In the previous section we showed that the expansion of spacetime causes light to be stretched in the same manner that space is stretching. We will now see that this leads directly to the Hubble law.
Hubble showed that the redshift of a galaxy, which he interpreted as its velocity, is proportional to its distance. Redshift, z, is given by
\begin{equation} z = \frac{\Delta\lambda}{\lambda}=\frac{\lambda_o-\lambda_e}{\lambda_e} \end{equation}We can write this expression the following way:
\begin{equation} z =\frac{\lambda_o}{\lambda_e}-1 \end{equation}From the previous section we know that we can write the wavelength ratio in terms of the scale factor. We will use that relation to replace the ratio of wavelengths to get
\begin{equation} z =\frac{S(t_o)}{S(t_e)}-1 \end{equation}Now consider a short time in which the spacetime does not expand by very much. Under these conditions the scale factor will be essentially constant and we can write:
\begin{equation} S(t_o) = S(t_e) +\Delta S \\ z =\frac{S(t_e)+\Delta S}{S(t_e)}-1 \end{equation}If the expansion over this time happens such that S changes by ΔS in a time Δt, then we can define the rate of change of the expansion:
\begin{equation} \dot{S}\equiv \frac{\Delta S}{\Delta t} \end{equation}And we can write our redshift expression as:
\begin{equation} z =\frac{S(t_e)+ \dot{S}\Delta t }{S(t_e)}-1 \end{equation}This can be simplified by separating the terms and canceling the common term of S(te). And since S(te) and S(to) are assumed to be essentially the same, we will drop the te and use simply S from now on. Then we may write:
\begin{equation} z = \frac{S(t_e)}{S(t_e)} + \frac{\dot{S}\Delta t } {S(t_e)} -1 \\ z = 1 + \frac{\dot{S}\Delta t } {S(t_e)} -1 \\ z \approx \left(\frac{\dot{S}}{S}\right)\Delta t \\ \end{equation}We have switched to an “approximately equal” sign because S is assumed to be nearly constant.
Next we can use our spacetime interval and the behavior of a light beam to express Δt in a more useful way. We use the fact that the spacetime interval traversed by a beam of light is always zero. Therefore:
\begin{equation} 0 = S^2(t)(\Delta r)^2 -(c\Delta t)^2 \end{equation}Rearranging the terms and taking the square root, we get:
\begin{equation} c\Delta t = S(t) \Delta r \end{equation}If we assume that Δt is small enough that S(t) remains approximately constant, then we can replace S(t) with So. Now we can solve this for Δt:
\begin{equation} \Delta t = \frac{S_o\Delta r}{c} \end{equation}Substituting this expression in for Δt, the redshift becomes:
\begin{equation} z \approx \left(\frac{\dot{S}}{S}\right) (\frac{S_o\Delta r}{c}) \end{equation}But SoΔr is the physical distance traveled (not the comoving distance) by the light during this brief time interval. So if we call the physical distance d (= SoΔr), then we can rewrite the redshift as:
\begin{equation} z \approx \left(\frac{\dot{S}}{S}\right) \left(\frac{d}{c}\right) \end{equation}The ratio in the first parentheses is a constant; it is the fractional amount by which the spacetime expands per unit time. In other words, it is the Hubble constant H0, and the expression is the Hubble law. We can move the c over to the other side of the equal sign to obtain the more familiar form, with velocity (v = cz) proportional to distance:
\begin{equation} cz \approx \left(\frac{\dot{S}}{S}\right)d\\ v = H_0 d \end{equation}The subscript zero on the Hubble constant reminds us that we are only considering time intervals for which the scale factor is approximately constant, and in particular we consider the time period very close to the present one. In this short time light will not have traveled very far, so the relationship should only be valid for the “local” Universe, with the exact meaning “local” depending upon how fast the scale factor changes. If we begin to look at galaxies over larger distances (meaning the light has been traveling for a longer time) then the approximation of a nearly constant Hubble constant is not necessarily valid and we might expect to see deviations from a strict proportionality. In fact, that is exactly what we do see, and the amount and nature of the deviations can tell us many things about the history of the Universe and its energy balance.