Gravity and Curves in Space

Factual Questions
21

I began to understand this when someone explained to me that space being “curved” is actually a bad description. Going back to the sheet analogy, you notice that the gravity well has concentric rings, which would correspond to the orbit of a satellite or other object. The object revolves round and round, but in the relativistic sense, it’s actually going in the only direction it can go. It’s moving perfectly “straight” relative to the gravitational field. We see it as curved because gravitational fields don’t enter into casual observation.

Or, to put it a different way… in a classical sense, an object can only move in a curved path when there is some force upon it. In the relativistic model, mass causes space to “curve”, so orbiting objects appear to travel a curved path even when no force affects them.

22

There’s a very precise meaning to curvature in mathematics. Most other descriptions are only analogies to help visualization. If two observers are moving in exactly the same direction and they get closer or further away from each other then space is intrinsically curved.

Don’t consider space to be extrinsically curved into some other dimension, instead, just think of the intrinsic structure of space as being warped. If parallel lines converge or diverge there must be something internal to spacetime that causes this. We call this thing curvature.

23

Thank you all for your kind answers. I wish I could say they helped me. I just can’t seem to get an analogy that resonates an actual meaning for me. I can parrot back the words, but I don’t actually understand them. This is my problem, not yours. I’m pretty limited - to three dimensions and Euclidian geometry. I did well in first and second year calculus, but never really understood its application to the real world. I guess I’m like a fourth grader who just can’t grasp fractions, except my block comes later. FWIW, as I mentioned earlier, I have this problem with a few other areas too.

Once again, thank you very much for your attempts, and I hope others less blocked in this area than I am will have benefitted from this thread.

24

There is a fundamental problem with the rubber-sheet analogy. Before applying the bowling ball draw a regular grid pattern on the sheet. Chuck in the bowling ball. Once you have done Mathochist’s projection so that you’re no longer looking at a trumpet shape but at a flat surface with a (now) distorted grid around the shadow of the ball. Now place the golf ball on the projected grid. It is on a flat surface, why should it move? The (3D) picture they always use implies that the golf ball will roll towards the bowling ball, downhill :smack: The distorted grid does suggest that if the golf ball moves then its path will be affected by the bowling ball’s presence (it may go into orbit) but what if it’s just sitting there? What’s missing from the explanation that always seems to be used is that the golf ball is moving – through time, doesn’t have much choice about that. Take away time and gravity stops working. Stupid rubber sheet pictures never seem to point that out.

Quietly waiting to find out just how wrong that is.

25

Yeah, I actually hate the rubber-sheet analogy, or at least the way it’s often presented, for just the reasons you point out: it completely misleadingly makes it seem as though one is invoking some kind of meta-gravity at a level which pulls the bowling ball into the rubber sheet and then the golf ball into the resulting indentation; the confusion is particularly dangerous as it doesn’t manifest its incoherence immediately, and thus people are apt to walk away from the analogy thinking they understand at some level, when the picture they’ve got is still not yet anywhere near right.

26

Well, I don’t know if I actually can add anything to this discussion, but that’s never stopped me from trying… :slight_smile:
I think that at the heart of this confusion lies the misunderstanding of intrinsic curvature, and the mistaken assumption that it is equivalent to an extrinsic curvature (I don’t know if this terminology is actually used) in a higher dimensional space.
Take, for instance, an A4 sheet of paper. It’s flat, as one can easily demonstrate by drawing all sorts of triangles and parallels on it, with 180 degree angle sums and parallels that always stay nicely away from each other.
Now you can do all sorts of things to that piece of paper, like for instance rolling it up into a cylinder, or some kind of U-shape, and those geometric figures you just drew won’t change, regardless of how you bend that paper through three dimensional space, because it doesn’t stretch (observably much). That’s what’s meant by a flat, two dimensional surface. What’s important here, I should point out again, is that this behaviour is simply because of the fact that the piece of paper isn’t intrinsically curved, regardless of its curvature in 3d space, so you can just as well do away with the latter. It’s a characteristic of the paper to be not curved, not one that has anything to do with whatever space it is embedded in.
But there’s one thing you can’t do to that piece of paper, and that’s bend it into a surface that does have an intrinsic curvature, like for instance that of a sphere – that’s why all maps are distorted, for instance. Yet, the surface of a sphere is just as two dimensional as the piece of paper is, and just as independently so from the space it’s embedded in. You can describe it without having to say anything about the ‘surrounding’ space, and you’re not omitting anything. But on that surface, you can’t draw a triangle with an angle sum of 180 degrees, and, for instance, if you start at the equator, and draw a line at a right angle to that (going north, for instance), and then start another line like that somewhere to the side of the first one, they obviously will meet (at the north pole), despite being locally parallel. That’s the property of intrinsic curvature, which doesn’t need any talk about being embedded in higher dimensional space.
But if that’s true for two dimensional spaces, then, via applying dimensional analogy, one can easily infer it being true for three dimensional space, even though it’s not easy to be visualized, which is where all those – usually not terribly good – analogies of rubber sheets and inflating balloons and the like stem from.
To employ one of those analogies for a minute nevertheless, imagine you were a two-dimensional being living on a sheet of paper; you only know the concepts of left, right, forward, and backward. It’s usually said that these beings lack the ability to tell if they’re on a curved surface or not because they lack the concepts of up and down, but that’s not so; they are perfectly able to tell whether or not they live on a curved surface without those concepts, using only what is familiar to them – by, say, drawing a triangle and measuring its angles. The intrinsic curvature is completely independent of the concepts of up and down, and indeed, they need not even exist.
And just the same goes for three dimensions.

27

Wait a second, I had a glimmer here. Let me propose a model, just for space, and see if it’s an adequate one.

Take a cube whose sides are made of elastic material. When sitting there normally, it looks like a regular cube, and goemetrically it operates like a normal cube. But bring the sides around, and not only do the elastic sides distort everywhere (so that you might very well have parallel lines crossing, for example), but *inside * everything is distored positionally as well! Or dimple that same cube with lots and lots of golf balls, some deep, some shallow. Everywhere they are, there are distortions not only at the edges, but in the areas around the edges.

Problem is, you still need edges, dammit! Otherwise why the distortion rather than just pushing aside? I mean, even assuming my analogy is right, which it probably isn’t.

28

Oy, I think there’s something to your idea for an analogy.

Having said that, I’m starting to think the best thing to do is drop analogies and just look at the facts–they may be clear enough by themselves! So what if we just say thae following. It’s possible for a three dimensional space to be such that, two things can start off from different points, going in the same direction, with neither changing direction (i.e. neither accelerating left right up or down, simply continuing to move forward)–and yet these two things can end up moving closer to or farther away from each other.*

Without trying to visualize this in some way that makes it “comprehensible” to the imagination, we can at least simply understand what the claim amounts to and accept that it’s true, not just theoretically, but based on actual observations of the way things behave in the space we live in.

Forgetting analogies for a moment, I wonder whether you find the bald, un-intuitivized claim itself particularly difficult or easy to accept.

-FrL-

*If I got this wrong, someone should say so, and my apologies if that’s the case

29

Everything you said is basically correct. I can’t speak to whether a lay audience could understand it better or worse than the other ways we’ve said it here, but it’s worth a shot.

30

I can accept it intellectually, i.e. I believe you are telling the truth. But I can’t understand it any better than if you said imagine a number system in which 1 = 6. As I said before, I believe this to be my limitation, not yours. But thank you for trying.

31

This is true. Let’s say that you start out in the Galapagos right on the equator, and I start out due north a ways in Minneapolis. We both face due east, so Greg Chamitoff up in the ISS sees us both looking in parallel directions. Then we start walking.

As we go, you just walk right along the equator all the way around the Earth. I, on the other hand start drifting south. Greg sees us moving closer and closer together until we pass each other (hi Frylock!) somewhere in the Atlantic, about a thousand miles south of the Ivory Coast-Ghana border. I keep going south until eventually I’m way out in the South Indian and you’re up on the equator about 1500 miles west of the Indonesian coast. Then I start moving back up. We pass each other again way out in the Pacific as I head from Australia towards Hawaii, and we finally get back where we came from, and start it all over again.

Along the way, Greg has seen us move together, pass, move out to the other side, and back again. If he didn’t know any better, he’d think that there was some invisible spring pulling us together until we overshot and it stretched out the other way and pulled us back together again, over and over and over. But we know it’s really because the Earth is curved, and we’re just walking straight forwards, following the curved surface.

The really weird thing is, spacetime is also curved, and you can do the same sort of thing in a gravitational field out in space instead of on the Earth’s surface.

32

Huh? You mean due east isn’t the latitude line? Why not?

33

Ah, because you’re going through the East and West Poles! I never thought about that convention before.

34

Because the Earth is curved :smiley:

More accurately, I start facing due east, but if I keep walking due east along the latitude line I’ll have to turn because of the Earth’s curvature. That is, latitude lines aren’t straight lines (except the equator).

Okay, maybe once I convince you of that fact, the other story will make sense. So go find yourself a globe. Don’t just imagine – get a hold of a real physical spherical map of the Earth. I’ll wait.

Got it? Good. Now take a length of string and put one end on Minneapolis (it’s at 45 degrees north and 92 degrees west). Put the other end on Lyon, France (at 45 degrees north and 4 degrees east). Now, if the 45th “parallel” were a “straight line” on the Earth’s surface (we say “geodesic” when we want to sound important) then when you pull the string taut between those two cities it’ll lie along the latitude line. Try it!

What happened? When you pull the string taut it curves up over the top of the globe! That is, the straight line path from Minneapolis to Lyon isn’t the latitude line. In fact, when you leave Minneapolis along the straight line you’re going more north than east.

If you couldn’t find a globe, here is a plot of the flight path from Minneapolis-St. Paul to Saint-Exupéry airports. It looks curved, but that’s only because projecting the map onto a flat rectangle distorts sizes the closer you get to the poles. Really that’s the path your string would lie on if you actually carried out the experiment.

As a side note, we call latitude lines “parallels”, when they’re anything but. They aren’t even straight lines. In fact, there aren’t any parallel straight lines on the Earth’s surface!

35

So are longitude lines straight?

36

Yes, longitude lines are perfectly straight north and south.

37

Well, thank heavens I got *something * right! :smiley:

38

Well, I think it could be best, for this case, for your purposes, not to try to understand how it could be, but rather, just understand what it means to say that it’s true. (I actually think you already understand that. I think you get that objects moving in a straight line don’t necessarily end up the same distance from each other. I think what you’re having trouble with is understanding how this could be.)

Since the description of our space that I gave can also be said about curved surfaces, scientists refer to our space as curved. I can see two ways to interpret this. We might interpret this as a hypothesis that explains this behavior in terms of 3-d space being curved within a higher dimensional space. Or else, we might interpret it, not as a hypothesis, but simply as a terminological stipulation. This property of the space we live in–the property that objects moving in a straight line might not end up the same distance from each other–we will term space’s “curvature.”

What scientists are doing, it seems to me, is less like the “hypothesis” and more like the “terminological stipulation.” And if I’m right about that, it might be best simply to drop the analogy that’s been discussed in this thread. What’s going on is called “curvature” but it might not really be referring to any kind of curvature you and I are familiar with in everyday life. It has certain properties that are like properties of curvature, but it’s not really the same thing. (Because for you and me in everyday life, curvature does require embeddedness in a higher dimensional space.) For that reason, I think it could be a good idea to drop the notion of “curvature” altogether, and simply look at the facts that term is supposed to stand for. Call it space’s “warpedness” or space’s “weirdness” if you like. :stuck_out_tongue:

-FrL-

39

And I do accept that it’s true. I can even say a few sentences to that effect. But unlike low level algebra, which makes sense down to the very core of my being and resonates with me such that I feel I truly understand it (even if I would only be able to say “because it *has * to work that way, don’t you see?!” - I’m not a good teacher!), or if-then-else type things, or any number of things that I actually do feel (perhaps incorrectly) that I actually understand, to me this one is more like the concept of a supernatural being or something like that.

I trust this stuff more than I trust that god exists because thus far this stuff has been shown to work pretty well, and because humanity had so many motivations for inventing religion that don’t apply to multi-dimensional space. But it still comes down to taking things on faith, and I’ve never been all that comfortable doing that. But fortunately I do have a happy capacity for ignoring things I don’t want to think about, so I think I’ll just ignore the whole business, except to nod my head sagely when I hear that “space is curved.”

40

Maybe I could have phrased that better. When I said “accept” I meant more like “understand”, though not quite that either.

I blame my training. Mathematicians presenting an argument will use “accept” in this sense. We know there’s a difference between “believe it’s true” and “accept the line of reasoning”, but we just gloss over it in our own language.