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According to Wikipedia, black holes emit Hawking radiation that corresponds to the spectrum a black body would emit at a temperature inversely proportional the mass of said black hole. Since the mass of a black hole is well defined (I believe), does this mean that the temperature of a black hole is also well defined?

Since the horizon of a black hole is an extended region in space, it means that the information of the temperature at some point in the horizon cannot travel faster than light to the whole horizon, I do not know how it would be possible for a black hole to have a well defined single temperature.

I am thus wondering if black hole have a well defined temperature, and if so, how is that possible that, for any observer, the temperature in the whole horizon space is the same?

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    $\begingroup$ Why do you think that the information of the temperature at some point in the horizon must travel faster than light to the whole horizon? Wouldn't your argument apply to the surface of a regular star? Having said that, in a view of any external observer, remote, hovering, or falling (as long as still external), information cannot travel along the horizon at all, because the speed of light at the horizon is zero. $\endgroup$ – safesphere Sep 27 '18 at 21:48
  • $\begingroup$ @safesphere not only would it apply to any star, it would apply to any object :) $\endgroup$ – Al Nejati Sep 27 '18 at 23:24
  • $\begingroup$ The case of a star or any object is quite different. They do not need to have a well defined temperature to begin with, so any tiny region of their surface can be at a different temperature than other ones. Black hole however, apparently need to have a well defined temperature since it must equal an expression that depends on the inverse of their mass $\endgroup$ – thermomagnetic condensed boson Sep 28 '18 at 5:45
  • $\begingroup$ Take an aluminium rod. If its temperature had to be equal to an expression inversely proportional to its mass, not only would its thermal conductivity need to be infinite, but heat propagation would need to happen at an infinite speed, which is of course impossible. My question is how do black hole can have such a property, if they really have it? $\endgroup$ – thermomagnetic condensed boson Sep 28 '18 at 5:50
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    $\begingroup$ The short answer is that changing the structure of spacetime in such a dramatic way would violate the assumptions under which black hole temperature was originally defined. The long answer requires a bit more physics and would probably be better served as its own separate question post. $\endgroup$ – Al Nejati Sep 29 '18 at 8:42

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