What would be the apparent velocity of an object approaching us at an actual velocity of $v$?

I know $cv/c-v$ is a possible answer but here's another argument – shouldn't the velocity be the Doppler shift

$$v\left(\frac{c+v}{c-v} \right)^{1/2}$$

times the actual velocity?

## Best Answer

Imagine a stealthed

^{1}alien vessel is approaching the earth with relative velocity $\beta = \frac{v}{c}$. At a distance of two light-years as measured in the Earth's frame^{2}the craft switches on its radio and send the Earth a ultimatum. We call this moment time zero in the Earth's frame.Then it turn the radio off until it reaches a distance of one light-year (again, as measured in the Earth frame), when it turns the radio on again to repeat the challenge.

Now consider the timing as measured in the Earth frame.

Earth bound observers see two messages from a single vessel at separated by a distance or one light-year that arrive with a time separation of $$\Delta t = t_{2'} - t_{1'} = \frac{1\,\mathrm{year}}{\beta} - (1 \,\mathrm{year}) \,.$$ From that they compute an apparent velocity for the approaching craft of $$ \begin{align} v_{app} &= \frac{\Delta x}{\Delta t} \\ &= \frac{1 \,\mathrm{lightyear}}{\frac{1\,\mathrm{year}}{\beta} - (1 \,\mathrm{year})} \\ & = \frac{c}{\frac{1}{\beta} - 1} \\ & = \frac{\beta c}{1 - \beta} \\ & = \frac{v}{1 - v/c}\,. \end{align} $$

For $v \ll c$ we get $v_{app} \approx v$, but as $v$ becomes a significant fraction of $c$ things change. In fact for $v = \frac{c}{2}$ we get $$v_{app}(0.5c) = \frac{0.5c}{1 - 0.5} = c \,,$$ and for approach velocities in excess of one-half c the apparent velocity exceed the speed of light.

This has been observed in jets directly strongly along our line of sight.

The effect is symmetric, the aliens see Earth approaching just as fast.

This effect is also separate of the Doppler shift.

There is an intimate relationship between this analysis and the "fast forward" (or "slow motion") effect on time as observed ahead or behind you when moving fast relative the thing you are observing.

^{(1)}Just so we can concentrate on exactly two signals, this does not affect the results.^{(2)}Radio observatories on Earth can measure the distance to the origin of the signal by direct parallax or by interferometric methods.