Rigid objects are effectively forbidden in special and general relativity, as a perfectly rigid object would transmit information faster than light.
However, what about a fixed-length loop of string? By fixed-length, I'm imagining that the loop is described by some curve $x^\mu(\lambda)$ such that $$\ell \equiv \int \sqrt{g_{\mu \nu} \frac{dx^\mu}{d \lambda} \frac{dx^\nu}{d \lambda}} d \lambda$$ is constant throughout the time evolution of the loop. I take $g$ to have signature $-+...+$. Phrased another way, I believe this is saying that the proper distance measured along the string is constant.
I am not familiar with the treatment of such loops in general relativity, or of the dynamics of strings. I am hopeful that they might still be consistent with special and general relativity, as maintaining the length of the string in a sense happens "locally", unlike maintaining the shape of a rigid object.
Is there anything forbidding such loops? If not, is there a nice Lagrangian treatment or way to write the stress-energy tensor of such an object?
