I finished reading Three Roads to Quantum Gravity (2001), by Lee Smolin, a theoretical physicist whose general sensibility I’ve always appreciated. I don’t always agree with his ideas, but I like the thoughtful way he expresses them. Smolin brings some philosophical thinking to his physics.
While he added a lengthy Postscript to the 2017 edition, the book is outdated both by time and by Smolin. In 2006 he published The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next, which explored issues in the practice of theoretical physics. But in 2001 he still thought string theory was (at least part of) The Answer.
Almost none of which is the subject of this post.
What is the subject is an idea Smolin discusses at length, the fairly recent result that the amount of information a volume of space contains is limited by how much information the surface of a boundary around that space can contain. It’s called the holographic principle.
The notion descends from, and is related to, thermodynamic notions about black hole entropy. The basic idea is simple. We describe black holes with only a few numbers: mass, spin, charge. As such, they were initially thought to have no entropy, since those numbers tell us everything we can know about them.
But if objects with entropy (such as hot gas) fall into the black hole, then it appears the entropy is lost and the overall entropy of the universe goes down — in violation of thermodynamics.
Jacob Bekenstein showed that black holes have maximum entropy — the most entropy that could be contained in the volume of the event horizon. He further showed that the area of the event horizon is proportional to the entropy, which gives us a way to know how much entropy is stored in the black hole.
This leads to the black hole paradox and eventually to string theory where it’s seen as a correspondence between, for instance, a 3D reality and a 2D surface surrounding that reality. (Or a 6D surface around a 7D reality.) The theory suggests our 3D universe could be a surface of a black hole we’re inside of.
Smolin, in 2001, suggests string theory is nascent and necessarily naïve. He compares the situation to how Einstein updated the equally naïve view of Newton’s that space and time are absolute (and separate). He feels that as string theory succeeds, it’ll evolve to a mature theory.
[The basic problem is that string theory is background dependent, but the consensus is that a true theory must be background independent. Newton is likewise background dependent while Einstein’s General Relativity is not.]
So the holographic principle means the surface has as much information as the volume it encloses.
I’d never given that much thought other than to be vaguely puzzled by it. But Smolin’s discussion got me thinking,… and now I’m very puzzled by it.
I mean… how does that work?
Imagine a sphere that encloses… something… and divide the surface of that sphere into tiny areas on the Planck scale. Call them bits or pixels.
Each bit is the smallest possible region of information. Reality doesn’t subdivide below that size.
Smolin likens it to a screen through which we view the something inside. Clearly any information we get (or send) is limited by the area of the screen. Smolin even calls the screen an information channel, and in that regard a limit to transmission makes perfect sense. But why does that limit the information the source can contain? Why can’t I receive a stream of data that sums to more than the channel can transmit at once?
Smolin also compares the screen to a computer display that’s programmed to give us a representation of the something inside — the point being that the something need not actually exist; the information could be entirely on the screen.
Which makes me very confused because the analogy seems directly contrary to the premise that the surface area contains all possible information about the volume.
If the screen is programmed to display something, and that display is only showing the something, where is the logic controlling the display? Isn’t that also inside the volume? What makes those pixels act like they do? Isn’t that extra information not shown on the screen?
So, firstly, I don’t see why the surface has to have all the information about the volume simultaneously. Why can’t it be sent out serially? Imagine the image on the screen rotating to present a new view.
(One analogy that occurs to me is of a memory chip. It may contain billions of bits, but it only has tens of connections into the chip. All those bits aren’t available at once, they have to be read out sequentially.)
Secondly, a Flatland that’s equivalent to a 3D space seems to lack the physical program to explain how pixels interact. Smolin’s analogies all seem to require additional structure (information) inside the horizon.
Thirdly, I don’t understand how 3D volume is accounted for. If I imagine a series of tightly nested spheres — as close together as the Planck areas — it seems the possible information has to exceed the surface area.
Imagine rays from each pixel to the center. The holographic principle seems to suggest those rays — which are as long as the radius — can only be one bit of information in the volume. The ray can’t have multiple bits along its length, because those would need pixels, and each pixel is already taken by a ray.
I do wonder if part of it is that not that every Planck volume inside the black hole corresponds to a Planck region on the surface. It may be that the surface bits associate to larger-grained volumes — particles, which are vastly larger than Planck volumes.
If the surface areas are “fully occupied” — each describes something different — the volume necessarily contains a black hole. If the something inside is not a black hole, then there is plenty of surface area to describe it.
It occurs to me this could depend on the black hole being a singularity with no real volume of its own. Then it makes sense the area of the event horizon relates to how much information fell in. The rays would go all the way to the center without hitting anything else.
There’s a lot more to Bekenstein bounds and horizons, but that’s food for another meal.
Stay unbounded, my friends! Go forth and spread beauty and light.