# Angular momentum limit of a Kerr black hole

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## Summary:

Could a rotating extended body exceed that limit?

## Main Question or Discussion Point

The Schwarzschild metric seems to model, for example, the earth’s gravity field above the earth’s surface pretty well, even though the earth is not really a golf-ball sized black hole down at the center. Can the same be said for the Kerr metric? Does it model a rotating extended body’s gravity field well above that body’s surface?

(I am wondering about the magnitude of the angular momentum of that extended body versus that of the Kerr black hole. Isn’t there a limit to how much angular momentum the black hole can have, and might the extended body have more than that limit? Or is that not possible?)

Dale

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PeterDonis
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The Schwarzschild metric seems to model, for example, the earth’s gravity field above the earth’s surface pretty well, even though the earth is not really a golf-ball sized black hole down at the center.
That's because of Birkhoff's theorem: any vacuum spherically symmetric region of spacetime must be a region of Schwarzschild spacetime.

Note that the Earth is not a perfect sphere, though, so the spacetime around it is not exactly modeled by Schwarzschild spacetime; there are deviations. (One of the deviations is that the Earth is rotating--see below.)

Can the same be said for the Kerr metric?
No. There is no analogue of Birkhoff's theorem for Kerr spacetime.

Another way of looking at this is: Kerr spacetime has no higher order moments (quadrupole and higher); this follows from the "no hair" theorem. But a real rotating object does have higher order moments, which must be reflected in the spacetime geometry around it. (Note that the Earth itself is an example of this: the Earth has a detectable quadrupole moment, which has been shown to affect the gravitational field and hence the spacetime geometry around it.)

exmarine