The Shape of Inner Space


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If you read the wikipedia history section on string theory, you’ll see that string theorists (or, at least the writers of the article), tie the historical roots of strings to Einstein’s General Theory of Relativity, and Heisenberg’s work in the 1940’s on the S-Matrix (scattering of particles in quantum mechanics). But, the real work on string theory didn’t start until the late 60’s and early 70’s as a way of explaining Hadrons (subatomic particles that feel strong interactions, like the electron and proton do). A few theorists continued working on the theory through the 70’s and early 80’s, but mostly it was ignored by other physicists until closer to the later 80’s and didn’t pick up real steam until becoming part of M-Theory in the mid-90’s. It wasn’t even mentioned in my physics classes when I was in university in 1983.

So, with all of my other recent ramblings on math and language, I started wondering what the fuss is all about, and I decided to ask for a book on superstrings for Christmas. What I received is Shing-Tung Yau’s and Steve Nadis’ The Shape of Inner Space: String Theory and the Geometry of the Universe’s Hidden Dimensions (Basic Books, 2010). The book starts out as part introduction, part string theory history and part autobiography (talking about Yau’s growing up in China and then going to university to study mathematics). The first 40-50 pages are actually very similar to what I’ve written in this blog in laying the ground work for differential calculus and integration, but it’s presented in a much more formal way. The math concepts go from Einstein’s work on gravity and the strong and weak force interactions, how the Calabi-Yau manifolds were developed, and how they tie the math world of geometry to real world physics, along with digressions on how early string theory introduced extra, non-existent particles as errors to the system. There were up to 5 different versions of string theory at one point, which were shown by Edward Witten to all be different aspects of a larger, more general concept he called M-Theory. (Witten never came out and explicitly said what the “M” stands for. It could mean “Membrane”, but I suspect it’s “My”.)

Yau is a mathematician, and his early theoretical work was on Eugenio Calabi’s Conjecture that certain versions of objects within so-called Riemannian space (named after Bernhard Riemann) can be constrained to fit specific requirements. These objects have more than 3 physical dimensions, and can’t be drawn on paper or created in the real world, but as purely theoretical ideas that doesn’t matter. Yau proved the Calabi Conjecture in 1977, introducing something called Calabi-Yau Manifolds. In effect, these manifolds are sections of a complex higher-dimensional object, and they have to fulfill Calabi’s requirements (Ricci Flat and vanishing first real Chern class).

Yau’s background is in geometry, and as such he sees gravity as a geometrical problem. Therefore, questions about astrophysics, as well as subatomic particle interactions are framed in terms of finding the right geometry (the correct shape) to quantify them and predict future outcomes. Meanwhile, physicists want more of an actual theory to explain real-world behavior, using math to support the theory. In effect, it’s like Yau has a massive police mug book and is showing individual mug shots to a witness and asking them if they see someone they recognize, while a police sketch artist draws a picture of the suspect from witness accounts that is then given to the physicists and THEY then go through the mug book. The suspect in this case is physics itself, and the race to develop a theory that links gravity to the weak (radioactive decay), strong (bond between protons and neutrons) and electromagnetic (interaction between charged particles) forces. While a general theory exists connecting the three non-gravitational forces, so far all attempts to add gravity to get a Theory of Everything have failed. Assuming that gravity is geometry, and that particle charge and mass can be mathematically predicted from a geometric analytic, then it may be safe to say that there’s a “shape” to the universe and that we can find which one it is if we assume that the shape isn’t just 3-dimensional (4D because we have to include time).

Yau presents String and M-Theory from the geometrical viewpoint, where it’s less about the physics of the real world and more about pure generalizations. The problem is that these “shapes,” or manifolds can come in a near-infinite variety, and when force-fit into the physics predict particles that don’t actually exist, or give incorrect values for particle mass and/or charge. Additionally, the math associated with many of the shapes is nearly-impossible to calculate, making them too hard to play with easily. Essentially, there are too many extra dimensions in the model, and the challenge is to find simplifications to bring the models down to 10 or 11 dimensions, and also make them easier to plug numbers into.

The book doesn’t resort to very much in the way of formulas, so in that sense the reader isn’t expected to wade through a sea of symbols. On the other hand, there is a lot of terminology, such as Ricci curvature, Chern classes, and Riemannian manifolds, that may go over most people’s heads. You do need some kind of a handle on university-level physics and math to follow the discussions, and if you have a recent university background then you may already know more about M-Theory than this book covers. One of my complaints about the book is that there are too many photos of physicists and mathematicians, and not enough illustrations to accompany the word examples. I have a pretty good idea of what Yau is talking about, and I don’t feel completely lost, but I’m not at the point where I could explain anything in my own words myself.

Having said all that, what is a manifold? More technically, a Calabi-Yau manifold, and how does it relate to physics? First, we need to ask “what is a dimension”? If you draw a straight line on a piece of paper, you have a space that allows one degree of freedom of movement over time (back and forth). If you stand on the line and rotate half way (90 degrees) and walk straight out from that line, you add a second degree of movement (left and right) to create a plane or sheet. If you lie on your back on the plane and walk straight away from it, you get a third degree of movement (up and down) and you can make a sphere, cube, or other infinite space, with respect to time. You now have 4 dimensions – the 3 physical ones plus time – that are at 90 degree angles to each other. And what relates the 3 physical dimensions is rotation. In my previous entries, I talked about how i, j and k are used as placeholders to move from one axis to another, with i^2 = j; j^2 = -1. If you add a fourth physical dimension, then j^2 = k; k^2 = -1. With manifolds, you can have up to 10 dimensions, where the first 4 describe physical space and the other 6 are motive forces. You don’t see the upper 6 because they’re really small, representing high-energy particles that don’t manifest themselves in your daily life. The example given in the book is of a 3-dimensional line that changes over time, but instead of having an infinitely small radius, the line does have some thickness, and is wrapped in around itself kind of like brain coral. If you take a multi-dimensional cross-sectional slice anywhere along the line, you need to unwrap it. Unwrapping signifies adding more energy (such as with the CERN accelerator), and if the extra dimensions represent fields that create particles, then in unwrapping the brain coral that is the cross section of the 3D line, you’re pulling the particles apart to determine what fields are needed to make them. And that implies that the equation involves space as well as energy.

One section of the history side of the book discusses the concept of entropy as it involves black holes. If black holes are singularities that nothing can escape, then anything that enters the event horizon of the black hole will lose all associated information representing that object. That is, there would be no way to work backwards from the energy signature of the black hole (or whatever was inside it) to reconstruct the original object. If that were the case, the black hole would have infinite entropy and all the math would break down. Hawking stepped in with his theory of Hawking Radiation, and the idea that some information is retained on the surface of the black hole, rescuing the math and implying that black holes have a kind of compacted geometry as well. The issue then, was to equate Einstein’s theory of relativity with the structure of the universe, the behavior of black holes, and how all of this ties gravity to the other three forces.

Yau and Nadis spend a lot of time talking about how String Theory (and, later M-Theory) grew up, and how that ties in with Calabi-Yau manifolds, but there’s almost nothing on the geometry of strings and branes themselves. Occasionally, they’ll say that strings are one-dimensional elements of a manifold, or that branes are the things that strings connect to if the string isn’t a closed loop, but beyond that, nada. If you really want to learn about M-Theory, you need to pick a different book.

Chapter list (and my summary comment)
1) A universe in the margins – Introduction
2) Geometry in the natural order – History of geometry
3) A new kind of hammer – Talking about analytic geometry and gravity
4) Too good to be true – Multi-dimensional spaces
5) Proving Calabi – Development of the Calabi-Yau manifolds
6) The DNA of string theory – String theory history
7) Through the looking glass – Relating manifolds to string and M-theory
8) Kinks in spacetime – The fact that string theory isn’t predictive yet
9) Back to the real world – Attempting to prove string theory in practice
10) Beyond Calabi-Yau – Trying to find non-CY manifolds that may also work
11) The universe unravels – Discussing vacuum energy and de Sitter space
12) The search for extra dimensions – Ongoing lab work to prove string theory
13) Truth, beauty, and mathematics – Trying to justify the continued use of manifolds
14) The end of geometry – Speculation on classical geometry being replaced with a quantum version that ties to quantum gravity
15) Another day, another donut – Trying to show continued relevance giving that M-Theory is still just a nice theory
16) Entering the sanctum – Yau’s poem on manifolds and space

It’s kind of strange. The last couple of chapters feel like handwringing to me, with Yau trying to keep the doors of string and M-theory, as well as analytic geometry, open for the future. The problem is that after 10-20 years of theoretical work, there’s been very little in the way of advances in picking a good Calabi-Yau manifold, or in finding non-Calabi-Yau’s that would also work. Or in making real-world observations that can only be explained by M-Theory. The entire point is that the Standard Model can’t tie gravity to the other 3 forces, and String Theory was developed to make a Theory of Everything. If theorists fail to get their TOE, then string, and/or M-Theory are deadends. On the other hand, the math behind the Calabi-Yau manifolds turns out to be really useful for solving certain math problems that are otherwise very difficult to approach, because you’re converting between different domains. Something hard in one domain becomes easier in another, and vice versa. So, what Yau keeps repeating is that the work on string theory so far hasn’t been a waste in any sense of the word, because it’s helped the field of mathematics, even if it turns out to be a bust in the field of physics.

Summary: The Shape of Inner Space is an attempt to explain what Calabi-Yau manifolds are, and how both the fields of mathematics and physics have been helping each other in making various advances for proving both manifolds and their application to quantum gravity. The concepts aren’t that easy to follow, and assume at least a solid high school calculus background, if not university-level math. In other words, if you already know what a Riemann surface is, you’ll be fine. Otherwise, this book is going to be a slow slog. Additionally, the discussion sticks mostly with the quest to find a Calabi-Yau manifold that fits what String Theory requires to work, without talking much at all about what the various kinds of strings and branes are, or what string “vibrations” are and how they’re supposed to produce different kinds of particles. Which means, if you want to learn about String Theory, or M-Theory, this is not the book for you. But, both theories use Calabi-Yau manifolds, and if you want to know how these manifolds tie in with analytic geometry, then The Shape of Inner Space is a decent overview of the topic.

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