General Relativity geodesics, killing vector, conserved quantities

[binbagsss]

Homework Statement

Homework Equations

The Attempt at a Solution

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Let $k^u$ denote the KVF.

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

$k^u=\delta^u_i$ , $V^u=(\dot{t},\vec{\dot{x}})$ so we get

$K= g_{ii}\dot{x^i}=t^{p_i}\dot{x^i}=K$ (1) is conserved.

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

ATTEMPT AT QUESTION:

I think this is a bad attempt but...

The curves of constant $x^i$ are given by: $\dot{x^i}=0$ , so above if $K=0$ equation (1) is trivially satisfied.

Any help appreciated, thanks .

 

[mark726]

Your attempt is a good start, but there are a few things that could be improved.

Firstly, when you say "Let $k^u$ denote the KVF," it's not clear what you mean by "KVF." Are you referring to the Killing vector field? If so, it's important to define what that is and how it relates to the geodesic equation.

Secondly, when you say "We have that along a geodesic $K=k^uV_u$ is constant," it's not clear what you mean by "along a geodesic." Do you mean that $K$ is constant along the entire geodesic, or just at a particular point on the geodesic? It's important to clarify this.

Thirdly, in your equation (1), it's not clear what you mean by $t^{p_i}$. Is this a typo? Do you mean $g^{p_i}$? Also, it's not clear what you mean by "conserved." Do you mean that $K$ is constant along the geodesic? If so, it's important to state that explicitly.

Finally, in your attempt at the question, you say that the curves of constant $x^i$ are given by $\dot{x^i}=0$. This is not quite correct. The curves of constant $x^i$ are given by $x^i = const.$, not $\dot{x^i}=0$. However, it is true that if $K=0$, then $\dot{x^i}=0$, since $K$ is the dot product of the Killing vector field and the tangent vector to the geodesic, and if $K=0$, then the tangent vector must be perpendicular to the Killing vector field.

So, to summarize, your attempt is a good start, but it could be improved by clarifying some of the points mentioned above. Also, it's important to make sure that you understand the definitions and concepts involved, and to state them explicitly in your solution.