We have had a related problem with the brachistochrone problem in Schwarzschild coordinates here. This problem here has struck me as rather odd, for this seems to refer to geodesics, where to me the . However, it has occurred to me there might be a problem here of some relevancy. Suppose we have a string in some orbit around a black hole. We then impose some condition that this string has end points attached to a fixed surface of some kind --- a Dp-brane? This string is then in an orbit around the black hole and will assume some configuration in the curved spacetime.
So to set this up we start with the line element, which gives the unit 4-velocity, $$ 1~=~\Big(1~-~\frac{R}{r}\Big) \Big(\frac{dt}{ds}\Big)^2~-~ \Big(1~-~\frac{R}{r}\Big)^{-1} \Big(\frac{dr}{ds}\Big)^2~-~r^2\Big(\frac{d\theta}{ds}\Big)^2~-~r^2sin^2\theta\Big(\frac{d\phi}{ds}\Big)^2, $$ where $R~=~2GM/r$. 394
We identify the $dt/ds$ with the energy $E$ and do some algebra to obtain an equation of motion $$ \Big(\frac{dr}{ds}\Big)^2~=~\frac{E}{m}~-~1~-~V(r) $$ for the potential function $$ V(r)~\simeq~\frac{1}{2}\Big(\frac{R}{r}~-~\frac{L^2}{2mr^2}~+~\frac{R}{mr^2}\Big) $$ The $L$ is the angular momentum of the object in orbit.
We now use this in the potential for the computation of the shape of the string $$ U_G~=~\int_0^\Delta V(r)\lambda\sqrt{r'^2~+~r^2}d\phi $$ $\Delta~=~D/r$ and $r’~=~dr/d\phi$ and employ the Lagrange multiplier $U_L$. The Euler-Lagrange equations are applied to the $$ {\cal L}~=~(V(r)~+~L)\sqrt{r’^2~+~r^2} $$
This is somewhat artificial for it assumes the string endpoints are held apart by some fixed distance. This could be relaxed by treating the endpoints are particle geodesic or orbits. In that case the angular separation of the fixed points varies and the problem is time dependent. This might be an interesting tool to use in some problems with strings and black holes. 246