The quantum mechanics of black holes

A black hole is a celestial and mysterious body. Einstein's theory of relativity taught us a lot about black holes, but many question marks remain about what goes on inside. It is likely that the picture will become clearer once a quantum theory of gravity is discovered, but until then we will content ourselves with the collision of quantum mechanics and general relativity around the event horizon. In this chapter we will discuss the paradox that emerged from this collision, the multitude of published solutions and the fascinating developments from the past two years

Black holes are mentioned in the traditional media most often in the context of general relativity. From a brief review of recent publications, one can find the black hole photographed in the center of the galaxy M87, which received significant resonance even outside the scientific community, or the black hole in the center of the Milky Way galaxy, which earned the astronomers who discovered its existence a Nobel Prize. From time to time black holes are mentioned alongside quantum mechanics, mainly in the scientific media, and usually around Hawking radiation and the information paradox that has seemingly disappeared. It is no coincidence that black holes are mentioned with low media prominence in these contexts. Even if the quantum description exists, it is only partial and complex for a reader who is not familiar with the details. It is important to emphasize that currently most of the descriptions are "semi-quantum", meaning they are not based on a quantum theory of gravity that can describe the microscopic structure of black holes. Answers to many questions are still not in sight and it is likely that we will not know what is happening inside the black hole until we have the "Torah of Everything" in our hands.

Black holes first appeared as a solution to Einstein's equation but were seen as nothing more than a mathematical solution. Over the years, astronomical evidence has been accumulated that revealed the prevalence of black holes in the universe and thus removed any doubt of their existence. Black holes were mentioned for the first time outside the framework of general relativity in a series of articles by the Israeli physicist Jacob Beckenstein and the British physicist Stephen Hawking between the years 73-75. Hawking's article discussed for the first time a quantum phenomenon around the event horizon that later received the name "the information destruction paradox". A paradox in physics indicates a lack of understanding and, in some cases, incorrect assumptions. It is important to remember that there are really no paradoxes in nature, but rather human failures in describing reality. Paradox actually calls for rethinking and searching for possible errors. Often a large number of solutions will be found and to know what the correct solution is it is necessary to conduct scientific experiments. When it comes to black holes, the number of possible experiments is very limited to almost zero, for now (gravitational waves may in the future teach him about gravitational corrections outside of relativity. In addition, systems simulating black holes have been recently studied). To understand the information destruction paradox, we will first need to understand what information is.

Information in quantum mechanics and black holes

Instead of quantifying information, it is convenient to quantify the lack of information, or the uncertainty. The common method for assessing lack of information is based on the von Neumann entropy. At the quantum level, information is encoded in the wave function. Assuming that the wave function is known in its entirety, the degree of uncertainty is equal to zero. The less we know about the system's wave function, the greater the degree of uncertainty, or entropy. In interlaced systems, maximum uncertainty exists when only one "side" of the interlaced system, or sub-system, can be measured. In the case of maximum uncertainty, the entropy is proportional to the number of degrees of freedom of the system, or to the number of possible states in which it can be found.

How does all this relate to black holes? In the XNUMXs Beckenstein and Hawking studied how black holes react to matter falling into them. The researchers demonstrated that the black hole responds similarly to thermodynamic systems. For this identity to hold, the entropy of the black hole must be proportional to the area of ​​the envelope, and not to the volume of the black hole, as has been commonly thought until now. Beyond that, the gravitational force induced by black holes should be parallel to the temperature, a fact that you will find useful later in the Unero effect, according to which accelerating bodies measure a different temperature at rest.

If indeed black holes are "hot" bodies, they radiate, but light cannot escape from the enormous force of gravity and leave the black hole. Hawking bypassed this obstacle thanks to a fundamental quantum process and was thus able to explain the thermal radiation emitted by the black hole. According to Hawking's model, the quantum fields outside the black hole are always noisy and occasionally in random processes pairs of particles are created from them and disappear immediately after their appearance. When this process occurs near the event horizon, there is a chance that one of the particles will be swallowed into the black hole and the other particle will manage to avoid being pulled in. The particles created by a random process are inherently entangled and since only one is outside the black hole, only "half" of the wave function of both can be measured. Uncertainty in the wave function carries with it an entropy greater than zero and because the entropy has left the black hole it should decrease its radius. Evaporation of black holes using Hawking radiation is very slow and the rate of radiation released from black holes decreases the larger the black hole is, but eventually if the black hole is inactive it will disappear completely.

Hawking's article concludes that the entropy resulting from the thermal radiation increases over time and the entropy of the black hole decreases until it disappears. Here actually lies the problem. The information encoded in the material that formed the black hole is fully known because black holes are formed by stars at the end of their lives. After the black hole has evaporated, only part of the information is accessible and the entropy is at its peak. Since the black hole completely evaporated, some of the information was lost along with it.

Is data loss really a problem? Many tend to equate the process of information loss with the process of combustion. Is information lost when a tree burns? Well the information only looks lost but in fact it is mixed up and changed. The amount of information remains the same and theoretically quantum mechanics teaches us that the information encoded in the log can be recovered if we follow the opposite physical process. This property is called "onetarity" and is a fundamental mathematical basis for quantum mechanics. Hawking showed that black holes violate the unitarity principle because some of the information swallowed by the black hole is disconnected from the universe, and perhaps even erased. Therefore, the reverse physical process is not possible because we only have partial information, the other part being swallowed up by the black hole. To be precise, the problem starts even before the black hole completely disappears, when the entropy curve of the Hawking radiation is greater than the entropy of the black hole. This area is particularly problematic because the total radiation is supposed to be intertwined with the black hole, but the black hole does not have enough information to be in a state intertwined with the radiation. The solution proposed by the physicist Don Page in the seventies is that the entropy of radiation actually decreases at late times and corresponds to the entropy of the black hole.

Solutions to the information destruction paradox

But proposals separately and reality separately. The real challenge of course is to show that indeed black holes obey this rule. Over the past 45 years, quite a few proposals have been published to solve the paradox:

1. The black hole didn't really disappear

The smaller the black hole, the greater the curvature around the event horizon. In the limit where the gravitational force around the event horizon is enormous, the effective theory on which Hawking was based is no longer valid. If Hawking's calculations are invalid, blackheads most likely do not evaporate at all. This hypothesis assumes that the evaporation process stops at the radius proportional to the Planck length. Unfortunately we will not be able to know if this hypothesis is correct without quantum gravity theory.

2. The information manages to leak out at the end of the black hole's life

Instead of the vaping stopping at some point, the information may simply leak out. Those who support this solution speculate that the leakage process is very slow, otherwise there is a problem with a Einstein limit describing the maximum amount of information that can be accommodated in a unit of area. In any case, the idea is again based on the assumption that a quantum theory of gravity exists.

3. The information manages to leak out at the very beginning

This is the preferred solution for physicists. The idea is based on the assumption that the laws of gravity play a more significant role near the event horizon. It is likely that corrections not taken into account in the calculations of Hawking and his colleagues will calibrate the entropy according to the Page curve.

4. Entropy was conserved in another universe

The idea that black people are an opening to a parallel universe is not reserved only for science fiction books. The idea tries to reconcile the destruction of the information by claiming that the information was not lost, but it just moved to another universe.

5. The information is in temporal entanglement of Hawking radiation

The information does not really disappear, but causes the Hawking radiation from the past to be intertwined with the one that will be projected in the future. There is no need to exaggerate here, it is clear that the solution is confusing and breaks any intuition about the arrow of time.

The first "real" solution to the information annihilation paradox is based on the principle of holography proposed by the Nobel Prize winning physicist Thoft and later deepened by the physicist Susskind under string theory. According to them, the information contained in any volume can be encoded only on the shell. The main use of the principle of holography was published in 97 by the physicist Juan Maldesana. Maldesena's paper, which later became one of the most cited papers in high-energy physics, showed that gravity theories with negative curvature (known as anti-de Sitter, or ADS) can be described by a fully quantum theory (in flat space) with conformal symmetry (known as CFT. Broadly speaking symmetry for stretching and contraction together with Lorentz symmetry) on top of the edge of the universe (in one less dimension). If all black holes can be described by a quantum theory that is dual to it, the paradox disappears because quantum theories are fundamentally unitary. Indeed, Maldesena showed that the Beckenstein-Hawking entropy can be calculated from the microscopic description of the dual quantum theory.

Netta Engelhardt's solution

Meanwhile the literature teaches us that duality only exists in spaces with negative curvature and there is little (if any) evidence for duality with positive curvature (one that matches our universe). Beyond that, physicists prefer to find direct evidence from the theory of gravity that the paradox does not really exist. In 2006, there was a positive development in this direction when the physicists Ru and Takeingi published a paper that corrected the von Neumann entropy on which Einstein and Hawking were based. The physicists took into account the principle of holography and developed an up-to-date formula for entropy in the presence of gravity.

The new expression was reproduced several times in a series of articles published until recently in 2019. Immediately afterwards, as part of a joint study by researchers from the United States, including Neta Engelhardt, a calculation was published for the first time that takes into account the corrected expression for entropy and shows that in fact the paradox does not exist, at least within the example presented in the article . More examples were published soon after in a series of papers by Maldesna and his collaborators in 2020. The researchers showed that the Roe-Takiengi entropy can be obtained even without assuming it. The method they used to calculate the entropy of radiation and black holes is called the "replication trick". This is a mathematical trick that, in simple words, calculates the entropy using a mathematical reproduction of the black hole whose replicas are connected by means of wormholes. In the end, it is not a model that describes reality and wormholes do not exist around the black hole. Still, it turns out that the trick works, the corrected entropy is revealed and the paradox disappears.

So have we solved the information destruction paradox? The answer to that is yes and no. Yes, because we are able to show that information does not disappear with the help of a macroscopic size (entropy), and not because we do not have a mechanism that explains how the information is rescued from the black hole. The quantum wave functions of Hawking radiation and the black hole will remain unknown until a quantum theory of gravity is found.