What are Hawking-type radiation?


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

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

Hawking radiation itself. Yes, in a series of posts about Hawking radiation, it seems fair 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 flows of positive and negative energy. Yes I said negative energy, we will see how it eats that. Of course, while we are divulging, we will explain the famous divulging image of the creation of particle/antiparticle pairs, in which one falls and the other leaves the black hole. We will see that the story that they usually tell us is not bad but it is not the complete one. In addition, we will discuss the essential elements of Hawking radiation and we will see with surprise that gravity does not matter, gravity is only a motivation, but it is not the essential reason.
Hawking-type radiation in a cosmological context. Historically, Hawking radiation was introduced into an expanding universe in which particles were created. Well, nothing, we will discuss this, especially the contributions of Leonard Parker.
Lastly, we will see the meaning of the proposals to simulate Hawking radiation in the laboratory. Analogue models of black holes are easy to understand and very instructive.

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

How about we start 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 into discrete energy levels. Perhaps that is why we associate quantum with discrete when this association is not entirely correct. In reality, a free electron, for example, an electron that does not interact with anything can have any energy, there is no discretization of the energy that is worth it. However, the same electron, if it 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 revolving around the proton, (actually they are the distances where it is more likely to find the electron in each of the energy levels).

What you have to keep in mind is that in quantum mechanics there is an indisputable requirement:

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

But the quantum does not only live on particles, in fact it almost never lives on them in the strict sense, 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 module, which represents the intensity of the field at that point, and a direction and sense.

This is the classic view.

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

The field has a series of 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 own particles associated with its mass, charges, spin, etc.

Ultimately, in quantum field theory, the quantum version of field theory, there is an indissoluble relationship 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 makes perfect sense only in theories in which spacetime is flat, that is, in theories in which gravity is not considered to exist.

Like good quantum systems, the fields have a state of minimum value of energy, 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 empty field and is represented by|0\rangle  .

What does it mean that the vacuum does not contain particles?

When we say that the vacuum contains no particles and is at the minimum field energy, we mean just that. That if we measure said state we will not find particles and the energy of said state will be the minimum possible, in usual fields we can say that they have null energy in their vacuum state (although this affirmation has a lot of crumb).

But, since quantum is evil and Machiavellian, when we are not measuring the vacuum it can be doing anything. In principle it can have fluctuations of energy in such a way that we can interpret, in terms of particles, that particle/antiparticle pairs are created (because the charges have to compensate and a particle and its antiparticle have opposite charges). So, these fluctuations appear and disappear in the vacuum as much faster as energy has the created couple. What quantum technology assures us is that we are not going to see these fluctuations so easily.

We can represent this with a Feynman diagram:

The void 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 into the void.

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

Can we see these fluctuations described as pair building?

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

The intense electric field separates the particles from the pair and gives up the energy necessary for them to come into existence. The electric field transfers the necessary energy to the vacuum so that it spits out these particles that literally come out of nowhere (understand that it is a poetic license).

There are other effects that reveal the structure of the void:

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

The change in the values of the coupling constants of the fundamental interactions.

Other effects.

This list is only for those interested to search the Naukas or the network to find out what this is all about.

A fundamental curioristy of the void

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

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