If A is the set of all events then we can see it as a fucntion from R[sup]4[/sup]–>A (where R[sup]4[/sup] is the set of real 4-tuples). Functions can be vectors once you define addition and scalar multiplication on them (curiousprincegeorge, didn’t do that so function is the correct term)
You’re saying it made sense what he was saying before in terms of events?
And he wasn’t getting confused with vector spaces?
Because that’s the matter at hand. I’m aware you could represent the sum of all events as a function of all dimensions.
That is exactly what I thought when I read that post. Once in awhile, though, those elegant, unrealistic models finds a use somewhere. Not sure this one will, but who knows?
Or a very large hydrogen nucleus.
Ok, give me a hint. I don’t even understand what I don’t understand here. Unless you mean the curvature is just a coincidental combination of some non-linear functions.
I think the idea he was searching for was that of a reference frame, which broadly speaking is an attempt ) to map all the events to R[sup]4[/sup].
I do not think he was looking for the concept of a vector space, because as I said what he said could’ve equally have applied to Newtonian physics and as the n-tuplets (x,y,z,t) representing time and postion in some reference frame don’t transform like vectors in Newtonian physics they cannot be thought of as representing vectors. In fact 4-postion vectors only really come into play when you’re talking about cooridnate systems with constant bases in Minkowski spacetime (i.e. fairly simple coordinate systems in special relatvity). For example it wouldn’t make much sense to talk about 4-postion vectors in general relativity.
Actually, no, it just has to be made of matter which isn’t wrapped around on itself in dimension Q (btw, in case that confuses anyone; I just made that name up :)).
It can be as big as you like in good old xyz.
Fine, point conceded then.
At least my other pedantry stood up…
The best analogy I’ve seen is from Einstein himself. Suppose your method of measuring distances is a whole bunch of little brass rods, all the same length. You have a large, flat table, and you start laying out brass rods on it. First, you put four together in a square (right angles at all the corners). Then, you take three more, and make another square next to the first one (again, all right angles). Then you take three more, and make another square, so the three squares form an L shape. Now, if you take a final two rods and put them down at right angles to other rods in the right place, you’ll find that they make a nice square, too, and finish off a neat little 2x2 block of squares. You can keep on doing this for as long as you like, and no matter how big an area you’re covering, you’ll still neatly cover it with squares. This is possible because the tabletop is flat, and in fact you can define a flat table as one for which this is possible.
Now suppose that instead of a flat tabletop, you’re laying out your rods on a dome. Well, now you’ll find that if you try to cover the whole dome with little squares, the squares get distorted. The distortion might be small enough to overlook at first, but eventually, it’s going to get very distorted indeed. From this, we can say that the dome is curved, and in fact, it is curved into another dimension, just like you’d expect.
But now let’s take another tabletop. It’s not domed, but it is heated nonuniformly. If you put a brass rod down on one of the hot parts of the table, it’ll expand a bit from the heat, and if you put one down on one of the cool parts, it’ll contract a bit. If you try laying out a grid on this tabletop, you’ll find that the grid gets distorted, which we’re using as our definition of “curved”. But it’s not curved into some other dimension; the curvature of this tabletop is an entirely two-dimensional phenomenon. With the right temperature distribution, you could make it have the same distortions as the dome, and if you didn’t know about temperature or height, you couldn’t distinguish between the two cases (remember, the rods are our method of measuring length, so we can’t tell they’re expanding, either).
So this is basically the same explanation I was once given for why space IS curved. But I’m not all that far off. Distances on the flat table aren’t linear from every place you look at it. But (here’s what I’m getting from this), they don’t have to curve out into another dimension to do that, there’s always enough room for them on the table because of this non-linearity of distance.
Or am I totally missing the point?
No you’re not missing the point at all, the idea that the shortest paths between points no longer behave like straight lines on a flat table (to carry on the 2-D analogy) is the basic motivation behind modelling gravity as the curvature of spacetime.
And again you’re perfectly correct in saying that they don’t have to curve out in to extra dimensions. The key idea here is that whilst we could view spacetime as being embedded in some flat, higher dimensional space, adding extra ‘embedding dimenions’ is simply not necessary and they would be a superfluous to the description (and if anything only act as an unneccesary complication).
The point of the ‘heated tabletop’ is to show a simple model of how you might want to model the tabletop as curved space even though you’d be very hard put to say that it actually curved into other dimensions.
I guess a way you could explain this is that:
(1) we are confined to the universe, so all we can really do to derive information about the curvature of space time is measure distances and angles inside the universe. Even if the universe is embedded in some other higher-dimensional space can never look outside the universe to see how it is “really” curved.
(2) There are geometric inconsistencies in the universe that are just like the geometric inconsistencies we would expect if the universe was embedded in a higher-dimensional space and not flat.
(3) But, as Einstein/Chronos points out, these geometric inconsistencies are also consistent with other models. Like, there being some kind of universal “temperature” that only affects length scales and varies from place to place. It is impossible for us to even in principle get data that would distinguish between a “embedded universe” model and a “varying length-scale” model, so it doesn’t make sense to say one or the other is “true”.
Incidentally, I once came upon this paper, which I think is really cool for showing the kind of geometric artifacts we’re talking about. It does GR calculations to show that, for example, the radius of the Earth is 2mm bigger than one would expect from it’s circumference. In an embedded universe model, you’d think that the Earth was pushing the universe into the extra space to make the radius a longer distance. In a varying length-scale model, you’d just think that the centre of the Earth is a “cold” region that cause the brass bars to shrink, requiring more of them to span the distance.
Thanks guys, and Chronos too. I didn’t get any formal physics education (was that phys ed class? I usually skipped that one) so getting these things straight is tough. The curved space concepts seem to abound, but they also imply that space must be curved into other dimensions (or at least I inferred that). This explanation actually makes things a lot clearer. And it’s not that difficult to compare to the readily observable world either. Things vary in density, weight, color, and all sorts of other ways, and although we need a different factor to measure them, we don’t have to think of bricks curving space because they weigh more than foam blocks.
Relativity just became a lot less strange to me. Quantum stuff though, that’s still weird.
Oh, yeah. Most people who work in relativity understand it, but for quantum mechanics… Well, even Richard Feynman said that “If you understand quantum mechanics, you don’t understand quantum mechanics”.
In terms of embedding, it’s worth pointing out perhaps that when general relativity was first formulated that it wasn’t actually known that you could embed spacetimes in to a higher dimensional space. A little research suggests to me that it wasn’t actually proven until 50 years after the appearance of general relativity. Whether or not you can embed a spacetime in a higher dimensional flat space is a point of mathematical curiousity with no direct bearing on general relatvity as a theory of physics.
Is the proof mathematical or observational?
Also, does this have to do with the genus of our 4D spacetime? (I assume if spacetime has “tunnels” in it, for example, then it can’t be embedded it in a flat higher dimensional space.)
You can embed 2-d spaces with tunnels in a 3-d space just fine.
I seem to recall that, for any metric, there exists some flat higher dimensional space you can embed it in, though it might take going up more than one dimension.
The proof is mathematical anddoesn’t have any direct bearing on the actual physics.
You can embed any Lorentzian manifold (i.e. any 4-d spacetime, including those that breach energy conditions, which I will come to in the next paragraph) in to a 10 dimensional pseudo-Euclidean manifold (i.e. a flat space of higher dimension).
This includes spacetimes with topological features such as wormholes. Traversable wormholes breach what are known as “energy conditions” which physicists impose on spacetimes in order to make sure that they are dealing with physically realistic spacetimes, but as I imply, the theorum treats spacetimes only as geometric and topological structures and is not concerned over whether they are physically realistic or not.
Wait, the higher-dimensional embedding space doesn’t even need to be Lorentzian? That is, you don’t need a minus sign in the metric for the time component? I hadn’t realized that.
I’ve taken Lorentzian manifold to mean a pseudo-Riemannian manifold with a metric of signature (1,3) (up to isomorphism), though you’re correct the more widespread terminology seems to mean any pseudo-Riemannian manifold of metric signature (1,n) (up to isomorphism yadda, yadda).
Using you’re (more correct) terminology, no there’s no assumption that the embedding space is Lorentzian. Just that it’s pseudo-Euclidean. However in order to embed a 4-D Lorentzian manifold such that it is a submanifold of the embedding space, you’re going to need a pseudo-Euclidean manifold of (n≥1,m≥3). So there’s at least one ‘timelike dimension’.
In fact your post has made me realize I erred a little (or a lot) with my last post. It’s infact it’s only been proved that youd need a minimum of a pseudo-Euclidean manifold of 90 dimensions (87 spacelike and 3 timelike) in order to embed all 4 -D Lorentzian manifolds such that they are submanifolds (the figure of 10 dimensions is for local isometric embeddings only).
Eh, that’s less than an order of magnitude off. That’s nothing, in cosmology.
IANAAstrophysicist, but this is the way I always thought of it. Imagine graphing a point in three dimensions, X, Y, and Z. With those coordinates, you know where the point is, but without time you don’t know when it was there. So to accurately plot a point in spacetime, you need that fourth coordinate. I could be way off, but that’s how I always explained it to myself.