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Lunch Break

How long could a black hole remain at the center of the earth? And make sure to read the comments.

By Ezra Klein  |  May 29, 2009; 12:30 PM ET
Categories:  Lunch Break  
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Is this another way of asking how completely full of crap the Vulcan planet-collapsing sequence in Star Trek was?

Posted by: Chris_O | May 29, 2009 12:48 PM | Report abuse

Comments are closed now so you get the answer.

All the weirdness around black holes happens near the event horizon. In the Schwarzschild metric, when you get more that a few Schwarzschild radii out, the effective potential is essentially the same as the one obtained from Newton's laws. So it will clean out its immediate neighborhood and then sit there and do nothing.

The Earth's core is solid iron. The Schwarzschild solution is spherically symmetric. So the hole it clears out will be spherical. This configuration of spherical shell with a point mass at the exact center is gravitationally stable. The hole will enlarge until the intermolecular forces holding the iron together are capable of supporting it against the pressure of the overlying rock. Then it will stop.

If you want interesting things to happen you need a rotating black hole since that will open an ellipsoidal hole, which is not gravitationally stable. In this sense, the question was not well posed.

Lots of talk in the comments about whether you could balance mass absorption and Hawking radiation. The answer is no. The temperature of a black hole depends on its surface gravity which depends on the inverse of its radius. The amount of matter it can absorb depends on its surface area (since that is where the matter must cross) and that depends on the radius squared. Consequently, as mass is absorbed and the radius grows, the capability to absorb more mass increases dramatically while the Hawking temperature decreases. As long as there is any mass around to absorb, Hawking radiation will always lose. People don't tend to realize how small an effect that actually is. Presumably, you could set an initial equilibrium with a magic value of the radius but that equilibrium would be very unstable. Throw in a single extra electron at some instant and the equilibrium is toast.

Posted by: evenadog | May 29, 2009 6:54 PM | Report abuse

re: lunch break

here is something very interesting to watch while eating lunch...
(the commercial only lasts for a few seconds)

Posted by: jkaren | May 30, 2009 12:01 AM | Report abuse

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