Posted by on Apr 12, 2013 in Blog, Science | 0 comments

Three kinds of space

Assume once more that space is homogeneous, at least at large scales; that is, for distances greater than 0.002 H_d (where H_d, the Hubble distance is roughly 14 billion light-years). We know that below these scales, the distribution of ordinary matter appears to have a correlation dimension of around 1.77±0.04, implying a fractal structure similar to that of diffusion limited aggregation (DLA). But if the universe has a homogeneous distribution of matter at large scales, then we might think about it like, for example, the paved surface of a asphalt road. Stand back, and the surface is uniform; look closely, and it is a fractal with small stones of many different sizes. Looking at the universe at a large scale, and assuming that it may have some curvature on this grand scale, the question is, what is that curvature?

There are various sorts of objects that have non-uniform curvature; for example, an American football or a rugby ball. These are more or less ellipsoids. The curvature is greater at the ends than in the middle. There are surfaces like the cone where the curvature is concentrated at the tip. There are surfaces like a torus where the curvature depends upon the path you follow over the surface.

However, none of these surfaces can apply to a universe that is homogeneous: the curvature would have to be the same everywhere (at least at a grand scale, as I say). This raises only three interesting alternatives: positive, zero, or negative curvature. Positive curvature corresponds to a closed and finite universe with a topology like a 3-sphere. This implies a negative total energy, in the context of the Friedmann equations that I’ve been working with in the last few posts. A zero curvature corresponds to ordinary, flat Euclidean space; and this would imply a null total energy. Finally, there is the case of negative curvature: this implies a hyperbolic and open universe with positive total energy.

What can we say about these different sorts of space? In a previous post, I went over the development of some flat and positively curved n-dimensional spaces. In that post, I neglected the development of any hyperbolic n-dimensional spaces. However, the model for a hyperbolic surface is mathematically very similar to that for a spherical one. The essence of the difference is easiest to see in the 2-dimensional case:

 x^2 + y^2 = 1

is the unit 1-sphere or circle, while

 x^2 - y^2 =1

is the unit 1-hyperbola.

One very interesting way to look at these curves is in terms of conic sections. If you have Wolfram’s CDF player, you can see a couple of demonstrations of conic sections below. These will allow you to play with the double cone in terms of various sections through it in three dimensions to see circles, ellipses, parabolae, hyperbolae, lines and points. You will also be able to get a feel for the two dimensional results. It is well worth installing the CDF player to be able to see this for yourself.

[WolframCDF source=”” CDFwidth=”685″ CDFheight=”700″ altimage=””]

[WolframCDF source=”” CDFwidth=”685″ CDFheight=”700″ altimage=””]

One exceptional way to tell the difference between being in a space that is either flat, positively or negatively curved has to do with the angles subtended by distant objects. For example, in a flat space, an object of length L at a distance D is at the base of an isosceles triangle of height L and base D. Knowing this, we can work out that the angle will be 2 ArcTan(D/2L) \approx D/L when L \gg D. That’s in a flat Euclidean space.

However, angles and triangles don’t quite work out this way in curved spaces. Let’s approach this idea through the concept of a stereographic projection, which can map the surface of an n-dimensional object onto an (n-1)-dimensional plane. To keep things very simple, I’ll begin by mapping a 2-dimensional circle and hyperbola onto a line.

Begin with a hyperbola:



In this figure, you can almost see the conic section quality of the hyperbola as I have included the asymptotes that the hyperbola is, in effect, sliced out of. Now, imagine that we have a small line segment, \Delta h on the surface of the hyperbola right where it intersects the abscissa. We could project this line segment onto the line shown at x=10, and for \Delta x small, we would simply find that the length projected is L = 10 \Delta x, by simple trigonometry. But this wouldn’t be true for line segments of length \Delta x placed anywhere else along the surface of the hyperbola. In fact, as we move the line segment further and further away from the origin, the projection becomes increasingly small.

If we take this hyperbola and turn it into a surface of revolution, we obtain a hyperbolic plane as the surface. This last figure continues to work as a means of projecting the hyperbolic plane onto a Euclidean plane stereographically. We could now imagine tiling the hyperbolic plane with some figure, such as a triangle or square, that would cover the surface. We would then get a projection of the tiles onto our plane at x=10. If you have the CDF player installed, then this next demonstration will show you the result of such a tiling in a stereographic projection.

[WolframCDF source=”” CDFwidth=”685″ CDFheight=”700″ altimage=””]

If you set the number of sides for the polygon at 3 (a triangle), the number of polygons at each vertex at 7 (the minimum), and hit the “Centralize” button, you’ll get a tiling with triangles that is relatively simple to visualize. Note the important feature that the projection of the triangles becomes smaller and smaller as the distance from the origin increases. Note that on the surface of the hyperbolic plane itself, each triangle is of equal size. More distant triangles occupy less and less area in the plane of projection.

The opposite picture arises for the projection of a circle onto our plane. The following very simple figure shows a unit circle centered at x=1 with a projection line at x=2. Now, assume a small line segment \Delta s on the surface of the circle near the abscissa around x=2. We project from the origin through this line segment, and again, using simple trigonometry, we expect a projected line segment of length about L = \Delta s. Again, as we consider other line segments of the same length on the surface of the circle, we’ll get larger and larger projections as we bring these closer and closer to the origin.

By turning the circle into a surface of rotation, we get a 2-sphere covering a globe, and we can now project a tiling on the 2-sphere onto the plane. If you, again, have the CDF player, the following demonstration will show the result of such a tiling (almost). The demonstration projects Platonic solids onto the plane, rather than a tiling of the sphere; but if you select the dodecahedron, you will see the projection of what is almost a tiling by pentagrams on the 2-sphere. What is important to note is that, in strong contrast to the projection of the hyperbolic plane, we see that the projected areas of the tiles increase dramatically as their location approaches the “North Pole”, or the origin of the projection.

[WolframCDF source=”” CDFwidth=”685″ CDFheight=”700″ altimage=””]

These observations go together with the concepts of an ordinary triangle (in Euclidean space), a hyperbolic triangle, and a spherical triangle. If you look at these topics in the links I’ve provided, you will immediately see that the hyperbolic triangles appear concave while the spherical triangles appear convex. Also, the sum of the angles in the hyperbolic triangle is less than \pi while that for a spherical angle is greater than \pi. This angular deficit or excess goes back to the curvature of the space that the triangle is embedded in. This idea is generalized as a Schwarz triangle.

By measuring the angle subtended by an object of known size at a known distance, it would therefore be possible to estimate the curvature of the space. For example, imagine that you are an astronomer and you know that some distant object, say a galaxy is precisely 10 light-years across and it is 1 million light-years away. From this, you can estimate the angle it should subtend in our sky by simple Euclidean geometry. If you measure the actual angle, you will either get what you expected, or a number that is lower or one that is greater.

If you were a cosmologist interested in the broad structure of the universe, you could do this same experiment for objects all over the sky. You could then come to a conclusion as to whether or not the universe was closed and finite (a sphere of positive curvature and negative total energy), open and flat (Euclidean and with 0 total energy), or open and infinite (a hyperbola with negative curvature and positive total energy).

So, is there anything in the sky that is at a known distance of cosmological scale and that has a known length that we could use to decide this question? The answer is “yes”: fluctuations in the Cosmic Microwave Background Radiation.



For a variety of reasons that I’ll explore in future posts, we can expect these fluctuations to be at about 13.8 billion light-years away and to be about 370,000 light-years across. By measuring the power spectrum of these fluctuations across the sky, one can arrive at a density in terms of angle:

CMBR Spectrum

CMBR Spectrum

The first peak is what is relevant for establishing curvature, and it is consistent with a flat universe. There are some issues though. It has been suggested that the higher angular variations are more consistent with a Poincaré dodecahedron than with a flat universe. Another possibility, with negative curvature, is a Picard horn.

This is an area of active research. More later…


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