Speed of particles on accretion disk next to black hole

Question

I have the following homework assignment:

A spaceship is stationary at the r-coordinate 10M outside a black hole of mass M. The spaceship contains a lab which is measuring the properties of the accretion disk which surrounds the black hole. What is the speed of the particles orbiting in the disk as measured in situ?

Possible answers are:

a) 0.99c b) 0.5c c) 0.35c d) 0.1c

I thought about solving this in the following manner:

Given that the movement happens on a plane, $dr=d\theta =0$, the four-velocity can be written as (using $G=c=1$) $$\tilde{u}=(\frac{dt}{d\tau},0,0,\frac{d\phi}{d\tau})$$ I know that $$\Omega=\frac{d\phi}{d\tau}=(\frac{M}{r^3})^{1/2}$$ and $$u^t=\frac{dt}{d\tau}=(1-\frac{3M}{r})^{-1/2}$$ which means $$\tilde{u}=((1-\frac{3M}{r})^{-1/2},0,0,(\frac{M}{r^3})^{1/2})$$ and that's it, I'm pretty much stuck in there. I don't know how to find the speed from this. I saw an example on Hartle's "Gravitation", where he uses $\frac{m}{\sqrt{1-V^2}}=E$, when calculating the speed of a comet, but I could not find a way to get there. Any help is appreciated.

Answer 1

$(M/r^3)^{1/2}$ is $d\phi/dt$, not $d\phi/dτ$. You can sanity-check your four-velocity by checking that its norm is $1$.

One easy way to get the speed is to use the fact that the dot product of four-velocities is the gamma factor. You don't need the $\phi$ component for this (except for the sanity check).