Page:VaricakRel1910b.djvu/7

Put $MN=u=\operatorname {th} \,^{-1}{\frac {v}{c}}$. From N we let fall the perpendicular NP upon MT, then we construct the perpendicular NR upon NP in N, then we lay off the line MS = 1 upon MT, and from S we let fall the perpendicular SR upon NR. If we additionally make NU = PS, then NU will be parallel to MT in the Lobachevskian sense, and this parallel encloses with the X-axis the angle $\varphi '$.

As point U always lies between S and R, it can be easily seen from the figure that in relativity theory $\varphi '$ is smaller than $\varphi '_{0}$ of ordinary mechanics.

In Einstein's formula for aberration,

$\cos \varphi '={\frac {\cos \varphi -{\frac {v}{c}}}{1-{\frac {v}{c}}\cos \varphi }}$

(32)

we denote the cosines on the left-hand side and in the numerator of the fraction by corresponding sines, then we square and get in this way after some transformations

$\sin \varphi '={\frac {\sin \varphi \cdot {\sqrt {1-{\frac {v}{c}}}}}{1-{\frac {v}{c}}\cos \varphi }}$

(33)

During the motion of earth in its orbit relative to the fixed stars as reference frame we have ${\frac {v}{c}}=0,0001$.^[1] For such small velocities we can neglect ${\frac {v^{2}}{c^{2}}}$, and then we obtain

$\sin \varphi '={\frac {\sin \varphi }{1-{\frac {v}{c}}\cos \varphi }}$

(34)

Formula (34) and formula (32), in which we will take $\varphi '_{0}$ instead of $\varphi _{0}$, give

${\frac {\sin \varphi '_{0}}{\cos \varphi '_{0}}}={\frac {\sin \varphi }{\cos \varphi -{\frac {v}{c}}}}$

(35)

from which we can, in accordance with the ordinary theory, easily find the formula

${\displaystyle }$

(36)

By comparison of formulas (33) and (34) we find

$\sin \left(\varphi '_{0}-\varphi \right)={\frac {v}{c}}\sin \varphi '_{0}$

(37)

However, if we take in Fig. 3 $OM'=u$, then we can put

$MM'=\sin \varphi '_{0},\ ON=\sin \varphi '.$

(38)

If would even be better, if Fig. 1 of my first treatise is used. Then we can take

$s=\sin \varphi '_{0},\ s'=\sin \varphi '.$

(39)

7. Light pressure

The following remarks are related to Einstein's formulas located in § 8 of his first paper on the relativity principle^[2]. We can easily see that

${\frac {E'}{E}}={\frac {A'}{A}}={\frac {\operatorname {ch} \,\left(u_{1}-u\right)}{\operatorname {ch} \,u}}$

(40)

and this can be interpreted, as earlier ${\tfrac {\nu '}{\nu }}$, by using Fig. 6.

For light pressure, Einstein gave the formula

$P=2{\frac {A^{2}}{8\pi }}{\frac {\left(\cos \varphi -{\frac {v}{c}}\right)^{2}}{1-\left({\frac {v}{c}}\right)^{2}}}$

(41)

We transform them at first into the form

$P=2{\frac {A^{2}}{8\pi }}\left(\operatorname {th} \,\ u_{1}-\operatorname {th} \,\ u_{2}\right)^{2}\cdot \operatorname {ch} \,^{2}u$

from which we easily obtain

$P={\frac {2A^{2}}{8\pi }}\left({\frac {\operatorname {sh} \,\left(u_{1}-u\right)}{\operatorname {ch} \,u_{1}}}\right)^{2}$

(42)

If we take a Lobachevskian right angled triangle with hypotenuse $u_{1}-u$ and an acute angle $\alpha =\Pi \left(u_{1}\right)$, then we obtain for the side adjacent to angle α, the expression

$\operatorname {sh} \,b={\frac {\operatorname {sh} \,\left(u_{1}-u\right)}{\operatorname {ch} \,u_{1}}}$

(43)

hence

$P={\frac {1}{4\pi }}\left(A\ \operatorname {sh} \,b\right)^{2}$

(44)

Agram, February 12, 1910

(Received February 18, 1910)

1. Brill, Einführung in die Mechanik, 1909, p. 206.
2. Ann. der Phys. 17, 913-915. 1905