[Atomic-S (OP)]

Let us consider two like cylinders, each composed of a dense (Pb, Pt, Os, Au, whatever) substance, each a good many (50?) kilometers in diameter, and a good many thousands of Km long, located out in space well away from other objects, and separated by a small (comparable to the diamter or less) distance, but not touching. Such objects will exert a significant gravitational attraction upon one another,  and would eventually hit one another except that let us assume that they are kept apart by a row of many springs. The springs will compress under the load, and the amount of the compression (the masses being in equibrium) indicates, to an observer at rest with respect to this system, the gravitational attraction.  Or more precisely, the gravitational attraction per unit length, because the number of springs is proportional to the length. (Adding more length of this system will not significantly change the compression of any one spring anywhere except perhaps near the ends).

Now consider an observer moving at a significant fraction of the speed of light, parallel to the axes.  In his reference frame, the cylinders are moving at high speed in the direction of their own length. Their masses are greater. Furthermore, the Lorentz contraction has made them shorter, meaning that not only their rest mass but also the additional relativistic mass is concentrated in a shorter length, so that the overall mass per unit length is doubly increased. On this basis we would conclude that the gravitational attraction per unit length ought to be increased proportionately; but obviously, the same springs are there, and the distance between them has not changed because the Lorentz contraction does not affect distance at right angles to the motion.

Of course, it can be argued that the Lorentz contraction has not only increased the mass per unit length (which increases gravitation), but has also increased the number of springs per unit length (which just counterbalances it).  That is correct, but we are still left with the portion of the  increase of mass per unit length which derives not from the changed geometry but simply from the motion

    m  =  m_0 / sqrt[1 - (v/c)^2]

Meaning, there is still an increase in mass without an equivalent increase in gravitational attraction.

?

[lightarrow]

Ok, there is an increase in relativistic mass, but how does gravitational field transforms relativistically from a reference frame to the other? Making an analogy with electric and magnetic fields (they transforms so they are not the same from one ref. to the oth.) it would seem it's not the same.

As far as I have understood, with moving objects, together with the normal field, there is also a "Gravitomagnetic"  field, which is the analogous of the magnetic field in electrodynamics.

I don't know much more than this, however.
(And I would like to know the answer!)

[lightarrow]

The force made by the springs transforms too. If it's F in the reference in which they are stationary, it becomes F/γ  in the reference where they are moving.
(γ = 1/√(1 - β^2).
I would bet that, using the equations of gravitational field's relativistic transformations (which I don't know), the overall gravitational field (that is, considering the gravitomagnetic field too) becomes 1/γ times the field in the stationary ref., so that it equals the spring's force again.

[Batroost]

Perhaps all you've shown is that the two observers would genuinely calculate different values for the Young's modulus of the springs...? I never heard anyone claim that this was relativistically invarient - but I might not have been listening?

[Atomic-S (OP)]

Regarding the invariance of Young's modulus: Let us consider an experiment between 2 parallel plates of insignificant mass, with a spring between them. Let the plates be brought together a little, compressing the spring to a certain force. Then let the spring be accelerated frictionlessly by means of some other agency, at right angles to the direction of compression, until it reaches a significant fraction of the speed of light. If the force changes with speed, the force exerted on the plates will change. Now let the plates be separated again, to their original spacing. Because the Lorentz contraction is not active at right angles to the spring's motion, the spring in that direction will return to its original length; but it will do so under a different force than that with which it was compressed. Now let the spring be stopped. The final result of this process is a cycle which violates the conservation of energy, the amount of work leaving the system being different than the amount entering. Conclusion:  the Young's modulus of a spring at right angles to the motion cannot change with speed.

[Batroost]

Then let the spring be accelerated frictionlessly by means of some other agency, at right angles to the direction of compression, until it reaches a significant fraction of the speed of light.

...is why this wouldn't be violating conservation of energy - you are doing work on the system by way of an external force.

But I'm not suggesting the Young's Modulus at right angles to motion is subject to Lorentz, I'm wondering if, in the situation you describe, the modulus of the springs is dependent on the relative motion of the observer?