Cosmological space-time YouTube

Deriving the metric from the assumptions

In special relativity, space-time, as well as space on its own, is flat. It can be characterised by the metric, $\eta_{\mu \nu}$, which had the particularly simple components, $$\eta_{\mu \nu} \rightarrow \begin{bmatrix}1 & 0 & 0 & 0\\0 & -1 & 0 & 0\\0 & 0 & -1 & 0\\0 & 0 & 0 & -1\end{bmatrix}$$

We can use our assumptions to construct a more general metric, $g_{\mu \nu}$, which includes a dynamic component, related to the expansion of the universe.

First of all, since space is flat at a given instant of time, all of the spatial components of the metric must be constants in space, although since the universe is expanding, they can still be functions of time, $$g_{m n} = g_{m n}(t)$$

Since space is isotropic (at a given instant), we can still find an orthogonal coordinate system, in which the metric has no cross-terms and since space is homogeneous (at a given instant), we expect the expansion rate to be the same everywhere (at a given instant). Thus, we can write $$g_{m n} = g(t) \delta_{m n}$$ where $\delta_{m n}$ is the Kronecker delta meaning that the components of the metric are then $$g_{\mu \nu} \rightarrow \begin{bmatrix}1 & 0 & 0 & 0\\0 & g(t) & 0 & 0\\0 & 0 & g(t) & 0\\0 & 0 & 0 & g(t)\end{bmatrix}$$

We have left the time components alone, and although a more general metric may have varying co-efficients in the time components, we can always choose a coordinate system in which $$\begin{align*}g_{00} &= 1\\g_{0 \nu} &= g_{\mu 0} = 0\end{align*}$$

By convention, we write the spatial components of the metric as $$g(t) = -a(t)^2$$ where $a(t)$ is called the scale factor.

The proper time between two events in coordinates with this metric is then $$\begin{align*}\mathrm{d}\tau^2 &= \mathrm{d}t^2 - a(t)^2 \mathrm{d}x^2 - a(t)^2 \mathrm{d}y^2 - a(t)^2 \mathrm{d}z^2\\&= \mathrm{d}t^2 - \left(a(t) \mathrm{d}x\right)^2 - \left(a(t) \mathrm{d}y\right)^2 - \left(a(t)\mathrm{d}z\right)^2\end{align*}$$

The physical distance between galaxies along the $x$-axis, say, is $$a(t) \Delta x$$ and we can consider it in two ways:

  1. We could consider $a(t)$ to be a dimensionless number, leaving, as usual, the units of length to be included in the spatial components of the coordinate system, $\Delta x$.
  2. Alternatively, a more useful approach, and one we shall use, is to consider co-moving coordinates, in which $a(t)$ has the dimension of length, meaning that spatial components become labels anddon't carry any information about physical distance. In particular $\Delta x$ doesn't change with time.

Since we are assuming that space is expanding, meaning that the $a(t) \Delta x$ is increasing over time, then if we accept that $\Delta x$ remains constant over time, it must be the scale factor is an increasing function of time. That is, $$a(t_2) > a(t_1), \text{ for all } t_2 \gt t_1$$

Co-moving coordinates

The standard model of space is that of a flat (infinite) rubber sheet that is expanding over time.

Suppose, at a given instant of time, we map a (Cartesian) coodinate system onto the sheet. We can then define a set of coordinates, called co-moving coordinates, which have the property that, once defined at a particular instant, expand at exactly the same rate as the rubber sheet (universe).

Figure 8.9 - Co-moving coordinates

Figure 8.9 - Co-moving coordinates

Consider a region of space as depicted in figure 8.9, where we have indicated that the region is expanding equally in all directions.

At a particular instant of time, we can impose a set of flat, Cartesian coordinates on the region, where the unit, or scale factor can be chosen arbitrarily, but once chosen, we can label some galaxies, say, $A,B,C$ in this coordinate system.

Distance in this unit could be defined, somewhat imprecisely, as the average number of galaxies between a pair of points.

We now demand that the coordinates are co-moving, that is, they expand at the same rate as the universal expansion rate.

Figure 8.10 - Co-moving coordinates (later)

Figure 8.10 - Co-moving coordinates (later)

The distance between a pair of points does not change in these units, since distance is roughly the number of galaxies distributed between.

Of course, physical distances have changed. The same region is now twice as large as before (the dashed circle in figure 8.10). But so is the co-moving coordinate system.

Nb. Figure 8.10 also indicates that, if measured in a co-moving coordinate system, the density of matter in a region doesn't change over time, whereas it decreases if measured in a physical coodinate system.

Co-moving coordinates are a convenience, allowing us to transfer the physical dimension of length into the scale factor in the expressions for the metric and the proper time.

This is the basic setup for most of observational cosmology.

Co-moving distance versus physical distance

For a concrete example, suppose that, in figure 8.9, the distance between galaxies $A,B$ at time $t = 0$, say, is measured in the co-moving coordinates to be $$\Delta x_{AB} = \left|\vec{x}_A - \vec{x}_B\right| = \frac{1}{\sqrt{2}}$$

Even though we haven't yet defined the scale factor, $a(t)$, explicitly as a function of time, we can say that the physical distance, $D_{AB} = D_{AB}(t)$, between $A$ and $B$ at time $t = 0$ is given by $$D_{AB}(0) = a(0) \Delta x_{AB} = \frac{a(0)}{\sqrt{2}}$$

Later on, in figure 8.10, the co-moving distance is still $$\Delta x_{AB} = \frac{1}{\sqrt{2}}$$ but the physical distance is $$D_{AB}(t) = a(t)\Delta x_{AB} = \frac{a(t)}{\sqrt{2}} \gt D_{AB}(0)$$

In general, we could have chosen any two galaxies, so we drop the subscripts. Also, since space is isotropic, we can, without any loss of generality, confine ourselves to galaxies along the $x$-axis, say.

In this context then, if the co-moving distance between two locations is $\Delta x$, then the physical distance, $D = D(t)$, is the product of the scale factor at time $t$ and the co-moving distance, $$D(t) = a(t)\Delta x \gt a(0)\Delta x = D(0)$$