What are Hawking-type radiation?

Hawking radiation is very often talked about, and many times heuristic explanations are given that although they capture part of the movie, they do not tell the whole story. I propose a series of posts where we will delve into the world of Hawking radiation and its equivalents in different fields. Our walk will go through the following corners:

1.- Discussion of the quantum vacuum. This is the essential element in the whole discussion about Hawking radiation, so we must dedicate some time to it.
2.- Creation of particles in “normal” situations. At this point we will stop to explain that there are Hawking-type effects due to accelerated observers in a spacetime where we have turned off gravity, the flat spacetime of special relativity.

3.-Hawking radiation itself. Yes, in a series of entries on Hawking radiation, it seems appropriate to talk about it. In this entry we will give and justify, with some formulas if necessary, the derivation of Hawking radiation and the appearance of positive energy and negative energy fluxes. Yes, I said negative energy, let's see how that eats. Of course, since we are disclosing, we will explain the famous disclosing image of the creation of particle/antiparticle pairs, where one falls in and the other comes out of the black hole. We will see that the story we are usually told is not bad but it is not the whole story. We will also discuss the essential elements of Hawking radiation and we will see with surprise that gravity plays little role, gravity is only a motivation, but it is not the essential motive.
4.-Hawking radiation in the cosmological context. Historically, Hawking radiation was introduced in an expanding universe in which particles were created. Well, we will discuss this, especially the contributions of Leonard Parker.
5. Finally, we will see the sense of the proposals to simulate Hawking radiation in the laboratory. The analog models of black holes are simple to understand and very instructive.

The main objective is to clarify many concepts, and some misunderstandings, about this very interesting phenomenon. I hope that there will be an interesting discussion on this subject since there are many members in this house who are very well prepared to contribute interesting points of view.

How about starting with the quantum vacuum?

Quantum mechanics tells us that the systems it studies can have different states and that in many circumstances these states are organized in discrete energy levels. Perhaps that is why we associate quantum with discrete when this association is not entirely accurate. In reality, a free electron, to give an example, an electron that does not interact with anything can have any energy, there is no discretization of the energy that is worth. However, if the same electron, meets a proton and forms a hydrogen atom, the system acquires discrete energy levels.

The figure shows the energy levels of hydrogen on the right and the orbits that would correspond to the electron in a semiclassical model of hydrogen, where the electron is circling the proton, (in reality they are the distances where it is more likely to find the electron in each of the energy levels).

What must be kept in mind is that in quantum mechanics there is an indisputable requirement:

Every quantum system has to have a well-defined minimum energy. This minimum energy identifies in systems such as atoms the fundamental level of the system.

But quantum mechanics does not live by particles alone - in fact, it hardly ever lives by particles in the strict sense of the word - but we also have fields. A field is an assignment of a certain physical property to each point in space. The electric field is nothing more than an assignment of a vector to each point in space. The vector will have a modulus, which represents the intensity of the field at that point, and a direction and a sense.

This is the classical view.

If we introduce a quantum view of the field two things happen:

The field has several allowed states and a minimum value of its energy.

The field can be interpreted as a system in which particles associated with the field appear. Each field has its particles associated with their mass, charges, spin, etc.

In short, in quantum field theory, the quantum version of field theory, there is an indissoluble relation between a field and its associated particles. For example, the electromagnetic field at the quantum level is associated with the presence of photons, which are its associated particles.

It is important to note the following:

This interpretation of quantum fields in terms of particles only makes full sense in theories in which spacetime is flat, i.e., in theories in which gravity is not considered to exist.

As good quantum systems the fields have a state of minimum energy value, in the vision in terms of particles this state is the one that does not contain any particle associated with the field. For that reason, this state is called the vacuum of the field and is represented by   .

What does it mean that the vacuum contains no particles?

When it is said that the vacuum does not contain particles and is at the minimum energy of the field we mean just that. If we measure this state we will not find particles and the energy of this state will be the minimum possible, in usual fields we can say that they have zero energy in their vacuum state (although this statement has a lot of clues).

But, as quantum is evil and Machiavellian, when we are not measuring the vacuum it can be doing anything. In principle it can have energy fluctuations in such a way that we can interpret, in terms of particles, that particle/antiparticle pairs are created (because charges have to be balanced and a particle and its antiparticle have opposite charges). So, these fluctuations appear and disappear in the vacuum as fast as the energy of the pair is created. What quantum science assures us is that we are not going to see these fluctuations just like that.

We can represent this with a Feynman diagram:

The vacuum has nothing but a pair is created, the top of the curve is a particle and the bottom is an antiparticle or vice versa, and then they disappear in the vacuum.

In this diagram, the only thing to understand is that the circle represents a particle/antiparticle pair, which appears and disappears in the vacuum as explained in the figure description.

Can we see these fluctuations described as pair creation?

The answer is yes. It is sufficient to apply a very large electric field to a vacuum state of a field. Since the vacuum is neutral the antiparticle-particle pairs that are created in the vacuum fluctuations will have opposite electric charges. If we plug in a very large electric field we will see vacuum particles appear. This seems like science fiction, but it is an effect known as the Schwinger effect. We can represent it with this Feynman diagram:

The intense electric field separates the particles of the pair and gives up the necessary energy for them to come into existence. The electric field yields the necessary energy to the vacuum so that it spits out these particles that literally come out of nothing (understand that this is a poetic license).

Other effects highlight the structure of the vacuum:

• Casimir effect.
• Lamb effect.
• The mass of protons and neutrons.

This list is only for those interested in searching Naukas or the web to find out what this is all about.

A fundamental curiosity about the vacuum

If we are in a world where there is no gravity, spacetime is flat, and we have several observers moving in a straight line and at constant velocity and one of them determines that the state of a certain field is vacuum, all the others will coincide with him.