The spectroscopic radial velocity is not exactly equivalent to~$dr/dt$, but \index{Light-pulse!in curved world}% the divergence is unimportant. A pulse of light emitted by an atom situated at $r = R\sin\chi$ at time~$t$ will reach the observer at the origin at time~$t'$, where by~\Eq{(67.5)} \[ t' = t + \log\tan(\tfrac{1}{4}\pi + \tfrac{1}{2}\chi), \] so that for the time-interval between two pulses \begin{align*} dt' &= dt + \sec\chi\, d\chi \\ &= \left(1 + \sec\chi\, \frac{d\chi}{dt}\right) \frac{dt}{ds}\, ds \\ &= \left(\sec\chi + \sec^{2}\chi\, \frac{d\chi}{dt}\right) ds, \quad\text{by~\Eq{(67.33)}} \end{align*} \PageSep{164} \index{Atom, time of vibration of!in de Sitter's world}% neglecting the square of the velocity of the atom. If $dt_{0}'$~is the time for a similar atom at rest at the origin, \begin{align*} \frac{dt'}{dt_{0}'} &= \sec\chi + \sec^{2}\chi\, \frac{d\chi}{dt} \\ &= \sec\chi + \sec^{2}\chi\, \frac{1}{R}\, \frac{dr}{dt}. \Tag{(70.4)} \end{align*} The first term represents the general shift to the red dependent on position and not on velocity. Assuming that it has been allowed for, the remaining part of the shift corresponds to a velocity of $\smash[b]{\sec^{3}\chi\, \dfrac{dr}{dt}}$ instead of~$\dfrac{dr}{dt}$. The correction is scarcely of practical importance. The acceleration $\frac{1}{3}\lambda r$ found in~\Eq{(70.22)}, if continued for the time~$R\chi$ taken by the light from the object to reach the origin, would cause a change of velocity of the order $\frac{1}{3}\lambda r^{2}$ or~$r^{2}/R^{2}$. The Doppler effect of this velocity would be roughly the same as the shift to the red caused by the slowing down of atomic vibrations. We may thus regard the red shift for distant objects at rest as \emph{an anticipation} of the motion of recession which will have been attained before we receive the light. If de~Sitter's interpretation of the red shift in the spiral nebulae is correct, we need not regard the deduced large motions of recession as entirely fallacious; it is true that the nebulae had not these motions when they emitted the light which is now examined, but they have acquired them by now. Even the standing still of time on the horizon becomes intelligible from this point of view; we are supposed to be observing a system which has \emph{now} the velocity of light, having acquired it during the infinite time which has elapsed since the observed light was emitted.