Do Black holes obey Thermodynamic Laws? - Hawking's Area Theorem

 ** Review of an extract from Hawking's phenomenal book, 'A Brief History of Time' **

Black Hole Thermodynamics: Explained

How is entropy related to the event horizon of a black hole?

The remarkable discovery made by Hawking and Bekenstein in the early 1970s, that black holes do abide by the laws of thermodynamics, was the closest to combining the two giants in physics - general relativity and quantum mechanics. This notion of black holes showing thermodynamically similar properties was one of the most powerful clues that gave the idea of Hawking radiation and the Black Hole Information Paradox, both brought up by Hawking himself.

Entropy and Information

Before directly jumping into black hole thermodynamics, we need to understand the definition and features of an unusual physical property, called entropy. What is entropy? Or what do we mean by the entropy of a system?

In simple words, entropy is the measure of disorder or chaos in a system. It has an alternate definition too - Entropy can be thought of as the amount of information, which a system possesses.

 For example, imagine a piece of solid ice cube, kept in a freezer. If you take its molecular structure you can see that all of its molecules are tightly packed in an orderly manner (uniformly distributed). We can say that the information the ice cube possesses will only give an idea of the structure or pattern in which the molecules are arranged, and maybe the molecular composition of each molecule.

Variation of entropy as time goes on

But suppose you take that ice cube out from the freezer and place it at room temperature. The tight molecular structure begins to loosen up as it adopts its liquid state. And as time goes on, the molecules get more and more loosened up and adopt a more disoriented or non-uniform state. 

Now, in the first case, when the molecules were more uniform and ordered, we say that the ice cube has less entropy (less amount of disorder or chaos). But in the second case, when the ice cube melts to form liquid water, the molecules become loose and more non-uniform, which means that now the system has more entropy (increased disorder) than in the first state. In other words, since all the particles are in any random position and all of the particles are in unique states, there is more information within the system, and more information means more entropy. 

In another case, if we consider a box filled with gas molecules, as time goes on, the position of the gas molecules is definitely going to be more spread out and uneven throughout the box, to a state of increased entropy.

In all these cases, we see that the entropy (or information) of a system always stays the same or increases, but it'll never decrease. And that's exactly what the second law of thermodynamics states - The entropy of a system will never decrease or go down. 

And since this is a universal law, it should be applicable everywhere in the universe, even in the case of black holes!! But do they?

What is entropy in the case of black holes?

Now that we know how the entropy of a system varies in time, let's take a look at the event horizon of a black hole, where matter is being torn apart into ionic particles under the extreme gravitational pull of the black hole. 

Image by David  Mark- Pixabay

But suppose a spaceship is falling into the black hole, crossing the event horizon. We know that the spaceship would be gone out of sight forever. We also know that according to thermodynamic notions, the spaceship would have a specific amount of entropy (or information) within itself, which is supposed to either stay the same or increase according to the 2nd law. However, as soon as the spaceship crosses the event horizon, the entropy associated with it would also be gone forever. This means that the object went from having high entropy to having no entropy at all (by disappearing into the black hole), thus violating the 2nd law of thermodynamics.

Hawking's Area Theorem

By Gifer

But this inconsistency was solved when Bekenstein argued that the area of the event horizon of a black hole increases as additional matter falls into it. Also, if two black holes collided to form a new black hole, the area of the event horizon of the new black hole would be larger than the sum of the parent black hole's event horizon area. Hawking noted that this property of black holes showed the area of the event horizon will always either stay the same or increase, but it can never decrease! 

This proved that entropy in classical thermodynamics and are of the event horizon of a black hole showed very similar features to each other, and concluded that entropy of the infalling matter is not lost - rather it is stored on the event horizon of the black hole. As more matter and radiation falls into the black hole, the area of the event horizon increases. This is known as the Area Theorem. 

This also gave rise to one of the sought to be solutions to the information paradox, which is known as the holographic principle.

Right after the discovery of the Area Theorem, Hawking doubted whether black holes could emit radiation, as it partially agreed with thermodynamics. This was when Hawking Radiation was discovered and proved that not only did black holes emit radiation, but it also has a temperature, which is inversely proportional to their mass!! 

This marked the huge discovery of Black Hole Thermodynamics, a milestone in the field of black hole physics because of its accurate consistency.

It was from this idea that we understood that black hole properties can be determined by its 3 features - its mass, charge and spin.

This series of discoveries were made by Hawking when his research was mainly focused on whether the universe had a Big-Bang singularity at the beginning, which was when he turned his research towards black holes!

To know more about entropy and black holes,  watch the video by Prof. Brian Greene