$L_3$ conserved on geodesics?

Let’s take a simple $E^3$ space with coordinates $(x,y,z)$ and metric tensor
$$ g = mathrm{d} x otimes mathrm{d} x + mathrm{d} y otimes mathrm{d} y + mathrm{d} z otimes mathrm{d} z $$

The geodesics are straight lines
$vec{r} = vec{r}_0 + vec{v}_0 t$. Let’s take a very specific straight line
$$ begin{aligned} x (t) &= 1 y (t) &= t z (t) &= 0 end{aligned} $$

There are many conserved quantities, let’s take for example $L_z = x dot{y} – y dot{x}$ (verifiably conserved).

Let’s now make new coordinates with the transformation
$$ begin{aligned} x &= r sin theta cos phi y &= r sin theta sin phi z &= r cos theta end{aligned} $$
for which
$$ g = mathrm{d} r otimes mathrm{d} r + r^2 mathrm{d} theta otimes mathrm{d} theta + r^2 sin^2 theta , mathrm{d} phi otimes mathrm{d} phi $$

And take the same line, just in the new coordinates, that becomes
$$ begin{aligned} r (t) &= sqrt{x^2 (t) + y^2 (t)} = sqrt{1 + t^2} theta (t) &= pi / 2 phi (t) &= arctan(y(t)/x(t)) = arctan(t) end{aligned} $$

The same quantity $L_z$ now becomes $dot{phi}$. As we can see, $dot{phi}$ is not a constant, because
$$ dot{phi} (t) = frac{1}{1 + t^2} $$

What went wrong?

• is the notion of a geodesic dependent on coordinates (that would be kinda icky, since coordinates are totally arbitrary, which would mean that geodesics of a certain space are totally arbitrary)?

• is $L_z$ no longer conserved, so the symmetries and conserved quantities depend on choice of coordinates (again, icky, …)?

• is $dot{phi}$ not equal to $L_z$? I obtained it from the Killing vector in Cartesian coordinates, $xi = x e_y – y e_x$, which transforms into $e_phi$.

• does the conservation of $L_z$ depend on reparametrization of the geodesic? If I take $y (t) = tan (t)$ instead, I get $phi (t) = 1$, which is conserved. Does that mean that symmetries and conservation laws depend on reparametrisation of a geodetic? Maybe there’s some caveat with Killing vectors and reparametrization.

Edit: as Michael suggested, the problem might be in mixing unit vectors and coordinate basis. To be clear, what I meant was that the Killing vector in coordinate basis in Cartesian is $x partial_y – y partial_x$, and the Killing vector in coordinate basis in spherical is $partial_phi$. However, to get the conserved variable, we have to do
$$
xi cdot frac{mathrm{d} vec{r}}{mathrm{d} t}
$$
and in spherical coordinates, that $cdot$ actually carries the additional baggage in terms of the components of the metric tensor, so specifically
$$
L_z = (r^2 sin^2 theta) frac{mathrm{d} phi}{mathrm{d} t}
$$

So for my specific example, I have $L_z = r^2 dot{phi} = left( sqrt{1 + t^2} right)^2 / (1 + t^2) = 1$. Hooray!