Special relativity puzzle relating to motion of a body when restrictions are placed on the Doppler shift of the body

[jcsd]

TL;DR Summary

A puzzle relating to motion of a body when restrictions are placed on the Doppler shift of the body as viewed by an observer

Hello

I have not posted here for a while, but just wanted to post this puzzle I devised. Was posted on Reddit, but no takers. I think the solution is interesting.

The Puzzle

You are flying around in space and you see your friend Steve chilling in an inertial frame in his own spaceship and you wish to go over to his spaceship to say hello. However if Steve sees you to be even slightly Doppler-shifted as you approach him he will not recognise you and he will accelerate off such that you can never catch him. Steve is also very impatient so you must minimize the time taken in his frame to reach him.

What speed (in Steve's frame) should your spaceship travel in to reach Steve?

Spoiler: Hints

There is no solution for 1 dimension of space, but you can just look at 2 dimensions of space as the 3D solution is the same.

The solution does not involve anything silly or designed to trick like travelling at the speed of light, undefined Doppler shift or Steve leaving his inertial frame.

 

[PeroK]

You could paint your spaceship black.

 

[jcsd]

I like the lateral thinking but not the answer I was thinking of!

Spoiler: Answer

Your path must be a logarithmic spiral in which you travel at constant speed in Steve's frame (or another space curve that will give the same answer). The time taken is proportional to your radial distance from Steve, and that proportion can be expressed as a function of the speed, differentiating the answer is c*sqrt(3/4)

 

[jcsd]

I like the lateral thinking but not the answer I was thinking of!

Spoiler: Answer

Your path must be a logarithmic spiral in which you travel at constant speed in Steve's frame (or another space curve that will give the same answer). The time taken is proportional to your radial distance from Steve, and that proportion can be expressed as a function of the speed, differentiating the answer is c*sqrt(3/4)

Oops in fact I've made a mistake and the puzzle isn't quite as interesting. the answer is just as close to c as possible.

 

[Orodruin]

Fastest approach speed is c. Just as it was on reddit.

Edit: In fact, this is very intuitive. As you get closer to the speed of light in Steve’s frame, the angle for a net shift of zero tends to align towards the direction of Steve. So it is quite natural that the max approach speed is c.