The Universe as a Hologram

There were quite a lot of Multiverse concepts that we have covered so far. Some of them are really involved, but I hope that my explanations provided a way to conceptually grasp these concepts. There remain only a couple of models which we are not familiar with. Today we shall be talking about a model which might seem completely improbable and we have to remember that it is barely taken seriously even by cosmologists themselves. In fact, it is not about the Multiverse proposals, which we were concerned with in other articles, but about the possibility of alternative reality, which is why it was included in Brian Greene’s book “The Hidden Reality” as one of such possibilities.

First, let us have a look at what a black hole was considered to be after Karl Schwarzschild provided his solution to Einstein’s General Relativity equations. That solution showed that there should be extreme configurations of matter according to the mathematics, the configurations which we now call black holes. Schwarzschild was concerned with simple non-rotating systems which all would have a so-called Schwarzschild radius, such that if the system is squeezed to this radius it has to become a black hole (at that time there was not a term black hole, which was introduced by John Wheeler decades later, and such objects were called “invisible stars” since it was properly hypothesized that they don’t emit light). We are not going to dig in deep for the details of black holes, but if you are interested in this subject I suggest you read Peter Cooper’s article on this topic. What’s of relevance for us here is that collapse of a star does not stop when the star reaches its Schwarzschild radius. Instead, it continues to collapse up to the moment when all of its mass is compressed to a point with a vanishingly small size, which we now call a singularity. But when our star reached its Schwarzschild radius it produced a region of space where this radius became a point of no-return. Albeit all the matter of our star is sitting at an infinitesimal point of space, a region around it up to the point of no-return – which we now call an event horizon – is also a part of the black hole. We will need to recall this concept later; a black hole does not have a physical boundary, so you could easily fall into it without noticing anything until you’ve been spaghettified.

The reason why black holes are the main concept of this topic is that they were long thought to violate the 2^nd law of thermodynamics which has been the main physical notion explaining why we perceive time following strictly one direction – from past to future. You never expect spilled water to appear exactly at a certain point, or a hundred tossed coins all to fall down either to the obverse or to the reverse side, or that the milk added to your coffee will separate out and return into the bottle. A sequence followed by any of these cases has to do with the 2^nd law of thermodynamics which says that the entropy of any given system can in no way remain constant, it always increases.

You might be wondering what entropy is. It is quite an involved concept but we can say that entropy is a degree of disorder. Suppose that two neighbors – Steve and Artur – have a conversation. Steve, not being a shy person, asks Artur why he’s been to his room and made disorder out of everything. Artur argues that he hasn’t entered his neighbor’s room for quite a while. Then Steve asks Artur to enter his room to be convinced that it is in complete disorder. When Artur comes in he doesn’t notice anything wrong in the room. What are you talking about – Artur asks – everything looks pretty ordered as always in your room. Steve answered: take a look at the bookshelf, the books were always in an alphabetical order, but now they aren’t! The socks in my wardrobe were always in pairs but now they’ve been put there completely haphazardly. Euros, dollars and yens were separated in my table, but now they are all placed together. If it was not you then somebody has broken in here!

What this tells us is that if there is a highly ordered system, almost any event taking place within it will introduce some disorder and will easily be noticed. Indeed, if Steve’s room were in complete disorder he would never notice those changes, but since it was highly ordered the changes were easily noted. Any series of events taking place within a system tends to increase the disorder – or the value of its entropy.

Now we return to the question of black holes. Why were black holes considered to break down the 2^nd law of thermodynamics? As we have mentioned, all the matter in a black hole is concentrated at the singularity – a point with an infinitesimally small size and an infinite density. There is just no place in a black hole where the events introducing any disorder would take place. So if you take a black hole into account, it was thought not to increase the entropy within itself, breaking down the 2^nd law of thermodynamics.

There is also a so-called no-hair theorem which is saying that you have to know only three parameters to entirely describe a black hole. These parameters are: its mass, electric charge and angular momentum. If you know the values of these parameters you know everything about a given black hole, this theorem suggested.

But there is one of John Wheeler’s students named Jacob Bekenstein who wasn’t willing to accept this dogma. Bekenstein became one of the main contributors to black hole thermodynamics. He proposed that black holes should contain entropy which would always increase in correspondence with the 2^nd law of thermodynamics. He suggested this based on the concept according to which an event horizon of any black hole can only get larger with time. Say, you have a pair of black holes orbiting each other, whose orbits gradually decay due to the gravitational waves they must produce. They will eventually merge together and the radius of event horizon of the resulting black hole must exceed the sum of event horizons of two initial black holes. This concept was introduced by Stephen Hawking in 1970s, and Bekenstein saw a connection between this concept and the way in which the entropy of any given system can in no way decrease or stay constant.

However, there was a concept which seemed to rule out the idea of black hole entropy. To contain any entropy a system must be of non-zero temperature. By zero temperature we mean absolute zero, which is 0 degrees Kelvin, or minus 273.15 degrees Celsius. But any object whose temperature is above absolute zero must give off light. For example, you see the Sun and other stars in the sky because they shine. Stars you see are typically ranging from nearly 2000 up to tens of thousands degrees Kelvin. Planets have much lower temperatures and you have no chance to ever see a planet’s light with your eyes. Instead, you see the light reflecting off a planet. But this doesn’t mean they do not shine. They do, but their temperature is not enough to produce visible light. The light they produce is located at the infrared part of the spectrum. Similarly, your body emits infrared light as well because the temperature of your body allows this. Decrease a temperature further and your object will produce radio waves, but until it gets to absolute zero it does emit light. But as we know black holes do not emit any light, and this was the main reason for the conjecture that black holes have no entropy whatsoever.

But here is where one of the most astonishing works of Stephen Hawking, taking the question of black hole radiation into consideration, comes into play. Let us have a look at what Hawking radiation actually is. I have to say that the following argument is not exactly what Hawking calculated, but it is much simpler to grasp and is quite close to the initial idea. According to Heisenberg’s uncertainty principle the very fabric of space and time must possess some fluctuations making energy density vary. At the places where the density becomes higher, a pair of electron-positron particles suddenly born for an infinitesimally short period of time, meeting each other and annihilating whereafter. If one of these particles has positive energy, the other one must possess the same amount of negative energy due to the conservation of energy. And according to the laws of quantum mechanics there is no experiment, even in principle, that would be capable of defining whether the particle’s energy is positive or negative.

Hawking considered quantum fluctuations but not in empty space, as was previously done, but in a close vicinity to an event horizon. What Hawking had found shook physicists to the core. When a pair of particles emerge close to event horizon it is possible for one of them to fall down into the black hole and for the other to escape black hole’s grasp. While the particle with negative energy is falling into the black hole another one with positive energy is flying away. And what’s particularly relevant for us here is that the particle with negative energy for an observer outside a black hole becomes a particle with positive energy for an unfortunate observer inside. So in this case the laws of physics still prevent the detection of a particle with negative energy. This work has shown us that black holes in fact do have temperature and, as Hawking also calculated, their entropy must be of non-zero value. What’s important about it is that according to Hawking’s calculations this value is determined by the area of an event horizon and is equal to the number of units whose length is the Planck length or 10 to the negative 35 meters. This might seem completely confusing that the entropy of a 3-dimensional object can be measured by its 2-dimensional area instead of its volume, but this is where we can see the first connection with the main topic of this article, holographic principle.

Now we have to update our notion of entropy. As we previously saw, entropy is a degree of disorder within a physical system, or equivalently, a number of system’s microscopic components’ rearrangements which do not lead to any macroscopic changes of this particular system. For example, if a number of water molecules inside your glass switch their positions, the substance of your water remains the same, and there could be any number of such rearrangements, which means that the entropy within this system is high. This is saying that entropy is a measure of informational difference between the data that we possess (a system’s macroscopic properties which can be known to a high level of precision) and data that we do not possess (information hidden in system’s microscopic properties). Let’s consider a situation with 1000 coins. If they are placed on either heads or tails haphazardly, the system has high entropy. You will not notice if one coin is going to be turned over from head to tail. However, if all of them are placed on either heads or tails you will obviously notice such a change.

Now let’s see how we can define the value of entropy within our system. If you toss only two coins, both of them can fall down either on heads or on tails after been tossed. In this case you have 4 possible configurations of your system: head and head, head and tail, tail and head, tail and tail. The number of configurations is obtained by multiplication of two possible configurations for the first coin and two configurations for the second one. If you have 3 coins, this number will be 8 and so on. For our example with 1000 coins we have 2 to the one thousand configurations. The entropy value of our system is given by an inverse function of the exponential function given above, namely logarithm. I won’t bother you with the mathematics here but all we need to know is that the logarithm of our function 2^1000 is just equal to 1000, so in this sense logarithms allow us to work with far more manageable numbers.

Then it might be not completely clear how we can figure out the amount of hidden information. It can be thought of as number of questions that are needed to be answered in order to obtain this information. If we toss 2 coins, there are 2 questions to be answered to achieve the needed information. Head for the first coin? Yes. Head for the second coin? No. This is where the term bit comes from. It is an acronym of “binary digit” and means that each of 2 questions can be answered as either yes or no. In computer world it is expressed as 1 or 0 which can be thought of as digital expressions for true and false. So for our system with 1000 coins there are 1000 questions needed to be answered for obtaining information about the system. In this sense entropy is the measurement of hidden information.

Let’s now return to the question of black holes. As Hawking established, the entropy value of a given black hole can be measured by the number of units whose area is equal to 10 to the negative 66 power. Given the information about questions needed to be answered as 1 or 0 we can imagine the picture of a black hole given below. The amount of entropy here is given by event horizon separated into Planck segments.

But a very significant question arises here: if the value of entropy is determined by event horizon’s area, where is this hidden information actually located? Is an event horizon just a convenient ‘instrument’ for measuring the value of entropy or does it actually become the storage place of our hidden information? Here we have to consider what is known as the firewall paradox. As we have seen, the boundary of a black hole, event horizon, should be a region where myriads of particles emerge, being pushed out thereafter. Although these particles lose most of their energy overcoming black hole’s gravitational pull, the closer we get to event horizon the higher energy those particles must possess. This means that an unfortunate person falling down into the black hole should be burned to death at the moment when he gets quite close to the event horizon. But General Relativity clearly says that you can’t notice your passing through an event horizon since you are free falling in this case and such a free fall is completely equivalent to floating in empty space. So we have come to a strange conclusion: if you are falling into a black hole you do not notice your passing through its event horizon and your feelings start noticing something unusual only when you approach singularity. On the other hand, if you are observing someone falling down into a black hole you clearly see that he is being exposed to radiation until he is burned to death. While this might seem nonsensical since you can’t be both alive and dead at the same time, this situation is not a usual one. Photons irradiating the body of our astronaut need time to reach our observer, and if you figure it out, the observer would not have enough time after that to jump into the black hole and to notify the astronaut that he is dead. Similarly, the astronaut will have no chance to notify the observer that he passed through the event horizon alive. So the answer to the question as to whose point of view is actually correct would be both, according to Leonard Susskind and some other physicists.What we’ve got here is that event horizon is to be considered as actual storage of information for an external observer.

Now, when we are familiar with entropy and its connection with black holes we can finally be concerned with the work of an Argentine physicist Juan Maldacena who is now professor of physics at the institute for Advanced Study in Princeton, New Jersey. This is where String Theory comes into play. Some aspects of this theory have been under consideration in previous topics so all the interested readers may have a look there. The details of Maldacena’s work are very complicated, but I will try to focus on aspects which are quite comprehensible. Firstly, Maldacena separated his mathematically constructed ‘Universe’ into two parts. One of them is taking only open strings into consideration, thus leaving gravitational interactions away. The other is concerned with closed strings which are capable of travelling through the bulk. That can be imagined as follows: you have a number of branes which are so close to one another that they form a monolithic plate. The larger the number of branes the higher gravitational force they possess. For an external observer this plate eventually becomes a black hole with some distinct properties, so we call it a black brane.

Recall that there are two types of strings in the String theory: open ones that are stuck to a brane being not capable to escape, and closed ones which have looped shape and can travel from one brane to another through hyperspace. And here is the surprise, when you examine the part with only open strings placed on the brane’s boundary you find that this description is equivalent to quantum field theory developed in the mid of XX century and described in 4-dimensional space-time. But when you examine the same system inside the bulk, gravitation of the black brane curves the space-time within this system making it 10-dimensional. The particular relevance of this result is that even though these two parts seem to be completely different, they describe exact same physics just from different points of view.

And this is saying that our quantum field theory, widely used for three non-gravitational forces, when applied to physical processes taking place at 2-dimensional boundary of a system describes exactly the same physics as the String theory applied to 10-dimensional Universe with gravity. This work has become one of the most important in past few decades and it was improved later by Edward Witten and other String theorists who developed a clear mathematical way of transmitting a result obtained in one part of the theory into the other.

This implies a striking implication. If two theories depict exactly the same picture of processes taking place on the boundary of a given system and processes occurring in the volume that this boundary encircles, it might be the case that those processes on the boundary trigger the ones inside. That is, if we imagine our Universe as a bubble, it has a 2-dimensional boundary and processes on that boundary lead to everything we see inside. This remains a highly controversial model but it is certainly one of the most astonishing findings in the String theory.

No one claims that our Universe must work as a hologram but this certainly remains one of the possibilities provided by the mathematics. Mathematics certainly remains our gateway to reality.

Thank you.