511 karma · joined July 21, 2012
ben@ben-johnson.me
That's another instance where the holographic principle applies. To hand-wave, the holographic principle applies to any theory describing a system, where you can make an equivalent description of the system using only information on its boundary. For example, in the black hole case, if you know the state of the system inside the event horizon, every particles' position and momentum in 3D, then you could describe how the state will evolve based on the usual laws of physics.
But by definition, that information is unavailable to an observer outside the event horizon. This presents a thermodynamic conundrum. We take as an axiom that in a closed system, total entropy can never decrease. Think of entropy as the compressibility of information. A string of a thousand 0 bits followed by a thousand 1 bits is highly compressible (low entropy/high order), while a perfectly random sequence of bits cannot be compressed at all (high entropy/low order). Equivalently in the physical world, an egg has somewhat low entropy, yolk encased in egg white encased in shell, a complete description is succinct. But if you scramble the egg, suddenly full knowledge of its state requires lots of information, you have to know the specific positions of every particle of yolk and white to have the whole picture.
We consider this an axiom because we want physical law to obey time-symmetry. In a deterministic universe, you should be able to take a full physical state and "run it in reverse" to get previous states. If you could decrease entropy, go from a globally random state to a globally ordered one, then you would lose information about the starting conditions. Just like matter and energy, information can never be destroyed (we suppose).
Okay so the universe taken as a whole is a closed system, with ever increasing entropy, i.e. over time more bits are required to understand its full state. Now toss your scrambled egg into a black hole. It crosses the event horizon and poof it's gone. Is information lost? Can we, in principle, run the universe in reverse and see an egg come back out? Where did those bits go?
The interesting thing about black holes is that all the information required to understand their behavior right there on the boundary of the event horizon. Specifically, the number of distinct physical states that a black hole can take on is proportional to its surface area, not its volume as you'd expect for an spherical region of ordinary space.
How exactly does that work? At this point my understanding sadly falls off. But I can point to this as a good starting point for the so inclined: https://en.m.wikipedia.org/wiki/Black_hole_information_parad...
The Ads/CFT correspondence is also a bit above my head, but it relates string theory to quantum field theory. It introces the anti-de Sitter space, which is a space in which geometry is non-euclidian, very similar to a hyperbolic space, where the boundaries are like asymptotes, you can get as close as you want but never reach it. The interesting thing is the geometry of that boundary space. For an appropriate Ads, its boundary has the same geometry as our 4D spacetime (it's a Minkowsy space, to be precise). The idea is that low-dimensional string-theory stuff happens in the interior (bulk) of the Ads space, and the boundary is the spacetime we know and love. The correspondence is that a complete theory of the boundary (QFT describing spacetime) can also completely describe the action in the bulk, and vice-versa. The mind bending thing about an Ads is that it's boundary is actually higher dimensional than the space as a whole. Analogous to how in linear algebra, you can project a space into a higher dimensional space without losing information.
Wow didn't mean for that to be so long. If anyone knows better than me and I've made a mistake please do point it out, I'm just a casual observer who loves getting in deep with this stuff.
EDIT: Oops, just now went back read your question and realized you probably knew all of that and I didn't even answer your actual question. Hopefully someone else finds this interesting! Specifically about the surface area thing, when something falls into a black hole from an outside view you never actually see it cross, as gravity increases on the object from your point of view it undergoes time dialation, redshifting the light it emits, and length contraction, flatting it along an axis normal to the black hole's surface. In the limit the object falls slower and slower, getting flatter and flatter, and dimmer and dimmer and ultimately "smeared out" across the surface, Hawking radiation emitted from some kind of virtual particle interaction I don't understand interacts with the object on the boundary, making the information recoverable from those interactions. Or something like that