Rigidly Re-Orienting An Extended Body

It's well known that special relativity does not permit a rigid disk (where "rigid" is defined in Born's sense) to change its state of rotation. This is most easily seen by considering a stationary disk, and noting that by setting this disk into rotational motion the circumferential distances between points of the disk must undergo length contraction, while the radial distances do not. Hence some mechanical deformation of the disk is unavoidable. It follows that, as Rindler says, "the motion of one point of a rigidly moving body determines that of all the others", because the rotational degrees of freedom of a rigidly moving object are restricted by the fact that its state of rotation cannot be changed. Every Born-rigid object has a certain fixed rate of rotation, which can never be changed, so its only "freedom of motion" is translational.

However, contrary to what one might suppose, it is still possible to re-orient a non-rotating rigid body. Given a disk initially stationary with respect to some system of inertial coordinates, we can (in principle) subject this disk to Born-rigid translational acceleration in some direction by suitably coordinating the accelerations of the individual parts of the disk. In this way we can impart to the disk some translational velocity v, and then we can cease the acceleration, leaving the disk at rest in some new inertial frame. We can then subject the disk to rigid translational acceleration in roughly the perpendicular direction (relative to the direction of the previous acceleration), and then follow this with linear acceleration opposite the first acceleration, so that the disk is again moving at the speed v (with respect to the original rest frame), but in a slightly different direction from its first motion. Repeating this sequence of disjoint linear rigid accelerations, we can cause the rigid disk to translate around the circumference of a circle, essentially maintaining the speed v for the entire journey, and ending back where it began. We can then linearly decelerate the disk back to a stationary condition with respect to the original rest frame.

Will the orientation of the disk be the same as it was originally? Since it's impossible to change the rotational state of a rigid body, and since this particular disk (by assumption) is not rotating, it might be tempting to think that its spatial orientation can never change. However, if we carry out the above procedure, we will find that the disk's orientation has changed - despite the fact that it has remained parallel to itself throughout the process. The re-orientation experienced by the disk is due to Thomas precession, a phenomenon that is essentially just a consequence of the aberration of angles with respect to different inertial reference frames. Consider a segment of the disk lying along the direction of the velocity v following the first linear acceleration. This segment is now at rest with respect to a new inertial frame, and the next linear acceleration will be applied in the sideways direction to all points of this segment simultaneously - but this is simultaneity based on the new rest frame. With respect to the original rest frame, the two ends of this segment are accelerated at different times, and consequently the segment is rotated (slightly) with respect to the original inertial frame. As the disk continues to be accelerated (parallel-transported) around the circular path, there is a cumulative re-orienting effect, with the net result that its final orientation differs from its initial orientation.

The total amount of Thomas precession experienced by each segment of the disk during one complete loop around the circular path is the same, but it does not accumulate at the same rate for each segment during a single loop. This is because the Thomas precession experienced by a segment depends on the orientation of that segment relative to the translational velocity vector. Hence while some segments on the disk are precessing faster than others at different times during the completion of the loop. This might seem to imply that the disk cannot be rigid after all. On the other hand, we are subjecting the disk to only translational Born-rigid acceleration at each instant - which is certainly possible to do - so we know the disk is rigid. There is actually no conflict here, because at different locations around the loop the disk has translational speed in different directions, and the aberration effects on the various segments of the disk vary in precisely the manner necessary to be consistent with the variations in the precession rates.

Of course, the path along which the disk is moved need not be circular. In general, the parallel transport of any rigid object around a closed loop will result in some amount of precession of the object. Taking this effect into account, we see that Rindler chose his words carefully when he said "the motion of one point of a rigidly moving body determines that of all the others", because the statement would be false if we substituted the word "position" in place of "motion". For a rigidly moving body we cannot infer the position of every point from the position of a single point (even if we also specify the time with respect to some suitable coordinate system), because the body can be oriented in different directions, and this orientation depends on the entire history of motion of the object. Two identically-oriented stationary objects, if subjected to different sequences of rigid motions (in two or three spatial dimensions) and then brought back to mutual rest, in general may have different orientations. Likewise for rotating objects we can introduce a phase shift in the angular positions of the objects versus time by transporting them around a closed loop.

This effect is closely analogous to the different lapses of proper time that can be experienced by two particles carried along different paths between two given events, despite the fact that each particle's proper time has everywhere coincided with the particle's instantaneous rest frame coordinate time (i.e., "parallel transport" of the time axis).

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