By the no hair theorem, a black hole is completely characterized by it's mass, charge and angular momentum. Therefore dipole and higher pole magnetic fields are completely expelled when a black hole is formed. (The magnetic field exponentially decays as electromagnetic fields radiate away the energy of the magnetic field.)
Assume you are falling into a supermassive black hole and you are carrying an ordinary bar magnet that has a dipole magnetic field. Wouldn't the no hair theorem imply that as you fall past the horizon, the magnetic field of the magnet you are carrying would be expelled or left behind at the horizon? Obviously, if this is true and if you are measuring the magnetic field of the magnet as you fall in, you will detect a change as the horizon is passed.
Since the black hole is supermassive there would not be any significant tide as you cross the horizon. And if you used a powerful rocket to hover just above the horizon until you turned off the engine and fell through the horizon, you would not be moving at the speed of light as you crossed the horizon, so you should have time to perform experiments.
So will you detect changes in the magnet's field as you cross the horizon? I fully expect the answer to this question is "No", but I am not sure and would like to understand why "no-hair" doesn't apply to this situation. It is still somewhat mysterious to me how magnetic fields are expelled from black holes when they are formed. This reference http://arxiv.org/abs/arXiv:1208.3487 , which is summarized by an answer to this question: When a neutral star with a magnetic field collapses to form a black hole, what happens to the magnetic field? shows how a magnetic field is expelled when the black hole forms.
However, the answer to this question: If a magnetic monopole falls into a schwarzchild black hole, what happens to the magnetic field? implies that if you fell in with a magnetic monopole, nothing would happen since a black hole can support a static magnetic charge. So it is only dipole and other higher pole magnetic fields that are not allowed for a black hole. In fact according to the answer to this question: What happens to an embedded magnetic field when a black hole is formed from rotating charged dust? a rotating charged black hole can have a quite complicated magnetic field associated with it.
ADDITION: Imagine observers with sensitive magnetometers in the last stable orbits around the black hole. They will never see the falling magnet cross the horizon - does that mean they will always measure the dipole field of the magnet? We know that because of the infinite gravitational time dilation at the event horizon, real photons get red shifted to longer and longer radio wavelengths and the rate of real photon emission god to zero, so from a visual (photon) point of view the falling magnet becomes invisible and frozen at the event horizon. But what about the virtual photons of the magnetic field? How are they affected? Surely we cannot see a permanent magnet stuck on the surface of the event horizon - that would give it "hair"! But how and where is the magnetic field radiated away? And why wouldn't the observer on the magnet detect something happening?
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