Coordinate-free proof of the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic

Question

I would like to know if the following proof of the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic is correct. I already found a coordinate-dependent proof but my rationale seems to be rather different from the coordinate-dependent proof.

Be $K$ a Killing vector and $\gamma$ the geodesic and $\dot{\gamma}$ its tangent vector. $g(\cdot, \cdot)$ is the scalar product based on the metric $g$. $\nabla_V X$ is the covariant derivative.

$$d g(K,\dot{\gamma})(\dot{\gamma}) = \nabla_\dot{\gamma}g( K,\dot{\gamma}) = g( \nabla_\dot{\gamma}K, \dot{\gamma}) + g( K, \nabla_\dot{\gamma}\dot{\gamma}) = g( \nabla_\dot{\gamma}K, \dot{\gamma}) $$

where the second term vanishes because of $\nabla_\dot{\gamma}\dot{\gamma}=0$ along the geodesic. The first term also vanishes since I can conclude from the coordinate-independent form of the Killing vector equation:

$$g(\nabla_V K,W) + g(V,\nabla_W K)=0$$

that

$$g(\nabla_\dot{\gamma}K,\dot{\gamma})=0$$

So $$d g(K,\dot{\gamma})(\dot{\gamma})=0$$

which shows the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic. Is this derivation correct?

Answer 1

I'll give another proof, just for fun. We know that in a pseudo-Riemannian manifold $(M,g)$, geodesics are critical points of the energy functional $$E[\gamma] = \frac{1}{2}\int_I g_{\gamma(t)}(\dot{\gamma}(t),\dot{\gamma}(t))\,{\rm d}t$$Since $K$ is Killing, the flow of $K$ (consisting of isometries) leaves the Lagrangian we're integrating invariant. So Noether's Theorem says that the charge $$\mathscr{J}(x,v) = \mathbb{F}L(x,v)\left(\frac{{\rm d}}{{\rm d}s}\bigg|_{s=0} \Phi_{s,K}(x)\right) = \mathbb{F}L(x,v)K_x= \frac{\rm d}{{\rm d}t}\bigg|_{t=0} \frac{1}{2}g_x(v+tK_x,v+tK_x) = g_x(v,K_x) $$is constant along geodesics. But $\mathscr{J}(\gamma(t),\dot{\gamma}(t)) = g_{\gamma(t)}\big(\dot{\gamma}(t), K_{\gamma(t)}\big)$, and so we're done. Here $(x,v) \in TM$ (which is to say that $x \in M$ and $v \in T_xM$) and $\mathbb{F}L$ denotes the fiber derivative of $L$.