Why is there an emergent quasi Lorentz symmetry in classical string lagrangian?

Question

For a chain with spacing $a$, the action is

$$S=\int dt\sum_i \left(\frac{M}{2}\dot\phi^2_i-\frac{k}{2}(\phi_{i+1}-\phi_i)^2\right)$$

which only has a translational symmetry. When you take the continuous limit, that is $a\rightarrow0$, $M/a=\rho $, the action of string becomes

$$S=\int dt \, dx \, \left( \frac{\rho}{2}\dot\phi(t,x)^2-\frac{T}{2}(\partial_x\phi(t,x))^2\right).$$

It is obvious that the action above has quasi "Lorentz symmetry" with $c=v_s$ where $v_s$ is the sound velocity. Why will this kind of Lorentz symmetry emerge in a totally classical mechanics problem?

Answer 1

I would say this is an accident of only considering the 'leading order' terms. There will be non-linear interactions in any real chain that will add extra terms to your Lagrangian that have no reason to be Lorentz invariant. (For example you had to linearize the restoring force acting on the links of the chain to write your original lagrangian down).

More to to the point, to actually make any measurements on this chain you need to interact with it. Those interactions will break the 'quasi Lorentz invariance'. After all, a chain is a material, and materials do have a preferred frame, one without bulk motion.