Conserving Quantity in Schwarzschild Metric

[binbagsss]

Homework Statement

Conserved quantity Schwarzschild metric.

Homework Equations

The Attempt at a Solution

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$\partial_u=\delta^u_i=k^u$ 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?

 

[Orodruin]

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.

 

[PeterDonis]

Moderator's note: moved to homework.

 

[binbagsss]

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.

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.