- #1

zimo123

- 18

- 0

## Homework Statement

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.

## Homework Equations

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.

[tex]\frac{\mathrm{d}\phi}{\mathrm{d}\tau}=\frac{\pm\Omega}{\sqrt{1-2M/R-R^2\Omega^2}}[/tex]

[tex]\frac{\mathrm{d}t}{\mathrm{d}\tau}=\frac{1}{\sqrt{1-2M/R-R^2\Omega^2}}[/tex]

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:

[tex]V=\frac{\vec{u_B}\cdot\vec{e_{\phi}}}{\vec{u_B}\cdot\vec{e_t}}[/tex]

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:

[tex]E=-\vec{p}\cdot\vec{e_t}[/tex]

Or for the momentum

[tex]P=\vec{p}\cdot\vec{e_r}[/tex]

Thanks for your help!