What is an “almost” Calabi-Yau manifold?
Thanks for straightening me out (I guess I was curved), Chronos.
Here’s a detailed discussion of Middle Earth flatness, taken from the textbook Chronos referred to: geometry - Is Tolkien's Middle Earth flat? - Mathematics Stack Exchange
I have to admit this as I do not understand the intricacies of Calabai-Yau manifolds or string theory I cannot give a good answer, but I understand that in order to recover some properties of quantum mechanics the background of string theory needs to actually be a (small) perturbation of a Ricci-flat Calabai-Yau manifold.
Fermilab has a video that may be more accessible for people who don’t wan’t to dig into high math.
There are 3 dimensions, and there are only 3. You got your length, your width, and your height. Build a fish tank with that and it will hold water. Build it with less, and you have a wet floor, add more dimensions and it won’t hold more water. Therefore there are 3.
People will tell you that something is one dimensional. It’s an analogy. It’s not really one dimensional. People will tell you that they are working on a problem with many dimensions. They are making an analogy. People will tell you that time is the fourth dimension. It’s not. Time is its own thing, it just reminds people of a dimension, and people have a habit of thinking that if one thing reminds them of another than those things are the same. It’s a well known human failing, called projection.
Physicists will talk about “n dimensional space,” and solve equations where space has more than 3 dimensions. This is useful mathematical poppycock. Useful because it solves problems, poppycock in the sense that you can’t store your hat there. There are making an analogy.
Things got off kilter when that schmuck wrote Flatland, and it’s been a mess ever since. Now, everybody who cares about something claims it’s a dimension so completists will be forced to consider it.
Here’s how that works:
I say the 4th dimension is beer. Because there is no fourth dimension and if there was you could never go there, you just have to take my word for it. Now everything needs to plotted with a length, width, height, and beerness. If you don’t use my fourth coordinate plane I claim your work needs updating, and by getting you to use my fourth coordinate plane, now your work really becomes all about what I want it to be.
That this is true can be soon in the proof that the Oscar’s and Grammies endup being about “Me too” and how bad Trump is instead of about film and music. They added a “beer dimension.” And got it to stick.
Not that there is anything wrong with “me too” any more than there is anything wrong with beer. It’s just that there is a time and a season to every purpose under heaven, and beer and string theory and time need to stay out of my coordinate math and Hillary Clinton should not be reading to me from an attack Trump book on the Grammys, and if she does she shouldn’t choose the one passage in the whole thing that demonstrates Trump is acting prudently (if you were Trump wouldn’t you be worried about people putting shit in your food?)
And in case the above came off as smart ass instead of dead serious, try this:
Beer has five dimensions
- Hops
- Yeast
- Malt
- Color
- ABV
Now using an N dimensional space where N equals 5, we can plot a coordinate map and plot it with all beers in the known universe. With this 5 dimensional map we can compare beers theorize about undiscovered beers, and solve problems with beers. What happens if you mix Smithwicks with Coors lite? You get Yuengling. The only way that we could have solved that was with our mathematical 5 dimensional coordinate space, because we know that in the real world nobody would ever dare mix the two.
Now do we apply this to the real world and assume that all objects have a length width depth hops level and ABV?
Actually I might be on to something.
You see, it’s just playing with dimension as an analogy.
You got your three. Three is the number of the count8ng and the number of the counting is 3. Thou shallt not count 2 unless that be immediately before counting 3. 4 thou shallt not count.
Who says science and scripture don’t agree?
There are a couple of definitions out there, but the usual one seems to be that Calabi-Yau manifolds are ones with a nowhere-vanishing holomorphic form \Omega with \Omega\wedge \overline{\Omega} proportional to the volume form. For a Calabi-Yau manifold, that constant of proportionality is actually constant; for an almost Calabi-Yau manifold, it just has to have absolute value 1 everywhere. In particular, they’re not necessarily Ricci-flat.
Don’t worry. It didn’t come off as smart in any way, shape, or form.
Yes, absolutely you could plot beers in a five-dimensional coordinate space as you describe. And those five dimensions are exactly as valid, for that purpose, as the three dimensions of length, width, and height are for purposes of determining the capacity of a fishtank. Neither is “poppycock”.
Than you understood and I was successful. No need to thank me.
I don’t think so. I could plot beer according to different criteria and use a different number of dimensions and come up with valid data. I am just using the conventions of plotting in a dimensional space as a tool. I could choose bar graphs or scatter grams or any other of a variety of ideas that might be equally valid tools. There is nothing intrinsic about dimensional space that is intrinsic to my beer analysis.
With the fish tank the dimensions are not arbitrary. They are intrinsic. Those three dimensions represent reality. There is no other to tell how much water the tank will hold without first filling it with something that does not boil down to length times width times depth.
So, in terms of being arbitrary or replaceable or made up (in other words poppycock) my beer example qualifies while my fish tank does not.
btw fish tank science is cool. I am convinced that you can teach all of of high school science em plus some college stuff, physics, math, chemistry, biology by making a kid run a reef tank from scratch.
Yes, you could, depending on what you’re interested in.
And there also, you could use a different number of dimensions, or a different set of them, if you were interested in some other properties of the fish tank. For instance, you might wonder, instead of how much water the tank will hold, how much floor space it will occupy, and only two dimensions are needed for that.
And in that case you still need to use spatial dimensions to calculate the footprint. There is no other way to do it. The spatial dimensions are intrinsic to its. We have gotten down to brass tacks and irreducible basic principles.
Spatial dimensions of which there are three and only three are intrinsic to to these sorts of problems. Because that is the world we live and navigate in we translate other problems into that useful environment in order to model them. Are models can use as many dimensions labeled however we want them too because they are just models. Depending on how good those models are they will be more or less poppycock, but poppycock nonetheless as they are imperfect translations of one thing into another.
The calculations based on the dimensions of the fish tank though are already in their own terms and are 5herefore more valid.
Hmm. There’s what you say and what actual working physicists say. Which to pay attention to? Such a problem.
And have I denied that spatial dimensions are often relevant? I’ve said repeatedly that they very often are. But sometimes other dimensions are relevant too, and sometimes they’re more relevant than the spatial ones.
For a concrete example, take the configuration space of a classical rigid body. It has six dimensions: three spatial dimensions and three angles of rotation. The latter are perfectly valid, even though they’re measured in different units than the former.
Well, that’s the thing: We don’t actually live in that kind of space. We live in a space that looks a lot like what you describe if you’re putting together a fish tank or walking slowly. If you’re working with GPS, MRI machines, superconductors, cosmic rays, or any of a larger variety of activities, that approximation is no longer sufficient. If you tell me that a fish tank measures 3 meters in length, that’s great if I’m standing next to you on earth, but I’m not going to get the same result if I’m travelling close to the speed of light and relativistic effects become significant. You might complain that that isn’t a case most people deal with in their ordinary lives, but that doesn’t make it any less valid or real. No one’s arguing that classical approximations aren’t useful, but they aren’t the full picture.
Chronos:
Let me try a different way. The spatial dimensions that are observable are real in that sense that we can interact with them, observe them, manipulate them, understand their rules, perform calculations on them and get good data, etc.
Now, as for a hypothetical fourth spatial dimension, we can do none of those things except for arguably performing calculations, whether or not these calculations are purely hypothetical is largely acedemic. But is it real? Is there really such a thing? If we found or created a fourth dimensional springboard, stepped on it and activated it, could it propel us into a fourth spatial dimension that we could observe and interact? Could we go our friends from the inside the way we can touch the center of a 2d circle from the third dimension? Is there anything there? Are there any actual 4d objects there?
Well, no. We can only go by the rules we know work in our space, and any 4d object wold have infinite mass compared to us, which would exert a force that would be… well catastrophic. Similarly there are no true 2 dimensional objects, either. Everything has 3 dimensions.
So, what separates a spatial fourth dimension from say… Middle Earth, or Hogwarts, or any other imaginary place? Wishful thinking because those are more likely. So, when we say a 4th dimension exists we are engaging in an act of faith. Occam’s razor suggests that we don’t believe in this notional place.
Similarly, these extra dimensions that string wrap themselves around in string theory, if we could shrink ourselves down could we go there and see them observe them, or act with them in any way?
Assuming these theories are correct and they hold up, and explain things, and are useful than possibly they are real in the sense that calculus is real. You can’t go to calculus or take some with you into the bathroom to wipe your ass with, or anything else. It’s just a set of rules that provides useful data by modeling the observable universe.
This is what I’m getting at. The notions that we use to model the universe are not the universe. It has no obligation to play by those rules and follow our models. Beer does not Become. Five dimensional thing because I model it that way.
A lot of physics gets confused by laymen such as myself, when physicists model something and people take it literally. Schroedinger was not being literal when he suggested that their were multiple cats or that the cat existed in some temporary undecided state until it was observed. He was pointing out the contradictions between the subatomic realm and pour observable universe, and suggesting that things were either very strange down there or we had got something wrong because it doesn’t make sense.
And this is part of the problem with quantum physics and us humans. We simply don’t have the tools to grasp it as it is. Our brains and biology don’t work that way. It works with concepts and rules that we are not adapted to deal with and understand as they are. To make sense of them we have to convert them into something else that we can understand. To translate them with math and make use of of things we do understand like spatial dimensions, and then we extrapolate extra dimensions because we can’t make it work using just 3. What we get is a translation and likely not a very good one. We are far off from the thing we are describing.
But extra dimensions and alternate universes are heady stuff, and people think these models are literal.
It doesn’t help when people who should know better claim they are real. I know, I know, calculus is real and a 5 dimensional signal model of beer that provides useful data is real, but neither is literal.
So, we have 3 dimensions, and a bunch of math that extrapolates others to model other things. Some of those other things are purely notional and the rest are likely being oversimplified by suggesting their are in other dimensions.
That’s what I was saying when I said that everything that goes beyond fish tanks is an analogy. And it is.
It depends on what context you are talking about.
The 11 dimensions of string theory may end up being “true” but right now they have absolutely zero experimental evidence that they exist.
We need people to work on theoretical physics, but at this point in time you shouldn’t “believe” in these dimensions, or think that they are real.
If you are talking in the domain of SR/GR there are 4 dimensions, that make up space-time and that has been demonstrated through 100s of experiments. With our current best understanding if the world it is safe to think that those dimensions “exist”.
M-theory and superstring theory would be elegant solutions, but until there is any experimental evidence that they do exist they are just an interesting idea.
Until any of these theories can apply the third tenet of the scientific and even design an experiment that can be conducted they are a concept.
But for the most part I think all of the parties are talking past each other here.
One group is using the informal definition as the minimum number of coordinates needed to specify any point within it, and others are talking more about parameter spaces or configuration spaces etc…
This is what Itself is talking about here.
But for the three spatial dimensions and three angles of rotation can be reduced to three spatial dimensions, where the 10/11/XX dimensions in string theory are not exactly like that.
But right now the idea that people are using an overloaded term seems to be the sticking point. It is shaky ground claim that anything “exists” with zero experimental or observational evidence confirm the existence of these extra dimensions.
If Chronos intended to claim that these dimensions are proven, and that you *should believe in them * we need a serious cite, but I think that a divergence in the base terms has resulted in the confusion.
If you’re trying to model the universe as just R^3 with a time component off to the side, we already know that doesn’t work: Special relativity introduces a coupling between the spatial and time dimensions, and geneal relativity requires some nontrivial curvature (if not topology). My point above was just that saying that there are three spatial dimensions because common experiences tell you so is not a valid argument. Physics is complicated stuff, and it doesn’t agree with everyday experience. We could be living in a Poincare sphere, on a 10+1 dimensional space filled with superstrings, or something else more complicated. We may not be, but the fact that we don’t see those effects in ordinary experience doesn’t somehow invalidate them, any more than the fact that driving to work doesn’t involve QED means that that theory is nonsense. If you ask a question about physics, you’re going to get an answer about physics.
4 Dimensional space-time has tested every test so far, even if it is incomplete and/or imperfect. 3-D space is a model of convenance when it is appropriate.
Please provide a cite that has demonstrated the existence of five-dimensional space or n-dimensional space as a as a physical entity.
I will repeat a cite that is a video I actually posted to help people visualize the extra dimensions, but is fresh in my memory for my claim.
Fermilab’s Dr. Don Lincoln, a codiscoverer of the top quark.
Particle Physics, when talking about what is “real” sets the the bar high.
“Evidence of a particle,” is p<0.003 (three sigma) and the standard to report a “discovery” is p<0.0000003 (five sigma).
The fact that a theoretical physics model may be pretty does not change that. We need people to work on theoretical physics but to claim that they are scientific claims is in error. These ideas haven’t even reached the point where we can even test them, let alone claim that they are descriptive of reality as best as we know it.