A star moves with speed $v$ in the $x$-direction in a reference frame $S$. When viewed in its rest frame $S^{\prime}$ it emits a photon of frequency $\nu^{\prime}$ which propagates along a line making an angle $\theta^{\prime}$ with the $x^{\prime}$-axis. Write down the components of the four-momentum of the photon in $S^{\prime}$. As seen in $S$, the photon moves along a line that makes an angle $\theta$ with the $x$-axis and has frequency $\nu$. Using a Lorentz transformation, write down the relationship between the components of the four-momentum of the photon in $S^{\prime}$ to those in $S$ and show that

$$\cos \theta=\frac{\cos \theta^{\prime}+v / c}{1+v \cos \theta^{\prime} / c}$$

As viewed in $S^{\prime}$, the star emits two photons with frequency $\nu^{\prime}$ in opposite directions with $\theta^{\prime}=\pi / 2$ and $\theta^{\prime}=-\pi / 2$, respectively. Show that an observer in $S$ records them as having a combined momentum $p$ directed along the $x$-axis, where

$$p=\frac{E v}{c^{2} \sqrt{1-v^{2} / c^{2}}}$$

and where $E$ is the combined energy of the photons as seen in $S^{\prime}$. How is this momentum loss from the star consistent with its maintaining a constant speed as viewed in $S ?$