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It looks like usual criteria (positivity of Hessian; what geometrically means a cancave of entropy) is no useful, becouse entropy is not additive and not extensive for black hole. Then what is the right criteria?

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It depends on the temperature difference betweem the BH and its environment. – Dilaton May 15 '13 at 10:47
And how the criteria is formulated mathematically? – drobnbobn May 15 '13 at 10:51

One can see that a black hole in thermodynamic equilibrium with its environment is always unstable by looking at the Hawking temperature of a black hole, given by

$$ T = \frac{\hbar c^3}{8\pi k_B M G} $$

The temperature being inversely proportional to the mass means that big black holes are cold, a black hole with the mass of the sun has for example a temperature of about only $4\cdot 10^7 K$. This is much colder than the empty space with a microwave background radiation of about $2,7 K$. So heat (=energy= mass) will flow from empty space which makes it even colder according to the equation for the Hawking radiation.

If the black hole is small and initially warmer than the empty space, heat flows out of the black hole which increases the temperature further.

This counterintuitive behavior means that the black hole has a negative heat capacity which means that a black hole in thermodynamic equilibrium is unstable.

Lenny Susskind explains these things in the second part of this lecture, my explanations follow what he said there.

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Dilaton, Schwarzschild black hole has a negative heat capacity, nevertheless well known that Schwarzschild black hole is thermodynamically (and dynamically) stable. I do not put in consideration quantum effects like Hawking radiation. – drobnbobn May 15 '13 at 23:37

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