Hawking's paradox

The essence of the problem that Hawking formulated is the following: during the formation and subsequent disintegration of black holes, information about their detailed composition is lost.

Infrared offset

To explain the essence of the paradox, consider electromagnetic waves. They come in different frequencies, and radio waves respond to the lowest frequencies. If you increase the frequency, it will be infrared radiation. Then we get the waves from the visible (light) spectrum. Further, outside the visible spectrum will be ultraviolet radiation, X-rays and, finally, gamma radiation.
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If we put a radiation source at some distance from any massive space object and follow the light emitted by it at a large distance from the center of gravity, we will see the so-called infrared displacement. The observed radiation frequency far from the gravitating body will be somewhat lower than that emitted in its vicinity. This is explained by the fact that the energy of photons (electromagnetic waves) is directly proportional to their frequency. The photon, as it overcomes the gravitational attraction, does the work, respectively, loses energy, so its frequency decreases.

For a body like Earth, this effect is rather weak, but measurable. However, for example, for a neutron star, the magnitude of the infrared bias can be quite large. In turn, for a black hole, this phenomenon reaches its extremum in the following sense. The fact is that a black hole has a so-called event horizon - a surface from which any radiation undergoes an infinite infrared shift. That is, if the radiation source is right on the horizon, then the field created by it you see is not changing in time: there is no radiation, no matter what distance from the horizon you would hang. The horizon is just that surface from which light (or any wave) cannot fly out.

“Hairlessness theorem”

Black holes are designed in such a way that they create exclusively stationary fields, even if they rotate around its axis (provided that their center of mass rests). The gravitational and electromagnetic fields created by them will not change in time. This statement is called the theorem about the absence of hair in a black hole. For stars, this is not so: they can create around themselves, for example, time-varying magnetic fields, even if their center of gravity is at rest. This is due to the fact that the charges inside the star make various motions, creating radiation. But a black hole does not create anything of this kind, even if a terrible movement of charges occurs under its horizon.

Let's set up a mental experiment: let's say we have two clouds of particles, one consists solely of protons and antiprotons, and the second is of neutrons. Something began at some point to compress these clouds. If their masses and torques were the same, then as a result we get two black holes that are absolutely indistinguishable from each other.

Hawking radiation

Stephen Hawking in the early 1970s showed that a black hole should emit radiation, but it has a fundamentally different nature compared to the classical radiation we talked about above. The radiation discussed above has sources, namely moving charges and masses. And Hawking radiation, one may say, has no source: it is not the result of any movement of charges. This radiation arises as a result of a change in the properties of a vacuum (amplification / amplification of zero-point oscillations) due to the collapse of matter into a black hole. Moreover, if charges and masses give rise only to electromagnetic and gravitational waves, then as a result of Hawking's quantum radiation, there can be a birth of electrons, positrons, protons and other particles.

So, black holes begin to produce various particles in their neighborhood. This radiation has a number of characteristic properties. First, it is stationary, that is, it changes in time very slowly, if the black hole is heavy enough and slowly loses its mass, giving birth to particles. Moreover, Hawking radiation has a thermal spectrum. That is, a black hole radiates as a normal source heated to some temperature — the shape of such a spectrum is characterized exclusively by the magnitude of the temperature.

An important feature of the temperature spectrum is that all characteristics of particles, except mass and charge, are emitted with the same probability. Roughly speaking, for example, any neutral particle and photon with the same energy are emitted with the same probability.

Paradox

Now we are ready to formulate what is the information paradox. Imagine that you have two clouds familiar to us, one of which consists of protons and antiprotons, and the other of neutrons. Imagine that something formed from them two stars - proton and neutron. And then, as a result of their burning, these stars emitted some part of their mass, but something remained in the form of a cold ball. Theoretically, based on the remnants of the evolution of stars, we can trace the history of each elementary particle that was part of the clouds. Of course, technically this is an incredibly difficult task, but here it is only a matter of principle. The difference in the case of black holes is that, first, we cannot seem to distinguish between two black holes — proton and neutron, as explained above. Secondly, temperature radiation without sources does not carry any detailed information about the composition of a black hole. Thus, according to the remnants of the evolution of black holes, even if their mass has completely passed into radiation, we, it would seem, are fundamentally incapable of restoring their origin.

Why is this a paradox? The fact is that the power of science lies in its predictive power. Science can predict that if you do this and that, you will get such a result with such probability and such precision, and express this statement quantitatively. And to check this or that experiment can any other scientist. It turns out that if information is lost, then in the presence of a black hole all this turns out to be wrong. Mathematically, this is expressed in the fact that the total probability of some processes may be unequal to one, even greater than one.

Criticism of the Paradox

However, all of the above was based on some qualitative reasoning. They all require a formal computational confirmation. These computational proofs of the paradox are formulated with such a low degree of rigor and with such a number of crude assumptions that it can be refuted with the same degree of rigor. Another thing is that many of the details of different processes that occur in the presence of black holes, remain unclear. And for that part of the scientific community, which believes that there is a paradox, its solution is the guiding star in the knowledge of the nature of black holes. It often happens in science that there are different points of view regarding a subject that is still poorly understood.

Emil Akhmedov , D.Sc. (Physics and Mathematics), Leading Researcher at the AI Alikhanov Institute for Theoretical and Experimental Physics, Associate Professor at the Department of Theoretical Physics, Moscow Institute of Physics and Technology, Associate Professor at the Faculty of Mathematics of the Higher School of Economics.