Page:VaricakRel1910b.djvu/4

$\operatorname {sh} \,Y\ \operatorname {ch} \,Z=\operatorname {sh} \,c_{1}$

(12)

and

$Z=\operatorname {sh} \,^{-1}c_{2}.$

(13)

Their center planes are the coordinate planes XZ and XY.

The Lorentz-Einstein transformation can be interpreted as a translation through the intersection line of these two equidistant areas.

To simplify matters we put Z = 0 and thus the Weierstrass coordinates of a point in the plane become

$\left.{\begin{array}{l}x=\operatorname {sh} \,\xi =\operatorname {sh} \,X\ \operatorname {ch} \,Y,\\y=\operatorname {sh} \,\eta =\operatorname {sh} \,Y,\\l=\operatorname {ch} \,r=\operatorname {ch} \,X\ \operatorname {ch} \,Y.\end{array}}\right\}$

(14)

The Lorentz-Einstein transformation

$\left.{\begin{array}{c}l'=-x\ \operatorname {sh} \,u+l\ \operatorname {ch} \,\ u,\\x'=x\ \operatorname {ch} \,u-l\ \operatorname {sh} \,\ u\\y'=y,\ z'=z=0.\end{array}}\right\}$

(15)

defines the motion along the distance line Y = b, having the X-axis as its center line. The parameter b is arbitrary.

The displacement by the distance s along that equidistant line is defined by the equations^[1]

$X'=X-{\frac {s}{\operatorname {ch} \,b}},\ Y'=Y.$

(16)

Here, s is the arc $MM'$ of that distance line. If u is its projection $NN'$ in the X-axis, then

$X'=X-u,\ Y'=Y,$

(17)

thus

$\left.{\begin{array}{l}\operatorname {sh} \,X'=\operatorname {sh} \,X\ \operatorname {ch} \,u-\operatorname {ch} \,X\ \operatorname {sh} \,u,\\\operatorname {sh} \,Y'=\operatorname {sh} \,Y.\end{array}}\right\}$

(18)

By multiplication of the first equations by

$\operatorname {ch} \,Y'=\operatorname {ch} \,Y$

it is given

$\left.{\begin{array}{rl}\operatorname {sh} \,X'\ \operatorname {ch} \,Y'=&\operatorname {sh} \,X\ \operatorname {ch} \,Y\ \operatorname {ch} \,u-\operatorname {ch} \,X\ \operatorname {ch} \,Y\ \operatorname {sh} \,u\\\operatorname {sh} \,Y'=&\operatorname {sh} \,Y.\end{array}}\right\}$

(19)

According to Fig. 2 we have in addition

$\operatorname {ch} \,OM'=\operatorname {ch} \,X'\ \operatorname {ch} \,Y',$

(20)

or

$\left.{\begin{array}{c}\operatorname {ch} \,r'=\operatorname {ch} \,(X-u)\operatorname {ch} \,Y=\operatorname {ch} \,X\ \operatorname {ch} \,Y\ \operatorname {ch} \,u\\-\operatorname {sh} \,X\ \operatorname {ch} \,Y\ \operatorname {sh} \,U.\end{array}}\right\}$

(21)

If we consider equations (14), then we can bring (19) and (20) into the form

${\begin{array}{l}x'=x\ \operatorname {ch} \,u-l\ \operatorname {sh} \,u,\ y'=y\\l'=-x\ \operatorname {sh} \,u+l\ \operatorname {ch} \,u,\end{array}}$

and those are of course the equations (15) of the Lorentz-Einstein transformation.

3. Local time

If M and $M'$ are two observers moving with uniform but different velocities, then each of them can claim with the same justification, the he is at rest relative to empty space. However, that the observer in M is using a different time calculation as the observer in $M'$, can be seen from the figure above, because r is different from $r'$. The hyperbolic cosine of the radius vector of the point, at which the observer is located, is according to our conception his local time. The conception of the new notion of time^[2] will be essentially simplified by our interpretation. It nearly acts intuitive. It only remains to investigate, if it is able to accomplish the same as the simple and suggestive coordinate transformation of Minkowski.

Point M can take any location on that distance line. We take it so that u will be equal to its abscissa, i.e., point M falls into the intersection point of the distance line with the ordinate axis, or we move the coordinate origin to $N'$.

The space-time transformation that is caused by a uniform motion

1. Concerning the transformation of the Lobachevskian plane see my relevant papers in "Rad jugoslavenske akademije" 165, 50-80, 236-344, 1906, or the short excerpt from it in Jahresber. d. deutsch. Mathematikerver. 17, 70-83, 1908.
2. Planck, Acht Vorlesungen über theoretische Physik, p. 117, says that this new conception of the notion of time surpasses in boldness everything previously suggested in speculative natural phenomena and even in the philosophical theories of knowledge: non-euclidean geometry, which only comes seriously into consideration in pure mathematics, would be child's play in comparison.