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I know of stars actually dying (exploding as a supernova) but I have never heard of what happen to black holes. Do they continue to live forever, waiting there for more matter to absorb? Do they explode at some point ?

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They evaporate. –  Stan Liou Feb 15 at 13:42
    
Hawkings new paper may be of interest for you arxiv.org/abs/1401.5761 –  samcarter Feb 17 at 18:08

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Like Afzaal Ahmad said black holes evaporate through Hawking radiation, proposed by Stephen Hawking in 1974. Black holes have a temperature just a little bit above absolute zero; for a black hole with the weight of the Earth it's about 0.02 K. (Such a black hole would be about 2.5 cm in diameter.) And black bodies radiate energy mainly in the form of photons as their temperature is larger, so the radiation for a black hole isn't very high: for the Earth-weigth black hole it's about $10^{-17}$ W. Larger black holes are even colder, and will radiate even less.

But how can the photon escape the black hole's gravity when we know light can't escape it? It happens through quantum tunneling. A photon is not just a particle, it's also a wave, and quantum mechanics shows a probability distribution for the location of a particle. The black hole's event horizon is a barrier for the photon, but it could exist beyond that barrier. Under these conditions the photon can tunnel through the event horizon and suddenly find itself outside of the black hole. (Actually tunneling through is not a good choice of words; the photon doesn't move through the barrier, it just suddenly is at the other side.)

So energy can and does radiate away from the black hole, but this happens so slowly that it may take a trillion times the age of the universe before the whole hole :-) has evaporated.

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Yes, everything that is in this universe has to say Good bye to its existence. That's what Black hole also has to follow.

Stephen Hawking gave this concept and it is known as Hawking Radiation, in which the Black Hole radiates energy because of the quantum mechanical process. But they donot stay there forever to consume more matter nor do they burst up. They just simply evaporate slowly; a concept of the Hawking's theory.

http://en.wikipedia.org/wiki/Hawking_radiation

http://curious.astro.cornell.edu/question.php?number=163

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