Posted by on Apr 14, 2013 in Blog, Science | 0 comments OK, my title is really corny. Call it geek humour of the worst kind, if you want. But you’ll see where I’m coming from soon enough.

In my last post, I was going over the three kinds of space that could be broadly consistent with a homogeneous distribution of matter; this leading to a criterion of uniform curvature. To review briefly, this yields either a universe of positive energy and negative curvature in the form of a 3-hyperbola or a universe of zero energy and zero curvature in the form of a Euclidean 3-space or a universe of negative total energy and positive curvature in the form of a 3-sphere.

In this post, I want to introduce the metrics for these spaces and to show how these metrics would impact the measurement of angles subtended by distant objects. Along the way, we’ll see a few other interesting features.

But first, I’m going to use polar or spherical coordinates for this procedure. There is a simple reason: it makes imminent sense to imagine ourselves at the center of our universe since that is our (non-Copernican) point of view. We look out from where we are placed along radial lines. If we can distinguish what we would see at some given distance along such a line, depending on the curvature of our universe, then this may be a way to tell what that curvature is. Also, I’m going to start with the metrics of a space and then work to the metrics of the three versions of space-time.

## Flat 2-space

For a simple, flat Euclidean space in two dimensions, we get a metric of the form $ds^2 = dx^2 + dy^2$

and of course, if we work in three dimensions, we’d have $ds^2 = dx^2 + dy^2 + dz^2$

In polar coordinates, this metric becomes $ds^2 = dr^2 + r^2 d\theta^2$

where $\theta$ is the polar angle. This corresponds to the construction of a circle, aka a 1-sphere, at a distance $r$ from the origin. If I consider the metric on a unit 1-sphere, it is just $ds^2 = d\theta^2$. For reference, I want to define this metric as $d\Omega_1^2 \equiv d\theta^2$. I can rewrite this metric for Euclidean space as $ds^2 = dr^2 + r^2 d\Omega_1^2$

In other words, what this is telling us is that as I look out into a Euclidean 2-D space I see circles of radius $r$ at a distance of $r$ away from me. Of course, this makes all the sense in the world, in a flat space. What else would the radius of those circles be, if not $r$? So, our flat 2-d space can be thought of as being constructed out of a densely nested set of concentric circles expanding out forever.

By extension, we can go to a Euclidean 3-space. Now, we will look out and at a distance $r$ we have a the surface of a 2-sphere of radius $r$ at each distance $r$ away from our view point at the origin of coordinates. To consider this construction, we first have to get the metric for the 2-sphere, and we are going to do this using the metric on the 1-sphere (aka the circle).

## The Spherical Plane, or 2-sphere

We made a flat Euclidean 2-space by filling it with circles of radius $r$ at each distance $r$. We get a 2-sphere, instead, by creating a finite space through a sequence of circles of radius $r sin \phi$ at a distance $r$. Frankly, this mathematical construct makes more sense we either set $r=1$ and only let $\phi$ run from $0$ to $\pi$ or set $r = \phi$ and let $r$ run from $0$ to $\pi$. The idea of the construct is simple enough though: as we look out on the surface of the 2-sphere at some distance $r$, we have a circle (1-sphere) of radius $sin r$. The following picture may make this clearer:

[You can click on the thumbnail for a larger version.] For a world constrained to live on the surface of the 2-sphere, distance away from the origin, which I’ve chosen to be the “West Pole” as it were, is measured like a radian distance on the surface. I’ve shown the size of the maximum circle at a distance of $r = \pi/2$ and the radius of that 1-sphere is $sin r$. In this way, the notion of distance and parametric angle are rather merged together subject to the constraint of a unit 2-sphere.

This construct is different than the case of the flat space in which the nested 1-spheres have a radius that grows linearly with distance; instead, the nested 1-spheres grow almost linearly to begin with (since $sin(r) \approx r$ for small $r$) but then they reach a maximum and finally begin to shrink back to $0$. If a 2-dimensional astronomer lived on such a world, he might confuse his world for a flat one because locally there might be little difference if he could see out only over a small fraction of his world. This would be very much like our ancestors back in the Middle Ages who thought that the Earth was flat because they could only see an extremely small fraction of the entire surface.

With the idea that distance on the unit 2-sphere is equivalent to a radian angle, the metric on the 2-sphere is $ds^2 = dr^2 + sin^2(r) d\Omega_1 ^2$

Note the difference between this metric and that of the one for the Euclidean space; namely, the 1-sphere part is multiplied not by $r$, but by $sin(r)$. This metric can be called $d\Omega_2 ^2$ in a notation consistent with labeling an n-sphere as $\Omega_n$. We’ll use this construction in the form to develop metrics in 3-dimensional spaces soon enough.

## The Hyperbolic Plane

Holding this model, we can construct a hyperbolic plane simply enough. Just as we look out into a 2-sphere and see a nested sequence of 1-spheres that have a radius of $sin(r)$ at a distance of $r$, for the hyperbolic plane, we look out and see a 1-sphere of radius $sinh(r)$ at a distance of $r$. If we designate the unit hyperbolic plane as $\mathcal{H}_2$ then its metric is just $ds^2 = dr^2 + sinh^2(r) d\Omega_1 ^2$

Again, a 2-dimensional astronomer on such a surface might not realize any difference between this and a Euclidean surface since $sinh(r) \approx r$ for small $r$. So if he could see only out to a small fraction of the total hyperbolic plane, he might think that it was flat.

## Extending to 3-space

Now, we can extend these concepts to 3-dimensional spaces that are either flat, spherical or hyperbolic. For the flat 3-dimensional space, we have for flat space (call it $E_3$) $ds^2 = dr^2 + r^2 d\Omega_2 ^2$

For the 3-sphere $ds^2 = dr^2 + sin^2(r) d\Omega_2 ^2$

And for the 3-hyperbola $ds^2 = dr^2 + sinh^2(r) d\Omega_2 ^2$

## The point at the East Pole

In each case, we look out to a distance $r$ and see objects at that distance on the surface of a 2-sphere. This is very much how our 3-dimensional world looks to us, in fact. We look out a few meters, and we see objects on a sphere of radius that many meters. We look out as far as our sun, and see the sun as if it were on a sphere of one astronomical unit in radius. So far, it seems as if our universe were flat. The question is, when we look out a distance of about 13.8 billion light-years, what are we seeing?

If our space were a 3-sphere, when we looked out a sufficient distance away, we would see a smaller set of 2-spheres. From a sufficiently great distance, we would, in effect, see the single point at the “East Pole” in every direction in the sky. Is this why we see the Cosmic Microwave Background Radiation everywhere? Yes and no… we have to add time into the picture to get to this view. So, hold your horses.

## Counting galaxies and density

Likewise, if our space were a 3-sphere, as we looked off into the distance, we would see a decreasing number of galaxies becoming increasingly magnified in size, assuming that the density of stuff, including galaxies, were uniform. This is different from a flat 3-dimensional space with a uniform density of galaxies. As we would look out to some great distance $D$, we would find a sphere of radius $D$ and we would count up a certain number of galaxies consistent with the overall density of them. But in a 3-sphere, we would eventually find some distance $D$ at which the number of galaxies began to shrink, and then approach $0$.

Conversely, if our universe were a 3-hyperbola of some form, we would look off into the distance and see a vastly increasing number of galaxies; their number would grow at an exponential rate in a way that would apparently exceed their local density. This is because the size of each more distant 2-sphere is expanding exponentially, since $sinh(r) \approx exp(r)/2$ for large $r$.

## Seeing only to a horizon

Of course, as 3-dimensional astronomers, we might be fooled into thinking that our universe was Euclidean if we could see only some small fraction of the total. How might that work? Well, imagine that our universe were expanding in time such that objects further than some distance $D_H$ had a velocity that exceeded that of light and they vanished behind a cosmic horizon. Imagine that this had been going on for some time so that the total radius of the universe were, say, $10 D_H$. In this way, the total volume of the universe would be roughly 1000 times more than what was visible to us. Of course, that volume would depend on the shape of the universe including what was invisible and behind the cosmic horizon; but you get the idea. Of course, this is just the situation we find ourselves in. $D_H$ is about 14 billion light-years and we can only extrapolate as to what is behind that horizon; just like a geographer in the Middle Ages would have had to extrapolate beyond his horizon on the Earth.

## A very simple view

Let’s go back to the simplest possible view:

Here are our three candidate spaces as forms of line. The flat line has no curvature and goes on forever. The circular line bends around in such a way as to “bite its tail” and form a finite set. The hyperbolic line goes on forever but has a bend; and someone moving on this line, in effect, finds that bend moving along with them. One is always, locally, at that bend. Look at this, think about this. Let it sink in.

## Angles subtended by distant objects

Meanwhile, imagine that we could find distant objects, say galaxies, that we knew at least an average size for. Say they were of length $L$ across and at some distance $r$. We can use our three metrics to consider what angle they might subtend in our sky, depending on the curvature of our space. Our metrics are constructed in such a way that they have a radial component and then an angular component; and for a distant object like a galaxy, the radial component is nil. The integral of $ds$ is just $L$ and we can consider that equivalent to a small angle $\Delta \theta$. For each of our metrics then, we get $\Delta \theta = \frac{L}{r}$

for flat space, or $\Delta \theta = \frac{L}{sin(r)}$

for positively curved space, or $\Delta \theta = \frac{L}{sinh(r)}$

for negatively curved space. So, for a flat space, the angle subtended is inversely proportional to the distance. For a positively curves space, the angle is inversely proportional to the sine of the distance; and for a negatively curved space, it is inversely proportional to the sinh of the distance. Since $sin(r)$ is always less than $r$, the angles are larger than in flat space. When $sin(r)$ begins to collapse back to $0$, we’d find very distant galaxies would begin to look very large indeed. Conversely, in a hyperbolic space, $sinh(r)$ is always greater than $r$, the angles are smaller than in flat space. Hence, if we could look far enough into cosmological distances, we could tell something about curvature in this way. So, we can count galaxies at some distance based on a notion of uniform density, and see differences. We can also measure the angles subtended, and find differences depending on the kind of space.

## Adding in a scale factor

So far, we’ve been looking at metrics for unit n-spheres, etc. We can scale these to any size by adding in a scale factor $a(t)$, where consistent with our previous models of expanding or contracting universes based on the Friedmann equations, we’ll let the scale factor be a function of time. This is easy to do by just multiplying the metric on the unit space by $a(t)$. We get for flat space ( $E_3$) $ds^2 = a^2(t) ( dr^2 + r^2 d\Omega_2 ^2 )$

For the 3-sphere ( $\Omega_3$) $ds^2 = a^2(t) ( dr^2 + sin^2(r) d\Omega_2 ^2 )$

And for the 3-hyperbola ( $\mathcal{H}_3$) $ds^2 = a^2(t) ( dr^2 + sinh^2(r) d\Omega_2 ^2 )$

This gives us time-dependent metrics for our three classes of space. As the scale factor is increased, the curvatures decrease for both the positively and negatively curved spaces. The curvatures are inversely proportional to the scale factor. On the grand scale of the entire universe, the curvatures become very small and very difficult to measure, especially if we factor in the notion that much of the universe is hidden behind a cosmic horizon beyond the Hubble distance.

## Now, to space-time

It is now very straightforward to construct three metrics for three space-times based upon our three candidate spaces. They are just obtained by including a factor of $-c^2 dt^2$ into $ds^2$. I’m going to work with $c=1$, so these space-time metrics are simply, for flat space ( $E_3$) $ds^2 = -dt^2 + a^2(t) ( dr^2 + r^2 d\Omega_2 ^2 ) = -dt^2 +a^2(t) dE_3^2$

For the 3-sphere ( $\Omega_3$) $ds^2 = -dt^2 + a^2(t) ( dr^2 + sin^2(r) d\Omega_2 ^2 ) = -dt^2 +a^2(t) d\Omega_3^2$

And for the 3-hyperbola ( $\mathcal{H}_3$) $ds^2 = -dt^2 + a^2(t) ( dr^2 + sinh^2(r) d\Omega_2 ^2 ) = -dt^2 +a^2(t) d\mathcal{H}_3^2$

All nice and compact. These are the space-time geometries that are strong candidates for our own universe, assuming homogeneous distributions of matter and energy. We can work out, for example, the paths of light in these universes, since these paths would be null paths; that is, $ds^2 = 0$ for light.

Likewise, we can figure out the space-like distances between any two objects, such as galaxies, at some time. We could place our comoving cosmological lattice on any of these structures, and find that each will be perfectly consistent with the Hubble law. In short, everything I’ve worked out so far concerning expansion of the scale factor as a consequence of the presence of matter or radiation applies to each of the three models directly.

A note on my title, you have to pronounce the $sinh$ function as “shine” as opposed to “sinch”. Get it? sinhy spheres = shiny spheres. Sinhy means they’re expanding exponentially. Good pun, eh?

Next, I’ll go on and work with these metrics in the context of general relativity and what that says about cosmology. 