Consider the thought experiment in the diagram below. According to the authors, the vector will rotate even in the rest frame of the moving train relative to the body of the train. Is it possible to consider the spin axis of the gyroscope represented by the vector as a simple rod and analyse the situation as a simple 4-acceleration problem? If, so would the rod rotate by exactly the same amount in the rest frame of the train, irrespective of its original orientation? I suspect there is a critical orientation of the rod that would be exactly parallel to the boosted acceleration, such that no rotation of the rod occurs. If so, this would suggest that an initial assembly of rods that are orthogonal to each other would not remain orthogonal, because they rotate at different rates. The above diagram comes from this paper http://www.relativitet.se/Webarticles/2006CQG-Jonsson23p37.pdf which is specifically about gyroscopes. Would we expect rods and gyroscopes to rotate differently is no torque is directly applied to them? One issue that is mentioned in the paper is that in certain reference frames, the proper centre of mass is no longer at the normal centre of mass of a perfectly balanced gyroscope. This might be a source of torque even when we think we are applying the acceleration to the centre of mass of the gyroscope in a torque free way.