What is the Schwarzschild metric in cylindrical coordinates? I was researching online for different metrics of spacetime out of curiosity, and I found one that was said to be Schwarzschild metric in cylindrical coordinates:
$$ds^2 = -\left(1-\frac{r_s}{r}\right)dt^2 + \frac{r}{r-r_s}dr^2 + r^2d\theta^2 + r^2 dz^2.$$
I cannot remember from which site it was or find it in my history, but when I started to try to understand it, I found it to be wrong.
I was amazed to see that there such metrics out there. I never learned string theory, but I also discovered that cylindrical metrics are used to model the spacetime around cosmic strings.
So does the Schwarzschild metric in cylindrical coordinates describe cosmic strings? What is the Schwarzschild metric in cylindrical coordinates?
Edit: I should have mentioned this before, but I did not. I am asking for the Schwarzschild metric in cylindrical coordinates specifically for a cylindrical source/mass aligned/centered along the $z$ axis.
 A: As you noted, that's not the Schwarzschild metric in cylindrical coordinates.
In spherical coordinates, where the corresponding Cartesian coordinates would be
$$(x,y,z) = r (\sin θ \cos φ, \sin θ \sin φ, \cos θ),$$
the metric is given by the line element
$$\frac{r}{r - r_s} dr^2 + (r dθ)^2 + (r \sin θ dφ)^2 - \frac{r - r_s}{r} (c dt)^2.$$
To express the metric in cylindrical coordinates, where
$$(x,y) = ρ (\cos φ, \sin φ),$$
set
$$(ρ,z) = r (\sin θ, \cos θ).$$
Then
$$ρ dρ + z dz = r dr, \hspace 1em z dρ - ρ dz = r^2 dθ, \hspace 1em ρ dφ = r \sin θ dφ, \hspace 1em r = \sqrt{ρ^2 + z^2},$$
and the line element becomes:
$$\frac{\sqrt{ρ^2 + z^2}}{\sqrt{ρ^2 + z^2} - r_s} \frac{(ρ dρ + z dz)^2}{ρ^2 + z^2} + \frac{(z dρ - ρ dz)^2}{ρ^2 + z^2} + (ρ dφ)^2 - \frac{\sqrt{ρ^2 + z^2} - r_s}{\sqrt{ρ^2 + z^2}} (c dt)^2.$$

There is also the underlying question of what the cylindrical *generalization* of the Schwarzschild metric is. That's the Kerr metric which, when given in Boyer-Lindquist coordinates, is:
$$ds^2 = Σ \left(\frac{dr^2}{Δ} + dθ^2\right) + \left(r^2 + a^2\right) (\sin θ dφ)^2 + \frac{r_s r}Σ \left(c dt - a \sin^2 θ dφ\right)^2 - (c dt)^2,$$
where the corresponding Cartesian coordinates are
$$(x,y,z) = \left(\sqrt{r^2 + a^2} \sin θ \cos φ, \sqrt{r^2 + a^2} \sin θ \sin φ, r \cos θ\right),$$
and
$$r_s = \frac{2GM}{c^2}, \hspace 1em a = \frac{J}{Mc}, \hspace 1em Σ = r^2 + (a \cos θ)^2, \hspace 1em Δ = r^2 - r_s r + a^2,$$
and this $r_s$ being the same as the $r_s$ for the Schwarzschild metric.

This describes a source with angular momentum $J$, mass $M$, in relativity, with $c$ being the in-vacuo light speed. You can work that what that comes out to in cylindrical coordinates, with the coordinates modified to the following form:
$$ρ = \sqrt{r^2 + a^2} \sin θ, \hspace 1em z = r \cos θ,$$
and use the cylindrical version of the Schwarzschild metric to check this against the $J = 0$ (and $a = 0$) case.
