General Relativity- geodesics, killing vector, Conserved quantity Schwarzschild metric

1. Jun 25, 2017

binbagsss

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

Conserved quantity Schwarzschild metric.

2. Relevant equations

3. The attempt at a solution

$\partial_u=\delta^u_i=k^u$ is the KVF $i=1,2,3$

We have that along a geodesic $K=k^uV_u$ is constant , where $V^u$ is the tangent vector to some affinely parameterised geodesic.

For example if we take the Schwarzschild metric, $K^u=(1,0,0,0)$ , $V^u=(\dot{t},\dot{r},\dot{\theta},\dot{\phi})$ so we get $K= (1-\frac{2GM}{r})\dot{t}$ is conserved, for example.

where dot denotes a derivative with respect to some affine parameter $s$

QUESTION:

What here is to say that $s$ is an affine parameter?
I.e- why is $V^u=(\dot{t},\dot{r},\dot{\theta},\dot{\phi})$ the tangent vector to an affinely parameterised geodesic?

Last edited by a moderator: Jun 25, 2017
2. Jun 25, 2017

Orodruin

Staff Emeritus
You assumed it to be an affine parameter and then wonder why it is affine?

If it is not an affine parameter $k^u V_u$ will not be a conserved quantity.

3. Jun 25, 2017

Staff: Mentor

Moderator's note: moved to homework.

4. Jun 25, 2017

binbagsss

ahh apologies got it yes, it is in the proof showing that $k^u V_u$ will not be a conserved quantity is a conserved quantity.