Two rockets are orbiting a Schwarzschild black hole of mass M, in a circular path at some location R in the equatorial plane θ=π/2. The first (rocket A) is orbiting with an angular velocity Ω=dΦ/dt and the second (rocket B) with -Ω (they orbit in opposite directions).
Find the speed of B as measured by A, whenever they meet. Equivalently, find the relative Lorentz factor.
Schwarzschild metric with dr=dθ=0.
The Attempt at a Solution
I have calculated the four-velocity components of each rocket in terms of M, R and Ω.
Obviously the other two components are zero.
According to my textbook (Hartle), observed quantities correspond to projections on the observers' basis vectors. I know I should compute the relative velocity with:
where e_t and e_Φ are the coordinate basis vectors of the observer (in this case of rocket A) and u_B the four-velocity of rocket B (with the -Ω). The problem is I don't understand why I should divide by the projection onto e_t. For example, when we calculate the measured energy, one only has to project on the relevant basis vector:
Or for the momentum
Thanks for your help!