In a different thread the Herglotz Noether theorem was brought up and it was mentioned that this theory implies it is impossible for a cylinder rotating about its vertical axis to remain Born rigid in a gravitational field even at constant altitude. This is an extension of the claim that a Born rigid rotating disc cannot have linear acceleration and remain Born rigid. Now I accept and understand that: 1) A non rotating disc cannot be spun up and remain born rigid during the angular acceleration phase. 2) A vertical rotating cylinder that descends in a gravitational field cannot remain Born rigid. What I do not understand is why a vertical cylinder rotating around its vertical axis cannot remain Born rigid if its rotation is constant and it remains at constant altitude in a gravitational field. To try and break the problem down to its simplest elements, I would like to initially discuss/analyse the dynamics of an infinitesimally thin disc in the y/z plane that is rotating about the x axis, that experiences constant proper acceleration in the x direction. First, in the context of artificial acceleration in flat (Minkowski) spacetime, the disc would appear to have increasing velocity in the x direction in a given inertial reference frame. To make discussion easier I will identify my statements/assumptions/considerations with letters. a) I think we can also assume that as the velocity increases in the x direction, the rotation of the disc will slow down in proportion to the time dilation factor. b) To an accelerating observer that remains at the centre of the disc, the rotation rate will appear to remain constant. c) In the inertial reference frame, the increasing linear velocity and corresponding angular velocity slow down of the disc, implies that the radius and perimeter of the disc has to increase if the perimeter length of the disc is to remain constant as measured by observers on the rotating disc. d) If (d) is true, then by the equivalence principle. this implies that a thin disc with constant rotation at constant altitude will expand or rip itself apart (if Born rigidity is not maintained). As far as I aware this effect as not been observed in everyday life or in a lab. e) It may be that the expansion in consideration (c) does not occur. Study of successive Lorentz transformations of a rod shows that if the first transformation (to S') is parallel to the rod (say in the x direction) and the second (to S'') is orthogonal to the rod (say in the y direction), then Thomas precession does not occur and the length of the rod as measured in S'' is independent of the velocity component in the y direction and equal to the length of the rod as measured in the S' reference frame. This implies that if the disc is already rotating in the y,z plane about the x axis, and the constant linear acceleration is along the x axis (or upwards in a gravitational field) then length contraction effects above and beyond those already present due the rotation of the disc, should not occur. OK, that's enough musings for now. Can anyone correct/expand on my thoughts here and explain what exactly happens in the special case of a infinitesimally thin disc that is rotating with constant angular velocity and linearly accelerated with constant proper acceleration along its rotation axis?