How Long is the Astronaut Gone in the Twin Paradox with Circular Motion?

[Urd]

Homework Statement

A variation on the twin paradox, with uniform circular motion. The traveling sibling moves so that his acceleration is g at all times, pointing to the centre of his circular path, constant velocity. There is given that the trip takes twenty years in the frame of the traveller. Then how long will the astronaut be gone as seen by the inertial observer?

Homework Equations

The Attempt at a Solution

I have an expression for the time elapsed (as seen by the inertial observer) in function of radius and velocity. And as I'm unable to calculate radius or velocity for this problem, I am stuck.

 

[Meir Achuz]

You have two equations:
2pi R=vT (T=20 years), and v^2=Rg.
Solve for R and v.

 

[Urd]

When I use these equations, the velocity is more then three times the speed of light?

 

[Meir Achuz]

The statement "his acceleration is g at all times" is ambiguous.
If it refers to his acceleration in the Earth system, you do get v~3c.
It must mean his acceleration in his rest system. In that case,
a in the Earth system is a=g(1-v^2/c^2), and
The centripetal equation becomes v^2=Ra=Rg(1-v^2/c^2), which eventually gives v<c.
Incidentally, if you work with LY (light years), then c=1 and g~1.