Why does the horizon area of a black hole never decrease?

[M. next]

Hawking said that the horizon area of a black hole never decreases and illustrated that in his Hawking Are Theorem:

dA/dt ≥ 0

Does anyone know why is it like that. Why doesn't the area decrease?

 

[stevebd1]

Hawking said that the horizon area of a black hole never decreases and illustrated that in his Hawking Are Theorem:

dA/dt ≥ 0

Does anyone know why is it like that. Why doesn't the area decrease?

According to the Second Law of BH Thermodynamics, in any classical process, the area of the event horizon does not decrease

$dA\geq 0$

nor does the black hole's entropy, $S_{bh}$ (the BH's event horizon area can remain stable in classical mechanics but will increase 1) if mass is added or 2) if spin or charge are reduced). The second law of black hole mechanics can, however, be violated if the quantum effect is taken into account, namely that the area of the event horizon can be reduced via Hawking radiation.


BH thermodynamics-

http://www.fysik.su.se/~narit/bh.pdf pages 9-13

http://edoc.ub.uni-muenchen.de/6024/1/Deeg_Dorothea.pdf pages 11-13

 

[M. next]

Thank you a lot for your reply. I now know why does it increase. You are saying if mass is added or if spin or charge is reduced. How was it known? If this is difficult to answer, then I ask why doesn't it decrease? What will happen if you took the other option IF THE AREA DECREASED what will happen? Will this violate something conventional or what do you say?

 

[PeterDonis]

What will happen if you took the other option IF THE AREA DECREASED what will happen? Will this violate something conventional or what do you say?

Hawking proved the area theorem for a classical black hole (i.e., one in which no quantum effects like Hawking radiation are operating) by a geometric argument which requires considerable groundwork to understand. But the gist of it is that the horizon is made up of outgoing light rays that just barely fail to escape to infinity, and the area of the horizon, roughly speaking, counts the "number" of such light rays that make up the horizon, assuming that they don't converge. Since nothing can escape from a classical black hole, the number of the light rays can't decrease; and Hawking's geometric argument showed that they can't converge. Putting those two things together establishes that the horizon area can't decrease.

 

[M. next]

Thank you very much for putting this in a simple way!

 

[PeterDonis]

the BH's event horizon area can remain stable in classical mechanics but will increase 1) if mass is added or 2) if spin or charge are reduced)

Or if spin or charge are *increased*. (The mass of a classical BH can't be reduced, so we don't have to consider that option.)