Electron-positron pair production cannot occur in empty space

Question

I'm trying to prove that the electron-positron pair production cannot occur in empty space. From the conservation of energy we have that $$\hbar \omega = 2\gamma mc^2+E,$$ and from the conservation of momentum we have $$\hbar k=2\gamma mv\cos{\theta} + p,$$ where $E$ and $p$ are the energy and momentum of some other particle being present in the process, respectively; $v$ is the velocity of electron and positron, and $theta$ is the angle at which they "fly away" (relative to the photon's path). I'm basically trying to prove that the mass of this other particle is non-zero. Since $c=\omega / k$, we have from the second eq. $\hbar \omega = 2\gamma mvc \cos{\theta} + pc$. So from both equations $$2\gamma mc(c-v\cos{\theta})=pc-E,$$ but $c-v\cos{\theta}>0$ thus $pc-E>0$ which is unexpected, to say the least. I cannot find an error in this reasoning and yet it yields nonsense. I don't understand why this fact cannot be proven this way.