# Is there a relativistic theoretical upper limit to tensile strength per unit density?

When considering structures that support mainly themselves with tensile strength (e.g. space elevators and rotating space habitats like rotating wheel space station, Stanford torus and O'Neill cylinder), a relevant property is the tensile strength of the building material divided by its density. For example, maximum possible radius of a rotating space habitat with a fixed simulated gravity (say, $$1g$$) grows linearly with said property.

It is interesting to note that this property has dimensions of velocity squared, which seems to hint that there might be a relativistic theoretical upper limit comparable with $$c^2$$.

Is there such a limit?

On the first glance, tensile strength per unit density seems unrelated to special relativity (SR). However, there is this hint from dimensional analysis and even more than that. Namely, looking microscopically, this property seems to correspond (up to dimensionless quantities) to force between particles times distance between them divided by their mass, which seems to similarly correspond to binding energy per mass and if this is around $$c^2$$ or higher, then it seems that trying to tear it apart would cause a pair production (similar to the one in QCD when trying to pry individual quark from a hadron) before reaching the limit imposed by the supposedly higher tensile strength because the state with "inserted" particles would have a lower energy.

However, the above consideration is very non-rigorous. Is there a theoretical result that imposes that kind of limit in a more rigorous way? I am looking for a "clean" result in SR which would be like Buchdahl's theorem in GR, in the sense of not assuming any particularities about available particles.

I think what you are looking for is the dominant energy condition (DEC), or one of its relatives. The DEC states that the matter density (in natural units) must be bigger than the pressure or tension: $$\rho \geq |p|$$. This is basically demanding that we cannot observe any matter fields moving faster than light. So here you would indeed have have the critical specific force $$c^2$$.
There are theoretical arguments bounding the speed of sound in hadronic matter to less than $$c/\sqrt{3}$$. This in turn implies a maximal specific tensile strength of $$c^2/3$$.
• The paper arXiv:1408.5116 seems to derive the bound on the speed of sound from a constraint on the sign of the trace $T^\mu{}_\mu$ which stipulates that $p\le\rho/3$ (in a spacetime with 3 spatial dimensions), giving the $c^2/3$ result without a need to consider the speed of sound. Jul 22 at 14:04