From N. Straumann, *General Relativity*

Exercise 4.9: Calculate the radial acceleration for a non-geodesic circular orbit in the Schwarzschild spacetime. Show that this becomes *positive* for $r>3GM$. This counter-intuitive result has led to lots of discussions.

This is one of those problems where I have absolutely no clue what to do. Since it says non-geodesic, I can't use any of the usual equations. I don't know what equation to solve. *Maybe* I solve $\nabla_{\dot\gamma}\dot\gamma=f$ with $f$ some force that makes $\gamma$ non-geodesic. But I don't know where to go from there if that's the way to do it.

Also any specific links to discussions?

Any help would be greatly appreciated.

EDIT: So I tried solving $\nabla_u u=f$ with the constraints $\theta=\pi/2$, $u^\theta=0$ and $u^r=0$. lionelbrits has explained I must also add $\dot u^r=0$ to my list. This all leads to $$(r_S-2Ar)(u^\varphi)^2+\frac{r_S}{r^2}=f^r$$ ($A=1-r_S/r$, notation is standard Schwarzschild) The problem with this is that the $u^\varphi$ term is *negative* for $r>3m$. So somewhere a sign got screwed up and for the life of me I don't know where it is. A decent documentation of my work: http://www.texpaste.com/n/a6upfhqo, http://www.texpaste.com/n/dugoxg4a.

This post imported from StackExchange Physics at 2015-01-06 22:42 (UTC), posted by SE-user 0celo7