# BH rotating at maximum speed and extremal BH

Academic problem: What should be the rotational speed of a planet at the equator such as there is no surface gravity? Plugging centripetal force=gravitational force...

$$mv^2/R=GMm/R^2$$

we get, assuming sphericity and uniform density

$$f=\sqrt{G\rho/3\pi}$$

For a black hole with effective density we get

$$\rho=3c^6/(32\pi G^3M^2)$$

Thus,

$$f=c^3/(4\sqrt{2}\pi GM)$$

Can this frequency be given a sense in the context of Kerr black holes (extremal case) and rotating black holes and/or gravitational wave frequencies? I feel I should protest since, as we know from black hole thermodynamics, surface gravity can not be reduced to zero...

What I did here is usually made with planets or stars in "academic problems". Is the calculation meaningful for black holes as well with suitable care?

• Your calculations are for Newtonian gravity, which is a good model for the gravity of a planet, but not a good model for the gravity around a black hole. To do this right you would need to use general relativity from the start. Mar 20 at 22:01