Black hole thermodynamics

Jacob Bekenstein and Stephen Hawking

Stephen Hawking, is going to have initial suspicion, of a relationship between the surface area of a black hole’s horizon, to the entropy of the black hole itself. This was the early 1970s. This was counterintuitive, indeed, since black holes, at the time, were believed to follow only 3 classical properties: 

• mass, 

• charge and 

• angular momentum. 

This is known as the no-hair theorem and there is nothing random about it. This result would be even more unexpected, since, for most physical systems, it was understood that the entropy would scale with the volume and not with the surface area.

However, Jacob Bekenstein, is going to propose otherwise. To Bekenstein, the black hole’s surface area, and it’s entropy, were the same property, perhaps, by some mathematical relation. However, Bekenstein also thought, that if the opposite were true, that black holes have zero entropy, than the 2nd law of thermodynamics, which states that entropy can only increase, would be violated. This result was highly unsatisfactory, as well as contradictory. For this reason, Jacob Bekenstein proposed that if the 2nd law of thermodynamics were to be preserved, than the black hole must contain some kind of entropy. Bekenstein thought that the source of that entropy was the surface area of the black hole itself. This proposal seemed to make sense to Bekenstein. Bekenstein constructed a mathematical relationship between the black hole’s surface area and it’s entropy, all the while, maintaining the 2nd law of thermodynamics. The evidence that there was an impressive relationship between the properties of black holes and the laws of thermodynamics was increasing. Bekenstein proposed that the entropy of a black hole was:

• Finite.

• Well defined.

• Proportional to the surface area of the black hole’s event horizon.

Jacob Bekenstein was able to combine his ideas with Stephen Hawking’s to give way to an exact equation that would calculate the entropy of a black hole. 

Black hole evaporation

The next logical step in our black hole investigation will take us to the proposal that: black holes radiate. It was 1973, and Stephen Hawking is taking a visit to Moscow, Russia. Hawking is going to be convinced by Yakov Zeldovich and Alexander Starobinsky, that, according to the Heisenberg uncertainty principle of quantum mechanics, black holes, that rotate, should emit a kind of radiation. The idea that Starobinsky and Zeldovich were attempting to convince Hawking of, was, that rotating black holes emit particles. Hawking, liked the idea in principle, however, didn’t completely agree with the mathematics on the emission itself. 

In fact, when Hawking ran the calculations himself, he found that even non-rotating black holes, should still continue to emit particles. The best part of his calculations, is that, the particles were being emitted in a way that did not violate the 2nd law of thermodynamics. The particles, themselves, that were being emitted, were a kind of radiation, that had a temperature that depended, on the mass of the black hole. Black holes with higher mass, would have radiation of lower temperature. According to quantum mechanics, there cannot be completely empty space. This follows from the uncertainty principle, since, all of the values of the fields in a given area of space were 0. This minimum amount of uncertainty that must exist (thus has to be greater than zero, as if it were zero in all fields, it would be in a state of perfect certainty) can be thought of as a quantum fluctuation. In other words, the vacuum of empty space, is not empty, thanks to quantum mechanics. A quantum fluctuation, would be a pair of particles, called virtual particles, one with positive and one with negative energy. This would be a particle and an antiparticle pair. They appear together, move apart, then come back together and annihilate with each other. The quantum fluctuations or pairs of virtual particles (call them what you will), will not live very long before they collide and annihilate. Virtual particles cannot be directly observed, however, their effects on ordinary particles can be measured to a pretty remarkable degree of accuracy. Now, the negative energy particle is doomed to fall into the black hole. It will be short lived. Since the virtual particle that fell into the black hole has negative energy, than the black hole will lose mass. However, what is possible (since the gravitational attraction of a black hole is so immense) is for a negative energy virtual particle to fall into a black hole, and become an ordinary particle. As for the positive energy virtual particle partner, it can either fall into the black hole, or, escape the black hole entirely. The positive energy virtual particle, after it departs from a negative energy virtual particle that has fallen into a black hole, as it propagates away from the black hole, would appear to have been emitted at the black hole’s event horizon. Thus, smaller black holes will be hotter, since, negative energy particles will have a shorter distance to travel, before they become a regular particle and be emitted from the black hole as radiation. The positive energy of the particles moving out of the black hole would be balanced out by the negative energy of the particles falling into the black hole. Also, since, thanks to e=mc^2, we know that mass and energy are equivalent, we know that the mass of the black hole is reduced when negative energy virtual particles move into the black hole. The black hole will lose mass, thus the size of the surface area of its event horizon will decrease, thus the entropy will decrease. However, the entropy of the emitted positive energy particles as radiation, will more than make up for this loss of entropy. Thus the 2nd law of thermodynamics, will never be violated.