Illustration by Nicholas Forder/Future Publishing/ Getty


Cogitating black holes

The Universe cannot always be understood through observation. Instead, physicists explore by devising thought experiments

by Michael Dine + BIO

Illustration by Nicholas Forder/Future Publishing/ Getty

Albert Einstein’s theory of gravitation, known as general relativity, is intimidating, even for highly trained theoretical physicists. In his theory, matter and energy cause space-time to curve. In most situations, this warping is so small as to be unobservable, even with powerful and sophisticated instruments. In fact, for many years after Einstein put forth his theory in 1916, there were only three situations in which small corrections to Newton’s classic laws of gravity (the force we feel here on Earth) could be observed: the bending of light by the Sun during a solar eclipse; a small anomaly in the motion of Mercury; and a small shift in the wavelength of light due to gravitation. Since that time, the situation has dramatically changed. General relativity has provided us with a framework for thinking about the Universe as a whole, and plays a role in much of what astronomers understand about stars. It even plays a role in the GPS system that helps us navigate the roads.

Einstein’s equations ultimately revealed a set of previously unknown, ultradense cosmological objects: black holes. The mathematics of Einstein’s equations showed that light starting inside the black hole could get only so far. That distance, known as the Schwarzschild radius, can be thought of as the surface of the black hole; this surface is known as the horizon, beyond which light cannot escape. Near and within the horizon, space and time are modified so violently that it even becomes tricky to figure out what is space and what is time.

No one could see inside this kind of object, but speculations on their nature date to the work of J Robert Oppenheimer (famed for his leadership of the atomic bomb project during the Second World War) and John Wheeler, a Princeton theorist who provided, among other things, the name ‘black hole’.

Over the past half-century, astronomers have found black holes in great numbers around the Universe. Some are the result of stellar collapse, and have masses typically a few times larger than that of our Sun. Much more massive ones exist at the centres of most galaxies, including our own. Smaller black holes are typically ‘seen’ as they swallow matter from companion stars; the large black hole at the centre of our galaxy was discovered through its effects on the motion of stars orbiting about it. We may never be able to literally peer inside a black hole, but knowledge of the cosmos and emerging theories of physics allow us to think through their nature; the modus operandi for this kind of exploration, the thought experiment, has been a cornerstone of physics since Einstein dramatically altered our understanding of space and time.

Einstein’s theory that the Universe is curved and time is relative has been subject to direct experimental and observational study for more than a century – but thought experiments played a major role, as well. One of the most famous thought experiments of all time juxtaposed Einstein’s general relativity, which looked at systems as large as the cosmos, with quantum mechanics, also referred to as quantum theory, which resulted from experimental studies of objects on the scale of atoms or smaller.

Prior to the emergence of quantum mechanics, physicists thought of atoms as something like billiard balls. In the pre-quantum or classical view, their motion was governed by Isaac Newton’s laws, which allow a person, given knowledge of the basic forces of nature, to predict the motion of the particles in the future. But quantum mechanics called this viewpoint into question. Instead, it suggested an alternative picture of reality, coded in the Schrödinger equation – which provided the probability, though not the certainty, that an electron would be located at a given spot at a particular point in time. It was the physicist Max Born who made the radical proposal that quantum mechanics predicted probabilities of various outcomes, rather than a single certain result. Critical to his assertion was a set of thought experiments. Born asked what Schrödinger’s equation would predict for the outcome of the collision between two atoms, or an atom and an electron. Newton’s billiard ball outlook holds only when the probability of one particular outcome is far larger than that of any other.

Thought experiments suggested the widely separated elements would still be entangled

The notion deeply troubled Einstein, provoking his complaint in a letter to Born in December 1926: ‘Quantum mechanics is certainly imposing… The theory says a lot, but does not really bring us any closer to the secret of the “old one”. I, at any rate, am convinced that He does not throw dice.’

In 1927, Werner Heisenberg summarised the distinctions between the physics of Newton and that of the Schrödinger equation in his uncertainty principle, which sets limits on what one can measure about a system. The location of a particle, would always be a question of probability, never a sure thing. He arrived at this principle by considering various thought experiments, where he asked how particular measurements might actually be performed. Einstein tried to demolish the quantum theory through sharp critique, continually challenging Niels Bohr, a Danish founder of quantum mechanics and a leader in the effort to interpret the theory with thought experiments similar to those of Born and Heisenberg. At first glance, these seemed to show that quantum theory and its probability interpretation did not make sense. The questions Einstein asked were often tough, but Bohr, sometimes after a prolonged period of thought, invariably found a way to resolve each paradox. One such experiment, known as the EPR paradox (for Einstein and his two assistants, Boris Podolsky and Nathan Rosen), involved the connections between two widely separated parts of a single system. Thought experiments suggested the widely separated elements would still be entangled, with one part of the system invariably providing information about the other. This was eventually turned into a real experiment, proving quantum mechanics correct.

So what does all this have to do with black holes? A real-world experiment sets the stage.

According to the rules of classical physics, an object with electric charge, like an electron or proton, emits light as it speeds up or slows down. Einstein understood that, in a similar manner, his general relativity would lead to waves of the gravitational field – gravity waves – when mass or other forms of energy sped up or slowed down. These waves, in turn, would push and pull on matter as they passed by. Because the gravitational force is so much weaker than electricity and magnetism, these effects would be minuscule, even when huge amounts of mass are involved.

The first experimental programme with any real hope to detect these tiny gravitational waves began in the 1990s, and was known as LIGO, for Laser Interferometer Gravitational-Wave Observatory.

The programme was based on an outcome of general relativity understood early on by Einstein: when two planets collide, the mass involved would be insufficient to perceptibly impact the shape of space-time. But when two superdense objects like black holes collide, they would distort space-time enough that the effect could be detected. According to Einstein’s theory, these waves, travelling through space from their source, would stretch the space around them, ever so slightly. Objects nearby would appear slightly longer and then slightly shorter, and then slightly longer again. This stretching and shrinking would alert us that the objects had been there at all.

Now, when I say slightly, I mean slightly. The LIGO gravitational-wave detectors are long metal tubes each 4 kilometres long. Waves from colliding black holes stretch and shrink these huge bars by about 10-18 cm, an amount 105 times100,000 times – smaller than an atomic nucleus. Put another way, as a fraction of its length, each bar changes by about a trillionth of a trillionth of its length.

Throw in tables, chairs, planets, other stars, and the black hole’s mass increases and its horizon area increases

Only over the past decade has the detector picked up gravitational waves from collisions of neutron stars and black holes. With this discovery, a whole new way to study the Universe has emerged.

Yet these experiments go only so far. Indeed, in a universe governed by quantum mechanics, there are aspects of black holes that are far from clear. Because, in Einstein’s theory, a black hole can’t emit light or transmit information in other ways, they are almost featureless. If you know their mass, their electric charge, and how fast they spin, you know everything you can possibly know about them. They may have arisen from the collapse of a complicated star, surrounded by planets with advanced civilisations, but when they formed, all of that information simply vanished. This is different from a fire or an explosion, where you might hope, with a huge amount of work, to reconstruct all the original information by looking through the ashes and the outgoing light and heat. In the collapse of a black hole, such reconstruction seems impossible.

This new visualisation of a black hole illustrates how its gravity distorts our view, warping its surroundings as if seen in a carnival mirror. The visualisation simulates the appearance of a black hole where infalling matter has collected into a thin, hot structure called an accretion disk. The black hole’s extreme gravity skews light emitted by different regions of the disk, producing the misshapen appearance. Created by NASA Goddard Space Flight Center/Jeremy Schnittman

One physicist who tried to glean more through thought experiment was the late theorist Jacob Bekenstein of the Hebrew University of Jerusalem. He noted an analogy between black holes and the second law of thermodynamics. The second law says that entropy – which is a measure of disorder – always increases. For black holes, there is also a quantity that always increases: the area of the black hole surface, its horizon. Whenever you add something to a black hole – say throwing in tables, chairs, planets, other stars – the mass increases and the area of the horizon increases. Bekenstein proposed a precise relationship between the black hole area and entropy, and suggested that black holes were actually thermodynamic systems with a temperature.

In physics, we think of temperature as a measure of the energy within some set of particles – atoms, molecules, photons. Yet, from the outside, we have no information about the black hole apart from some gross properties such as its mass, and we certainly can’t identify things like particles.

It was Stephen Hawking who, in the early stages of his career, finally discovered the sense in which black holes have a temperature. Hawking had an interest in extreme situations in general relativity, such as the earliest instants after the Big Bang and the interior of black holes. Now thinking about the behaviour of particles such as electrons and photons near the horizon of a black hole – thought experiments again – he realised that black holes are not really black; they radiate particles now known as the ‘Hawking radiation’. This is an intrinsically quantum phenomenon. The uncertainty principle permits brief violations of energy conservation in ordinary space-time. As a result, for an extremely short time, a particle and its antiparticle (in the case of an electron, for example, the antiparticle has the same mass but the opposite electric charge, known as the positron) can appear, even in a complete vacuum, and then annihilate each other and disappear again. For us, there is no observable consequence because energy is conserved.

But Hawking realised that some of these flickering particles could borrow some of the enormous energy of the black hole and become real. If produced near the horizon, one of these virtual particles could fall back into the black hole while the other escapes. Hawking found that the particles were emitted just as they would be from an object with the temperature predicted by Bekenstein. (The radiation from an object with a given temperature is called ‘blackbody radiation’ and has characteristic features; the most dramatic example is the Universe itself, whose temperature is 2.7 degrees Kelvin).

In short, the black hole appears to be a much more complicated object in a quantum world than in a classical one. In the quantum world, there’s a lot going on inside. The black hole in the quantum universe is not static. As it emits particles, it gradually evaporates, eventually disappearing altogether.

For a black hole formed in the collapse of a star a bit more massive than our Sun, the time for the entire object to evaporate is very long – about 1067 years, far, far longer than the present age of the Universe. But we can contemplate smaller black holes, which might be disappearing today. At the end of their lifetimes, there would be a large burst of energy. Astrophysicists are currently searching for this possibility. But we’d have to be quite lucky to find such a thing and, so far, there is no evidence for black holes of this size.

Hawking’s theoretical discovery of the Hawking radiation, possible through thought experiment, was a major accomplishment. It brought general relativity and quantum theory together in a remarkable way. But performing still another thought experiment, Hawking was puzzled by features of this radiation – or more precisely, its lack of features. Critical to Born’s probability interpretation of quantum mechanics was that something always happens. If you add up the probabilities for anything that may happen, you will find that the total probability is one. This can be formulated as a statement about information: if one knows everything one can know about a system at one time, one can know everything about it at later times. But this did not seem to be the case for radiation from black holes.

These ideas may be unfamiliar – indeed they are unclear to many physicists, so it is worth elaborating a bit. The fact that the probability of all outcomes is one is illustrated by a familiar pastime. If you enter your state or national lottery, you focus on your chances of winning. If you buy one ticket and there are 10 million lottery tickets sold, your chances of winning the jackpot are 1 in 10 million. That’s a really minute chance. But I either win or lose the lottery: the chance of winning or losing is 100 per cent.

What does it mean for information to disappear? Of course, we all forget things, lose records of various types, or deliberately shred or burn papers. But we believe that with enough patience and resources, we could reconstruct this information. The amount of information in a system (or the Universe) doesn’t change, though much of it may be hard to access. For a complicated system, like a collapsing star, there is a lot of information – an unimaginably large amount. In classical physics, there would be the positions and velocities of all the nuclei and electrons. In quantum mechanics, there are complicated relations between all of them; one can’t give the probability that one particle is at a point without specifying also the probability of finding all the other particles at particular places as well.

There is a situation where black holes could exist and quantum mechanics could make sense: string theory

So a collapsing star contains a huge amount of information. Thanks to Hawking, we know that, if the star is heavy enough, it forms a black hole and then slowly evaporates, emitting radiation. The vast amount of information that was contained in the initial star has been reduced to just the temperature of a warm body. Hawking, in his 1976 paper, argued that the information was simply lost. Quantum mechanics, he asserted, breaks down near black holes.

Many leading theorists have struggled to resolve the puzzles raised by this thought experiment. Some have argued that, indeed, one has to redo quantum mechanics or general relativity to resolve Hawking’s paradox. Others have been more sceptical of Hawking. Perhaps, for example, the evaporation of a black hole is like a lump of ash from the burning of a log in a fireplace. Surely the laws of quantum mechanics don’t break down when an object burns? In that case, the resolution of the puzzle is that the outgoing radiation is not exactly that of a black body because subtle connections between the outgoing photons remain intact. But it was soon realised that the answer to Hawking’s question about the black hole problem could not be so simple; the structure of space and time makes it hard to understand how such correlations might arise. There were other proposals, none very satisfying. Perhaps Hawking was right: just as Newtonian physics was usurped by quantum mechanics and general relativity on large or tiny scales, something had to give here as well.

It turns out that there is a situation where black holes could exist and quantum mechanics could make sense: string theory. String theory, also emerging from thought experiments, replaces the particles of quantum mechanics with one-dimensional strings. That concept has provided at least a partial resolution of the puzzle. Two theorists at Harvard University – Cumrun Vafa and Andrew Strominger – building on the work of the late Joseph Polchinski, of the University of California at Santa Barbara, were able to understand the temperature of certain idealised black holes in quantum mechanical terms. In other words, the information, at least for these idealised systems, somehow survives, evading Hawking’s paradox.

But while this result settled the question in an abstract way, it left many physicists dissatisfied. Because the calculation is done in a situation that doesn’t much resemble an astrophysical black hole, it is hard to figure out just what went wrong with Hawking’s argument.

There remains something important about the way general relativity works that we don’t yet fully understand. It may be that the rest of the story will be rather mundane, but it seems likely that fully resolving these questions will yield dramatic new insights into the quantum nature of space-time, and might answer some big questions we have about the Universe as we observe it. One of the biggest puzzles in our current understanding of nature is that most of the energy of the Universe – about 70 per cent – exists in a strange form with negative pressure, known as the dark energy. But it is very hard to understand why there is so little of it.

It is conceivable that a thought experiment resolving Hawking’s puzzle might provide some clues. The most radical possibility is that space-time is not the basic arena for the phenomena of nature. A being living in a crystal, for instance, would experience something like space-time, but would have a very different character. Condensed matter physicists would say that space-time is emergent. The basic underlying entity might be something else entirely. Perhaps one day our science and technology will be so advanced that actual experiments will reveal what it is – but, until then, thought experiments involving black holes, among other phenomena, will have to light the way.

Adapted excerpt from the book This Way to the Universe by Michael Dine, published by Dutton, an imprint of Penguin Publishing Group, a division of Penguin Random House LLC. Copyright © 2022 by Michael Dine

15 November 2022