# Israeli Physicist Finds Key to Solving Stephen Hawking’s Black Hole Paradox

How Netta Engelhardt of MIT and colleagues found a way to reconcile the basic theories of physics, Quantum physics and Einstein’s theory of relativity. You’re about to discover what ‘unitarity’ is

Modern physics is based on two major theories. Quantum physics describes events at the “micro” level, of subatomic particles; Einstein’s theory of relativity describes events at the macro level, of large, heavy bodies. Each theory has been supported by much observation; their validity has also been empirically tested by the technology that they enabled, such as lasers and processors at the micro level, and satellites and GPS systems at the macro level.

The two theories act at what appear to be completely different magnitudes: electrons and light waves, versus stars and galaxies. But there is at least one place in which the two theories meet. Or, more accurately, where they collide. That place is the black hole.

The 'information paradox' remains one of the most significant land mines on the path to attaining one of physics’ major objectives

In 1974, following discoveries about black holes by the Israeli scientist Jacob Bekenstein, the then-young physicist Stephen Hawking published a dramatic article that cemented his status as a scientist of the first rank. Hawking demonstrated that regarding black holes, one of the two major theories leads to an error.

According to his calculations, the radiation emitted by the hole is not a function of the material the hole swallows, and therefore, two black holes that formed by different processes will emit the same exact radiation. This meant that the information on every physical particle swallowed into the black hole, including its mass, speed of movement, etc., disappears from the universe.

But under the theory of quantum mechanics, such deletion is impossible.

This is the “information paradox,” aka the “black hole information paradox,” which remains one of the most significant land mines on the path to attaining one of physics’ major objectives: the development of a general theory of nature that encompasses both quantum mechanics and the general theory of relativity.

Now an intriguing twist has been proffered by Professor Netta Engelhardt, an Israeli-born physicist at the Massachusetts Institute of Technology, with colleagues. She believes they have made monumental progress towards the heart of the problem. She and colleagues were even awarded the New Horizons in Physics Prize for 2021, one of the most prestigious prizes granted to young researchers in the world of physics, for complex, original calculations that bring the world of science closer to resolving the paradox posed by Hawking.

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In practice, Engelhardt and her colleagues calculated the quantity of information amassed in black holes at various stages of their lives, and for the first time, managed to reconcile the effects of the two major theories of physics in their calculations. They then published their work in the Journal of High Energy Physics.

To understand the pathway they present, we need a short course on quantum mechanics, to understand what the information that is destroyed, according to Hawking, and why this destruction creates a paradox.

**The missing information **

Quantum mechanics describes phenomena that we don’t necessarily understand intuitively, such as superposition, which states that a particle may be found in two different places simultaneously; or entanglement, which states that two particles can constitute a single system, so that if we affect one part of it, then the second part will be affected as well – even if it is located at a distance light years away. This theory rests on the shoulders of a principle known as “unitarity.”

What is unitarity? Professor Hagai Eisenberg of the Racah Institute of Physics at the Hebrew University explains. “Any shift from one state to another, for instance from left to right, or from the before-change to after-change, can be described in mathematical language. In quantum mechanics, the mathematical description of advancing in time is unitary.”

Think of it this way: If we turn a steering wheel to the right, we can still easily turn it left; but if we light a match, we can’t carry out the opposite action, and turn a burning match into a non-burnt match. According to quantum mechanics, Eisenberg says, moving ahead in time is always reversible, meaning unitary, and every mathematical expression of advancing in time also has a mathematical expression that describes the opposite action.

In the terms of quantum mechanics, there’s nothing to prevent a burned match from becoming an unburnt match.

This is where black holes come into the picture. These cosmic phenomena are created when a star exhausts its nuclear fuel. Due to the force of gravity, the star implodes. In sufficiently massive stars, this collapse compacts the star’s material into a single point. Due to the colossal gravitational force exerted by the compacted point, almost nothing can escape or be emitted from it, not even light. The black hole is tiny, but exerts a colossal gravitational pull, and therefore both the laws of quantum mechanics and the laws of the theory of relativity are relevant to it.

Hawking discovered that despite this colossal gravitational force, some radiation “escapes” from the black hole, eventually leading to the hole’s evaporation. Based on the theory of relativity, according to Hawking, the radiation emitted by the hole is solely thermal, and does not contain any information about the material that has been swallowed up inside. In other words, the radiation cannot teach about the particles inside the hole. That being the case, when the black hole evaporates, all of the information that had been contained in it disappears. This is an irreversible process, and it therefore violates the principle at the basis of quantum mechanics.

This is the source of the collision between the two theories at the event horizon, creating the paradox.

The scientific world went mad . The Stanford physicist Leonard Susskind even wrote a book entitled: “The Black Hole War: My Battle With Stephen Hawking to Make the World Safe for Quantum Mechanics.” The physicist John Preskill from Caltech made a bet with Hawking that he was wrong (they decided that the loser would send the winner an encyclopedia, meaning an object from which it is possible to extract information).

Hawking continued to work on the paradox, and eventually agreed that it was not possible for information to disappear (and Preskill received an encyclopedia about baseball). Nevertheless, the question of how to settle the dispute of these two major theories of physics in black holes remained unanswered.

Relating to Engelhardt’s new concept, Professor Micha Berkooz of the Department of Particle Physics at Weizmann Institute explained, “There is a contradiction between the theories [quantum mechanics and the theory of relativity], both of which are cherished and both of which work well. For a long time, there was an argument in the community on whether black holes violate unitarity, but today, it seems they do not lose information. So quantum mechanics works, but it is not clear what is missing in the theory of relativity in the context of calculating gravitation in black holes, and this is a fundamental question. Engelhardt’s work demonstrates how it is possible to calculate the dynamics of black holes, with gravitational force, as well”- in other words, adhering to the laws of the theory of relativity.

**The event horizon**

Engelhardt grew up in Jerusalem. In 1998, her family left Israel for the U.S. for her parents’ careers.

“As a child, I loved reading,” she says. “But I didn’t speak any English. When we moved to the US, we had only taken a handful of Hebrew-language books with us. So I read everything in the house, until there was just the one last book left. It was ‘A Brief History of Time’ by Stephen Hawking”

Engelhardt was nine when she read the book, having been left with no other choices. “And the rest is history,’ she chuckles.

Engelhardt and her colleagues’ innovation involves using a measure called entropy to examine whether black holes lose information.

Entropy essentially boils down to ignorance about some element of the system, the researcher explains. Before the first black hole formed, it was theoretically possible to know everything about the universe, and at that point the entropy was zero. If a star collapses and becomes a black hole, Engelhardt says, entropy results from ignorance of the black hole interior. As more matter falls into the black hole, its entropy increases.

According to quantum mechanics, when the black hole evaporates and disappears, it is not possible for the information that was amassed within to have been lost; which in turn would mean that information must leave it as radiation leaks out, and that is how the entropy is brought back down to zero. The unitary process is preserved.

However, according to calculations based on the theory of relativity, even after the black hole evaporates, entropy remains as it was, at its peak.

“This is effect is quantifiable,” explains Engelhardt. “Quantum mechanics says that the entropy begins at zero, rises, and then goes back to zero, whereas the theory of relativity says that as you start at zero, entropy rises to a certain value as the black hole forms, and remains at that value when it evaporates.”

Engelhardt and her colleagues carried out a calculation that factored in both quantum mechanics and the theory of relativity, without assuming a unitary state, and demonstrated that after the black hole dies down, entropy declines. In other words, the information is preserved.

In order to do so, they had to first calculate the area of a particular surface in the black hole. Calculating this area is critical, since the entropy in the black hole depends on the area, a connection first discovered by Jacob Bekenstein and further refined by Shinsei Ryu and Tadashi Takayanagi. If a black hole swallows an object, then the entropy of the black hole increases, since all of the information about the object becomes blocked off from its exterior. However, this swallowing up also increases the area of the black hole - after all, now it contains more material. Therefore, there is a connection between the area and the entropy.

The outside and inside of a black hole are determined by the event horizon - a sort of imaginary spherical envelope that surrounds the black hole, in which all of the material penetrating it, including the fastest particles - particles of light - enters the hole’s field of gravitational pull and can no longer extricate itself from it. The event horizon is, then, a defined borderline: a moment before it is crossed, the light particle can extricate itself from the gravitational field, but if the particle has crossed that boundary, even by a fraction of a millimeter, it will never be able to extricate itself.

However, we are referring here to an exclusively theoretical concept, explains Engelhardt, for in order to know if a particle of light has crossed the event horizon, we must wait and see if it has been extricated from the gravity field of the black hole, or not. The problem is that the waiting period is unlimited, and can theoretically be eternal.

As a measurable alternative to the event horizon for calculating the area of black holes, Engelhardt used the concept of a quantum extremal surface, which she defined quantitatively for the first time together with Dr. Aron Wall of Cambridge University in 2015.

In order to understand this complex concept, imagine a ball that has a flashlight or a point of light at its center. The more distance the rays travel from the light source, the more distant they grow from one another, therefore the area between the rays also grows.

So far, this is altogether intuitive. But if there is a bend in the space-time, the rays will grow more distant from the center, but the area between them will shrink, in other words they will grow closer to one another. That is what happens in a black hole. Eventually, the rays of light converge, until a spacetime singularity forms in which they are still separate rays, but they collapse one on the other. The space is so bent that the rays get closer to one another even when they have been launched in different directions.

“The critical issue,” says Engelhardt, “is that each ray has an exact point of equilibrium at which it is no longer growing more distant from the other rays, but is still not getting closer to them.” If we connect all of these points of equilibrium of all of the rays in a star, we will receive a new borderline, an ‘envelope’ that is the extremal surface. When you take quantum influences into account, calculation of the quantum extremal surface makes it possible to calculate the area of the black hole.

From their breakthrough related to understanding the importance of the quantum extremal surface until they resolved the mystery, Engelhardt and her colleagues worked practically without stopping. “The whole story went on for three weeks, in which we worked around the clock,” she recalls. “I got very little sleep during that time; between our collaborators on the west coast — Don Marolf and Henry Maxfield — and the Ramadan fast that my Princeton collaborator Ahmed Almheiri was observing, a lot of work was being done in the evenings and at night.”

The researchers were able to calculate the area of the quantum extremal surface — the classical gravitational contribution to the entropy — as well as the entropy related to all of the particles swallowed into this area (quantum entropy). They tracked the evolution of these two indices of entropy together, as well, and found that as time passes, it diminishes and eventually registers zero. Unitary theory won out, and a significant part of the paradox was illuminated.

“I remember the moment that we saw the entropy coming back down,” Engelhardt recalls. “I have a very distinct memory of driving home, and I was thinking to myself, wow, this could be it. We’ve seen that unitarity can be consistent with gravitation.’ It was very exciting.”

Berkooz adds: “The paradox led to an understanding that we have to amend the theory of relativity, and that is what Engelhardt is doing. The picture isn’t yet complete, but now there are several methods with which it is possible to understand where Hawking’s calculations were incorrect, and that the radiation is not only thermal. Her work is very important because it shows how it is possible to track, using gravitation, all of the quantum conditions of the black hole as it is dying out, and to ascertain using gravitation that information is not being lost. Engelhardt is not changing Einstein’s formulas, but beyond from his formulas there is a substantial array of tools that describe how to deal with more complicated situations. Her work is a very significant enhancement that has released the cork in our understanding of how to implement these tools for the study of dying black holes.”

Engelhardt says that the discovery (which was also reached independently by the physicist Geoff Penington of UC Berkeley, who won the same award) hinges on the interaction between gravitational entropy and quantum entropy.

“Each measurement, examined independently, does not demonstrate preservation of information; you need both of them,” she stresses. “Now what’s left is to understand how the information emerges from the hole and is not destroyed, and why the quantum extremal surface successfully incorporates unitarity. That is something that we still don’t understand. But we do know which questions we have to ask in order to make progress on the information paradox. We’re able to ask more precise questions, and we have a much better idea of where we’re going.”