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While he added a lengthy Postscript to the 2017 edition, the book is outdated both by time and by Smolin.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.

∇

September 6th, 2021 at 7:56 am

Something a little different today: A triple header!

September 6th, 2021 at 8:25 am

As I understand it, all the theories Smolin discusses are quantum theories (correct me if that’s wrong!), so I think it’s best not to think in terms of discrete “bits” in the mathematics, even if that might be what Smolin says in a popular account. It’s qubits and other Hilbert spaces and a continuum of states. Measurement results can be discrete valued, 0 or 1, in a quantum theory, but states are not.

Holography is a very strong constraint on the mathematics used to model the world, which I think of as quite closely analogous to complex analyticity. There is no empirical reason for such a strong constraint on the mathematics, but if we assert it

a priorithen, indeed, in such models the state on a boundary determines the state everywhere else. I find it hard to see that, however, as more than a claim about the models.September 6th, 2021 at 9:04 am

Smolin’s work is in string theory, LQG, and quantum gravity in general, so he’s definitely on the quantum side of things. The holographic principle is a huge deal with AdS/CFT, so yeah, it does seem we’re in the quantum realm. That said I’ve never

heardanyone refer to qubits when talking about either the Bekenstein bound or the holographic principle. Perhaps it’s just assumed?But bits or qubits, it’s still single pieces of information (at least from what I can gather), and the idea that the surface area of a volume limits the information within the volume is a hard one for me to swallow. The image of rays from each area to the center… or even trying to imagine a nest of tightly packed spheres… in both cases the interior seems capable of containing much more than the surface area.

I’ve decided it must be that many many Planck areas work together to define actual matter which is much less dense than a black hole would be. I keep thinking about the vast difference between Planck size stuff the and atomic nucleus (let alone the whole atom). I further assume that most Planck volumes within the horizon must be empty.

(But what do I know. I only recently realized that the surface area of circles, spheres, squares, cubes, and torii, is the derivative of the volume. It’s been right in front of my face all this time, and I never noticed.)

September 6th, 2021 at 12:26 pm

I meant to add that, from what I’ve read, a lot of this has to do with black hole thermodynamics. (The idea that hot gas falling into one seemed to represent a big decrease in entropy.)

September 6th, 2021 at 8:30 am

From what I understand about black holes, the information on its event horizon is frozen there due to time dilation, at least from the perspective of outside observers. That might be why all of its information can be, has to be, on its horizon. It has to do with its one way causality.

I think the next logical step is to say that the event horizon is therefore the maximum amount of entropy for its given space. Add additional entropy and the overall volume increases along with the surface area of the horizon. You can’t have a space with more entropy, a black hole being the most information dense thing that can exist.

Though the jump from there to the holographic principle is one I’m still pretty blurry on myself.

September 6th, 2021 at 9:15 am

I’ve thought the same thing, about frozen information on the boundary, especially after the idea was a key point in Pohl’s

Beyond the Blue Event Horizon. One consideration, though, is that as a distant observer watches something fall into a black hole, the light we see from it is shifted more and more downwards in frequency/energy. Pohl’s image of Broadhead’s lover forever frozen on the horizon is a false one. The image would have shifted down into infra-red, radio waves, and invisibility as photon energy reaches zero. (Those photons have to work their way out of the gravity well. The closer the horizon, the harder that is.) And the whole thing about the horizon is that someone falling through itoughtto notice exactly nothing.Yes, the Bekenstein bound is exactly that assertion that [A] a black hole contains the maximum amount of entropy any volume can contain and [B] the horizon surface area is directly proportional.

But as you say, the jump to the holographic principle is a puzzling one. And it’s not just black holes. Anyone in accelerated motion has a Bekenstein bound behind them. So long as you are accelerating, there is a region of spacetime where no information from that region can ever reach you. But you have to

keepaccelerating, which is a bit of a trick.September 6th, 2021 at 11:18 am

Hmmm. That condition may be met by the ongoing expansion of the universe. Maybe that’s the rung to the holographic principle? (Still blurry for me though.)

September 6th, 2021 at 12:24 pm

It’s not a whole lot sharper on this end. I never spent any time trying to figure it out. Smolin in this book, and in

Time Reborn(which I’m reading now), talks a lot about how a cosmological theory can’t be from the perspective of some putative observer outside the universe. He avers there is no such observer possible — by definition the universe is everything. Our cosmological theories must include us (and everything else) as part of them.So the putative view from outside the universe that might indeed see the expanding boundary as some sort of 2D surface containing everything in the 3D volume isn’t (according to Smolin) very meaningful. But that is the scenario I’ve heard floated.

The Bekenstein bound seems to be simply that even light can’t catch up to an accelerating body, and there does seem some connection with the expansion of the universe where the most distant points, from our perspective, are moving faster than light. As you know, that expansion will put everything except our galaxy permanently out of sight. Astronomers of the distant future will be correct in assuming they live in an “island universe.”

September 6th, 2021 at 12:44 pm

Okay, now here’s a thought: If light from inside an expanding universe can’t reach the boundary (or vice versa), how can the surface contain all the information in the volume? I think that, as Peter Morgan above suggests, there must be some severe constraints on how and where this applies.

Its connection with string theory makes me wonder if it’s one of those things that’s only true in an AdS space.

September 7th, 2021 at 3:37 am

If entropy is the degree of freedom then how the event horizon can be considered as dense. Entropy is information by Hawkins then what about the entropy of the event horizon These are my doubts as a beginner The answer should be simple not like complex analysis singularity etc because a student should understand it Please do not take the quote of Feynman here. Quantum mechanics is not understandable etc Sorry the author I am not underestimating your writing but my curiosity to learn made me to post like this. The science should be popular Quantum mechanics should not be an exception !!

September 7th, 2021 at 8:22 am

I’m afraid that right now quantum mechanics

isan exception. What little we understand is in the math, but no one knows what that mathmeans. About the only way to gain any understanding right now is by learning the math. There are no simple answers, sorry.I’m not sure I understand your question. Entropy isn’t so much information, but

lackof information — what wedon’tknow about a system. We characterize a hot gas only by temperature and pressure, not by information about every gas molecule. We characterize a black hole by mass, charge, angular momentum. That’s all the information we have about them. But they can hide the information of millions or billions of stars. Thus they are high entropy.The event horizon isn’t really a thing — more a location in space where there is a boundary between your ability to stay in this universe or fall into the black hole. It doesn’t have entropy on its own.

September 10th, 2021 at 9:31 pm

I will disturb you because I want to learn it and teach my student this subject and I will try my level best to popularise it I am behind the maths of QM and successfully studied tensors differential equations Schordinger equation preliminary So in future I will disturb you now and then Thank you so much to find time for us

September 11th, 2021 at 9:11 am

I am always happy to (try to) answer questions!

September 11th, 2021 at 11:34 pm

Thank you