain the corresponding metric for the angle $d{\varphi }_1,d{\varphi }_2,d{\varphi }_3$ between consecutive tangents (or osculating planes or osculating spaces) and the arc $\tau$:

Now we can pass to the limit $c=\mathrm{\infty }$:

For the Hamilton hodograph of velocity

we evidently have the three-dimensional matrix

corresponding to the four-dimensional matrix $D$, and for the radii of the first and second curvature:

Template:Pagenum Since we have now

it follows: The world lines of constant curvatures correspond to a constant magnitude of the first acceleration in Template:Scian mechanics, and furthermore to a common helix or its variations: a circular line or straight line as hodograph of the velocity. In particular, the types of § 6 correspond to:

(A) $\frac{1}{{\Re }_1}\ne 0,\frac{1}{{\Re }_2}\ne 0,$ $𝔳(t)$: helix; $x(t)$: uniform rotation around $z$, free fall along $z$, to which also a uniform translation along arbitrary direction might be added.

(B) $\frac{1}{{\Re }_1}\ne 0,\frac{1}{{\Re }_2}=0,$ $𝔳(t)$: Circular line; $x(t)$: uniform rotation around $z$, to which also a uniform translation along arbitrary direction might be added.

The case ${\mathrm{R}}_1=i{\mathrm{R}}_2$ does not have a Newtonian analogue because of the unequal order of ${\mathrm{R}}_1$and ${\mathrm{R}}_2$ in $c^2$.

(C) $\frac{1}{{\Re }_1}=0,\frac{1}{{\Re }_2}=0({\dot{𝔳}}^2\ne 0),$ $𝔳(t)$: Straight line; $x(t)$: free fall along $z$, to which also a uniform translation along arbitrary directions might be added (side piece to hyperbolic motion).

(D) ${\dot{𝔳}}^2=0,\frac{1}{{\Re }_1}=0,\frac{1}{{\Re }_2}=0,$ $𝔳(t)$: Point; $x(t)$: uniform translation.

Of course, these results could also have been obtained as trajectories of infinitesimal transformations of the Galilean group (from the Lorentz group by $c=\mathrm{\infty }$). From the things previously said one can see in addition, as to how the passage to the limit from differential geometry of the hyperbolic $S_3$ to the Euclidean $S_3$ is carried out, if one considers the $xyzt$ as Template:Pagenum homogeneous coordinates in hyperbolic $S_3$, which are connected by the relation $x^2+y^2+z^2-c^2{t}^2=-c^2$.

Hyperbolic motion and its generalization.

For vectors rigidly connected to the tangent, we have found in § 7:

in which the $\eta$ satisfy the differential equations:

and the $b_2,b_3,b_4$ are certain arbitrarily chosen unit vectors, which are also rigidly connected to the tangent, and which only have to satisfy the condition that they are perpendicular both to the tangent and among each other. We want to consider in advance the case

thus plane curves; then one finds

i.e. when mutually perpendicular fixed directions on the plane of the curve are analogously substituted for $c_3$ and $c_4$, it follows

Then, and only then the two types of curves considered in § 7

and

Template:Pagenum are identical. Hyperbolic motion, for which additionally $\frac{1}{{\mathrm{R}}_1}=\text{const}$, therefore belongs to both classes of curves.

We have seen, that the reference system

in the general case is to be understood as the system comoving with the light-point, since to any radius vector $y-x$ fixed in it, for which ${\eta }_0^{(1)}\equiv 0$, there belongs a point $y$ which is steadily at rest in it. It is predictable, that in such a system the fields have a preferred shape. It should be now

then it follows by (28a)

in which we recognize the vector arising in the Schwarzschild formulas (see [14], § 4). Because they read

If we no choose $W=\frac{\partial y}{\partial s_x}$, which is proportional to $\frac{\partial y}{\partial s_x}$, as time axis, i.e. we transform the reference-point to rest, then with respect to a world line of the reference-point given in such a way, it follows that the electric components of the field alone are different from zero, though the magnetic ones will vanish. Again, the reciprocity of the representation between $x$ and $y$ only holds for the special case of hyperbolic motion. It can be furthermore seen, that these curves will sustain $y$ when $x$ is steadily Template:Pagenum approximated by hyperbolic motions (i.e. the curvature circle), and if one combines the infinitesimal pieces of the corresponding circles (of the respective family belonging to $x$) that are going through the locations of $y$, as well as the earlier curves $y$ are sustained when the general osculating “hyper helix” (A) is used.

It may be still remarked, that the two types of Template:Sc's rigid body allow for the representation

$y=x+{\eta }_0^{(2)}{b}_2+{\eta }_0^{(3)}{b}_3+{\eta }_0^{(4)}{b}_4$

Template:Optional style|(I)

$y=x+{\mathrm{\Lambda }}^{(2)}{c}_2+{\mathrm{\Lambda }}^{(3)}{c}_3+{\mathrm{\Lambda }}^{(4)}{c}_4$

Template:Optional style|(II)

which distinguish themselves from the types studied here by the allocation between the $y$- and $x$-curve. With respect to Template:Sc's body this is done by spacelike vectors, while here this is done by minimal vectors.

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