Ehrenfest's Paradox: Resolving the Kinematical Solution

[DrGreg]

For reference some previous threads which included discussion of quotient spaces and rotating disks:

• Large rotating disk, particularly post #17 onwards
  
• analogy between centrifugal force and gravity

 

[DrGreg]

Coincidentally, this has just cropped up in another thread:

When expressed in cylindrical polar coordinates, these are known as Born coordinates
$$ds^2 = -\left( c^2 - \omega^2 \, r^2 \right) \, dt^2 + 2 \, \omega \, r^2 \, dt \, d\phi + dz^2 + dr^2 + r^2 \, d\phi^2$$

so I might as well point out that if you "complete the square" on this formula you get$$ds^2 = -\left( c^2 - \omega^2 \, r^2 \right) \left( dt - \frac{\omega r^2}{c^2 - \omega^2 r^2} d\phi \right)^2 + dz^2 + dr^2 + \frac{r^2}{c^2 - \omega^2 r^2} \, d\phi^2$$The expression$$dz^2 + dr^2 + \frac{r^2}{c^2 - \omega^2 r^2} \, d\phi^2$$represents the non-flat metric of the quotient space. The same technique applies to any stationary spacetime.

Ref: Rindler (2006), Relativity: Special, General and Cosmological, 2nd ed. p. 198