Obtain the Lorentz transformations that relate the coordinates of an event measured in one inertial frame $(t, x, y, z)$ to those in another inertial frame moving with velocity $v$ along the $x$ axis. Take care to state the assumptions that lead to your result.

A star is at rest in a three-dimensional coordinate frame $\mathcal{S}^{\prime}$ that is moving at constant velocity $v$ along the $x$ axis of a second coordinate frame $\mathcal{S}$. The star emits light of frequency $\nu^{\prime}$, which may considered to be a beam of photons. A light ray from the star to the origin in $\mathcal{S}^{\prime}$ is a straight line that makes an angle $\theta^{\prime}$ with the $x^{\prime}$ axis. Write down the components of the four-momentum of a photon in this light ray.

The star is seen by an observer at rest at the origin of $\mathcal{S}$ at time $t=t^{\prime}=0$, when the origins of the coordinate frames $\mathcal{S}$ and $\mathcal{S}^{\prime}$ coincide. The light that reaches the observer moves along a line through the origin that makes an angle $\theta$ to the $x$ axis and has frequency $\nu$. Make use of the Lorentz transformations between the four-momenta of a photon in these two frames to determine the relation

$$\lambda=\lambda^{\prime}\left(1-\frac{v^{2}}{c^{2}}\right)^{-1 / 2}\left(1+\frac{v}{c} \cos \theta\right)$$

where $\lambda$ is the observed wavelength of the photon and $\lambda^{\prime}$ is the wavelength in the star's rest frame.

Comment on the form of this result in the special cases with $\cos \theta=1, \cos \theta=-1$ and $\cos \theta=0$.

[You may assume that the energy of a photon of frequency $\nu \mathrm{~ i s ~ h}$ momentum is a three-vector of magnitude $h \nu / c .]$