# Inextensible filament structure in gravitational field

Let $$\gamma$$ be a curve lying in a gravitational field with each point moving as an independent point mass. Formally,

$$\begin{equation} \partial^2_t \gamma(u, t) = -\nabla U(\gamma(u, t)), \end{equation}$$

where $$U(x)$$ is the gravitational potential at some point $$x$$.

Question: Excluding the trivial examples listed below, is there a feasible gravitational field and an initial configuration of the curve (initial positions and velocities at each point) such that it will evolve as an inextensible filament, meaning $$\partial_t|\partial_u \gamma| \equiv 0$$?

One can imagine setting up a system of equidistantly spaced objects along this curve, which would work as information or energy transmitters. Assuming the curvature of the filament would be small relative to the spacing, the distance between these objects would remain constant in time. Or if a space cable of some sort would be constructed in this way, it will not experience any internal tension.

The following examples may clarify the problem.

Examples: The trivial example here is a circular ring orbiting one mass body. For example, planetary rings are very close to being circular, even though the individual particles may have a slightly eccentric trajectories (for interesting discussions see this or this). Another trivial example would be any curve in a massless universe. Counter examples: An elliptic ring or a space elevator like line segment are not valid examples as these structures are subject to an internal tension. This is why an extremely strong material needs to be developed in order to make the space elevator possible. 