Blackholes and 2nd Law of Thermodynamics

[JustinLevy]

Is it possible to deduce that a black hole must have something akin to Hawking radiation due to the 2nd Law of Thermodynamics?

Let us consider for now a purely classical black hole. The event horizon is effectively a 'light diode' in that if light can pass through in a particular direction, it can not pass through in the opposite direction. Therefore if we place a (again, purely classical) black hole in a vacuum with a thermal bath of photons, it will take away thermal energy from the surrounding area, creating a thermal gradient in the bath ... which in turn would allow energy to be extracted from the thermal bath.

This seems to violate the 2nd Law of thermodynamics, unless we consider (still purely classical) black holes to be objects of zero kelvin temperature with infinite heat capacity. Since we can classically consider the formation of a black hole from a collapse of finite temperature material, that seems to be an invalid consideration unless the creation of a black hole is violently endothermic. Since black holes of mass M (and no spin or charge) classically settle on only one solution, their entropy would be zero so we can't satisfy the 2nd law via the usual way endothermic reactions do.

It seems to me that the 2nd law of thermodynamics alone demands that black holes have thermal radiation. I don't think one could derive much or any specifics, let alone the details that is Hawking radiation. Yet it appears black hole evaporation is necessary for the 2nd Law of Thermodynamics?

Does anyone have some comments on this? I believe I remember reading a similar argument in a 'undergraddy/simple math intro gr' book, but can't find it currently. If you could point me to a more indepth textbook that discusses this, that would be wonderful.

 

[Bacat]

According to this website (http://www.math.ucr.edu/home/baez/physics/Relativity/BlackHoles/hawking.html), the theory of Hawking radiation depends on quantum theory. Thermodynamics does not depend on quantum theory.

I don't think that the 2nd law is strong enough, alone, to derive Hawking radiation. But it is does provide theoretical support for the quantum theory.

 

[JustinLevy]

According to this website (http://www.math.ucr.edu/home/baez/physics/Relativity/BlackHoles/hawking.html), the theory of Hawking radiation depends on quantum theory. Thermodynamics does not depend on quantum theory.

Yes, of course. On both accounts. (Although I find entropy in 'classical' thermodynamics is less satisfying than the deep connection it obtains in statistical thermodynamics, let alone in combination with quantum mechanics.)

I don't think that the 2nd law is strong enough, alone, to derive Hawking radiation. But it is does provide theoretical support for the quantum theory.

I'm not claiming we can derive Hawking radiation from thermo alone (actually I already stated that I don't think one could derive specifics of the thermal radiation). All I'm saying here is that it seems a classical black hole (due to its "light diode" property) and the second law of thermodynamics do not appear to be compatible. If we take the 2nd law as a given then this appears to demand some kind of thermal radiation.

Here, let me try to focus this discuss as much as possible.
1] Would a polarization independent "light diode" necessarily violate the second law of thermodynamics?
2] What properties would a "light diode" have to have in order to be compatible with the second law of thermodynamics?
3] Is a classical black hole compatible with whatever you answer for #2 ?

To help discussion, my current level of understanding (remember, I'm asking the questions because I do not know the answers, so please don't focus on these ... it is merely to show where my current understanding is)
1] I do not believe so. For example if the device necessarily always transferred enough heat (through energy other than light) against the light diode direction to prevent violation of thermodynamics.

2] The device must either gain entropy in the transfer of light, or transfer heat against the direction of light via other energy, or some combination of both. I do not see another possibility.

3] I do not believe a classical black hole is compatible with my answer to #2, as nothing can transfer from the inside of the horizon to the outside (this 'diode' is not limited to light), and the classical black hole appears to have no available phase space accept one configuration and thus zero entropy.Is this understanding correct? If not, please explain to me the correct answers.

 

[Bacat]

I think the problem is with the definition of a light diode. Entropy in the universal system increases, but may not in a local system. I think you could also posit some interesting quantum effects (since the scale of such mass under such intense gravity is very small) as a result of the black hole. Furthermore, if the mass is compressed to less than the Planck length, all bets are off and the system can basically do whatever it wants without consequence (by the Uncertainty Principle).

I haven't taken the time to see if any of the above applies to this problem, but if you're interested in reconciling entropy with black holes, this is one possible route.