system
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human
What is the temperature of a black hole?

From my understanding, as matter heats up atoms move more freely.  But with a black hole, there is (I'm assuming) little-to-no movement due to the extreme gravity, so does that inhibit temperature or are there other components that effect the temperature?
gpt
[TL;DR]

It depends on just what you mean, but the usual definition of black hole temperature comes from Hawking radiation, in which case if we express our mass in multiples of solar masses (so the sun would have M = 1, a star ten times as massive would have M = 10), the temperature is

T = (0.0000000617/M) Kelvin.

So, for example, a black hole the mass of the sun would have a "temperature" of approximately 0.0000000617 Kelvin. One that was ten times as massive would have one tenth the temperature.

[Comment Proper]

When dealing with curved spacetimes, you have to be careful with the definition of things like "temperature". When dealing with black holes in particular, you also have to be careful about how you interpret questions that you expect naïvely to depend on the behavior of stuff inside the event horizon.

One of the difficulties here is that we can't speak to the microscopic behavior of matter inside of the black hole. We do not have a consistent model for describing quantum systems in such spacetime regions, and any proper microscopic description is going to require a quantum theory. Thus, we can't really talk about temperature in the sense of "motion of particles".

However, we *can* apply quantum field theory in a blackhole spacetime provided, roughly, that we stay away from the singularity and our field doesn't carry too much energy. In particular, if we're interested in describing events outside of the event horizon, we can use these techniques as a good first approximation. We can also apply quantum statistical mechanics/thermodynamics to this process. When you do this, you find that if an observer falling across the event horizon sees themselves in a vacuum, an observer far from the black hole must see a flow of thermal radiation coming from the black hole. This radiation has the same spectrum as a blackbody source of some definite temperature, so we can associate it with a temperature. It turns out that, for a black hole of mass M, the temperature T associated with this radiation is

T = (ħc^(3) / [8πGk_*B*_])/M,

where ħ is the reduced Planck's constant, c is the speed of light, G is Newton's gravitational constant, and k_*B*_ is Boltzmann's constant. That factor is just a constant that we can evaluate, and it comes out to roughly

T = 1.227\*10^(23) kg K / M = (0.0000000617/M) Kelvin,

where the second form comes about by expressing M as a multiple of the solar mass.