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Chapter 18: Problem 30

Consider two radual light s?gnals (null geodesics) received at the spatial origin of coordinates at times \(t_{0}\) and \(t_{0}+\Delta t_{0}\), emitted from \(x=x_{1}\) (or \(r=r_{1}\) in the case of the flat models) at time \(t=t_{1}

Short Answer

Expert verified
After a series of derivations from the properties of null geodesics and the scale factor for an expanding universe, we find that the observer experiences a redshift, i.e. 1 + z = \( \frac{a(t_{0})}{a(t_{1})}\) in the case of an expanding universe.

Step by step solution

01

Identify given relations

In this case, the radial light signals received are null geodesics, which means that their interval is zero. From this assumption and the definition of a geodesic, we can set up the following relation: \(ds^2=0=-dt^2+d\rho^2\), where \(\rho=a(t)x\). \[ds^2 = c^2dt^2 - a(t)^2dr^2 = 0\] Hence, we can derive from this that: \[dt= a(t)dr\] when considered for the two times, \(t_0\) and \(t_1\).
02

Derive relation between times and radial light signal

Knowing that \(dt=a(t)dr\), we can integrate both sides over the time interval \(\Delta t\), resulting in \[\int_{t_1}^{t_0} dt = \int_{r_1}^0 a(t) dr\] By a change of variables as \(a(t)dr=dr'\), we get \(t_0-t_1=\Delta t_0= \int_{r_1}^0 dr' . \) Or similarly, integrating the differential dt from \(t1\) to \(t0\) will yield \(\Delta t_0\). This also corresponds to the travel time of the light signal.
03

Derive Redshift

Redshift \(z\) is defined as the change in wavelength or time interval observed as a result of the expansion of the universe. Therefore, the redshift is related to the time intervals as \(1+z = \frac {\Delta t_0} {\Delta t_1}\). We can understand the terms \(\Delta t_0\) and \(\Delta t_1\) as the time interval between the reception of two light signals at the observer and the time interval between the emission of these two light signals at the source, respectively. As a result of the universe's expansion, the light signal is stretched causing the observer to see the signal 'redshifted'.
04

Conclude Result

The scale factor \(a(t)\) characterizes the size of the universe and it increases with time in an expanding universe. As \(a(t_0) > a(t_1)\) because \(t_0>t_1\), it means the Universe expands as time goes from \(t_1\) to \(t_0\), and from the result at step 3, we know that the redshift \(z\) is greater than 0 as \(\Delta t_0 > \Delta t_1\). Combining these results, we can conclude that the observer experiences a redshift in the case of an expanding universe and is given by \[1+z=\frac{a(t_{0})}{a(t_{1})}\].

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