# General Relativity: Particle velocity travelling on Schwarzchild orbit

1. Mar 6, 2012

### stongio

1. The problem statement, all variables and given/known data

What is the speed of a particle in the smallest possible circular orbit in the Schwarzschild
geometry as measured by a stationary observer at that orbit? Note: The orbit in
question happens to be unstable.

2. Relevant equations

Normalization condition:

$\textbf{u}_{obs}(r)$$\cdot$$\textbf{u}_{obs}(r)$=$g_{αβ}$$u^{α}_{obs}(r)$$u^{β}_{obs}(r)$=-1

Schwarzchild Metric...

3. The attempt at a solution

So from the Normalization condition and the Schwarzchild metric it is easy to find the four-velocity of the observer, because the spatial components are zero. My question relates to the observer's measurement of the particle's velocity. If his four-velocity is
${u}^{a}$=[1/$\sqrt{1-2m/r}$,0,0,0]
then, how does observe the particle's velocity?
An answer I found somewhere said

$\frac{u^{2}}{1-u^{2}}$=$u^{a}u^{b}(g_{ab}+u_{a}u_{b})=(u^{a}u_{a})^{2}-1$

But I have no idea how to arrive at this relation, and why it relates to the observer's measurement of the particle. Is it possibly a derivation of a relation between t and the proper time?

Also, this might seem very rudimentary but I am not sure why we are able to write:

$u_{a}u_{b} \ast u^{a}u^{b} = (u^{a}u_{a})^{2}$

I'm very new to general relativity, and would really appreciate the help!
Thanks a lot!