The Black Hole Information Paradox and the emergence of Black Hole Thermodynamics

Can our Universe possibly be a projection of a 2 dimensional surface? - Like a hologram. What if there is a limit to the amount of information that can be stored in a finite space? Such situations might actually be true as we shall explore in this article, the basic concepts of Black Hole Thermodynamics and the Black Hole Information Paradox. Already equipped with our previous knowledge on Black Holes, we can now study some of the fundamental problems that they pose to our current theoretical models. The solution to those problems then include some of the most strange possibilities regarding our Universe like the "Holographic Principle" and "Bekenstein Bound" whose hint I gave in the first two sentences of this article. 

              A quick recap from the previous article - The definition of Black Hole was given as that point in space where the gravitational attraction is so strong that not even light can escape from it. This point of no return is actually a sphere surrounding the central singularity and is called as the Event Horizon. If you read the previous blog title carefully enough, I included the term - "Recycle Bins of the Universe". However, I left you with no explanation of why I called them that way. Rather, the reader might have pictured them as just "bins" of the Universe. They just suck in everything and anything that falls into it is lost forever or in other words "deleted". This implies that anything which falls into the Black Hole is lost from us forever, never to return. One can also state that any information that goes into the Black Hole is deleted and cannot be retrieved. A law in Quantum Mechanics states that Quantum Information can never be destroyed (somewhat analogous to the law of conservation of energy). Initially, you might think this is the Black Hole Information Paradox we are talking about. But, the answer is no. The fact that any information which goes into the Black Hole is lost from "us", does not mean that it is destroyed. It still resides into the Black Hole, thus preserving the law. The real problem arises when we consider a popular effect known to be associated with  Black Holes - "Hawking Radiation", proposed by Stephen Hawking. He studied the mathematics of Quantum Field Theory in presence of a Black Hole and discovered that due to quantum effects near the Event Horizon, a Black Hole actually radiates energy in the form of virtual particles or photons. Hence, it turns out that something can actually escape from a Black Hole's event horizon. This radiation causes the Black Holes to lose mass and actually increases their temperature. Thus, it is also called as "Black Hole Evaporation". After trillions of years, they completely radiate away their mass and die out in a massive explosion, thus leaving no clue of their existence or what went into them during their lifetime. The radiation from the Black Hole is completely random. This right here is a contradiction in the conservation of information because now the information is  "deleted" from the Universe in a true sense.

              Hence, there was a contradiction in the two results given by the same set of laws. The conservation of information (or Unitary Law) given by Quantum Mechanics and the Hawking Radiation derived from Quantum Field Theory didn't agree with each other. In such cases, there are three possible ways to resolve the contradiction, either to abandon the pre-existing results or modifying the theory to remove the conflict or finally introducing a new theory to explain the results.  The Black Hole Information Paradox arose from the most fundamental theories of Quantum Mechanics and Quantum Field Theory, which are rigorously tested and are proven to hold true. Thus, the first option of discarding these theories didn't seem viable and modifying them involved many complications and unnecessary assumptions. Physicists were left with no option but to formulate a mechanism to somehow preserve the lost information. One of these mechanism was quite bizarre and resulted from the Einstein - Cartan Theory and seeks the help of a famous hypothetical concept of "wormhole" and "white-hole". It states that the information which falls into a Black Hole doesn't get eternally trapped inside. Rather, it travels through a wormhole and enters an entirely different Universe from a "White Hole". This Universe is inaccessible to us, but the information does remain preserved. This solution was strongly supported by Stephen Hawking. On the other hand, theoretical physicist John Preskill believed that the information must escape the Black Hole. This led the physicist Gerard 't Hooft to formulate a mechanism by which information which falls into a Black Hole does not enter the singularity but remains on the surface area of Event Horizon. This information distorts the Event Horizon and can possibly leave its imprint on the Hawking Radiation which escapes from there.

              There were many such theories proposed but they included greater assumptions without much proof. The most genius solution for the paradox was recognized by Jacob Bekenstein. The proposed solution not only aimed to solve the paradox but it also introduced numerous other predictions mainly the link between Thermodynamics and Black Hole and the Holographic Principle. In the previous blog article, the thermodynamic entropy was introduced as a measure of disorder or randomness in a system. However, there exists another way by which entropy can be expressed. It is proportional to the amount of information present in a system. Bekenstein found a strange correlation (possibly an analogy) between the surface area of the event horizon of a black hole and the thermodynamic entropy of any system. It is well  known that the second law of thermodynamics dictates that total entropy (S)  of a closed system should always increase (or remain same) with time and never decrease. Furthermore, Stephen Hawking proved that the surface area of Event Horizon should always increase (or remain same) with time and infalling matter and never decrease. The equation for change in entropy of a thermodynamic system is given as:

                                                        dS = dU / T  

where dS is change in entropy, dU is change in internal energy of the system and T is the temperature. On the other hand, the equation for change in the surface of area of event horizon is given as :

                                                                  dA = dM / Ɵ

where dA is change in surface area of event horizon, dM is change in mass of the Black Hole (or infalling mass) and Ɵ is the thermodynamic entropy of infalling matter. This close resemblance between the surface area of event horizon and entropy could not be a coincidence. The link between the two was further confirmed by Stephen Hawking's paper on the "Hawking Radiation" which was published a year after Bekenstein's. The process of a black hole emitting radiation suggested that Black Holes have a certain temperature and thus a thermodynamic entropy, as opposed to the "No-Hair Theorem" which suggested that they exhibit only three observable characteristics namely - Mass, Charge and Angular Momentum. Eventually, Bekenstein showed mathematically that the entropy of a Black Hole is proportional to the surface area of its event horizon and thus originated the beautiful subject of Black Hole Thermodynamics. But Bekenstein didn't halted here, he knew from the Information Theory that the entropy is also directly proportional to the amount of information. He mathematically constructed an imaginary Black Hole from idealized elementary particles, each possessing one "bit" of information. From the previous correlation between entropy and surface area, he deduced that information falling into Black Hole remains on the surface area of a Black Hole. Precisely, it was proved that the surface area stores only one "bit" of information per "planck area" of the event horizon. Planck area is the smallest possible surface area of any physical space. In other words, the infalling information (or entropy) was stored in one planck area of the event horizon. It was as if the surface area of event horizon was like a television screen divided into numerous pixels, each storing a bit of information. The result can be generalized to any finite physical space. This limit of information which can be stored in a volume of space is known as "Bekenstein Bound".

Entropy stored on surface area of event horizon.

              

              Gerard 't Hooft discovered a strange implication from Bekenstein's work. Logically, one would expect that the amount of information which can be stored in a finite space should depend on the volume of that space. We measure quantities like milk in a container, air in a room, or water in a tank in terms of volume (litres, cubic cm, etc.). But from Bekenstein's proposal it was shown that the amount of information depends on the surface area of the space. It was as if the information which we perceive in three-dimensional space is actually a projection of information stored on a two dimensional surface. This idea was picked up by Prof. Leonard Susskind and further interpreted using the String Theory, which led to "The Holographic Principle". It stated our entire Universe is a 3-D projection of a 2-D surface, like a Hologram. The idea profoundly described the interactions of elementary particles in 3-D space in terms of a 2-D space without introducing any of the fundamental forces. This actually solved the problem of Quantum Gravity and the theory marked its name amongst many other contending theories, which we shall explore in a separate post.

An Illustration of our Universe as a Hologram.

              

              The entirety of this blog article may seem like a fragile framework constructed from mere theories with no practical base. However, the theories themselves are results of two most foundational and experimentally tested pillars of Physics - General Relativity and Quantum Physics. We have seen how far one can ponder using the powerful tools of mathematics and theoretical physics, and predict the properties of an object which we can only see as a blurry image. The theoretical idea of a Black Hole wasn't confirmed until centuries later, we captured its image. Similarly, the solutions to Black Hole Information Paradox and Black Hole Thermodynamics shall remain just theories until time yields us with the correct picture. It may happen that neither of the above theories are true. Nevertheless, the elegant theoretical adventure about solving the paradox and obtaining an unexpected link between Black Hole and Thermodynamics in the process will always remain imprinted in the history of theoretical astrophysics.

- Thank You!