So, do I understand you correctly: relative to the "static observer", the rods are both moving "sideways", as if the x-axis of a graph were moving toward +y and

So, do I understand you correctly: relative to the "static observer", the rods are both moving "sideways", as if the x-axis of a graph were moving toward +y and the y-axis were moving toward +x? In that case, the "static observer" would see the point of contact moving at very nearly sqrt(2) times the speed of light (along the x-y diagonal). But that's okay, because no physical object (or bit of information) is being carried along at that intersection point. (If you put some little spindle at the intersection to be carried along, you'd suddenly have Problems.) On the other hand, from the reference frame of one moving bar, the intersection point appears to be moving almost infinitely fast! If the "static observer" measured speeds u and v for the two rods (as fractions of light speed, so u and v are both close to 1), then an observer moving with one rod will see the intersection point move at speed v/sqrt(1-u^2). If we write u = 1 - ε for some very small ε, then the intersection point's speed in this frame is roughly v/sqrt(2ε). (The intuitive reason for this result is interesting: an observer on one moving rod will consider the other rod to be almost parallel, at a very small angle (rather than perpendicular), and still rushing toward her at nearly the speed of light.)