Authors: Arpan Dey and Sanchari Sen

"How wonderful that we have met with a paradox. Now we have some hope of making progress." - Niels Bohr

### Abstract

This article is about the black hole information paradox, a fundamental paradox in theoretical physics. In this article, we discuss entropy, information and the information paradox itself, as well as possible solutions of the paradox and the holographic principle. The information paradox is significant because it arises out of the "combination" of general relativity and quantum mechanics – the two pillars of modern physics. Further research on the information paradox might reveal the link between these two theories and might lead to a theory of everything – a single theoretical framework that would be valid for all possible interactions in this universe. Understanding this paradox would be a crucial step toward unifying quantum field theory and general relativity. And although there have been some great insights into the subject, the paradox has not been resolved completely.

### Entropy and the Second Law of Thermodynamics

The concept of entropy is one of the most important concepts in physics. Simply put, the more the disorder, the more the entropy. As per the second law of thermodynamics, the entropy of a closed system must always either remain the same or increase. Any machine, for instance, must always waste some energy on its functioning and become less efficient over time. The idea of a perpetual motion machine is not just an engineering problem – it is prohibited by the laws of physics. There is a fundamental limitation to the efficiency of anything. No process is possible whose sole result is the transfer of heat from a colder object to a hotter object. For this to occur, some external work must be done on the system. (This is obvious, since everything tends to achieve an equilibrium state. Heat always flows from a hotter object to a colder object.) This second law of thermodynamics is perhaps the most "sacred" law of physics. To quote Arthur Eddington:

If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations — then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.

We've found no violation of the second law. Nor do we expect to find one.

Figure 1: Entropy increases as a system evolves from an ordered state to a disordered state.

### What is Entropy, Exactly?

Entropy can be defined as the number of ways a particular state can be achieved. More precisely, entropy is the number of ways a state can be achieved, by taking into account spatial configurations, velocity configurations, and energy distributions. In our universe, it is clear that entropy increases with time. It apparently seems that the entropy increases in this universe because the universe is expanding and more and more spatial configurations become possible. However, entropy doesn’t depend solely on the number of different spatial configurations possible. Even if the universe were contracting, the overall entropy would have increased due to the fact that every object in the universe will continue emitting photons of light, which contain energy. These photons travel throughout the universe and disperse its energy everywhere. The number of possible energy distributions go up, and so does the entropy.

It seems reasonable to assume that entropy is a measure of disorder but that is not exactly so. Some people will tell you if you have an untidy room, it's in a high entropy state, whereas a neat and clean room will have a low entropy. That's true, but that's not a very accurate description of the concept of entropy. The following example might be more helpful. If we have five differently colored balls and two jars, then the lowest entropy state would be achieved if we keep all five balls in one jar and thus, no balls in the other. This state is not an equilibrium configuration, obviously, for the concentration of the balls is higher on one side. (An equilibrium state can’t be achieved in this case, since that would require each jar to have 2.5 balls in it!) If we decide to keep two balls in one jar and three in the other, by taking into account the five different colors, there are many different ways we could achieve this state. Thus, in this case, the entropy of the system is more than in the previous case, when there were only two possible ways to achieve our state (that is, either put all the balls in the first jar or in the second jar, while keeping the other jar empty). And in this case, a near-equilibrium is attained (which means this is a more stable state). Everything in this universe tends to gain stability. Entropy is increasing in our universe is itself a testimony to the fact that we started off in an orderly state, that is, at low entropy. If we go by the Big Bang theory, at the very beginning, a tiny fluctuation in the vacuum gave rise to an infinitely dense point that began to expand to form this universe. The Big Bang state must have been symmetrical. It was just a point. And you can’t define asymmetry for a point. Yet that stage was clearly not in equilibrium with the “nothingness” (that is, vacuum). Thus, entropy was low then, for it was a fundamental state. (There are not many ways you can achieve a fundamental state.) Over time, with its expansion, the universe has advanced more toward equilibrium, and has lost symmetry in the process. The basic idea is that the four fundamental forces were actually unified into a single "superforce" just after the Big Bang. The temperature was extremely high then. As the universe expanded with time, everything went away from each other and the universe cooled down. Symmetry was broken and the "superforce" gradually broke down into the four forces we see today. Breaking of symmetry (which introduces differences in the system) is essential for stability. Water is, by itself, symmetric (there is no particular asymmetry in it, it just assumes the shape of its container). No matter from what angle you view the water, it appears the same. Now cool it. You get ice, which is not perfectly symmetric. Ice is a crystal, and it is not symmetric. Ice will not look the same from different angles. We know that hotter things have more energy, and are less stable. Everything wants to lose energy and gain stability. For that, it needs to cool down, and break symmetry in the process. Symmetry only exists at high energies. Let’s return to our original discussion of entropy. As of today, the most accepted view is that the universe is expanding continuously and its entropy is increasing. More the expansion, increasing is the disorder, the information contained, and the number of ways a particular state can be achieved. During this expansion, some particles may come closer and will have gravitational attraction between them, giving rise to stable behavior. Galaxies are formed in those regions where the regional density is high. Entropy may not increase at that particular portion of space, but there is an increase in entropy in its surroundings so overall entropy still increases.

### What is Information?

What exactly do we mean by information here? We can think of information in terms of reduction of uncertainty. If someone is about to send us a message, we are uncertain about what the message would say. But once we read it, the information in the message reduces our uncertainty. The information (I) in the message can be defined as the logarithm, to the base two, of the total number of messages that might possibly have been sent (M). The logarithm is taken to the base two, since according to the model we're discussing here, one gets information by asking questions that can have two answers: true or false (yes or no). If the answer to a question can be either true or false, then the answer can be assumed to contain one bit of information. For answers that can’t be defined using just true and false, we may assume that the information associated with it is just the number of questions one needs to ask whose answer can be either true or false, to have enough data to make a successful guess at the answer. For example, the answer to "What is your name?" can't be true or false. In such cases, we ask questions whose answers are yes/no, like "Does your name start with A?," "Is your name five-lettered?" etc., and information, as we just discussed, is the number of such questions we need to ask so that we can narrow down the possibilities and make a successful guess at the answer (in this case, the name).

The more different the objects you are trying to describe are, the more the amount of information you need to completely describe the objects. For example, suppose A and B are similar in all respects except one. Then, you need to only describe either A or B and mention this one difference. If A and B are different in two aspects, you need more information to describe A and B, and so on. In some sense, information is responsible for the knowledge of the existence of the different objects in the universe. It is the information of the arrangement of electrons and number of protons and neutrons in an atom that defines whether the atom is a hydrogen atom or a carbon atom (etc.). Different arrangements of these atoms will give rise to different forms of the same element. For instance, carbon can exist in two main forms: graphite or diamond. 11111111 and 0000000 contain less information than 01100011111111011100000000101. You can describe 11111111 as “eight ones.” This is not possible for 01100011111111011100000000101. Now consider another aspect of information. Redundancy. “If you can read this message then you are just like everyone else” is just “If u cn rd ths msg u r jst lke vryne lse.” It is sufficient to convey the message, grammatical formalities aside. Thus, English (like most other languages) is a redundant language.

Entropy and information are not independent concepts: they are highly interlinked. According to the information theory of entropy, entropy is the measure of the average amount of information needed to describe an event, while considering all possible outcomes. The more the entropy, the more the information you would need to describe the system, since it will be a disordered and random system.

### The Black Hole Information Paradox

According to quantum field theory, energy can be temporarily "borrowed" from a particle. According to Heisenberg’s uncertainty principle, the errors or uncertainties in position and momentum OR energy and time must always be greater than or equal to a constant. This means we can never calculate a particle’s position and momentum simultaneously with perfect accuracy (and the same goes for energy and time). As per Heisenberg’s uncertainty principle, the greater the amount of energy you "borrow," the sooner you must return it. The exchange of photons gives rise to electromagnetic forces. Virtual photons can pop out of nowhere around an electron, by "borrowing" some of the electron’s energy. If there is another electron near the virtual photon, it will absorb the photon. Thus, essentially, some energy and momentum is exchanged between the electrons, causing them to repel each other, since the second electron, on gaining energy, will move away from the first one. Note that in the above case, the exchange of the photon gave rise to the force of repulsion between the two electrons. And this is why like charges repel each other. A photon can also materialize into an electron and its antiparticle, the positron. This process is called pair production. Here, electromagnetic energy is converted into matter.

Vacuum is not an empty space. The amount of energy is randomly and temporarily changing at every point in space. Or in other words, what we call vacuum is teeming with particle-antiparticle pairs (called virtual particles) that pop up, and after a very short amount of time, annihilate each other and just vanish. These particle-antiparticle pairs emerge spontaneously in vacuum, and this phenomenon is called vacuum fluctuation. The quantum fields (there is an underlying field for every particle) are randomly fluctuating, and this is what gives rise to these virtual particles. In other words, particles are nothing but vibrations in a field. And also, the existence of virtual particles, although not directly perceptible, gives rise to certain interesting effects (like the Casimir effect), and such effects have been observed in the laboratory.

Figure 2: Virtual particles in vacuum.

So particle-antiparticle pairs can pop up in vacuum. However, they must soon annihilate each other. Now consider that a particle A and its antiparticle B emerges near the event horizon (roughly speaking, the boundary of the black hole) of a black hole in such a manner that A falls inside a black hole, while B remains outside the black hole. Now, before the particles have a chance to annihilate each other, A is sucked in by the black hole’s gravitational pull. But then, who annihilates B? We lose all information about A, and the entropy of the universe decreases, violating the second law of thermodynamics. Jacob Bekenstein showed that the event horizon increases in area whenever an object falls into the black hole. And this area represents the entropy of the black hole. And since black holes have entropy, Hawking concluded that black holes emit radiation (Hawking radiation). After a very long amount of time, the black hole evaporates away completely.

Figure 3: Pair production at the event horizon of a black hole.

So, information, in some sense, helps us tell things apart. On the other hand, black holes suck in things and crush them into the singularity. Or in other words, the difference between a pen and a pencil, as they fall into a black hole, is lost in the sense that inside a black hole, we have no means to differentiate a pen and a pencil. It’s impossible for us to obtain information about anything that has crossed the event horizon and moved into the black hole. As far as we know, there is no way to retrieve information from a black hole. If you fall through a black hole, your existence is simply "deleted." This means that the information associated with you is lost. (In fact, over time, when every star in our universe will either become a black hole, or be sucked into one (and all planets will be destroyed), if the information associated with objects that fall into black holes is indeed lost, all the information associated with this universe will be deleted entirely.) But what exactly is paradoxical about all these? Well, information can’t be destroyed. This is a fundamental law. Every object in the universe is composed of particles with unique quantum properties. And no matter how much we try to destroy these objects, the quantum information related to them can never be deleted. In theory, it is even possible to recreate the object. (Like in quantum teleportation, you can, in theory, transfer all the quantum information about an object to a different place, where it can be reassembled.) But we can't retrieve information from a black hole. However, the information must be present somewhere. Oh well, maybe we can’t perceive this information. But it exists inside the black hole alright. After all, no one has ever been inside a black hole. But once we learn that even black holes are not permanent, they radiate energy, the possibility of the information resting peacefully in there gets ruled out, too. So, is the information preserved in the Hawking radiation? We don’t have sufficient evidence in support of the theory that the Hawking radiation contains the information about the objects that fell into the black hole. No matter what goes inside the black hole, you can’t identify it just by looking at the Hawking radiation. And after the black hole evaporates away completely, where does the information go? This is the information paradox.

### Entanglement Entropy

Let’s first understand the problem in a more rigorous manner. When an object falls into a black hole, it leaves some traces. Apparently it seems that black holes violate the second law of thermodynamics. When an object disappears into a black hole, its entropy is gone. Then Stephen Hawking and Demetrious Christodoulou proved that in various processes, such as black hole mergers, the total area of the event horizons never decreases. Then Jacob Bekenstein proposed that a black hole has entropy proportional to the area of its horizon. According to Bekenstein, when matter falls into a black hole, the lost entropy is taken care of by the increase in black hole entropy. These were very crucial steps that helped physicists achieve a better understanding of the black hole information paradox.

Then Don Page took a very important step forward. He claimed that, according to quantum entanglement, there is a link between the emitted radiation and its origin in the black hole. If you just look at the radiation, it will appear random. But when you consider the system in its entirety (the black hole plus the emitted radiation), maybe you will be able to retrieve the information. Page calculated the entanglement entropy (the entanglement between the black hole and the radiation). At the beginning, the entanglement entropy is zero, since there is no radiation to get entangled with. And at the end, if information is preserved, the entanglement entropy should be zero again, since there is no longer a black hole.

Figure 4: Page’s curve, according to which the entanglement entropy increases up to a point, after which it starts decreasing due to the release of information inside the black hole. Entanglement entropy measures the amount of missing information.

### The Holographic Principle

Gerard ’tHooft demonstrated that particles falling into black holes cause gravitational deformation and "bumps" on the event horizon. This could contain the information of the particles. This means, essentially, the black hole can be treated as a hologram. This is because the information of the three dimensional object falling into the black hole is stored on a two dimensional surface: the event horizon.

Figure 5: A two dimensional surface can contain all the information of a three dimensional world.

According to modern physicists, the universe can also be a hologram. We can consider gravity as nothing but a projection of quantum mechanics in a higher dimension. This is called the holographic principle (first proposed by Juan Maldacena). Holography can explain and link two entirely different kinds of theories in physics. If you work out the details of any one of these theories (the easy-to-handle one), you automatically know everything about the other theory. Holographic duality is also referred to as AdS/CFT correspondence, where AdS refers to Anti de-Sitter spaces (AdS are spaces with negative curvature), while CFT refers to Conformal Field Theory. All this is technical, but the concept is really simple. Imagine a sphere. CFT is related to the boundary, while the AdS space sits inside the sphere. Everything in the AdS space has a counterpart in the CFT boundary. It is crucial to understand that a hologram is two dimensional, but it can contain all the information about all three dimensions of the object it represents. A three dimensional universe contains black holes and stuff governed solely by gravity, whereas the two dimensional boundary of this three dimensional universe contains ordinary particles governed solely by standard quantum field theory. This means it is possible to relate gravity to a quantum field theory which has no gravity, and this may prove to be of great help when trying to reconcile general relativity and quantum field theory. The AdS/CFT correspondence can also offer a possible solution to the black hole information paradox. The information is contained in the two dimensional event horizon of the black hole. There is no gravity in the two-dimensional description of the black hole, so information is preserved in the black hole.

Figure 6: According to the holographic principle, all information is always stored in the boundary, and the entanglement entropy remains constant.

### Other Solutions

When things are massive as well as tiny (consider the singularity in a black hole), both general relativity and quantum mechanics must be taken into account, but they are incompatible. If we try to unify them, we get infinities and other nonsensical results like a probability greater than one. It is clear that our present understanding of the world is incomplete. Spacetime is supposed to be smooth, according to general relativity. But if we zoom in, really deep, we will see what has been called the quantum foam. We will reach a point where the familiar notions of space and time simply don’t work.

According to string theory, every particle is made up of ‘strings,’ and the different modes of vibrations of these strings correspond to the different particles. Roughly speaking, there is a smallest possible distance, distances smaller than which can’t be probed and don’t exist (in some sense). Strings are not point particles, they are smeared out, and by putting a lower limit to the distances that can be probed, we can avoid the conflict between general relativity and quantum mechanics, which arises when we magnify spacetime to very small scales. String theory offers some possible solutions to the information paradox.

A fuzzball is considered to be a true quantum description of a black hole. Fuzzballs aren’t empty pits (like traditional black holes). Rather, they are packed full of strings. They have a surface just like any other star or planet. And they emit heat in the form of radiation. Now to consider the idea of fuzzballs as a potential solution, one needs to give up the idea of event horizons and singularities - quite a big leap of faith. There cannot be a vacuum at the horizon, or it’ll lead back to the information loss problem through Hawking radiation. Instead, there are hairy structures in the horizon that hold on to the information as they fall inside the black hole. So no information is lost beyond the horizon because there is no “event horizon”.

Figure 7: Fuzzballs, packed full of strings, are essentially the quantum mechanical description of a black hole.

Another probable solution to this paradox can be achieved by bringing wormholes into the picture. Wormholes, also called Einstein-Rosen bridges, are hypothetical structures connecting different points in spacetime. In 1992, physicist Don Page, a former graduate student of Hawking, started by looking at quantum entanglement, which is when distant particles have their fates linked. Page measured the amount of entanglement by calculating the "entanglement entropy" (a measure of the amount of information contained in the entangled Hawking radiation). A wormhole is a connection between the interior of the black hole and the radiation outside, like a bridge. A second surface, the boundary beyond which nothing can escape, appears inside the event horizon of a black hole. Threads from a wormhole connect that surface to the outside world, entangling information between the interior of the black hole and the radiation leaks at its edges.

### Conclusion

There have been some great insights into the subject of black hole information paradox recently, and it is clear that the information of an object falling into a black hole is not lost. But the information paradox has not been resolved completely.

### References

Physics Videos by Eugene Khutoryansky, YouTube. Entropy is not disorder: micro-state vs macro-state. https://youtu.be/vX_WLrcgikc

Physics Videos by Eugene Khutoryansky, YouTube. Does gravity decrease entropy?. https://youtu.be/gJivcAAbE0A

Jon Cartwright. PhysicsWorld. https://physicsworld.com/a/information-paradox-simplified/

Jacob Bekenstein. Scientific American. https://www.scientificamerican.com/article/information-in-the-holographic-univ/

George Musser. Quanta Magazine. https://www.quantamagazine.org/the-most-famous-paradox-in-physics-nears-its-end-20201029

Jennifer Ouellette. Quanta Magazine https://www.quantamagazine.org/how-fuzzballs-solve-the-black-hole-firewall-paradox-20150623/

T. Padmanabhan. Combining general relativity and quantum theory: points of conflict and contact. https://cds.cern.ch/record/522187/files/0110046.pdf

Paul Sutter. A spiderweb of wormholes could solve a fundamental paradox first proposed by Stephen Hawking. https://www.livescience.com/black-hole-paradox-solution