# 15.E: Relativistic Forces and Waves (Exercises)

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15.1 Two spaceships are connected by a strong cable. Both ships are initially at rest, and ignite their engines at the same time. Their accelerations are identical at every point in time. Does the cable stay intact? *Hint*: use a spacetime diagram to analyze what happens.

15.2 In classical mechanics, the Doppler shift in the wavelength of an object moving towards the observer at

speed u is given by equation (9.7.1) repeated again below:

\[\lambda_{\mathrm{obs}}=\frac{v-u}{v} \lambda\ \label{15.E.1}\]

where \(v\) is the speed of the wave (usually sound). To get this result, we compared what happens with the source and the wave in a fixed time interval \(\Delta t\). As you now know, this result cannot hold at speeds close to that of light, because in that case there will be a significant effect due to time dilation. In this problem, we’ll, therefore, redo the calculation to account for relativistic effects.

We consider a distant source of light that moves with velocity \(\boldsymbol{u}\). At time \(t = 0\) (for both the source and the stationary observer), the source emits a signal (this could be a wave crest, but the argument holds for any signal). A time \(\Delta t^{\prime}\)) later, as measured on the clock moving with the source, the source emits a second signal, see Figure 15.E.1.

- Determine the time interval \(\Delta t\) between the emission of the first and second signal as measured on the clock of the stationary observer.
- Determine the change in distance \(\Delta x\) between the (stationary) observer and the (moving) source in the time interval between the two signals, as measured by the stationary observer.
- Now determine the time interval \(\Delta t_{obs}\) between the arrival of the first and second signal at the location of the observer.
- From your answers at (a-c), show that the observed frequency \(\nu_{obs}\) is related to the source’s frequency \(\nu_{s}\) through \[v_{\mathrm{obs}}=\frac{\sqrt{1-u^{2} / c^{2}}}{1+(u / c) \cos \theta} v_{\mathrm{s}}\label{15.E.2}\] Note that equation (\ref{15.E.2}) reduces to (\ref{15.E.1}) in the case that the source is moving radially away from the observer.
- From equation (\ref{15.E.2}), find the expression for the (relativistic) transverse Doppler shift for the case that the source is moving in a direction perpendicular to the observer’s line of sight (i.e., \(\theta = 90^{o}\)). How can you tell that in this case, the Doppler shift is exclusively due to time dilation?