What Would Happen If a Coin-Sized Black Hole Were Placed at the Earth's Center?

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Feb. 10 2014 7:39 AM

What Would Happen If a Coin-Sized Black Hole Were Placed at the Earth's Center?

This question originally appeared on Quora.

Answer by Frank Heile, Ph.D. in physics from Stanford University:

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The Earth would be destroyed, but the whole planet would not be swallowed up by the black hole. A black hole with a Schwarzschild radius of about a centimeter, which would make it about the size of a coin, would have about the same mass as the Earth. The reason the Earth will be destroyed but not simply swallowed up is because the Earth will be resisting the black hole in at least two ways.

The explosion

First of all, not all of the Earth would simply be sucked into the black hole. When the matter near the black hole begins to fall into the black hole, it will be compressed to a very high density that will cause it to be heated to very high temperatures. These high temperatures will cause gamma rays, X-rays, and other radiation to heat up the other matter falling in to the black hole. The net effect will be that there will be a strong outward pressure on the outer layers of the Earth that will first slow down their fall and will eventually ionize and push the outer layers away from the black hole. So some inner portion of the core will fall into the black hole, but the outer layers, including the crust and all of us, would be vaporized to a high temperature plasma and blown into space.

This would be a gigantic explosion—a significant fraction of the rest of the mass of the Earth matter that actually fell into the black hole will be converted into energy. For astrophysical black holes, up to 40 percent of the rest mass of the accreted material can be emitted in radiation. This radiation will be absorbed by the outer layers of the Earth and will vaporize them. Examples of this kind of dramatic matter to energy conversion are quasars. Quasars are the most luminous objects in the universe, and they are powered by matter falling onto a supermassive black hole. So there will be plenty of energy available to blow off the other layers of the Earth—and they will escape! For example, when the black hole is first placed at the center of the Earth, the first thing we would all notice is that gravity increased by (only) a factor of two on the surface of the Earth (assuming the black hole had the same mass as the Earth). However, the escape velocity of an object only increases as the square root of the mass, so the current 11 km/s escape velocity on the surface of the Earth will only increase to about 16.8 km/s. A very significant fraction of the mass of the Earth will become a vaporized hot plasma and will be going faster than that when it passes the radius of what used to be the surface of the Earth.

The accretion disc

Secondly, the Earth is rotating, so by conservation of angular momentum, when a significant amount of mass has started to fall into the black hole, the mass will also begin rotating at a higher and higher rate. (Imagine the ice skater pulling in her arms to rotate faster.) This angular momentum will tend to slow down the fall into the black hole and will eventually result in something like an accretion disc around the black hole. This will also limit the fraction of the Earth that will fall into the black hole and will greatly increase the time it takes for the black hole to consume whatever fraction of the mass of the Earth it will consume. The reason for the delay is that the accretion disc has to use friction to transfer angular momentum from the innermost portion of the disc to the outer edge of the disc, where it will cause material to be ejected from the vicinity of the disc—carrying away angular momentum. The lower angular momentum near the center will allow that innermost material to fall into the black hole.

In fact, even though the Earth only rotates once per day, the angular momentum of the Earth is huge. There are limits to how much angular momentum a black hole can have—roughly the maximum angular momentum is where the "surface" of the black hole (if it had a surface) would approach the speed of light. Trying to make a small (two Earth mass) black hole with all of the Earth's angular momentum would mean that the surface would have to travel at about 109 times the speed of light. So most of the mass of the Earth would have to be used to carry away almost all of the original angular momentum of the Earth in order to keep the black hole below its angular momentum limit.

How long?

But what if there is no explosion and no angular momentum to stop the surface from falling in to the black hole? How long would it take for the Earth to "fall" into the black hole? Well, imagine that somehow, magically, all the mass of the Earth just became a black hole at the center of the Earth and that you were standing on the North Pole (with no angular momentum) in a space suit (since you are now in a vacuum). How long would it take until you are spaghettified as you fall into the black hole? We can get an approximate answer by using Newtonian gravitation instead of general relativity, which is what is really needed for motion into or near a black hole. According to Newtonian gravity, it would take approximately 15 minutes to fall into the black hole (see the calculation on WolframAlpha). For a black hole with twice the mass, it would take 10 minutes to fall into the hole. So the more accurate general relativity answer may be slightly different, but the time for the surface to fall in will be something close to 10 to 15 minutes. This would be the time as measured by you as you fall into the hole. For someone on the moon watching you fall, the gravitational time dilation will make it look like you are falling slower and slower when you get very close to the black hole, so it would look like it would take forever to hit the black hole horizon. However, for you, falling in, it will be all over in approximately 10 to 15 minutes or so from your point of view. Similarly, if there were no explosion and no angular momentum that would retard or prevent the swallowing up of the Earth, then it would take about 10 to 15 minutes for the whole Earth to fall into the new black hole at the center of the Earth.

A more exact numerical answer of how long the explosion would last and what faction of the Earth would be absorbed versus being blown away would require a supercomputer running relativistic hydrodynamic codes to simulate this very complicated explosion. I will leave that as an exercise for the reader.

And for Star Trek fans, the "red matter" black hole could have destroyed Vulcan, but it would not have just "sucked" or "collapsed" the planet into the black hole.

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