Imagine that light from a galaxy is emitted at some time, temitted (te for short). At some later time, tobserved (to for short), the light is observed at a second galaxy lying a comoving distance R from the first. The spacetime interval traveled by the light is described by the spacetime interval we introduced earlier for a flat, homogeneous spacetime. For this case it is better to write the interval in spherical-polar coordinates instead of Cartesian coordinates (you will see why next):
\begin{equation} s^2 = S^2(t)\left [(\Delta r)^2 + (r\Delta\theta)^2 + (r\sin\theta\Delta\phi)^2 \right ] -(c\Delta t)^2 \end{equation}We can choose our coordinates so that the second galaxy, where the light is observed, is at the origin. That will make the path of the light completely radial, so that there is no change in either Θ or φ. Under this simplification the angular terms are zero and the interval becomes:
\begin{equation} s^2 = S^2(t)(\Delta r)^2 -(c\Delta t)^2 \end{equation}Since we are imagining the path of a beam of light, we know that the total spacetime interval will be zero. This is always true for photons traveling in spacetime. Thus we can write:
\begin{equation} 0 = S^2(t)(\Delta r)^2 -(c\Delta t)^2 \end{equation}Now we can rewrite the equation so that it gives the comoving distance traveled, Δr, in terms of the corresponding time to travel that distance, Δt, and the scale factor, S(t):
\begin{equation} \frac{c\Delta t}{S(t)} =\Delta r \end{equation}This expression is true for small parts of the path, but we cannot simply plug in the total comoving distance traveled, R, and the total time traveled because, as the photon travels, the scale factor is constantly changing. Is there a value of the scale factor that is appropriate for the entire journey? No, but for some tiny interval, where both Δr and Δt are very small, the scale factor is essentially constant. This suggests that one way to correctly evaluate our expression is to add up all such tiny contributions along the light travel path:
\begin{equation} R= \Delta r_1 + \Delta r_2 + \Delta r_3 + … + \Delta r_i +…\Delta r_N \end{equation}Or in terms of the other side:
\begin{equation} R= \frac{c\Delta t_1}{S(t_1)}+\frac{c\Delta t_2}{S(t_2)}+\frac{c\Delta t_3}{S(t_3)}+…+\frac{c\Delta t_i}{S(t_i)}+…+\frac{c\Delta t_N}{S(t_N)} \end{equation}These sums can be written more compactly using summation notation. This notation is shorthand to indicate that we are going to add up some number of terms. We use the letter sigma (Σ), the Greek form of ‘s’ for “sum.” The individual terms are notated with an i and the total number of terms in the sum is N. So the sum will add up the terms from i = 1 to N1.
The sum of the Δrs becomes:
\begin{equation} \sum_i^N \Delta r_i \equiv \Delta r_1 + \Delta r_2 + \Delta r_3 + … + \Delta r_i +…\Delta r_N \end{equation}and the sum of the terms on the other side of the equation becomes:
\begin{equation} \sum_i^N\frac{c\Delta t_i}{S(t_i)} \equiv \frac{c\Delta t_1}{S(t_1)}+\frac{c\Delta t_2}{S(t_2)}+\frac{c\Delta t_3}{S(t_3)}+…+\frac{c\Delta t_i}{S(t_i)}+…+\frac{c\Delta t_N}{S(t_N)} \end{equation}Combining these two, we have a way to write an overall expression, summing over the whole path, as the Universe expands:
\begin{equation} \sum_i \Delta r_i = \sum_i\frac{c\Delta t_i}{S(t_i)} \end{equation}Such a sum will take into account the changes of the scale factor if each time element Δti is small enough so that the corresponding scale factor S(ti) of the Ith term is very nearly constant over that time interval. Adding the small contributions from all of the tiny terms in the sum we will give us the total comoving distance.
As we have written above, the term on the left of the equal sign is by definition the total comoving distance traveled, R. In addition, the speed of light, c, can be factored out of the summation on the right hand side, because it is a constant that occurs in every term. So we can simplify by writing:
\begin{equation} R = c\sum_i^N\frac{\Delta t_i}{S(t_i)} \end{equation}This expression gives the total comoving distance the light travels. It does this by adding up all the tiny distances Δri = Δti / S(ti) traveled in some tiny interval of time Δti. During these time intervals the scale factor of the spacetime (in other words, of the Universe) is S(ti) and is assumed to be approximately constant.
Now consider a single wavelength of light. The front end of the wave will be emitted from the source at a time te and arrive at the observer at a time to. The back end of the wave would be emitted by the source at a slightly later time, te + δte and arrive at the observer at a time to + δto. The times δte and δto are both tiny, but not zero, and they are not necessarily the same. Since the comoving distance between the galaxies is the same for both light waves—remember that the comoving distance has no dependence on the expansion of the spacetime—the sum above applies to the comoving distance traveled in terms of either the front or the back of the wave.
We will now break this down and consider the sum from the front and the back of the wave separately. First we will write a sum over interval between the emission of the front of the light wave in the first galaxy and the observation of the front of the light wave by the observer in the second galaxy:
\begin{equation} R = c\sum_i^{N^{\prime}}\frac{\Delta t_i}{S(t_i)} \end{equation}For the back of the light wave we have:
\begin{equation} R = c\sum_i^{N^{\prime \prime}}\frac{\Delta t_i}{S(t_i)} \end{equation}Here the sum runs from the interval between the emission of the back of the light wave in the first galaxy and the observation of the back of the light wave by the observer in the second galaxy.
The sums are essentially identical, but with a subtle difference. We have used N’ and N’’ as the limits on the sums to remind ourselves of this difference, namely that the second sum is taken over a slightly different spacetime interval than the first sum. In both cases, the total number of terms (N’ or N’’), is large enough to make the individual time intervals in the sum small enough that the scale factor S(ti) remains constant during each time interval.
These two expressions are a bit complicated, so we should break them down. The first says that the total comoving distance traveled by the front of the light wave is given by summing up all the tiny distances traveled by the light during many tiny time intervals. The total time spans from when the front of the wave is emitted at the first galaxy to when it is observed at the second galaxy. The scale factor is different for each of these little time intervals, but remains constant during any given interval. This way of computing the total comoving distance allows us to take into account the differences in the scale factor over the light’s journey. The second expression says exactly the same thing, except that now we are finding the comoving distance traveled by the back of the light wave, not the front.
Since the back of the light wave is emitted an instant after the front of the light wave, and it is observed an instant after the front is, the two sums are not identical. Using N’ and N’’ for the number of terms sums is a way of reminding ourselves of this subtle but important difference, though it otherwise has no mathematical significance.
The two comoving distances are the same because the expansion of the spacetime does not affect comoving distance: all information about the expansion is contained within the scale factor. Since the two expressions evaluate to the same value, R, we can set them equal to each other.
\begin{equation} c\sum_{i=1}^{N^{\prime }}\frac{\Delta t_i}{S(t_i)} = c\sum_{i=1}^{N^{\prime \prime}}\frac{\Delta t_i}{S(t_i)} \end{equation}Now consider these two sums carefully. They are nearly, but not quite identical. The first has an extra little bit during the interval δte before the second one begins, or in other words, before the back of the light wave is emitted. Likewise, the second summation has an extra little bit during the interval δto, after the first summation ends, due to the extra time required for the end of the light wave to reach the observer. We have used an N’ as the limit for the second sum to remind us of this difference.
We can pull the extra term in each sum out of the summation notation and write it explicitly. We can also cancel the common factor of c. In that case we have the following
\begin{equation} \left [\sum_{i=1}^{N}\frac{\Delta t_i}{S(t_i)}\right ] + \frac{\delta t_e}{S(t_e)} = \left[\sum_{i=1}^{N}\frac{\Delta t_i}{S(t_i)}\right ] + \frac{\delta t_o}{S(t_o)} \end{equation}Now we use the same number of terms, N, in each sum because the sums are over exactly the same time spacetime interval. The extra bit in each sum, not contained in the other, is written out explicitly. The quantities in the square brackets are obviously identical, so after canceling them, the only terms left are the extra terms from the beginning and end of the sums. This relates the intervals for the emission and detection of the wave in terms of the scale factors at the times of emission and absorption:
\begin{equation} \frac{\delta t_o}{\delta t_e}=\frac{S(t_o)}{S(t_e)} \end{equation}The ratio of the intervals at the observer’s end and the emitter’s end is the same as the ratio of the scale factor at the time of observation and the time of emission. The time interval between the passage of the front of the wave and the back of the wave is the period of the wave. Recall from Chapter 2 that the frequency of a wave f is the inverse of the period (so f = 1/δt here). Writing our expression in terms of frequency we get:
\begin{equation} \frac{f_o}{f_e}=\frac{S(t_e)}{S(t_o)} \end{equation}This is the relation we were looking for. It expresses the observed frequency of light in terms of the emitted frequency and the amount by which the spacetime has expanded while the light was traveling. It is called the cosmological redshift. Note that it has nothing to do with Doppler shifts, but only with the fact that we are in a spacetime that is expanding; it is the stretching of spacetime itself that stretches the wavelength of the light.
If we wish, we can write the cosmological redshift in more familiar wavelength terms. We just have to substitute for frequency using f = c/λ:
\begin{equation} \frac{\left(\frac{c}{\lambda_o}\right )}{\left (\frac{c}{\lambda_e}\right )} =\frac{S(t_e)}{S(t_o)} \end{equation}Canceling the factor of c and rearranging, we get:
\begin{equation} \frac{\lambda_o}{\lambda_e}=\frac{S(t_o)}{S(t_e)} \end{equation}This form shows explicitly that the light is stretched by exactly the amount by which the spacetime has expanded while the light was traveling. Notice that we have not yet put gravity into our spacetime at all. It is not gravity but the stretching of spacetime itself that creates the cosmological redshift.
The equation above relates the wavelength of light from a source at the time it was emitted and observed to the scale factor of the Universe at the time the light was emitted and observed. If we want to rewrite this in terms of a redshift, we can use the definition of redshift (from Chapter 3):
\begin{equation} z = \frac{(\lambda_{o} - \lambda_{e})}{\lambda_{e}} = \frac{\Delta\lambda}{\lambda_{e}} \end{equation}where Δλ is the change in wavelength. First, express the observed wavelength in terms of the emitted wavelength and the change in wavelength, then separate the terms, then finally substitute for redshift:
\begin{equation} \frac{\lambda_{o}}{\lambda_{e}} = \frac{\lambda_{e}+\Delta\lambda}{\lambda_{e}} \\ =\frac{\lambda_{e}}{\lambda_{e}}+\frac{\Delta\lambda}{\lambda_{e}} \\ =1+\frac{\Delta\lambda}{\lambda_{e}} \\ = 1 + z \end{equation}So we have:
\begin{equation} 1+z=\frac{S(t_o)}{S(t_e)} \end{equation}