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6. Redshifts (INCOMPLETE)

We introduced expanding, non-Euclidean spacetimes in the previous chapter. Now we begin to work out observational consequences of living in such a spacetime. In this set of exercises and the next set we will derive Hubble's Law, \(v=H_0 d\). In fact, we will derive a more general version of it valid for arbitrarily large distances. First we will focus on the left-hand side of the equation and then we will focus on the right-hand side.

In Minkowski space, the invariant distance, \(ds\), between spacetime point \((t,x,y,z)\) and another one at \((t+dt,x+dx,y+dy,z+dz)\) is given by:

\[ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2\]

We call it invariant because it is invariant under Lorentz transformations. Physically, for \(ds^2 > 0\), \(ds\) is the length of a ruler with one end on each of the two space-time points, if the ruler is at rest in a frame in which the two space-time points are simultaneous. For \(ds^2 < 0\), \(\sqrt{-ds^2}/c\) is the amount of time that elapses on a clock that freely falls from one space-time point to the other (that is, in a frame attached to the clock in which the two space-time points are at the same point in space).

In spherical coordinates this becomes:

\[ds^2 = -c^2 dt^2 + dr^2 + r^2 \left(d\theta^2 + \sin^2\theta d\phi^2\right) \]

The invariant distance in a homogeneous and isotropic Universe can be written as:

\[ds^2 = -c^2 dt^2 + a^2(t)\left[\frac{dr^2}{1-kr^2} + r^2 \left(d\theta^2 + \sin^2\theta d\phi^2\right)\right] \label{eqn:FRWds} \]

We are now ready to derive the relationship between redshift and expansion. By expansion we mean the increase in the scale factor over time. By redshift we mean the stretching of the wavelength of light as it travels across the expanding universe from emission in the past to reception today.

We use the symbol \(z\) to denote redshift and define it as:

\[z \equiv (\lambda_{\rm received} - \lambda_{\rm emitted})/\lambda_{\rm emitted}\]

We will derive the relationship between redshift and expansion for light that leaves an object at rest at time \(t_e\) and is observed today by an observer at rest at time \(t_r\). We'll get there with the following two exercises.

Note

By "at rest" we mean at rest in the coordinate system that has an invariant distance of the form in Equation \ref{eqn:FRWds}. This frame is called "cosmic rest." Note that for a frame that is moving with respect to cosmic rest, slices of constant time would be different, and we'd no longer have this simple form where the scale factor only depends on the time coordinate.

Box \(\PageIndex{1}\)

Exercise 6.1.1: Show that if \(\Delta t_e\) is the time interval between emitted pulses (as measured by a stationary observer located where the emission is happening) and if \(\Delta t_r\) is the time interval between reception of first pulse and second pulse (as measured by a stationary observer located where the reception is occurring) then \(\Delta t_r/\Delta t_e = a(t_r)/a(t_e)\). To do so, use conformal time defined by \(d\tau = dt/a(t)\) and draw the pulse trajectories on an \(x\) vs. \(\tau\) diagram.

Box \(\PageIndex{2}\)

Exercise 6.2.1: Imagine propagation of electromagnetic waves. Use the above result to show that the wavelength of the waves emitted at time \(t_r\) and observed at time \(t_e\) is stretched so that:

\[\begin{equation*}
  \begin{aligned}
        z \equiv (\lambda_{\rm received} - \lambda_{\rm emitted})/\lambda_{\rm emitted}  = a(t_r)/a(t_e) - 1
  \end{aligned}
\end{equation*}\]

Box \(\PageIndex{3}\)

Exercise 6.3.1: The most distant object for which a redshift has been measured is called a gamma-ray burst and it has a redshift of \(z=8.2\). By what factor has the universe expanded since light left that object?

We have just worked out the amazing fact that if we can identify a spectral line and measure its wavelength, then we can directly determine how much the universe has expanded since light left the object. We will use this later to derive Hubble's Law \(v=H_0 d\).

Before closing this section, we remind the reader of how redshifts due to the Dopper effect are related to speed -- a result we will use later in deriving Hubble's Law. If a source is moving away from you at speed v and emitting pulses with period \(T\), then the second pulse has to travel a distance \(vT\) further to get to you than was the case for the first pulse. So it's arrival will be delayed by a time \(vT/c\). Thus the period for the arriving pulses is \(T(1+v/c)\). Since wavelength is proportional to period this means the wavelength is stretched by a factor of \(1+v/c\), which means, by definition of the redshift \(z\), that \(z = v/c\). Note that our derivation has ignored the effect of relativistic time dilation. If the source had period \(T\) at rest, then if it were moving with speed \(v\) with respect to us, in our frame the period would be stretched to \(\gamma T\) and so the complete expression for \(z\) from the Doppler effect is \(z = \gamma (1+v/c)\). But we are only interested in this expression for small \(v/c\), for which \(\gamma \sim 1.\)