Monday, December 15, 2014
From Plato's Cave to the Holographic Principle
by Tasneem Zehra Husain
Remember Plato's allegory about the cave? Prisoners, chained inside a cave, sit facing a blank wall with a fire lit behind. All they know of the world is through shadows cast on the wall, by whatever it is that moves between them and the fire. The entirety of their knowledge is constructed from observations of these moving silhouettes. For them, reality consists of flat images, devoid of color and and (three-dimensional) form.
But of course the fallacy must be exposed, and so one prisoner somehow breaks free of his shackles. He turns and sees the fire, and the objects that cast the shadows. Suddenly, he is confronted with things far more complex than he could have conceived, with qualities he lacks the vocabulary to describe. Should he venture out of the cave, his confusion and disorientation increases by several orders of magnitude. Bathed in light and color, he is assaulted by the unfamiliar sensory richness that surrounds him. Were he now told that he had been harboring a delusion his entire life, and that this is in fact reality, he would have a hard time wrapping his mind around it.
The point of this story, of course, is that we are prisoners of our experience. Imagination helps us explore extrapolations and combinations of the familiar, but what if there are things that lie beyond our ken? Who's to say that what we perceive isn't just a sliver of the whole truth? Plato's millennia old allegory remains relevant, because even now we are haunted by the insecurity that we might be missing out - that the universe is more than we can know. So here's an interesting twist: what if our perception adds a dimension, instead of slicing it out? How could that happen? Let me give you an example.
About twenty years ago, stereograms were all the rage. On the surface, these ‘Magic Eye' pictures were merely repeated patterns, but if you stared at them long enough and in the right way, a three-dimensional image would pop out of the paper. In case you haven't seen these before, here's one you can practice on.
The only advice I have to offer, if you're new to this, is that it generally works best if you hold the paper (or screen) relatively close to you. Beyond that, it just takes some patience. As with all illusions, once you've seen through it, it's much easier the second time around. (Hint: this particular stereogram hides a single word.)
But how do these images work? The answer lies in the way we perceive depth. As we look out onto the world, both our eyes form individual images, from their own spatially separated viewpoints. (To compare the difference between the image formed by one eye and the other, try holding a pencil up, a foot or so in front of you. First close one eye, and then the other. The pencil appears to move.) The brain processes both these images and combines the information to form a judgement about depth.
Stereograms create the illusion of depth by tricking the brain. Because of the repeated patterns, the eyes might each be looking at two distinct points, but be confused into thinking that they are the same. The brain, as it processes the images from each eye, assumes those two points should overlap and, as a result, conjures up an illusion of depth. So human physiology leads us to add a perceived dimension, even when it is not physically present.
Holograms are another example. Even though the mass produced ones on currency and credit cards aren't the most impressive specimens of their kind, they do give you the general idea - a three-dimensional image seems to emerge from the card (or bank note), and often the image changes as you look at it from different angles (giving the illusion of ‘truly' being three-dimensional rather than flat.) Holograms are produced in various ways, but the general principle is that, in addition to the light rays reflected off the surface of an object (which is what we traditionally capture in a ‘flat' photograph), the interference pattern of these reflected rays is recorded as well. The combination of the two enables us to reconstruct a solid form: the hologram.
Of course there are the ‘Star Trek' type holograms as well, which are generated by lasers and not printed on paper, but a detailed explanation of how those are constructed would take me further afield than I want to go right now. The point of all this is simply to convince you that the manner in which information is encoded determines, to a large extent, the way in which it is interpreted. And the reason I want to convince you of this, is so I can introduce you to the holographic principle.
One of the more daring and fascinating conjectures of recent years, the holographic principle does not, contrary to popular opinion, say that the world is just a holographic illusion. It makes a more subtle and interesting statement than that.
The first hints came from black holes. Black holes, as you know, are massive sinks in space-time. Like giant vacuum cleaners, they suck in everything - including light - that comes within their reach. The boundary that defines the extent of a black hole's reach is called a horizon. Outside of this, we are safe, but once we venture onto the boundary, we have no choice left - falling in is the only option. There are so many wonderful puzzles and mysteries pertaining to black holes that we could happily spend years in contemplation (and many physicists do!) but for our present purposes, we just need to know a few basic facts, which I will lay out now.
Black holes emerged as an initially unwelcome solution of Einstein's general relativity. To describe a black hole completely, we need to specify only a few external quantities - its mass, electric charge and angular momentum. These cosmic shredders destroy all the information that ventures past their event horizons, rendering it all equal. All memories of whatever it was that fell in, are eviscerated. And since black holes can not be distinguished by their internal contents, they have very high entropy.
Entropy essentially counts the ways you can rearrange the constituents of a system, without changing the overall picture. The more specific and ordered a system is, the easier it is to spot a switch. So, a high degree of order corresponds to low entropy. In systems that are chaotic and random, such exchanges often go unnoticed, leading to the oft-quoted statement that entropy is a measure of disorder. Since all identifying markers are ripped off the particles that fall inside them, black holes are systems with extremely high entropy. In fact, while they are conventionally associated with inescapable gravitational fields, black holes can equally well be defined as objects with maximal entropy.
For ordinary objects, entropy appears to scales with volume, which sounds about right; as volume increases, you have more space in to which information can be stuffed. But black holes forge their own rules. About forty years ago, Hawking and Bekenstein discovered that the entropy of black holes was not proportional to their volumes, but instead the area of their horizons. So, the maximum entropy that can theoretically be crammed into an object is determined by its boundary and not its interior. It is as if the information inscribed upon the walls of a room suffices to recreate, in complete detail, all that unfolds within it. Doesn't that sound familiar?
Inspired by this discovery, the holographic principle says that the information contained within a volume can be represented as a hologram of the surface. Notice that this does not imply, in any way, that what unfolds within the room is an illusion, or that the information encoded on the walls is in some way more fundamental than that which fills the room - the statement is simply that the two descriptions are equivalent.
Here's the really thrilling part: If this idea is true, the physical laws that play out in our universe should be encoded somewhere in a ‘bounding' space that has one less dimension. [For a square, the ‘enclosed volume' is the 2-dimensional quantity we commonly refer to as area, and the boundary is the 1-dimensional (length of the) perimeter; for a cube, the volume is 3-dimensional, and the boundary - the sum of the areas of all six faces of the cube - is a 2-dimensional quantity. This relationship continues in higher dimensions. If the enclosed volume is n-dimensional, the boundary will be (n-1) dimensional.] So, it should be possible to formulate a dual set of physical laws in a world with one fewer dimension, that manage to fully describe our universe. In form, these laws will not necessarily be similar to those we commonly use, nor will the lower dimensional universe be a trivial (or obvious) ‘restriction' of ours - a stereogram does not, at first glance, look like a two-dimensional object we recognize - but all the necessary information about our higher dimensional goings-on will be contained therein, and there will be a definite way to extract it.
Apart from the fact that the idea has undeniable aesthetic and intellectual appeal, it could also be extremely useful. This is not universally true, but as a general rule of thumb, it is simpler to solve a system of equations in lower dimensions than in higher. So, if we come up with holographic reformulations of physical laws as we experience them, it might well be that these new equations are simpler to solve than those we currently know, and we might be able to solve problems that, in their current form, have us stumped.
The holographic principle was just a conjecture until about two decades ago when Juan Maldacana, with his newly minted PhD, came up with a beautiful concrete realization. Describing it properly would require introducing elements of string theory, which I don't have the space to do here, and so I will do something better - point you to Maldacaena's own explanation. Schematically, what he did was to show how physics in a five-dimensional (saddle-shaped) universe, with gravity, is equivalent to physics in a four-dimensional universe, without gravity. Many problems that were difficult to solve in one formulation could be handled in the other, and their results ‘translated' back to the first.
Of course neither of the universes Maldacena postulated are ours. They are models, mathematical toys, but they are a proof of concept. Work on this conjecture continues, and several other examples were proposed in the years that followed. There is, however still some way to go before we arrive at a map that connects a universe that is recognizably ours, to a less familiar, mysteriously dual, lower dimensional spacetime. But we continue, tantalized by the thought that the laws of physics could have been written - perhaps more elegantly than they appear to us - in fewer dimensions than we experience.
Posted by Tasneem Zehra Husain at 01:10 AM | Permalink