One reference claims the Sun also exceeds its Kerr limit, but that’s based on a calculation that assumes a uniform density and rate of rotation. The accepted value of the Sun’s angular momentum is: J_S = 1.92×10^{41} [https://arxiv.org/abs/1112.4168 ] which is 20% of the Kerr limit.
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J = (G/c) M_E ≈ 7.9×10^{30} should be: J = (G/c) M_E^2 ≈ 7.9×10^{30}
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"The Earth has more angular momentum than any Earth-mass black hole could have!" Cool! This becomes a bit less surprising if we note that the black hole would have a radius of just 9 millimeters.
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Yeah, it’s not all that surprising in retrospect, but given that stars that actually *do* collapse into black holes can also be close to the limit, it gives some sense that extremal Kerr holes aren’t really all that exotic. Having that much specific angular momentum isn’t rare.
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The bound isn't even just meant to be for a black hole. So that's kind of weird... Heuristically: assume a toy universe containing only Earth. Final state should be a Kerr BH. Some mass could radiate away to infinity. Assuming axial symmetry, angular momentum is conserved (1/2)
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Since final state satisfies m^2>J and initial "Earth" had same J and possibly more mass, it should also satisfy the inequality. (Which of course it doesn't, as you pointed out) (2/2)
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There must be a scifi story here, where some people on earth survive Earth being crushed to a black hole, because they happen to be standing on a piece of rock which would drive the new black hole over the limit and get thrown out.
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What about Earth-mass Neutron star?
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