GNU Astronomy Utilities


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9.1.1 Distance on a 2D curved space

The observations to date (for example the Plank 2013 results), have not measured the presence of a significant curvature in the universe. However to be generic (and allow its measurement if it does in fact exist), it is very important to create a framework that allows curvature. As 3D beings, it is impossible for us to mentally create (visualize) a picture of the curvature of a 3D volume in a 4D space. Hence, here we will assume a 2D surface and discuss distances on that 2D surface when it is flat, or when the 2D surface is curved (in a 3D space). Once the concepts have been created/visualized here, in Extending distance concepts to 3D, we will extend them to the real 3D universe we live in and hope to study.

To be more understandable (actively discuss from an observer’s point of view) let’s assume we have an imaginary 2D friend living on the 2D space (which might be curved in 3D). So here we will be working with it in its efforts to analyze distances on its 2D universe. The start of the analysis might seem too mundane, but since it is impossible to imagine a 3D curved space, it is important to review all the very basic concepts thoroughly for an easy transition to a universe we cannot visualize any more (a curved 3D space in 4D).

To start, let’s assume a static (not expanding or shrinking), flat 2D surface similar to Figure 9.1 and that our 2D friend is observing its universe from point \(A\). One of the most basic ways to parametrize this space is through the Cartesian coordinates (\(x\), \(y\)). In Figure 9.1, the basic axes of these two coordinates are plotted. An infinitesimal change in the direction of each axis is written as \(dx\) and \(dy\). For each point, the infinitesimal changes are parallel with the respective axes and are not shown for clarity. Another very useful way of parameterizing this space is through polar coordinates. For each point, we define a radius (\(r\)) and angle (\(\phi\)) from a fixed (but arbitrary) reference axis. In Figure 9.1 the infinitesimal changes for each polar coordinate are plotted for a random point and a dashed circle is shown for all points with the same radius.

gnuastro-figures/flatplane

Figure 9.1: Two dimensional Cartesian and polar coordinates on a flat plane.

Assuming a certain position, which can be parameterized as \((x,y)\), or \((r,\phi)\), a general infinitesimal change change in its position will place it in the coordinates \((x+dx,y+dy)\) and \((r+dr,\phi+d\phi)\). The distance (on the flat 2D surface) that is covered by this infinitesimal change in the static universe (\(ds_s\), the subscript signifies the static nature of this universe) can be written as:

$$ds_s=dx^2+dy^2=dr^2+r^2d\phi^2$$

The main question is this: how can our 2D friend incorporate the (possible) curvature in its universe when it is calculating distances? The universe it lives in might equally be a locally flat but globally curved surface like Figure 9.2. The answer to this question but for a 3D being (us) is the whole purpose to this discussion. So here we want to give our 2D friend (and later, ourselves) the tools to measure distances if the space (that hosts the objects) is curved.

Figure 9.2 assumes a spherical shell with radius \(R\) as the curved 2D plane for simplicity. The spherical shell is tangent to the 2D plane and only touches it at \(A\). The result will be generalized afterwards. The first step in measuring the distance in a curved space is to imagine a third dimension along the \(z\) axis as shown in Figure 9.2. For simplicity, the \(z\) axis is assumed to pass through the center of the spherical shell. Our imaginary 2D friend cannot visualize the third dimension or a curved 2D surface within it, so the remainder of this discussion is purely abstract for it (similar to us being unable to visualize a 3D curved space in 4D). But since we are 3D creatures, we have the advantage of visualizing the following steps. Fortunately our 2D friend knows our mathematics, so it can follow along with us.

With the third axis added, a generic infinitesimal change over the full 3D space corresponds to the distance: $$ds_s^2=dx^2+dy^2+dz^2=dr^2+r^2d\phi^2+dz^2.$$It is very important to recognize that this change of distance is for any point in the 3D space, not just those changes that occur on the 2D spherical shell of Figure 9.2. Recall that our 2D friend can only do measurements in the 2D spherical shell, not the full 3D space. So we have to constrain this general change to any change on the 2D spherical shell. To do that, let’s look at the arbitrary point \(P\) on the 2D spherical shell. Its image (\(P'\)) on the flat plain is also displayed. From the dark triangle, we see that

gnuastro-figures/sphericalplane

Figure 9.2: 2D spherical plane (centered on \(O\)) and flat plane (gray) tangent to it at point \(A\).

$$\sin\theta={r\over R},\quad\cos\theta={R-z\over R}.$$These relations allow our 2D friend to find the value of \(z\) (an abstract dimension for it) as a function of r (distance on a flat 2D plane, which it can visualize) and thus eliminate \(z\). From \(\sin^2\theta+\cos^2\theta=1\), we get \(z^2-2Rz+r^2=0\) and solving for \(z\), we find: $$z=R\left(1\pm\sqrt{1-{r^2\over R^2}}\right).$$The \(\pm\) can be understood from Figure 9.2: For each \(r\), there are two points on the sphere, one in the upper hemisphere and one in the lower hemisphere. An infinitesimal change in \(r\), will create the following infinitesimal change in \(z\):

$$dz={\mp r\over R}\left(1\over \sqrt{1-{r^2/R^2}}\right)dr.$$Using the positive signed equation instead of \(dz\) in the \(ds_s^2\) equation above, we get:

$$ds_s^2={dr^2\over 1-r^2/R^2}+r^2d\phi^2.$$

The derivation above was done for a spherical shell of radius \(R\) as a curved 2D surface. To generalize it to any surface, we can define \(K=1/R^2\) as the curvature parameter. Then the general infinitesimal change in a static universe can be written as: $$ds_s^2={dr^2\over 1-Kr^2}+r^2d\phi^2.$$Therefore, we see that a positive \(K\) represents a real \(R\) which signifies a closed 2D spherical shell like Figure 9.2. When \(K=0\), we have a flat plane (Figure 9.1) and a negative \(K\) will correspond to an imaginary \(R\). The latter two cases are open universes (where \(r\) can extend to infinity). However, when \(K>0\), we have a closed universe, where \(r\) cannot become larger than \(R\) as in Figure 9.2.

A very important issue that can be discussed now (while we are still in 2D and can actually visualize things) is that \(\overrightarrow{r}\) is tangent to the curved space at the observer’s position. In other words, it is on the gray flat surface of Figure 9.2, even when the universe if curved: \(\overrightarrow{r}=P'-A\). Therefore for the point \(P\) on a curved space, the raw coordinate \(r\) is the distance to \(P'\), not \(P\). The distance to the point \(P\) (at a specific coordinate \(r\) on the flat plane) on the curved surface (thick line in Figure 9.2) is called the proper distance and is displayed with \(l\). For the specific example of Figure 9.2, the proper distance can be calculated with: \(l=R\theta\) (\(\theta\) is in radians). using the \(\sin\theta\) relation found above, we can find \(l\) as a function of \(r\):

$$\theta=\sin^{-1}\left({r\over R}\right)\quad\rightarrow\quad l(r)=R\sin^{-1}\left({r\over R}\right)$$\(R\) is just an arbitrary constant and can be directly found from \(K\), so for cleaner equations, it is common practice to set \(R=1\), which gives: \(l(r)=\sin^{-1}r\). Also note that if \(R=1\), then \(l=\theta\). Generally, depending on the the curvature, in a static universe the proper distance can be written as a function of the coordinate \(r\) as (from now on we are assuming \(R=1\)):

$$l(r)=\sin^{-1}(r)\quad(K>0),\quad\quad l(r)=r\quad(K=0),\quad\quad l(r)=\sinh^{-1}(r)\quad(K<0).$$With \(l\), the infinitesimal change of distance can be written in a more simpler and abstract form of

$$ds_s^2=dl^2+r^2d\phi^2.$$

Until now, we had assumed a static universe (not changing with time). But our observations so far appear to indicate that the universe is expanding (isn’t static). Since there is no reason to expect the observed expansion is unique to our particular position of the universe, we expect the universe to be expanding at all points with the same rate at the same time. Therefore, to add a time dependence to our distance measurements, we can simply add a multiplicative scaling factor, which is a function of time: \(a(t)\). The functional form of \(a(t)\) comes from the cosmology and the physics we assume for it: general relativity.

With this scaling factor, the proper distance will also depend on time. As the universe expands (moves), the distance will also move to larger values. We thus define a distance measure, or coordinate, that is independent of time and thus doesn’t ‘move’ which we call the comoving distance and display with \(\chi\) such that: \(l(r,t)=\chi(r)a(t)\). We thus shift the \(r\) dependence of the proper distance we derived above for a static universe to the comoving distance:

$$\chi(r)=\sin^{-1}(r)\quad(K>0),\quad\quad \chi(r)=r\quad(K=0),\quad\quad \chi(r)=\sinh^{-1}(r)\quad(K<0).$$

Therefore \(\chi(r)\) is the proper distance of an object at a specific reference time: \(t=t_r\) (the \(r\) subscript signifies “reference”) when \(a(t_r)=1\). At any arbitrary moment (\(t\neq{t_r}\)) before or after \(t_r\), the proper distance to the object can simply be scaled with \(a(t)\). Measuring the change of distance in a time-dependent (expanding) universe will also involve the speed of the object changing positions. Hence, let’s assume that we are only thinking about the change in distance caused by something (light) moving at the speed of light. This speed is postulated as the only constant and frame-of-reference-independent speed in the universe, making our calculations easier, light is also the major source of information we receive from the universe, so this is a reasonable assumption for most extra-galactic studies. We can thus parametrize the change in distance as

$$ds^2=c^2dt^2-a^2(t)ds_s^2 = c^2dt^2-a^2(t)(d\chi^2+r^2d\phi^2).$$


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