Suppose I had two identical cubes, Cube A and Cube B, of volume V moving toward each other with velocity v, say on the x axis, as viewed by a third party observer, C on the x axis as well.
I know that transverse length contraction does not occur as shown in a proof by contradiction given in Morin Ch.11 Problem 1. However, take B's frame; B sees A contracted along the direction of motion (longitudinal contraction). B is somehow able to measure the volume of A.
My question is: what would this volume V' be? Would it still be V? If so, this must mean that with one of the directions which has contracted will cause a transverse direction to expand? Or is V' less than V, and only contraction occurs?