An easy way to think about the entropy of black holes is to consider that entropy represents the loss of free energy – that is, energy that is available to do work – from a system. Needless to say, anything you throw into a black hole is no longer available to do any work in the wider universe.
An easy way to think about the second law of thermodynamics (which is the one about entropy) is to consider that heat can’t flow from a colder location to a hotter location – it only flows the other way. As a result, any isolated system should eventually achieve a state of thermal equilibrium. Or if you like, the entropy of an isolated system will tend to increase over time – achieving a maximum value when that system achieves thermal equilibrium.
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If you express entropy mathematically – it is a calculable value and one that tends to increase over time. In the seventies, Jacob Bekenstein expressed black hole entropy as a problem for physics. No doubt he could explain it much better than I could, but I think the idea is that if you suddenly transfer a system with a known entropy value past the event horizon of a black hole, it becomes immeasurable – as though its entropy vanishes. This represents a violation of the second law of thermodynamics – since the entropy of a system should at best stay constant – or more often increase – it can’t suddenly plummet like that.
So the best way to handle that is to acknowledge that whatever entropy a system possesses is transferred to the black hole when the system goes into it. This is another reason why black holes can be considered to have a very high entropy.
Then we come to the issue of information. The sentence The quick brown fox jumped over the lazy dog is a highly engineered system with a low level of entropy – while drawing out 26 tiles from a scrabble set and laying them down however they come delivers an randomly ordered object with a high level of entropy and uncertainty (to the extent that it could be any of a billion possible variations).
Throw your scrabble tiles into a black hole – they will carry with them whatever entropy value they began with – which is likely to increase further within the black hole. Indeed it’s likely that the tiles will not only become more disorganized but actually crushed to bits within the black hole.
Now there is fundamental principle in quantum mechanics which requires that information cannot be destroyed or lost. It’s more about wave functions than about scrabble tiles – but let’s stick with the analogy.
You won’t violate the conservation of information principle by filling a black hole with scrabble tiles. Their information is just transfered to the black hole rather than being lost – and even if the tiles are crushed to bits, the information is still there in some form. This is OK.
But, there is a problem if in a googol or so years, the black hole evaporates via Hawking radiation, which arises from quantum fluctuations at the event horizon and has no apparent causal connection with the contents of the black hole.
A currently favored solution to this problem is the holographic principle – which suggests that whatever enters the black hole leaves an imprint on its event horizon – such that information about the entire contents of the black hole can be derived from just the event horizon ‘surface’ – and any subsequent Hawking radiation is influenced at a quantum level by that information – such that Hawking radiation does succeed in carrying information out of the black hole as the black hole evaporates.
Zhang et al offer another approach of suggesting that Hawking radiation, via quantum tunneling, carries entropy out of the black hole – and since reduced entropy means reduced uncertainty – this represents a nett gain of information drawn out from the black hole. So Hawking radiation carries not only entropy, but also information, out of the black hole.
But is this more or less convincing than the hologram idea? Well, that’s uncertain…
Further reading: Zhang et al. An interpretation for the entropy of a black hole.