In the conclusions to their paper, AMPS mention another way of seeing this information storage limit in action. Why don’t similar arguments imply that firewalls should appear at the Rindler horizons we met in Chapter 3 for accelerated observers? The answer is that Rindler horizons have infinite area and entropy, unlike a black hole horizon, and therefore ‘their quantum memory never fills’. Rindler horizons, in other words, never evolve to become old and can always support any information demands made of them to preserve the integrity of the vacuum.
Holography offers a means to save the Equivalence Principle and render the horizon safe, while also saving quantum mechanics and preserving information. The basic idea is that since the interior of the black hole is dual to the exterior; the early Hawking radiation, R, and the interior particles, A, are really the same thing. Crazy as it sounds, this is the way the firewall problem is avoided in holography. In processing the early Hawking radiation, R, to check its entanglement with B, Bob inadvertently destroys the entanglement with A. This creates a kind of mini-firewall that is just violent enough to prevent Alice from measuring entanglement between A and B, but not so violent that it destroys the interior of the black hole.
The world as a hologram
Spacetime holography was first presented by Gerard ’t Hooft in 1993, and further developed a year later by Leonard Susskind. They presented it as an integral part of their black hole complementarity idea but stressed that it probably ought to be of more general applicability. That’s to say, holography should be a universal feature of Nature, regardless of the black hole issues that originally motivated it. The holographic principle as currently understood even goes so far as to suppose that the entire world as we perceive it is a hologram.43
A hologram as conventionally understood is a representation of a three-dimensional object constructed from information stored on a two-dimensional screen. If you’ve ever seen a hologram, you’ll know that they can look remarkably real. You can walk around them and view them from all angles as if they were the real three-dimensional objects themselves. Now imagine a perfect hologram. What sort of a thing would that be? A hologram would be perfect if every bit of information necessary to reconstruct the three-dimensional object was also encoded on the two-dimensional screen that carries the holographic data. This is reminiscent of the Bekenstein entropy of a black hole, which says that the information content of the black hole can be computed by considering only the two-dimensional surface of the event horizon.
Now, as we discussed in Chapter 9, a black hole has the largest possible information density of any object and, since the information it stores is given by the surface area of the horizon, it follows that there can be no more information inside any region of space than can be encoded on the boundary of that region. This realisation led ’t Hooft and Susskind to argue that the information content of any region of space is encoded on the boundary to that region. The reason we discovered this first by thinking about black holes is that the boundary is exposed for anyone hovering outside the black hole to explore, in the form of the hot membrane close to the event horizon. In everyday life, well away from any black holes, it is rather less obvious how we could gain access to this holographically encoded information since we cannot ‘cut out a piece of empty space’ to reveal that the information in the interior is encoded on its surface.
Holography, then, is a perfect example of complementarity in action. There are two entirely equivalent descriptions of anything and everything, and this is an essential feature of all of Nature and not just black holes. Black holes are the Rosetta Stone, which has introduced us to a new language; an entirely different yet perfectly equivalent description of physical reality. One description resides on the boundary of any given region of space, and the other resides more conventionally in the space internal to the boundary. The implication is that our experience and existence can be described with absolute fidelity in terms of information stored on a distant boundary, the nature of which we do not yet understand. This sounds utterly bonkers, but clinching evidence supporting the idea comes from the most highly cited high-energy physics paper of all time.
Maldacena’s world
Scientific citations count how many times a research paper has been referred to in the literature. Naturally enough, the most important papers tend to get the most citations. Ranked number 13 of all time§ is Stephen Hawking’s 1975 paper, ‘Particle Creation by Black Holes’. The discovery of dark energy is up there, with the two key papers reporting the evidence ranked 3 and 4, and the papers announcing the discovery of the Higgs boson at the Large Hadron Collider are ranked 6 and 7. Top-ranked of all time is a paper written in 1997 by Argentinian physicist, Juan Maldacena titled, ‘The Large N Limit of Superconformal Field Theories and Supergravity’.44 With almost 18,000 citations to date, it is the paper that has, more than any other, changed the face of theoretical physics over the past 25 years. It is also the paper that provides the strongest evidence supporting the idea that the holographic principle is true.
The universe Maldacena considered is not the one we live in, but that’s fine. It is common for physicists to build models of the world with some simplifying features. The real world is complicated, and it’s often useful to do calculations in a pretend world in which things are simpler. The skill is to pick a simple world which delivers enhanced understanding while not being too unrealistic. Engineers make simplifying assumptions when designing things like aircraft and bridges, notwithstanding the fact that the stakes are rather higher. Importantly, Maldacena’s world was not specifically chosen because it supports holography. Rather holography was a feature that popped out of the mathematics.
Figure 13.2. The Poincaré disk projection of a two-dimensional hyperbolic space. Despite appearances, the solid line from A to B on the left figure is shorter than the dashed one. You can see this by counting triangles. On the right is M. C. Escher’s Circle Limit I. All the fish are of the same size and shape, and the lines are shortest lines. The patterns provide us with a visual representation of the metric of the space (we can ascertain distances by counting fish, or triangles) just like the squares on graph paper illustrate the metric of Euclidean space. (Right: M.C. Escher’s Circle Limit I © 2022 The M.C. Escher Company-The Netherlands. All rights reserved. www.mcescher.com)
We can capture the essence of Maldacena’s work by considering a two-dimensional toy universe.¶ The space of the toy universe does not have the geometry of ordinary flat space; rather its geometry is hyperbolic. Figure 13.2 shows a beautiful representation of two-dimensional hyperbolic space, known as the Poincaré disk. This projection was widely employed by the Dutch artist M. C. Escher, and we also include his well-known Circle Limit I. Rather like a Penrose diagram, the space represented in these projections is infinite and there is a great deal of distortion to bring infinity to a finite place on the page. Escher’s fish, for example, are all the same size as they tile infinite hyperbolic space. They appear smaller towards the edge of the disk, which represents infinity, because we are shrinking down space as we head outwards from the centre. The Poincaré disk projection is also a conformal projection, which means that the shapes of small things are faithfully reproduced (for example, the fish-eyes are always circular).