The angular velocity ω of the wheel, due to impossibility of the wheel rim to exceed the speed of light c, cannot be higher than the value of c/R. The closer is the angular velocity ω of rotation of the wheel to the value c/R, the closer is the linear speed of the rim of the wheel to the speed of light c. In the absence of slippage between the rail and the drive wheel, the speed of movement of the rail relative to the trolley in the reference frame K is equal to the linear velocity v of the wheel rim and, with ω approaching the value of c/R, also tends to the speed of light c.

Suppose now that on the trolley next to the wheel rim there is a generator of light pulses and on the drive wheel there is a sensor, which, when passing by the generator, produces a signal of the generator actuation. In this case, with each complete revolution of the wheel the generator emits a light pulse. If the wheel makes n revolutions per second, the frequency of light pulses of the generator is equal to n. The linear speed of the wheel rim and the speed of the rail relative to the trolley in this case can be expressed by the frequency of light pulses in the form v = 2πnR. The critical frequency N of light pulses is equal to c/2πR. Consider now the motion of the trolley in the reference frame K', rigidly linked to the rail. The trolley speed in this reference frame K' (relative to the rail) is equal to the value v.

In the reference frame K' the frequency n' of the generator light pulses is equal to , and, with the trolley  speed v approaching the light speed c, tends to  the value of   , which with v= c is zero.

The light signal frequency n' tending to zero means termination of the drive wheel rotation in the reference frame K', consequently, in the reference frame K' the trolley moves along the rails with slipping. However, the effect of the wheel slipping on the rail is not relative but absolute, that is why in the reference frame K' slipping cannot occur because it disagrees with the initial condition of the absence of slipping in the reference frame of the handcar.

The solution of the paradox is that while rotating the spinning rim experiences the Lorentz  contraction. The rim "shrinks", and the radius of the rotating wheels reduces to the value  . For this reason, when the trolley speed v approaches the speed of light c, the frequency n' of the light pulses in the reference frame K' tends not to  , but to  .

R E F E R E N C E S

  1. Ø. Grøn  Space geometry in rotating reference frames: a historical appraisal in G.    Rizzi and M. Ruggiero. Relativity in Rotating Frames.  ISBN: 1402018053 Springer (2008)
  2. Ø. Grøn, Relativistic description of a rotating disk, Am. J. Phys. 43, 869 (1975).
  3. Ø. Grøn, “Optical Appearance” of a Rolling Ring, Int. J. Th. Phys. 22, 821 (1983).

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