Black hole information paradox (part two)

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. 

This 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.

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Transcript

Black hole information paradox (pt2)

Hello everyone, welcome to the eleventh 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.

Welcome to part two on the black hole information paradox.

Before exploring its solutions, let me summarise the paradox itself. 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. 

Conservation of information (i.e. unitarity) underpins much of our current understanding of the natural world. If physicists have to abandon it, there will be serious consequences for earlier work. Many laws will have to be reconsidered in this new context. Therefore, most of the proposed solutions to the black hole information paradox aim to explain how information is conserved, rather than how it can be lost.

Gerard t’Hooft first proposed the holographic principle in 1993, then in 1997 Leonard Susskind developed it within the context of string theory. String theory, one of the main contenders for a theory of quantum gravity, suggests that every particle is a one-dimensional, vibrating string, whose different vibrations determine the type of particle. The holographic principle describes a theory of quantum gravity where all the information about a volume of space is completely encoded in its boundary. Think of it like a sphere with a map plastered on its interior, and this 2 dimensional map describes all the 3 dimensional happenings inside the sphere. This should sound familiar: Hawking and Bekenstein suggested that the entropy of a black hole is encoded in the surface area of its event horizon. 

In 1998, Juan Maldacena further used the holographic principle to conjecture AdS/CFT correspondence. AdS/CFT correspondence describes a holographic mapping between a quantum theory of gravity, specifically string theory, and a non-gravitational quantum field theory. On the AdS side you have the theory of quantum gravity which exists in an asymptotically Anti de-Sitter space, i.e. a hyperbolic space with constant negative curvature that obeys a set of conditions. CFT stands for conformal field theory, which is a type of quantum field theory that behaves the same at all scales. Quantum field theories are a way of combining quantum mechanics with relativity in order to describe how particles interact using fields. Importantly, the CFT lives in one fewer dimension than the AdS quantum gravity, and the mapping between them is holographic. Put simply, a lower-dimensional CFT without gravity lives at the boundary of a region of space with gravity (a gravity which can be described using quantum mechanics), and the CFT encodes all the information of the space within it. AdS/CFT correspondence has passed many significant tests e.g. symmetry groups and entanglement calculations. 

The correspondence is used to show how information is indeed conserved during black hole evaporation. Regardless of gravitational effects happening in the black hole’s interior, its boundary, without gravity, holds the information. But what I find most exciting is its generalisation to our entire universe. Perhaps we, and everything around us, are holograms of information encoded in the 2-dimensional boundary of the universe. In a lower dimension, many particles are interacting in complex and intricate ways, maybe as vibrating strings, and they project their emergent phenomena, including gravity, into this 3 dimensional universe. Our consciousnesses interpret our world as concrete and physical, because we evolved that way, but in reality, everything is holographic. 

Of course this is all highly theoretical. AdS/CFT correspondence is useful when studying black holes, and systems without gravity, but our universe is probably not an Anti de-Sitter space, because the negative curvature of an AdS space would produce an attractive universe, whereas our universe is expanding. Despite the lack of AdS/CFT’s applicability to our universe, there is a lot of work around a holographic universe. Some physicists are investigating irregularities in the Cosmic Microwave Background in order to find evidence.

Since the 1998 proposal of AdS/CFT correspondence, most physicists believe that information is preserved in black hole evaporation, but there are conflicting views as to how. Considering the relevance of quantum gravity in the paradox, it is unsurprising that the dominant views are split between the two main groups of quantum gravity theorists: string theorists and the loop quantum gravity community. Most string theorists think that Hawking radiation is not precisely thermal, i.e. extremely small details of the radiation encode information about the black hole’s interior. This perspective places more importance on the event horizon than the singularity. In contrast, loop quantum gravity theorists focus on resolving the problems of the singularity. Their solutions are often called ‘remnant scenarios’ because in them, information does not leave the black hole gradually, but remains in the interior until the very end of evaporation.

Let’s take a quick look at some specific solutions to the paradox. The small-corrections resolution suggests that fine-grained features of radiation are lost in calculations but ultimately preserve information. The strong-quantum-effects resolution says that an understanding of quantum gravity is needed to understand the final stages of evaporation, where the paradox will unravel. The soft-hair resolution (physicists like talking about a black hole’s hair!) suggests that ‘soft’ particles with no rest mass, like photons and gravitons, store information. Some physicists believe that information is irretrievably lost; others that Hawking radiation stops just before the black hole reaches the Planck size and information remains inside the leftover large remnant. Still others think that information is stored in a ‘baby universe’ separate from ours, information is encoded in correlations between the past and future, or information is transmitted through quantum channels. 

There exists an enormous variety of potential solutions. None are conclusive.

In some areas, there have been exciting developments, such as work led by Ahmed Almheiri using quantum extremal surfaces. First, though, you need to know about the Page curve. In the 1990s, Don Page proposed that quantum entanglement could solve the black hole information paradox. Quantum entanglement is when two particles are connected in such a way that any measurement done on one instantly reveals information about the other. According to Page, quantum entanglement could encrypt information so that the emitted Hawking radiation maintained a link to the black hole’s interior. Measuring only the radiation wouldn’t reveal enough information to reconstruct the black hole’s past, but considering the radiation, the black hole, and how they were entangled, could. ‘Entanglement entropy’ represents the total amount of entanglement between the black hole and its radiation. Before evaporation begins, the entanglement entropy is zero, and by the end, the black hole has fully evaporated, so it is zero again. In between, the entanglement entropy follows the Page curve. This is an inverted V shape. It rises, then at the Page time, about halfway through the black hole’s life, it starts to drop.

But there is no reason why the curve should drop at that point. Halfway through its life, a black hole is still very big, and on those macro scales, it should keep obeying the normal laws of physics. Physicists generally believe that strange laws of quantum gravity only take effect in extreme situations, like at very high energies or in very small spaces. Somebody needed to calculate entanglement entropy and see if it followed the Page curve.

Then somebody did. Work, beginning in 2018, led by Ahmed Almheiri of the Institute for Advanced Study, used AdS/CFT correspondence and the concept of a quantum extremal surface to calculate entanglement entropy over time. Think of the quantum extremal surface like a bubble within the black hole, separating its interior into two sections. The team’s calculations showed that early in the evaporation process, entanglement entropy rose (as expected from Hawking’s calculations) but then a quantum extremal surface materialised, leading to a drop in entropy. It matched the Page curve! 

This conclusion relied on the AdS/CFT correspondence, so next they considered black holes more generally. Without going into the highly complex maths, a path integral can be thought of as being the weighted sum of all paths that can be taken, perhaps by a particle, through space-time. In classical mechanics, a particle goes from point A to point B along one path. But in quantum mechanics, it takes many paths at once. Path integrals are hugely useful in the study of quantum mechanics. A gravitational path integral is used to describe gravity. Its definition and use is contentious, but you can think of it as describing the variety of possible shapes that space-time can take. The shape of space-time is responsible for gravity.

A very twisted shape, produced from the gravitational path integral, could produce wormholes between replica black holes. Replicas are a mathematical trick to simplify computations, but the gravitational path integral does not distinguish real black holes from replicas. Information could escape a single black hole as radiation through these wormholes. Ultimately, the team’s calculations mapped the Page curve for a more generalised black hole, and heralded a possible end to the black hole information paradox. Unfortunately, many physicists dislike how the team used so many idealisations and approximations, and warned from over-interpreting the replica trick. As exciting as the concept of wormholes are, the black hole information paradox is still very much unsolved.

Over two episodes, I started by defining information in quantum mechanics as the descriptive properties of a particle, which, when accessed, reduces the number of possibilities for that particle’s state. The fundamental principle of unitary states that total information in a system - the universe is also a system - must not decrease. If information decreased, you would not be able to predict, even in theory, the past states of the particles in that system, and you should always be able to do this. In black holes, this breaks down. The Hawking radiation leaving the black hole - initially found to allow the black hole’s thermodynamic properties - seems not to encode enough information for anyone to predict, even in theory, what fell into the black hole. You cannot reconstruct the past of the matter escaping the black hole once it has completely evaporated.

In response to the dreadful implication of the combination of general relativity and quantum mechanics inside black holes, the AdS/CFT duality, an extension of the holographic principle, was conjectured. It allows information to be conserved on the black hole’s boundary, regardless of how information might behave in its interior. However, the AdS/CFT duality is a tool, not a solution: it is mainly used to investigate quantum effects in non-gravitational spaces. 

There are a lot of proposed solutions to the black hole information paradox, all with nuanced mathematics and physical concepts. Some of the biggest ideas include quantum corrections and remnant scenarios. Recently, progress has been made on calculating entanglement entropy in order to reproduce the Page curve, and that work has led the research team to gravitational path integrals and wormholes. No solution has been conclusive. But with each, we get a glimpse of new physics, new aspects of our reality.

Black holes are some of, if not the most, mysterious objects in the entire universe. They are a collection of extremes - size, density, gravity - pulling physics to beyond its limit, breaking our dearest laws and reminding us how much there is left to understand. It is ironic, then, that this reminder of our ignorance comes in the form of an information paradox. 

Thank you for listening.


References:

• https://arxiv.org/pdf/1501.00007.pdf
• https://www.quora.com/What-is-AdS-CFT-correspondence-intuitively
• https://www.sciencedaily.com/terms/holographic_principle.htm
• Holographic principle - glossariumBITrihttp://glossarium.bitrum.unileon.es › principio-holografico
• https://www.britannica.com/science/quantum-field-theory
• https://physics.stackexchange.com/questions/122540/why-are-anti-de-sitter-spaces-so-interesting-when-we-believe-the-universe-is-exp
• https://www.sciencedaily.com/releases/2017/01/170130083231.htm
• https://blogs.scientificamerican.com/observations/have-we-solved-the-black-hole-information-paradox/
• https://en.wikipedia.org/wiki/Black_hole_information_paradox
• https://www.quantamagazine.org/the-most-famous-paradox-in-physics-nears-its-end-20201029/