Gravity and Curves in Space

Factual Questions
1

I am not a physicist, nor do I play on on TV. But I’ve “known” for years that space is curved, and I’ve many times seen the dimpled plot that supposedly illustrates this.

The only problem is, I really don’t understand this in the least. I mean, the kind of illustration they show is like the surface of a swimming pool. But isn’t space more like the middle of a swimming pool, without edges? Curves are a surface phenomenon.

I can see how you might have a wave that passes through the entire depth of a pool, so that there would be a vertical wall of more densely packed water preceded or followed at some point by either shallower or less densely packed water. But, a) how can you have shallower when space is limitless, or more or less densely packed when space is mostly a vaccuum, and b) why would you describe that as a curve? Why would you say that an orbit “thinks” it’s moving in a straight line, but the line it’s moving on is actually curved?

Can someone(s) who actually understand(s) this stuff explain it in terms that a bright first grader could understand?

BTW, I may not be able to get back to my email for about 12 hours, so thanks in advance for any help you might be able to give.

2

IANAP either, but when I see the ‘bowling ball on a sheet’ illustration I think ‘They’re only showing one plane.’ So imagine superimposing a second image on top of the first, only 90° north. And another one 33.547896° west. And another one… So the depression completely surrounds the ‘sun’.

Anyway, that’s the way I think about it.

3

Sonofabitch…I get it now. Next time you’re back in L.A., I owe you a beer. Sonofabitch…

4

Johnny has one part of it, but let me expand on his point because it’s related to what I’m saying in the thread about spheres: mathematicians and physicists talk about intrinsic properties of geometry, not properties of their embedding.

What does this mean here? You’re looking at the dented rubber sheet and thinking about how it sits in three dimensions. The problem is that you should be thinking of just the two-dimensional sheet itself.

Okay, how does this strike you? Imagine the sheet is not only rubber, but it’s transparent. There’s a bright light way above it, shining straight down onto a featureless white background below, and casting shadows of things moving around on the sheet.

There’s a bowling ball sitting on the sheet, denting it, and casting a big shadow on the ground. This represents the Sun. There’s also a golf ball rolling around representing the Earth. It makes a little dent, but we’ll ignore that. What we’re interested in is the path its shadow traces out on the ground. As it moves along the sheet, it curves around the dented surface. So what do we see on the ground? We see the small shadow circling the big shadow, just as the Earth circles the Sun.

So what if we forgot the sheet above and just looked at the ground? We’d see shadows moving around, and we might come up with a theory of 2-d gravity to explain how these shadows “attract each other”.

To get to physical reality, we need some sort of 3-d “rubber space”, which we can’t visualize deforming. Johnny’s suggestion works for a single object like the Sun, but it gets to be a big problem in more realistic situations. Luckily, the math can handle all this stuff for us.

While I’m on the subject, though, I may as well point out another feature of the rubber-sheet model. Why does the path curve? This gets confusing because you actually have to think of there not being gravity around the sheet (except that it makes the objects dent the sheet…). Objects actually move along straight lines on the curved surface.

What does this mean? If you zoom in close enough to the surface, it looks like a flat plane (same as the curved surface of the Earth does). So if you’re taking the place of an object, you just put your blinders on and walk straight forward. Never turn your head. Whatever direction your nose points: that’s the direction you go. It’s the curvature of the rubber surface that makes your shadow on the ground curve.

Make sense?

5

Imagine that, if in space, and ignoring limitations of travel and expansion of the universe, if you could go in one direction for a very long time, you might end up in the same place where you started from. That’s curvature, just like you can travel around the equator on the 2D surface of the Earth because the Earth’s surface is curved.

6

That’s… a kind of (global) intrinsic property that one might call “curvature” in ordinary language, but it’s not quite the same as the (local) intrinsic concept of curvature which is more important here.

Consider various non-Euclidean properties of the surface of the Earth: triangles on its surface have angles which don’t add up to exactly 180 degrees [consider one formed by the north pole and two points on the equator], two people travelling in straight lines in initially “parallel” directions can come to be travelling towards or away from each other [again, consider starting from two points on the equator and moving north], etc. These are the sorts of things which are most relevant here. And it is conceivable to have these properties without having any “wrap-around”; illustrating via embedding into Euclidean space, consider the surface of a parabaloid or such things.

On the other hand, the game of Asteroids, where one wraps from the top to the bottom and left to the right, has none of these properties. The Asteroids world doesn’t have the relevant kind of curvature, it just has “wrap-around”. It is locally Euclidean not only in the sense that things locally approximate Euclidean space, but in the stronger sense that things locally are Euclidean space.

7

Hm, that post could probably stand to be improved, from a pedagogical point of view. Well, until that happens, it should at least get the basic idea across.

8

[QUOTE=dre2xl]
Imagine that, if in space, and ignoring limitations of travel and expansion of the universe, if you could go in one direction for a very long time, you might end up in the same place where you started from. That’s curvature, just like you can travel around the equator on the 2D surface of the Earth because the Earth’s surface is curved.
[/QUOTE]

NO! That’s absolutely NOT curvature!

You only think of that as curvature because the embedding of the surface into the plane has to be curved to do that. Curvature is the tendency of initially parallel straight lines on the surface to diverge.

Take a sheet of paper and roll it up into a cylinder. This embedding of the cylinder into space is curved (obviously), but intrinsically, the cylinder is flat. If you draw parallel straight lines on the surface before you roll it up, they’ll still be parallel straight lines on the cylinder because the cylinder has no intrinsic curvature.

9

[QUOTE=Indistinguishable]
Hm, that post could probably stand to be improved, from a pedagogical point of view. Well, until that happens, it should at least get the basic idea across.
[/QUOTE]

Actually, that tendency for initially-parallel straight lines to converge that you were talking about is exactly what curvature is. You’re just looking at its effects over long distances, where you’ve added up the infinitesimal effects of local curvature along the two paths.

10

Johnny L.A., that’s sort of something I can actually visualize, although to be honest, I don’t see how it applies to planets and stuff.

dre2xl, I don’t think that would happen if I achieved escape velocity and headed out of the solar system, so I don’t think the analogy holds. Don’t we have Voyager going out into the galaxy going on forever and ever?

[QUOTE=Mathochist]
Johnny has one part of it, but let me expand on his point because it’s related to what I’m saying in the thread about spheres: mathematicians and physicists talk about intrinsic properties of geometry, not properties of their embedding.
[/QUOTE]

That thread is so far over my head that I drown in the first three posts.

[QUOTE=Mathochist]

What does this mean here? You’re looking at the dented rubber sheet and thinking about how it sits in three dimensions. The problem is that you should be thinking of just the two-dimensional sheet itself.

[/QUOTE]

Why? Earth moves through three dimensions and occupies three dimensions. Four if you want to include time. But I’ll settle for three. So space is everywhere around us, just like water in a swimming pool. How can you fold or curve water in a swimming pool, except on an edge - they are two dimensional concepts, or at least edge concepts. Space doesn’t have any edge as far as I know.

I understand that large circumference values look straight to someone on the circumference, but that doesn’t make it so. If I draw a circle on a sheet of paper, I don’t say the paper has curved and that the edge of the circle is straight. I say that the edge of the circle is an arc and the edge of the paper is straight.

[QUOTE=Mathochist]
Okay, how does this strike you? Imagine the sheet is not only rubber, but it’s transparent. There’s a bright light way above it, shining straight down onto a featureless white background below, and casting shadows of things moving around on the sheet.

There’s a bowling ball sitting on the sheet, denting it, and casting a big shadow on the ground. This represents the Sun. There’s also a golf ball rolling around representing the Earth. It makes a little dent, but we’ll ignore that. What we’re interested in is the path its shadow traces out on the ground. As it moves along the sheet, it curves around the dented surface. So what do we see on the ground? We see the small shadow circling the big shadow, just as the Earth circles the Sun.

[/QUOTE]

Could you please explain what you’re analogizing here? Because I don’t see any big rubber plastic tarp with huge recessed lighting shining down on the solar system, the galaxy, or the universe.

[QUOTE=Mathochist]
[
So what if we forgot the sheet above and just looked at the ground? We’d see shadows moving around, and we might come up with a theory of 2-d gravity to explain how these shadows “attract each other”.

[/QUOTE]

Okay, but that’s not what we see. We see bodies in three dimensions being attacted to one another.

[QUOTE=Mathochist]
To get to physical reality, we need some sort of 3-d “rubber space”, which we can’t visualize deforming. Johnny’s suggestion works for a single object like the Sun, but it gets to be a big problem in more realistic situations. Luckily, the math can handle all this stuff for us.

[/QUOTE]

Well, here’s the weird thing. The Sun* is * a single object, and about as realistic as a thing can get. And I personally can’t handle the math - as soon as I hit set theory, I went down in flames. I was born to do low level algebra.

[QUOTE=Mathochist]
While I’m on the subject, though, I may as well point out another feature of the rubber-sheet model. Why does the path curve? This gets confusing because you actually have to think of there not being gravity around the sheet (except that it makes the objects dent the sheet…). Objects actually move along straight lines on the curved surface.

What does this mean? If you zoom in close enough to the surface, it looks like a flat plane (same as the curved surface of the Earth does). So if you’re taking the place of an object, you just put your blinders on and walk straight forward. Never turn your head. Whatever direction your nose points: that’s the direction you go. It’s the curvature of the rubber surface that makes your shadow on the ground curve.

Make sense?
[/QUOTE]

Honestly, I wish I could say it did; I’m not trying to be snarky or willfully ignorant. I just don’t understand, and maybe I’m simply not capable of understanding - I’ve found that to be the case in a few areas; no matter how many times I’ve been told or explained to, I can’t wrap my mind around it.

Let me ask what is perhaps a simpler question. In both the science and the sci-fi I’ve read, they’ll talk about gravity waves and/or variations in gravity. The two variables I know about in gravity are mass of the objects and distance between the objects. So when they talk about variations, are they talking about changes in mass and/or distance, or are they talking about a change in the strength of the force itself, and if so, is there any theory as to how that could be? I remember this bothering me first when I read about mapping mountains in the bottom of the ocean by gravity variations and coming to the eventual conclusion that this must refer to distance variations rather than actual changes in the laws of gravity. On the other hand, in the Honor Harrington sci-fi series, they use gravity waves as a considerable aid to propulsion, and I suspect there they may be postulating areas where gravity itself is more powerful than elsewhere.

Again, I’m not sure when I’ll get a chance to return, so I thank you all!

11

[QUOTE=Mathochist]
Actually, that tendency for initially-parallel straight lines to converge that you were talking about is exactly what curvature is. You’re just looking at its effects over long distances, where you’ve added up the infinitesimal effects of local curvature along the two paths.
[/QUOTE]

Wait a minute. Aren’t parallel lines by definition lines that never converge and aren’t skew? Forget non-Euclidean geometry, about which I know absolutely nothing, I’m talking about Euclidean.

And Indistinguishable, the properties you’re describing are true because you’re plotting two dimensional figures onto a sphere. But space is a) infinite and b) three-dimensional. An infinite cube or sphere. There are no edges to distort.

I’m so confoosed. :frowning:

12

[QUOTE=Oy!]
Wait a minute. Aren’t parallel lines by definition lines that never converge and aren’t skew? Forget non-Euclidean geometry, about which I know absolutely nothing, I’m talking about Euclidean.(
[/QUOTE]

That’s one use of the word; it’s a global definition (you can’t tell whether two lines are parallel, in that sense, just by looking at what’s happening over here in a small neighborhood). But there’s a different, more local use of the word, which you surely have some intuition for as well: two lines which point in the same direction, so to speak.

I’m not sure what you’re talking about with “There are no edges to distort”, and nothing I said is particularly restricted to two-dimensional or finite cases. [If you can handle it, imagine the three-dimensional universe which is the surface of a ball in four-dimensional Euclidean space [just upping the dimensions from Earth]; this will have the same curvature properties I was talking about]

13

I’m not sure if this posted before or not, but, well, this may double-post:

[QUOTE=Oy!]
Wait a minute. Aren’t parallel lines by definition lines that never converge and aren’t skew? Forget non-Euclidean geometry, about which I know absolutely nothing, I’m talking about Euclidean.(
[/QUOTE]

That’s one use of the word; it’s a global definition (you can’t tell whether two lines are parallel, in that sense, just by looking at what’s happening over here in a small neighborhood). But there’s a different, more local use of the word, which you surely have some intuition for as well: two lines which point in the same direction, so to speak (this needs to be formalized a bit more, but that’s the intuition). The key to curvature is that straight lines which point in the same direction over here might not still point in the same direction over there, despite being straight. If that’s confusing, it’s because you’re so used to implicitly thinking of Euclidean space; the best intuitive analogy is generally to think of the surface of the Earth intrinsically (rather than thinking about it as embedded in some other three-dimensional space. Pretend you could never leave the surface, that nothing exists except the surface).

Incidentally, I’m not sure what you’re talking about with “There are no edges to distort”, and nothing I said is particularly restricted to two-dimensional or finite cases. [If you can handle it, imagine the three-dimensional universe which is the surface of a ball in four-dimensional Euclidean space [just upping the dimensions from Earth]; this will have the same curvature properties I was talking about]

As I said, the thing to keep in mind is that, as I think may be throwing you when you protest that space is “An infinite cube”, the idea that straight lines which start out pointing in the same direction have to keep pointing in the same direction is not an a priori necessary truth; it is an assumption. And, empirically, that assumption has not turned out to be true of the space in which we live. But that is no problem; it just means you were using a naive model of space before, which has to be modified. It is precisely as though you lived on the surface of the Earth and thought “Well, any three straight lines must form a triangle whose angles add up to 180 degrees”, and then one day actually carried out the experiment, and discovered you were wrong.

14

[QUOTE=Oy!]
\Why? Earth moves through three dimensions and occupies three dimensions. Four if you want to include time.
[/QUOTE]

Because the rubber-sheet model describes things moving in two dimensions. As I mention later, physical reality has things moving through three dimensions, but the visualizations there get a lot harder. So we make a simpler model (one dimension less) and visualize that to give us an intuition about how things behave in the more complicated reality.

This is related to what I say later about zooming in on the surface. The problem here is that “straight” on a curved surface means “not turning left or right, but following the surface”. If you do this on the Earth you’ll get great circles that travel all the way around the Earth. The equator is one, and each meridian is one, but latitude lines other than the equator are not great circles.

To see why this is the case, imagine following the 89th parallel as it winds around the north pole. To follow it on the surface of the Earth, you’ve got to keep turning left as you go. If you just walked straight ahead (on the Earth’s surface) you’d go further and further south until you got to the 89th south parallel and then you’d go back north.

In short, since we’re inside a curved spacetime, the lines we see as “straight”, never turning right or left, would look curved from the outside.

The rubber sheet is the universe, only it’s a 2-dimensional universe to make things easier to visualize.

No, they’re taking about waves in the rubber sheet itself. As an object moves in the sheet its dimple moves, and that sends off little ripples. And since the sheet is a stand-in for spacetime, this means that spacetime itself is waving.

That’s sort of the problem: curved spaces require non-Euclidean geometry. In fact, Einstein’s general theory of relativity linking differential geometry to as basic a part of physics as gravitation was the death-knell for Euclidean geometry.

15

Oy, the dimpled plot that is like the surface of a swimming pool is throwing you. They mean it a different way from the way you’re taking it, and they probably didn’t say so, or at least not clearly.

Never mind that it is more like the surface of the pool, and space is more like the middle.

Imagine a sculpture that looks like the dimpled plot, so it’s a rounded sort of a funnel, like an “innie” belly button, or the bell of a trumpet. That curved surface is meant to be a two dimensional map, a slice through space. If you are holding the trumpet in your hands and fooling around with the geometry of it, you are dealing with a two dimensional representation of real space, even though it takes three dimensions to have this representation. The kind of thing you’re supposed to do with it is, for example, laying down strips of tape to see how they appear to run off at funny angles, or driving a little toy car around on it, or measuring the area of a circle you draw on it. You will find that the kind of plane geometry you learned in, what, 8th grade works very strangely when you draw it on the bell of a trumpet.

So, yes, the real universe feels like it has 3 dimensions, but you’d need 4 dimensions to be able to make a model of it to play 3D geometry games on and see they turn out funny.

16

Oy

Your post number 10 leads me to ask whether you’ve been introduced to the ideas in the book Flatland.

That book may help you intuitively grasp how it could be that three dimensional space can be curved. (I don’t think that specific issue is mentioned in that book, but it gives you the resources to imagine how it could be so.)

In post 10, you were stumbling over the analogy between space and a rubber tarp, it appears to me, because the tarp’s surface is two dimensional while space is three dimensional. You were having trouble seeing how anything about the surface of the tarp could be analogous to a three dimensional volume like space.

One way to understand the analogy is to think of three dimensional areas (like space) as though they were “surfaces”* of four dimensional volumes. The tarp is a two dimensional surface curved in a three dimensional space. Similarly, a three dimensional surface can be curved in a four dimensional space.*

Well, that’s the intuition the tarp analogy relies on. As footnoted below, there are problems with this–ones touched on in previous posts in this thread–but it should do for starters. Is this the kind of thing you’re finding yourself getting hung up on?

-FrL-

*As people have said above, space’s being curved doesn’t actually require that it be embedded in a four dimensional space. But let’s not worry about that yet.

17

[QUOTE=Frylock]

*As people have said above, space’s being curved doesn’t actually require that it be embedded in a four dimensional space. But let’s not worry about that yet.
[/QUOTE]

It seems to me it does. It’s came up here before, I remember and exchange between WarmGun(?) and Chronos I believe. I didn’t think Chronos made his point well. Or maybe (wait, probably!) I don’t understand these things but if dots on a balloon (A 2D world) exemplify three dimension reality shouldn’t the nature of our 3D world represent a fourth dimension reality? The argument was that (dis-regarding the strings and wires) you could not arrange balls in our 3D space and move them in a way that would mimic our cosmos (Every ball appearing to be in the center of the ball with the others moving away from it).

18

It’s easy to see that no embedding is required; take your favorite curved surface embedded in Euclidean space, subtract the rest of the Euclidean space; presto, a curved universe not embedded in anything.

Less cheaply, but essentially the same thing, suppose the game of Asteroids was programmed a little differently, to behave like the surface of a 2-sphere; just to make things easier to visualize what this would mean, imagine that it has the camera always tracking your ship exactly, fixing it in the center of the screen always oriented to face towards the top; at any given moment, thinking of your rocket’s location as Ecuador and the direction it is facing as North, you can press up to move North, right to move East, down to move South, left to move West, etc. The difference from the existing game of asteroids would be that, as you moved, lines which appeared straight when projected on the screen would change to no longer appear straight on the screen, and so on. Clearly, we could program such a game, and clearly, the geometry of the gameplay universe would be curved/non-Euclidean. But in what reasonable sense is the two-dimensional universe visible in the game embedded in some abstract three-dimensional Euclidean space? Why assign any significant ontological status to anything other than the actual objects of the game?

That brings us to the most general argument, which is… there’s no need for an embedding, because we can mathematically describe curved spaces without resorting to describing them via an embedding in Euclidean space.

19

[QUOTE=Indistinguishable]
That brings us to the most general argument, which is… there’s no need for an embedding, because we can mathematically describe curved spaces without resorting to describing them via an embedding in Euclidean space.
[/QUOTE]

Exactly. The mathematics of GR talks about intrinsic properties of the spacetime manifold, and not about extrinsic properties of its embedding in any higher-dimensional space.

This is the answer to Don’t fight the hypothetical and his questions about whether a 2-d balloon means talking about the 3-d space around it: we don’t talk about the space around it because that’s not what the mathematics describes.

20

[QUOTE=Indistinguishable]
at any given moment, thinking of your rocket’s location as Ecuador and the direction it is facing as North, you can press up to move North, right to move East, down to move South, left to move West, etc.
[/QUOTE]

A misguided departure from the classic tradition on my part; obviously, the more Asteroids-style control scheme would have left and right rotating your orientation rather than translating your position (while up and down would merely increase and decrease your thrust, respectively, in the direction you face). I hang my head in shame. :frowning: