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I would guess it would result in considerable damage to the earth
Also remember there is no mechanism for the rapid loss of the earth's angular momentum which would be a very rapid rotation of the earth's mass if only a few millimeters diameter.
My back of an envolope calculation says it would be rotating at 10^15 rpm with a surface speed of 1.7c.
Just a thought. If by some bizarre glitch, the Earth turned into a black hole (of the same mass as the Earth)and we then waited for the Sun to go red giant, what would happen?
Lee, any BH that is formed anywhere other than at the poles would not fall in a straight line to the centre because of the Coriolis force. If formed on the equator, for example, it has a significant tangential velocity. Also remember there is no mechanism for the rapid loss of the earth's angular momentum which would be a very rapid rotation of the earth's mass if only a few millimeters diameter. I am not sure whether all the earth's mass would be accreted or whether some would remain in orbit. I don't know how you can be so sure without a simulation or elaborate calculation. Anyway it's all speculation (I hope).
Lee, the centre of gravity of the whole system will not change. The centre of rotation will be the same so it will tend to look like the BH will spiral in. This is no normal BH but one that has a tiny mass by comparison with "normal" BHs.
I rather hoped that the Solar systems salvation would be that the Earth mass BH would have a short life but the Hawking equation predicts 5.668638e+50 years.For it to start to acreate mass presumably it would have to begin life with a surface temperature of less than 300°K, this would correspond to an initial mass of 4.090677e+17tons.
"[...] so there would be a shift in the barycenter away from Sol and towards Earth". No there wouldn't, Lee. The Centre of Gravity of the total system will stay the same as would the total angular momentum. These can only change by outside influence.
The center of mass of a two-particle system lies on the line connecting the particles (or, more precisely, their individual centers of mass). The center of mass is closer to the more massive object
My figure for the rotational speed may be somewhat in error as I took the the radius of an Earth mass BH as 0.5 cm whereas Hawking quotes 0.8864564 cm, perhaps someone would care to repeat the calculation with greater precision.
My figure for the rotation of 10^15 rpm seems to coincide but as for the surface velocity I can only agree oops!.