### Physics Q&A #1: Curved Space

Welcome to my premier post of Physics Q&A, where I answer physics questions sent in by readers in order to help give you all an understanding of how the universe works. This week I'll be covering the question of what it means for space to be curved. I was also planning to cover the four fundamental forces here, but it got long enough I decided to just put it up and get at that in the next few days.

Q: What does it mean for space to be curved?

Warning: High math content below. Feel free to skim down to where I've marked "*****" if that's not your thing.

Before getting into this, it's helpful for you to understand what a metric is. Basically, it's a mathematical tool we use to measure distances in space. It takes a form like the following for normal, Cartesian space:

ds^{2} = dx^{2} + dy^{2} + dz^{2}

You might recognize this as being similar to the Pythagorean theorem for three dimensions, and that's no coincidence. For short intervals, you can calculate the distance between two points from the metric by:

Δs^{2} = Δx^{2} + Δy^{2} + Δz^{2}

In fact, in Cartesian space, this works exactly even for large distances, but this isn't the case in general. Most of the time we have to integrate along a line segment in order to find the length of it. For an example of a more complicated line segment, take a look at the metric for polar coordinates:

ds^{2} = dr^{2} + r^{2}dθ^{2}

Once we expand to three dimensions, we can add in another angle, φ, to get the spherical metric:

ds^{2} = dr^{2} + r^{2}dθ^{2} + r^{2}sin^{2}θdφ^{2}

Then, for brevity's sake, as we won't be doing many complicated things with the angles here, we combine the two angles into a single angular portion Ω, getting our final spherical metric:

ds^{2} = dr^{2} + r^{2}dΩ^{2}

This is the basic form metric generally used by astronomers for flat space. However, it turns out we can modify it to cover curved space as well by changing the r^{2} factor into some other function of r, which we'll call f^{2}(r). We could choose virtually anything for it, but observationally we have a few limitations. Basically, on a cosmic scale, space appears to be roughly the same in every direction (it's isotropic), and we would expect it to appear the same from when viewed *from* any point (it's homogenous).

Actually deriving the possibilities for f(r) takes some work that you probably don't care about, so I'll just skip to the three solutions:

f(r) = r - This standard, flat space.

f(r) = R*sin(r/R) - This is known as positively curved space

f(r) = R*sinh(r/R) - This is known as negatively curved space

(In case you aren't familiar with it, sinh is the hyperbolic sine defined by sinh(x) = (e^{x} - e^{-x})/2)

It turns out that if R is sufficiently large, then for small values of r, the metric for positively-curved and negatively-curved space behaves just like for flat space, so the difference wouldn't be perceptible to us. Note that when r becomes a significant portion of R, f(r) for positively-curved space increases more slowly than for flat space (angular distances correspond to less physical distance), while for negatively-curved space, f(r) increases faster (angular distances correspond to more physical distance).

***** (High-math content ends here)

So now the question is, what does space in these three different models actually look like? It's very difficult to picture in your mind how 3-dimensional space can warp, but if we bump it down a level, to 2-dimensional space, we can use a trick called "embedding" it into 3-dimensional space. What we do here is just take a 2-dimensional membrane and somehow bend it around so it has the curvature we want.**Case 1: Flat Space**

This is the simplest case. Here, we just have a flat plane representing our space:

Everything in here is perfectly sane. If you draw a circle with radius r, you'll measure its circumference to be 2πr. Nothing interesting to see, move along.**Case 2: Positively-Curved Space**

A bit more complicated, but still manageable. In this case, space ends up being represented as the surface of a sphere:

Now things are a bit more strange. If you draw a circle of radius r, you'll find that its radius is a bit *less* than 2πr. Additionally, the entire space is finite, and you can walk all the way around it given enough time. It's kind of like being on the surface of Earth if you pretend "up" and "down" don't exist. This type of space can be extended to three dimensions rather easily; just imagine that you can travel in space in any direction, and if you go long enough, you'll loop around to where you started. Additionally, at half that distance, no matter what direction you traveled, you'll end up at the same point, polar to where you started.**Case 3: Negatively-Curved Space**

Here's where things go to hell. Negatively-curved space is a bitch, and we frankly have no way to give an entire representation of it using our tool of embedding it in higher-dimensional space. But, we can give a few local representations. The notable feature to negatively-curved space is that it has what's known as a "saddle" shape locally:

As you can see, if you travel off in the x-direction either way, space curves up, while if you travel off in the y-direction either way, space curves down. This has the result of stretching out space, so if you were to draw a circle of radius r, you'll find that it will have a circumference greater than 2πr.

Now, unfortunately we run into problems when we try to find a representation of negatively-curved space that has constant curvature everywhere. A simple hyperbolic representation which I used to generate the graph above doesn't have constant curvature everywhere. Another common negatively-curved surface is what's known as the pseudosphere:

The problem with using a pseudosphere to represent space is that it isn't isotropic - one dimension extends out to infinity and goes imaginary at one point, while the other wraps around. However, it does have constant negative curvature. In the end, it turns out to be impossible to find a representation of isotropic negatively-curved space in this manner.

If we try to extend things to three dimensions, things get even worse. With our 2-dimensional membrane, we could have the x- and y-directions curve opposite ways, but if we add another dimension, we have no opposite left for it to curve in. To resolve this, we have to go up to at least 5-dimensional space to embed it. This allows us to have each dimension's curving go in a direction 120 degrees off from the other two dimensions:

This allows us to give a partial representation of how it looks (I'll spare your minds the 5-d graph). Unfortunately, we still run into a problem with coming up with a representation for infinite, isotropic, constant-curvature space. The lesson we have to take from this is one of two things:

1. An embedded representation might not be necessary. It could be that space just has a fundamental law about how distances in it work determined by the metric, and embedding is just a sometimes-convenient way of looking at it.

2. Negative space just won't work. This would be the case if the real way things were made in the meta-universe was with our universe being embedded in some higher-dimensional Cartesian space. It's possible that things would work out this way, but we can never know for sure. This possibility might imply that our universe can't be negatively curved, but we can't know for sure.

## 4 comments:

Okay, naive question here that may indicate my failure to grasp this, but does this then mean that in negatively curved space that if you traveled long enough in one direction, you could ultimately end up back at the spot that you started? It certainly appears that way from the 2D renderings that you've done...

Ah, no, that's one of the problems I noted with the pseudosphere. In actual negative space, it extends infinitely in any direction, so there's no way you can loop around. This part actually works with the saddle I showed in the first graph (formula: z = x^2 - y^2), but the problem with that is though it has negative curvature everywhere, the curvature isn't constant.

If you want to go and make a figure with constant negative curvature, you end up with a pseudosphere. However, this has the problem that it isn't homogeneous (space doesn't appear the same at every point), and that it's possible to loop around if you go in certain directions. The reason we can't accept it looping around goes back to the metric derived in the first part (the sinh(r)^2 coefficient, specifically). Since this goes off to infinity as r does, it implies that space can never curve back on itself.

OK, let's see if I got this right (regardless of whether I get it or not, I like this post).

Flat space - no loop around, but the universe can appear both homogeneous and isotropic.

Positively curved space - loop around occurs, and the universe can appear both homogeneous and isotropic.

Negatively curved space - no loop around, and the universe can appear either homogeneous or isotropic, but not both.

Is that a fair summary, if horribly simplified?

If what I've said is roughly true, then I can think of a way to test which type of universe we live in (warning: it's a little impractical).

Send a spacecraft away, very quickly. If it comes back, space must be positively curved. As it travels, have it take pictures and send those pictures back to us. If everything continues to look the same while it's travelling, then space is both isotropic and homogenous, so it can't be negatively curved.

Like I said, probably impractical. Am I missing something critical other than the ridiculously long time-scales required for my experiment?

Thanks again for posting this. I've almost thought up an astrophysics question, I just need to figure out how to word it as something other than a list of keywords to be deciphered.

Flat space - no loop around, but the universe can appear both homogeneous and isotropic.

Positively curved space - loop around occurs, and the universe can appear both homogeneous and isotropic.

Negatively curved space - no loop around, and the universe can appear either homogeneous or isotropic, but not both.

Is that a fair summary, if horribly simplified?

That works for positive and flat space, but not quite for negative space. The trick with it is that mathematically, we can indeed construct negatively-curved space that's both homogeneous and isotropic, we just can't embed it in a Cartesian coordinate system. It is possible to embed it in other types of space, such as Minkowski space (where distances are calculated a bit differently); it helps for the math, just not for visualizing it.

Here's a way to look at how it works on small scales: If you travel out in the X direction, then space in the Y and Z directions relative to you will:

Flat: Stay constant

Positive: Contract slightly

Negative: Expand slightly

A few other ways to look at it:

If you create a large sphere in the space, then directly measure the volume inside it, you'll find that it will be:

Flat: Equal to 4/3 * pi * r^3 (standard formula)

Positive: Less than 4/3 * pi * r^3

Negative: Greater than 4/3 * pi * r^3

If you draw a giant triangle in space, then measure the angles of it and add them together, you'll get:

Flat: Precisely 180 degrees

Positive: More than 180 degrees

Negative: Less than 180 degrees

As for testing it, we actually have done some direct measurements on it, mostly analogous to measuring a giant triangle. The problem is that the results are all very close to flat, so we can't say for sure whether space is perfectly flat, a bit positive, or a bit negative, as these options are all within the margin of error.

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