Angular velocity fixed at birth

Because of the Born rigidity of black holes, it follows directly from a consideration of the Ehrenfest paradox[5], that the angular velocity of a black hole can never be changed - it is fixed at birth. To explain this a little more fully, imagine a disk of radius \(R\) rotating with constant angular velocity \(\omega\).

Now imagine a reference frame fixed at the centre of the disk, The relative velocity of any point on the disk is given by \(\omega R\). So the circumference will undergo Lorentz contraction by a factor of

\[\sqrt{(1-(\omega R)^2/c^2)}\]

whereas the diameter does not.

So we have

\[\frac{circumference}{diameter}=\pi\sqrt{(1-(\omega R)^2/c^2)}\]

This is paradoxical for a truly rigid body and so in the general case, the body must deform. In the special case of a black hole, because of Born rigidity, deformation is not possible, and hence, the only alternative is that \(\omega\) is fixed. (Thanks to Wikipedia for this explanation)

**When black holes increase in mass, they must also increase in angular momentum in order to keep the angular velocity constant as it grows.**