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.

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