4-velocity in §2.8 of Misner, Thorne and Wheeler At section §2.8 of their "Gravitation", Misner, Thorne and Wheeler calculate the energy of a photon emitted at the rim of a turning centrifuge in a Mössbauer effect experiment. In the emitter's frame, they say the photon's energy is Ee = –pue, where p is the 4-momentum of the photon and ue is the emitter's 4-velocity.
What is the emitter's 4-velocity doing there? Shouldn't it be the velocity of the photon?
 A: This is pretty standard notation, but it can be a little bit of a tricky concept the first time you run into something like this.
In the emitter's frame, and using natural units, the four-momentum is $\mathbf{p}=(E_e, E_e)$ where the energy in this frame is $E_e$. We can clearly get this by multiplying $E_e=-(E_e,E_e)(1,0)=-\mathbf{p} \mathbf{u_e}$
The weird thing about this is that $-\mathbf{p} \mathbf{u_e}=E_e$ is an invariant, even though in general $E$ is not an invariant in general. In other words, not all frames will agree that $E_e$ actually represents the energy of that particle, but all frames will agree that an energy detector moving at $ \mathbf{u_e}$ will measure the number $-\mathbf{p} \mathbf{u_e}$ for the energy of that particle. In frames where $\mathbf{u_e} \ne (1,0)$ this number $E_e$ will not be the actual energy of the particle, but due to length contraction and time dilation of the measuring device, it is what will be measured.

What is the emitter's 4-velocity doing there?

It four-velocity of the device measuring the energy of the photon. In this case the emitting atom.
You will see this kind of construction often. Basically you will obtain an invariant measure of some quantity where the measurement is invariant because we have specified the motion of the measuring device using some timelike four-velocity.
