Page:DeSitterGravitation.djvu/25

we find

$n't'=nt-{\frac {an\beta q}{c}}\sin \ nt,\quad n'=n{\sqrt {1-q^{2}}},$

or putting

$\epsilon ={\frac {an\beta q}{c}},$

we have

$\phi -\epsilon \ \sin \phi =n't',$ $x'_{1}=a\ \cos \ \phi ,\qquad y'_{1}=a\mathrm {K} \ \sin \phi .$

(43)

If the orbit in ($x,y,z,t$) is a "perturbed" ellipse, the formulæ of transformation are more complicated, but the differences between the real motion and pure elliptic motion remain of the second order and periodic in terms of $t',$ if they were so in terms of $t$.

16. One of the consequences of the principle of relativity is that it must be impossible by observations on bodies belonging to one and the same system to detect a motion of the whole system. Suppose in the system of reference ($x,y,z,t$) the Sun to be at rest, and a planet to describe a circle with uniform velocity. This, of course, is a dynamically possible state of motion under the new law as well as under the old. In the system ($x',y',z',t'$) derived from ($x,y,z,t$) by a Lorentz-transformation with the axis ($\alpha ,\beta ,\gamma$) and the modulus $q$, the Sun and the planet have a common velocity $-cq$ in the direction ($\alpha ,\beta ,\gamma$), and the relative orbit is no longer circular, but is defined by (43). Let, again, the orbital plane be chosen as plane of ($x,y$) in the first system of reference, and let the axis of $x$ be perpendicular to the axis of the transformation. The observer belonging to the system has no means of ascertaining the position of the plane of ($x',y'$), he can only observe the plane of the orbit, i.e. the plane of $\left(x'_{1},\ y'_{1}\right)$. To him the velocity of the Sun is thus in the direction ($\alpha ',\beta ',\gamma '$), where—

$\alpha '=0,\qquad \gamma '={\sqrt {1-\beta '^{2}}}$ $\beta '=\cos(\mathrm {A} +\mathrm {I} ),{\text{ if }}\beta =\cos \ \mathrm {A} ,$

or

$\beta '\mathrm {K} =\beta {\sqrt {1-q^{2}}}$

Let the observer be on the Sun, and let a signal be sent him, from the planet every time when this latter crosses the axis of $y$.

In the system ($x,y,z,t$) the intervals between these crossings are equal, and also the times required by the signal to reach the observer are equal: he will observe signals at equal intervals.

In the system ($x',y',z',t'$) the intervals between the crossings are unequal, and the aberration-times are unequal, and these two effects must cancel each other.

The times of crossing are—

$\phi ={\frac {\pi }{2}},\qquad {\frac {3\pi }{2}},\qquad {\frac {5\pi }{2}},\qquad {\text{ etc.}}$