Why not consider two pancakes instead? I.e., two extremely Lorentz contracted protons (two pions would be even better, but that leads to more indirect measurements). The observed experimental results of this is that all internal "springs" will break, due to the creation of a large amount of low-mass particles and anti-particles. 

The relativistic Hook equation is a possible first step of string theory, which have been much studied for almost half a century. I am not in position to seriously digest the results of these studies; my guess is that many will survive, and someday be described as developed too much ahead of their time.

My main point is that the question you pose is much to complicated for serious answers. Could you please first provide us with your answer for the case of non-relativistic motion?

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Ok, for small deformation we can use linear dependence deformation from stress. After collision end we will see vibrations in each cube, so kinetic energy loss. Currently I try to consider nonlinear deformations.

Minas, this is well known equation with known solution, but this is not in the scope of my question. To solve it just imagine charged ball as in Thompson atom model and point like electron inside it. For Lorenz force we will have E=k*r. Then you can integrate energy expression.

It is really not relevant. We know the Hooke's law, for example in its simplest form F=-kx. Now, to see it from the frame moving with the velocity v all you have to do is to apply Lorentz transformations to F and x. 

If you want to consider a collision of two massive bodies with the relative speed close to c, I can assure you that the bang is not described by Hooke's law. The energies of atoms would be much higher than the potentials wells, no tiny collective displacements required for Hooke's law will happen. So, this would be highly speculative. 

As for the nonrelativistic collisions and the deformation they cause, that's out everyday experience including the sound when I type this. 

Igor, if only life would be so simple to transform force and coordinate :) I definitely mean not one mass on the spring end, but elastic body deformation. It looks like in SR case we should have speed of light in addition to speeds of sound in the body.

There are some papers on the subject in Google search. 

Yet, let's see how \omega (t - x/s) transforms to \omega' ( t' - x'/s'). 

s' = c (s- \beta c ) / (c - \beta s). This seems reasonable. If \beta = 0 then the speed of sound is the same. If the observer moves with s, she does not see any wave, if the observer moves with c, he sees the wave moving with -c, because everything moves with -c for him. I see no speed of sound plus speed of light. 

My point is that you'll not have t+/-x/s, but new physics with new equations. Imagine piece of rubber, floating in the space and oscillating. I want be sure, that this elastic body has surface and is more or less stable.

I don't think that is a good example. You can consider the piece of rubber in the frame of its center of mass and, then, if you want to see it from other frames, all you have to do is to apply Lorentz transformation.

But I understand what you want. You want to consider a problem when the velocities of parts of the rubber with respect to each other are relativistic. This problem is speculaitve if you apply it to rubber, because at these relative velocities the atoms would hold together. There is not a potential well deep enough. Probably, this is still applicable to something like neutron stars. One has to make estimates.

And, of cause, the elastic force, like many forces in classical mechanics are "the spooky action at distance". They are not Lorentz covariant, despite their electromagnetic nature.  That is a general problem of any potential V(x), unless it is derived directly from the all Lorentz covariant equations for all fields and particles.

Look the string theory. It is exactly the system you are asking about...

Due to locality of relativity theory you have to consider Hook law in differential form for continuous elastic medium. It could be considered as a sort of field theory.

E.g. one could consider 3 scalars \phi^a (a=1,2,3). The lines (in 4-dimensional space-time) \phi=const represent the world lines of medium particles. One could construct different action functionals, which correspond to different state equations of elastic media.

I have considered relativistic elastic media in the paper "String fluids and membrane media". See the publication attached.

Article String fluids and membrane media

Mikhail, I think your paper is the best answer to my question. Thank you very much!.

Btw, I like also Katanaev's big book :) and sent him a list of about 100 misprints, which were corrected then. And it looks, that I was discussed some topics with Volovich in ITEF in 80-th.

Nobel Prize for Gravitation!

Who is the smartest to explain Gravitation?

900 Nobel Laureates and they were not able to explain Gravitation, the most important force in nature.

In my view the graviton must have a speed higher then the speed of light; the gravitons with the speed of light will not escape a black hole.

General relativity, LQG, String theory, Quantum gravity theories are wrong theories because are limited to the speed of light and do not explain Gravity.

“I am the first who Understood and Explained Gravitation with high speed gravitons v = 1.001762 × 10^17 m/s, with Negative Impulse, Negative Mass and Negative Energy”                                                       Adrian Ferent

https://www.researchgate.net/publication/287331589_Ferent_Gravitation_theory

Please read the paper, may be helpful for you.  A new approach to the correction of

Galilean transformation.

@Sheng Liu: the limit m --> 0 exists in Galilean space time but not in Minkowsky: here we have 3 disjoint domains: time like (massive particles), light like (massless), space like (spin). The light cone is crucial. With c --> infinity you go from Lorentz- to Galilean relativity, but the reverse does'nt exist. 

@ Eugene Prokhorenko

Could you please tell me the motivation behind this study?

In SR paradoxes you will see, that straight rod become curved, but can it be just geometric effect without elastic force?

Could you please provide some relevant reference for it?

Taylor, Wheeler, Spacetime Physics. Paradox of barn and rod (I only have russian edition)

thanks for the information Eugene. I guess the book is available in English too.

@ Eugene,

Are you trying to check that Hook's law in relativistic case will yield curved deformations? Problem unsolved yet?

Eugene> In SR paradoxes you will see, that straight rod become curved,

There is a discussion about this here on ResearchGate, see the link. I don't think a more detailed and realistic description of the rod and its dynamics will make much difference. F.i., the movement of a one-dimensional object will swipe out a 2D surface in spacetime; it may look like quite different 1D-objects when viewed from different equal-time frames. 

When I am out fishing, and stick an oar in the water, it looks broken. That does not require a more detailed theory of oars to explain.

But the relativistic description of extended objects is an interesting topic on its own. For thin, one-dimensional objects, an invariant description is in terms of the 2D-worldsheet it traces out in space-time. That is very similar to the treatment of strings (without the additional constraints imposed by quantization), but with a different model for its mechanics. Perhaps just a little more general action? 

https://www.researchgate.net/post/Is_a_rigid_rod_moving_perpendicularly_to_x_axis_in_the_standard_Special_Relativity_axes_convention_parallel_to_the_moving_system_x_axis

Vikash, it looks like SR here somehow flows to GR! Or we just see, that SR with constant gravity "g" is sorta broken theory.

@ Eugene, 

I am finding this problem interesting. Do you mean that the load in an accelerated frame will yield a different force (due to different g and m) which could explain curved appearance of the rod?

Eugene> SR with constant gravity "g" is sorta broken theory.

Yes, I think you only consider g constant over distances h such that gh/c2 infinity with M/r2 constant.

Yes, I mean g constant. Even in this case I'll have events horizon from the GR point of view. Other way is to consider weak gravity on flat background, but still this is good locally, not globally, The same as for weak tension on long rod will give you big deformation.

O. Gron, Covariant formulation of Hooke's law, Am. J. Phys. 49, 28-30 (1981)

K. Kassner, unfortunatly there are no eq. of motion given. It looks like we need to consider non-inertial coordinate systems, making things more complicate.