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What We Don't Know

‘What We Don’t Know’ is a podcast series that explores the boundaries of human knowledge, investigating the unanswered questions and theories that unravel them at the frontiers of science. During this podcast I hope to get you interested in new areas of science, maths and technology, teaching you about existing concepts and igniting a curiosity for the things we have yet to know.

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What We Don't Know

Black hole information paradox (part one)

February 28, 2022 • Lana Howell • Season 1 • Episode 10

Different areas of physics seem to be incompatible inside black holes. When combining general relativity, thermodynamics and quantum mechanics, you get a paradox, which suggests that our knowledge of these areas is flawed. A solution, whatever it may be, would irreversibly shake up our understanding of the physical world. It may rewrite fundamental laws. It may unveil a new theory of quantum gravity.

In this episode I will unravel what ‘information’ means in physics and how it relates to unitarity. I’ll explain how black holes are formed and their key features, why Hawking radiation was proposed, and how Hawking radiation violates this unitarity. This should reveal what the black hole information paradox is. Then in the next episode, we’ll examine potential solutions to the paradox.

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Black hole information paradox (pt1)

Hello everyone, welcome to the tenth episode of ‘What We Don’t Know’, a podcast that explores the boundaries of human knowledge, investigating the unanswered questions and theories that unravel them at the frontiers of science. During this podcast I hope to get you interested in new areas of science, maths and technology, teaching you about existing concepts and igniting a curiosity for the things we have yet to know.

Out in the universe exist astronomical objects called black holes, so massive and condensed within such a small point that their gravitational field limits even the emission of light. If you fall in, there’s no getting out - at least according to Einstein’s theory of general relativity. But in the 1970s, Stephen Hawking showed that particles do escape from the black hole. Not only this, but their emission seems to violate one of the dearest laws in physics: the conservation of information. When your particles finally escape the black hole, nothing about them can tell an observer that it was you who fell in; that information is lost. This creates the black hole information paradox.

In this episode I will unravel what ‘information’ means in physics and how it relates to unitarity. I’ll explain how black holes are formed and their key features, why Hawking radiation was proposed, and how Hawking radiation violates this unitarity. This should reveal what the black hole information paradox is. Then in the next episode, we’ll examine potential solutions to the paradox, with particular focus on AdS/CFT correspondence (ie. the holographic principle) and recent work by Ahmed Almheiri.

What information does an astronaut contain? The build of their space-suit, the appearance of the person, their personality. Information about what they ate that morning is settled in their stomach. Information about their mood is swirling in their brain. The standard unit of information is the bit. Consider all the possibilities for the mood of the astronaut: excited, bored, terrified about the black hole in front of them. An observation - such as measuring their heart rate - contains one bit of information if it cuts the space of these possibilities in half. Two bits cut it by a factor of four, three by a factor of eight, four by a factor of 16, etc. Following this pattern gives us the equation probability = (½)^I, where I equals the number of bits. This can be rearranged to give the standard equation I = -log2(p). You do not need to remember this, I just think it’s a neat equation to intuitively calculate how many times you have cut the number of possibilities in half.

In general, we think about information on a macro scale. Quantum information, however, refers to information about the properties of a particle, like its position, velocity, or spin. Spin is the intrinsic angular momentum of a particle.

One interpretation of quantum mechanics (the Copenhagen interpretation) states that information about a quantum system is encoded within its wave function. This wave function is a mathematical expression which describes the state of the system. For example, it tells you the position of a particle. There are equations which tell you how the state of this system changes over time; these are called evolution equations, and a standard example is the Schrödinger equation. Applying information from a particle’s wave function to its evolution equation should allow predictions about what properties that particle will have in the future, i.e. after interacting with that photon, it will move to that position.

Also, these evolution equations are time-reversible. This means that the process can run backwards through time. Having measured the wave function for a particle, you should be able to calculate where it was in the past. This has the important consequence that two different states cannot become identical later, because you would not be able to find which state they were in originally by running their evolution equations backwards.

Combining these two principles about the evolution equation - that it can predict future states and run backwards - results in the principle of ‘unitarity’, which states that information must be preserved. It cannot be permanently deleted from the universe. The astronaut could fall into the sun and become unrecognisable, but the information in their particles should, in principle, allow us to reconstruct them.

How is a black hole born? From a star.

Stars are essentially big balls of hot gas. For most of their lives, they exist as stable yellow stars, like our sun. Under the gravitational attraction of their immense mass, they want to collapse inwards, but these gravitational forces are balanced by the energy release of hydrogen to helium nuclear fusion. In this delicate balance they shine for millions of years, until the hydrogen runs out. If the star is big enough, it’ll become a red supergiant, where it lives a while longer fusing heavier elements, but eventually all the nuclear fuel will run out. Cue a rapid collapse, leading to a supernova which expels most of the star’s matter into a nebula, whose materials can form new stars and planets like our earth. The star’s dead core continues to collapse. If it is big enough it will become a black hole.

This black hole becomes smaller and denser, warping space-time around it, distorting its gravitational field. Eventually, anything trying to escape the gravitational pull of the black hole will have to travel at the speed of light. As the radius becomes even smaller, it will have to travel above the speed of light, which, according to Einstein’s theory of special relativity, is impossible. In other words, nothing can escape. Not even light. It is truly a black hole, a box locked by the strongest key: the laws of gravity.

The boundary of a black hole is called the event horizon. It corresponds to the wavefront of light that hovers at Schwarzschild’s radius, just failing to escape. You may also have heard of the singularity before. This is the ultimate place of the unknown, a point where the finite mass of the dead star is compressed to a zero volume, giving the point infinite density and therefore a spacetime with infinite curvature. Here, our physics conventions truly break down.

But for now we’re going to focus on the event horizon. In 1970, physicists discovered that the surface area of the event horizon increases whenever matter or radiation falls into the black hole, and if two black holes collide and merge, the surface area of the resulting event horizon is greater than the sum of original areas. This discovery implied there is a similarity between the event horizon of a black hole and the concept of entropy in thermodynamics. Entropy is a measure of the disorder of a system, i.e. the lack of knowledge about its precise state. The first law of thermodynamics states that a small change in entropy is accompanied by a small change in the energy of that system - they are proportional - and the factor of proportionality is temperature. James M. Bardeen, Brandon Carter and Stephen Hawking considered the similarity between event horizon and entropy, and found a law for black holes that was similar to the first law of thermodynamics. It related a change in the mass of a black hole to a change in the surface area of its event horizon, where the factor of proportionality was surface gravity.

Surface gravity is the strength of the gravitational field at the event horizon. It is the same at all points of the event horizon, just like temperature is the same everywhere in a body at thermal equilibrium.

According to the second law of thermodynamics, which states that the total entropy of a system cannot decrease, black holes must have entropy. Otherwise, one could throw an object into the black hole, where its entropy would become zero, thus decreasing the total entropy in the universe. But what did the ‘entropy’ of a black hole refer to? It was originally unclear how the definition of entropy, as a lack of knowledge about a system’s state, applied to a black hole. Jacob Bekenstein resolved this using the no-hair theorem. The idea that a black hole ‘has no hair’ refers to how the final state of the black hole is characterised only by its mass, angular momentum and electric charge. Many different arrangements of matter, therefore, could have collapsed to form a given black hole. A red supergiant full of nickel and a wriggling ball of red pandas are very different, but could both produce the same black hole. Bekenstein said that the logarithm of the number of possible initial states could represent the entropy of the black hole. Note the connection between entropy and information: information measures how much you have eliminated the number of possibilities of a system’s state, while entropy measures how many possible states there are.

Already you might notice a problem. A given black hole, with its limited number of defining characteristics, might have been formed from many different initial states. Does this mean black hole formation is not time-reversible? Does it already contradict our beloved unitarity? Actually, it’s fine for now, because the ‘lost’ information about the black hole’s original state could be hidden somewhere inside the black hole, unable to be accessed by outsiders, but still in existence.

Unfortunately, the finite entropy-surface area correspondence implies a finite temperature-surface gravity correspondence, which implies that a black hole can be at thermal equilibrium at a non-zero temperature. This sent off alarm bells in the minds of many physicists of the time.

Fortunately, in 1974 Hawking resolved this paradox through his discovery that black holes emit particles at a steady rate, just like an ordinary hot body. He explained the process in his 1977 journal article ‘The Quantum Mechanics of Black Holes’. Quantum mechanics suggests that virtual particle-antiparticle pairs are constantly popping into existence then quickly annihilating with each other. They’re called ‘virtual’ as opposed to ‘real’ particles because they cannot directly be detected. Sometimes virtual particle-antiparticle pairs materialise near the event horizon of a black hole. They could both fall in, or both escape to infinity, or one particle could fall in while the other escapes to infinity. Escaping to infinity means the particle travels to outside the gravitational range of the black hole, so will not be pulled in. The particle that does fall in, i.e. the lost particle, could be interpreted as a particle coming out of the black hole but moving backwards through time, then, at the point of original materialisation, the gravitational field causes it to move forwards through time. The black hole is essentially emitting particles. This is Hawking radiation.

Hawking radiation can also be derived by applying normal quantum theory to matter in the curved space-time of a black hole. It is therefore a beautiful result of quantum physics, general relativity and thermodynamics.

As the black hole emits particles, its mass and size decreases, so the thickness of the event horizon barrier decreases, so it becomes easier for particles to escape. This means that the rate of emission increases until the black hole completely evaporates, leaving only Hawking radiation behind.

Now we reach the problem. Theoretically, any particle with a certain energy, spin and charge could be emitted as Hawking radiation, so a black hole could spit out a perfect replica of the astronaut who fell in, or a collection of red pandas. Every quality other than energy, spin and charge is random and equally probable. This results in Hawking radiation being nearly entirely thermal, because the largest number of combinations corresponds to a thermal emission spectrum. Particles have a fixed energy (and thus mass, through the energy-mass equivalence of e=mc2), a fixed spin (i.e. angular momentum) and fixed electric charge because these quantities are coupled to long-range fields, so the black hole preferentially emits certain particles depending on these quantities.

This means that after a black hole has completely evaporated, the remaining Hawking radiation tells you only mass, charge and spin. Nothing about what produced the black hole or what fell in since. Many different stars and falling objects could have become this Hawking radiation. Many different initial states could have become this final state, so the process is not time-reversible. Quantum information about the states of particles before they entered the black hole seems to be lost - not temporarily locked inside the black hole, but permanently gone from the universe. Black hole evaporation seems to break the principles of unitarity.

Let me summarise. When particles escape from a black hole via Hawking radiation, they only contain information on the mass, spin and charge of the black hole’s original material. Other information, that is needed to reconstruct the black hole’s past, seems to be lost permanently. This breaks the fundamental principle of unitarity which says that total information must be conserved, creating a paradox. Because the physical theorems involved do not work together, it seems that we have got something wrong about these important laws, or are missing a deeper truth beneath them.

Join me in the next episode for part two on the black hole information paradox.

Thank you for listening.

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Lana Howell

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