No, see, I understand all this.
Let’s try a different approach: In what 1-Dimentional geometry would you use to describe a circle? If you have to involve a warped topology to achieve this, I’m honestly curious as to what sort of geometry would be used. Or, In other words, how does one achieve a circle using 1D geometry?
There are no “higher dimensions”, a dimension is just a measuring tool.
Care to expound?
We can clearly demonstrate three spacial dimensions. We can measure these dimensions using tools, but it is a real, physical fact we live in a space comprised of at least three dimensions.
You can’t. No such thing exists.
How do you achieve a 4D hypersphere using 3D geometry? You can’t.
No dimension can express a higher dimension. But that is entirely different from saying that a dimension cannot be curved, warped, or otherwise deformed within itself.
Ok, I see what you’re saying. Ignorance fought.
But then, calling a 1D line within a positive curvature topology a “circle” is kind of a misnomer, no? I get the correlation, but circle as defined by conventional geometry is inherently 2D.
Anyway, I concede.
*That might be negative curvature, actually.
[As an aside: terminology-wise, the 1D example has no curvature. It does has a property of its topology called “periodic boundary conditions”. For properties that really are curvature, thought, the story of not needing higher dimensions stays the same.]
If I wanted to define this 1D universe without reference to a higher dimension I could do so with, “Picture a straight line with periodic boundary conditions.” No reference to a circle is needed. You’ve just got a 1D system, and going one way eventually brings you back.
If I wanted to define that 1D universe using everyday language, I might try invoking some more familiar systems, like a circle (or an oval or a square or any closed loop form, but let’s stick with circle). I could correctly say, “Picture the 1D surface of a circle.” That surface is a 1D space. It has the right (topological, not curvature) property of periodic boundary conditions. I managed to define this 1D space in terms of the surface of a 2D object (arbitrarily picked here as a circle), but I didn’t need to.
If someone uses your quoted language of “calling a 1D line with [periodic boundary conditions] a circle”, they have spoken incorrectly. They could have correctly said “a 1D line with [periodic boundary conditions] is (the same as) the surface of a circle.” But the surface of a circle isn’t a circle.
Side-scrolling video games with periodic boundary conditions make for a good mental exercise. Picture a Super Mario Bros. (NES) behavior except where if you pass the castle at the end, you end up back at the beginning of the level. If we lived in that 2D world, we would certainly have no problem drawing 2D circles starting from childhood. And if we wanted to describe to someone else in our world a 1D surface with periodic boundary conditions, we would not look to the circle. We’d just say, “Draw a line that way until it reconnects (naturally!) with itself, and imagine you live on that 1D surface.” We’d already be used to periodic boundary conditions, so it would be weirder to try to construct anything like that from subspaces of a circle.
Yeh, I believe I was right the first time—positive curvature. I blame lack of sleep.
nods
Okay. So, in this construct, ideas of area contained “outside” (or, “within” for that matter) of this periodic boundary would be meaningless since there is no higher dimension for it to exist in.
However, this does lead to the question, what’s causing/caused this curvature in the first place?
(Which brings us back to “What’s gravity?” Full circle, so to speak.)
Topology? What causes math is an age-old debate, with no answers. Where a mathematical object lives is something that no “common sense” approach has any chance of handling.
Yep. Exactly.
I vote that that gravity and math share an apartment at:
3.14159… Pi Circle
Fundamental Plenum, State of Mind
And, of course, mine was a rhetorical question.
With no easements or in-roads. You’re just somehow… there.
Not sure if this part has already jumped back to gravity. If you mean for the 1D line case, there is of course no cause. It just is. Our Super Mario Bros. denizens would playfully imagine living on 1D lines that (naturally) have periodically boundary conditions, and one day someone would say “Now imagine that the line… GOES ON FOREVER!!!” and everyone’s minds would be blown, and they’d ask how that could be and what could cause it.
(On preview I see you were being rhetorical. I’ll leave this in anyway.)
We live in a (3+1)D space. If you put mass somewhere, the space gains curvature (fully describable in (3+1)D). Objects move along “straight lines” in the space always, but since we’re in a “curved spacetime” with (3+1)D, we need to be careful what we mean by “straight line”. Formally, objects follow paths known as geodesics, which you can just think of as the shortest paths between points. The geodesic connecting two points will look different in curved space from flat space.
We have an elegant set of equations that define the curvature you get by placing a given amount of mass somewhere. It turns out that mass isn’t the only thing relevant, and things like momentum density and pressure also affect the curvature.
That’s the full story. We don’t know why placing a mass into this spacetime causes curvature. It just does, and we can fully describe the curvature it causes. Maybe that’s it – it just is. Or, maybe there’s something deeper that we’ll find (and in turn, we’ll move the “why” question another turtle down).
There is talk up-thread about particles, but that’s unrelated (as far as we can show) to this geometric approach. Maybe someday someone clever will write down a complete particle-based formulation of gravity, and maybe someone will eventually show that that and the geometric approach are equivalent in some way. None of that has happened yet.
[And ontologically, even though the geometric approach (General Relativity) is extremely successful, we should stay on our toes not to conflate our description with “what actually happens” which we can never know.]
Being wrong is not the same thing as failing to make a positive contribution to the advance of science. Both Priestly and (especially) Descartes (no way is “DeCarte’s” correct) made huge contributions to the advancement of science (and via the “false” theories you mention, in particular, although in other ways too). The history of science is almost entirely the history of the perceived defects of false theories providing both the motivation and teh conceptual groundwork for the development of more sophisticated ones that are less-false.
Right, it is bad advice for scientists (although advice that sadly many of them seem to take, in practice), and, if it is not directed at scientists it amounts to “Stop questioning my authority, you peasants!”
Well, even if we live in the 1D world of the circle’s surface, it’s possible for the circle to exist, with a dimension we can’t experience directly, but could infer.
In the case of the circle, the only feature to infer is periodicity; it’s a pretty trivial case.
In the case of living in the surface of a sphere, it would look much the same as living in a plane, except there would be a more complicated kind of periodicity. We could measure angles and note that a triangle’s angles add up to more than 90 degrees, and make other measurements, and a real genius might see the two dimensions, imagine a third, and conceive of a sphere in 3D. He’d have to be quite the genius.
In our world, we have 3 readily apparent spatial dimensions. It’s a lot easier for us to look at the difference between two and three, and figure out just what measurements to make to see whether space is curved in a fourth dimension, and to conceive of that extra dimension. We have our experience from cartography and mapping the globe onto flat paper as a guide. Plus we had a couple mathematicians who at about the same time posited different antitheses to one of the axioms of geometry (given a line and a point, there is exactly one line through the point, parallel to the first line) and worked out the math for it, and found out that the geometries so defined make a certain kind of sense, internally. Later physicists figured out that they might even apply to our universe.
Exapno said we can’t conceive it, but he meant “naturally”. With enough thought, we can get our heads around the idea, and see how it looks from a lot of different viewpoints, and make excellent predictions about it. We just don’t get a natural, intuitive conception for it.
No argument, but Priestly fought Lavoisier’s advancements tooth and nail. Basically, science advanced because the young turks were attracted to the more productive new theories, and the old guard died out.
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Doh! Of course. Thanks for the corrections!
Priestly certainly discovered a lot of important properties, and figured out how to isolate oxygen (without quite realizing what he had). His clinging to his theory kept him from making more significant discoveries. It worked because phlogiston theory is numerically equivalent (though inverse) to valence theory. (And the inverse is arbitrary too, I believe.) Descartes’s flirtation with vortices was pretty much a waste of his time and that of anyone trying to make sense of it. All of which is tangential to my point, because I turn around and admit the usefulness of making models even when we don’t have enough data to build a good one. So, really no point quibbling about this. And yes, even wrong models can help.
No argument.
I disagree. It amounts to “don’t use goto’s until you understand why not to use goto’s. There are lots of good reasons, but first you have to have a much better understanding of programming in general before we can discuss that in much detail.”
Of course, that brings to mind one of my favorite math teacher’s gems. Someone would ask a pointed question, and he’d make a bit of a sidebar. He’d say that there are two good ways a teacher can say “I don’t know” without admitting ignorance: (1) It’s beyond the scope of the course, and (2) You wouldn’t understand it if I told you. Then he’d say “But regarding your question, IT’S BEYOND THE SCOPE OF THE COURSE, AND YOU WOULDN’T UNDERSTAND IT IF I TOLD YOU!” That’s the guy who used to call us dirty sunny beaches (since he’d never swear in class.)
Well, it’s the same here, except we admit we don’t know. Newton admitted he didn’t know, but that didn’t make his equations any less useful.
It’s training wheels, not “bad advice to scientists” or argument from authority.
On the topic of curvature: One analogy Einstein himself used, is to picture a tabletop covered with a bunch of little metal rods, all of equal length. Arrange four rods into a square. Now add three more rods, to make a second square next to it. Continue adding more squares until you’ve covered a significant area of the tabletop. If your table is flat, then your grid will be nice and regular. If it’s not flat, then you’ll get distortions: Either some of the squares will be flattened, or some of them won’t meet up where they’re supposed to. We could in fact use this as a definition of curvature: A surface is curved iff a grid of squares on it is forced to be irregular.
But now suppose that the table is heated, with hot spots and cold spots. Rods sitting on a hot spot of the table are expanded slightly by the heat, and rods on cold spots are contracted (of course, we’re assuming that the rods are the only way we have of measuring the surface). But this means that our grid won’t be entirely regular any more: Rods of varying lengths will introduce distortions. Which, in turn, means that, by our definition of “curved”, the tabletop isn’t flat any more. But it became curved without needing any higher dimension to “curve into”.
That’s a good analogy, hadn’t heard that one before. Thanks, Chronos.
A point about embedding: exactly how a lower dimensional space is embedded in a higher dimensional space is extra information on top of the information contained in the lower dimensional space and it’s redundant information if the lower dimensional space already tells you everything you want to know.
Secondly we’re talking about spacetime (3+1 dimensional space) and a spacetime metric can’t be isometrically (let alone smoothly) embedded into any Euclidean space due to the fundamental difference between the two. The Nash embedding theorem covers Riemannian manifolds, whereas spacetime belongs to the broader class of pseudo-Riemannian manifolds. There is an equivalent theorem for pseudo-Riemannian manifolds, but the upshot is that to embed 3+1 dimensional spacetime in a higher dimensional pseudo-Euclidean space, you’ll find the index of the embedding space needed is greater than 1 (i.e. more than 1 “timelike dimension”) unless certain causality conditions are obeyed.
Fascinating, Asymptotically fat, I’d never heard of those theorems. That does seem to rather put the kibosh on the notion of embedding our spacetime into a higher-dimensional “simple” spacetime.