Centrifugal Acceleration GR: Formula for .8c, 1m Radius

[grav-universe]

I swing an accelerometer around my head with a constant relative speed of v and a radius of r. I want the exact formula with highly relativistic effects, so v = .8 c and r = 1 meter, say. What proper acceleration will the accelerometer read?

 

[PeterDonis]

Take a look at the Wikipedia page on Born coordinates:

http://en.wikipedia.org/wiki/Born_coordinates

They give a formula for the proper acceleration of a "Langevin observer", which corresponds to the accelerometer being swung around your head. They give it in terms of the angular velocity $\omega$, but using the formula $v = \omega r$, it is easy to come up with a formula in terms of the velocity $v$:

$$a = \frac{- v^2}{r \left( 1 - v^2 \right)}$$

This makes sense; it's just the standard Newtonian formula for centripetal acceleration, with an extra relativistic factor of $\gamma^2$.

 

[grav-universe]

Thank you PeterDonis.