\Section{70.}{Properties of de Sitter's spherical world} \index{de Sitter's spherical world}% \index{Recession of spiral nebulae}% \index{Red-shift!in nebulae}% \index{Sitter@de Sitter's spherical world}% \index{Spectral lines, displacement!in nebulae}% \index{Spherical world}% If in~\Eq{(67.33)} we write \[ r = R\sin\chi, \] we obtain \[ \left. \begin{gathered} ds^{2} = -\gamma^{-1}\, dr^{2} - r^{2}\, d\theta^{2} - r^{2}\sin^{2}\theta\, d\phi^{2} + \gamma\, dt^{2} \\ %[** TN: Hard-coded length] \llap{\text{where}\rule{0.87in}{0pt}} \gamma = 1 - r^{2}/R^{2} = 1 - \tfrac{1}{3}\lambda r^{2}, \end{gathered}\right\} \Tag{(70.1)} \] and the customary unit of~$t$ has been restored. This solution for empty space has already been given, equation~\Eq{(45.6)}. We have merely to substitute this value of~$\gamma$ in the investigations of \SecRefs{38}, \SecNum{39}, in order to obtain the motion of material particles and of light-waves in de~Sitter's empty world. Thus \Eq{(39.31)}~may be written \[ \frac{d^{2}r}{ds^{2}} - \frac{1}{2}\, \frac{\gamma'}{\gamma} \left(\frac{dr}{ds}\right)^{2} - r\gamma \left(\frac{d\phi}{ds}\right)^{2} - \tfrac{1}{2}\gamma\gamma' \left(\frac{dt}{ds}\right)^{2} = 0. \] Whence \[ \frac{d^{2}r}{ds^{2}} = -\frac{\frac{1}{3}\lambda r}{1 - \frac{1}{3}\lambda r^{2}} \left(\frac{dr}{ds}\right)^{2} + r(1 - \tfrac{1}{3}\lambda r^{2}) \left(\frac{d\phi}{ds}\right)^{2} + \tfrac{1}{3}\lambda r(1 - \tfrac{1}{3}\lambda r^{2}) \left(\frac{dt}{ds}\right)^{2}. \Tag{(70.21)} \] For a particle at rest \[ \frac{dr}{ds} = 0,\quad \frac{d\phi}{ds} = 0,\quad \left(\frac{dt}{ds}\right)^{2} = \gamma^{-1}. \] Hence \[ \frac{d^{2}r}{ds^{2}} = \tfrac{1}{3}\lambda r. \Tag{(70.22)} \] Thus a particle at rest will not remain at rest unless it is at the origin; but will be repelled from the origin with an acceleration increasing with the distance. A number of particles initially at rest will tend to scatter, unless their mutual gravitation is sufficient to overcome this tendency. It can easily be verified that there is no such tendency in Einstein's world. A particle placed anywhere will remain at rest. This indeed is necessary for the self-consistency of Einstein's solution, for he requires the world to be filled with matter having negligible velocity. It is sometimes urged against de~Sitter's world that it becomes non-statical as soon as any matter is inserted in it. But this property is perhaps rather in favour of de~Sitter's theory than against it.