Ehrenfest Paradox and its resolution

This video explains the Ehrenfest Paradox and its resolution. Four years after Einstein had proposed his Special Theory of Relativity, Paul Ehrenfest came up with a paradoxical situation on application of Lorentz contraction to the problem of rigid rotating objects. This was following Max Born's work on rigid bodies in Special Relativity. The original paradox dealt with a rigid rotating cylinder but most of the further works have replaced the cylinder with a rigid rotating disk. The inertial observer would measure a ratio of Pi between the circumference and the diameter of the disk. But when the disk rotates with a constant angular speed, this ratio would change due to Lorentz length contraction of the periphery of the disk while the radius would remain constant. This is a paradoxical situation and Ehrenfest felt that there could be an issue with the applicability of Special Relativity to the rigid bodies in rotational motion. Einstein gave a heuristic explanation in the following years through which he explained that the geometry of such rotating objects is non-Euclidean and therefore the measurements carried out by non-inertial observers in the rotating frame would vary with the measurements made by inertial observers of the disk. This resolved the paradox but a lot of questions surrounding the rotating disk have kept the scientists busy over the last more than 100 years.
For over 2 millennia, mathematicians have tried to prove Euclid's fifth postulate also known as the parallel postulate. During such efforts, mathematicians realised the possibility of existence of non-Euclidean geometry such as Elliptical geometry, spherical geometry and hyperbolic geometry. Some of the derivatives of the fifth postulate include the postulate of triangles whereby the sum of the angles of the triangle is always 180 degrees. This condition is valid only in Euclidean geometry. In non-Euclidean geometry, this is not true. Similarly, the ratio of the circumference to the diameter of a circle is not Pi in non-Euclidean geometries. Einstein's argument to resolve the paradox involved these considerations thereby arriving at a circumference larger than the one the disk has when at rest.

Born rigidity provided a set of stringent conditions and constraints to decide the rigid body kinematics within Special Relativity, under linear acceleration of objects. Paul Ehrenfest had in fact used Born Rigidity to propose his paradox while adopting it to the problem of rotating rigid objects with circular acceleration.
Special Relativity does not allow perfectly rigid objects. Even when an object is rotated from rest to a constant angular speed, Born rigidity is violated. Thus, Born rigidity can be applied only when the object is rotating and is further accelerated. Under these conditions, two methods are possible to accelerate objects towards rotary motion.
The first method involves accelerating with respect to the elements that make up the rigid body in such.a way that all elements are simultaneously accelerated in the non-inertial frame. This was shown to deform or break up the rigid body.
The second method involves accelerating the rigid body uniformly in the inertial frame. In this case, each element of the rigid body in their own frames would have a time difference between accelerations of its two ends. This time difference results as an equivalent distance to the observer in the frame of the rotating disk thereby resulting in the elongation of the periphery of the circular disk. This proved that Einstein's resolution of the paradox indeed was correct and the geometry of a rotating disk is indeed non-Euclidean.
The video discusses all these issues in great detail. After introduction of the Paradox, the video discusses Einstein's heuristic argument to resolve the paradox. Then non-Euclidean geometry is introduced starting from Euclid's postulates. The video then goes on to discuss the resolution of the paradox in great detail. The video also discusses why special theory of relativity does not allow for perfectly rigid bodies. In the context of resolution of the paradox, the video discusses the differences between 'seeing' and 'observing' or measuring in the context of the problem by comparing what different observers could see/measure of the rotating disk. Some of the questions raised by the paradox and how it inspired the development of General theory of Relativity is also touched upon in this video by mentioning the equivalence principle and the non-Euclidean curve nature of spacetime due to gravity. The point that the Ehrenfest paradox straddles the boundary between the Special and the General theory of relativity is highlighted. Some of the research problems related to the Ehrenfest paradox that have kept scientists busy for over the last 100 years are mentioned in the video.